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https://en.wikipedia.org/wiki/Cousin%27s%20theorem | In real analysis, a branch of mathematics, Cousin's theorem states that:
If for every point of a closed region (in modern terms, "closed and bounded") there is a circle of finite radius (in modern term, a "neighborhood"), then the region can be divided into a finite number of subregions such that each subregion is interior to a circle of a given set having its center in the subregion.
This result was originally proved by Pierre Cousin, a student of Henri Poincaré, in 1895, and it extends the original Heine–Borel theorem on compactness for arbitrary covers of compact subsets of . However, Pierre Cousin did not receive any credit. Cousin's theorem was generally attributed to Henri Lebesgue as the Borel–Lebesgue theorem. Lebesgue was aware of this result in 1898, and proved it in his 1903 dissertation.
In modern terms, it is stated as:
Let be a full cover of [a, b], that is, a collection of closed subintervals of [a, b] with the property that for every x ∈ [a, b], there exists a δ>0 so that contains all subintervals of [a, b] which contains x and length smaller than δ. Then there exists a partition of non-overlapping intervals for [a, b], where and a=x0 < x1 < ⋯ < xn=b for all 1≤i≤n.
Cousin's lemma is studied in Reverse Mathematics where it is one of the first third-order theorems that is hard to prove in terms of the comprehension axioms needed.
In Henstock–Kurzweil integration
Cousin's theorem is instrumental in the study of Henstock–Kurzweil integration, and in this context, it is known as Cousin's lemma or the fineness theorem.
A gauge on is a strictly positive real-valued function , while a tagged partition of is a finite sequence
Given a gauge and a tagged partition of , we say is -fine if for all , we have , where denotes the open ball of radius centred at . Cousin's lemma is now stated as:
If , then every gauge has a -fine partition.
Proof of the theorem
Cousin's theorem has an intuitionistic proof using the open induction principle, which reads as follows:
An open subset of a closed real interval is said to be inductive if it satisfies that implies . The open induction principle states that any inductive subset of must be the entire set.
Proof using open induction
Let be the set of points such that there exists a -fine tagged partition on for some .
The set is open, since it is downwards closed and any point in it is included in the open ray for any associated partition.
Furthermore, it is inductive. For any , suppose . By that assumption (and using that either or to handle edge cases) we have a partition of length with . Then either or . In the first case , so we can just replace with and get a partition of that includes .
If , we may form a partition of length that includes . To show this, we split into the cases or . In the first case, we set , in the second we set . In both cases, we can set and obtain a valid partition. So in all cases, and is inductive.
By open induction, .
|
https://en.wikipedia.org/wiki/Tolerance%20relation | In universal algebra and lattice theory, a tolerance relation on an algebraic structure is a reflexive symmetric relation that is compatible with all operations of the structure. Thus a tolerance is like a congruence, except that the assumption of transitivity is dropped. On a set, an algebraic structure with empty family of operations, tolerance relations are simply reflexive symmetric relations. A set that possesses a tolerance relation can be described as a tolerance space. Tolerance relations provide a convenient general tool for studying indiscernibility/indistinguishability phenomena. The importance of those for mathematics had been first recognized by Poincaré.
Definitions
A tolerance relation on an algebraic structure is usually defined to be a reflexive symmetric relation on that is compatible with every operation in . A tolerance relation can also be seen as a cover of that satisfies certain conditions. The two definitions are equivalent, since for a fixed algebraic structure, the tolerance relations in the two definitions are in one-to-one correspondence. The tolerance relations on an algebraic structure form an algebraic lattice under inclusion. Since every congruence relation is a tolerance relation, the congruence lattice is a subset of the tolerance lattice , but is not necessarily a sublattice of .
As binary relations
A tolerance relation on an algebraic structure is a binary relation on that satisfies the following conditions.
(Reflexivity) for all
(Symmetry) if then for all
(Compatibility) for each -ary operation and , if for each then . That is, the set is a subalgebra of the direct product of two .
A congruence relation is a tolerance relation that is also transitive.
As covers
A tolerance relation on an algebraic structure is a cover of that satisfies the following three conditions.
For every and , if , then .
In particular, no two distinct elements of are comparable. (To see this, take .)
For every , if is not contained in any set in , then there is a two-element subset such that is not contained in any set in .
For every -ary and , there is a such that . (Such a need not be unique.)
Every partition of satisfies the first two conditions, but not conversely. A congruence relation is a tolerance relation that also forms a set partition.
Equivalence of the two definitions
Let be a tolerance binary relation on an algebraic structure . Let be the family of maximal subsets such that for every . Using graph theoretical terms, is the set of all maximal cliques of the graph . If is a congruence relation, is just the quotient set of equivalence classes. Then is a cover of and satisfies all the three conditions in the cover definition. (The last condition is shown using Zorn's lemma.) Conversely, let be a cover of and suppose that forms a tolerance on . Consider a binary relation on for which if and only if for some . Then is a tolerance on as a binary relation. The map |
https://en.wikipedia.org/wiki/National%20Treasures%20of%20Japan%20%28statistics%29 | This table shows the number of National Treasures of Japan in each prefecture grouped by type of the property. Gold colored cells mark prefectures with the largest number of National Treasures for the given category (column).
References
National Treasures of Japan |
https://en.wikipedia.org/wiki/Albert%20Leon%20Whiteman%20Memorial%20Prize |
The Albert Leon Whiteman Memorial Prize is awarded by the American Mathematical Society for notable exposition and exceptional scholarship in the history of mathematics.
The prize was endowed in 1998 with funds provided by Sally Whiteman in memory of her late husband Albert Leon Whiteman. Originally it was awarded every 4 years with the first prized handed out in 2001. Since 2009 the prize is awarded every 3 years and carries a prize money of $5000.
Past recipients
2001 Thomas Hawkins
2005 Harold M. Edwards
2009 Jeremy Gray
2012 Joseph Dauben
2015 Umberto Bottazzini
2018 Karen Parshall
2021 Judith Grabiner
See also
List of mathematics awards
References
External links
Albert Leon Whiteman Memorial Prize on the website of the American Mathematical Society
Awards of the American Mathematical Society
Triennial events
History of mathematics
Awards established in 2001
American history awards
2001 establishments in the United States |
https://en.wikipedia.org/wiki/Stably%20finite%20ring | In mathematics, particularly in abstract algebra, a ring R is said to be stably finite (or weakly finite) if, for all square matrices A and B of the same size with entries in R, AB = 1 implies BA = 1. This is a stronger property for a ring than having the invariant basis number (IBN) property. Namely, any nontrivial stably finite ring has IBN. Commutative rings, noetherian rings and artinian rings are stably finite. Subrings of stably finite rings and matrix rings over stably finite rings are stably finite. A ring satisfying Klein's nilpotence condition is stably finite.
Notes
References
Ring theory |
https://en.wikipedia.org/wiki/6-polytope | In six-dimensional geometry, a six-dimensional polytope or 6-polytope is a polytope, bounded by 5-polytope facets.
Definition
A 6-polytope is a closed six-dimensional figure with vertices, edges, faces, cells (3-faces), 4-faces, and 5-faces. A vertex is a point where six or more edges meet. An edge is a line segment where four or more faces meet, and a face is a polygon where three or more cells meet. A cell is a polyhedron. A 4-face is a polychoron, and a 5-face is a 5-polytope. Furthermore, the following requirements must be met:
Each 4-face must join exactly two 5-faces (facets).
Adjacent facets are not in the same five-dimensional hyperplane.
The figure is not a compound of other figures which meet the requirements.
Characteristics
The topology of any given 6-polytope is defined by its Betti numbers and torsion coefficients.
The value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, and is zero for all 6-polytopes, whatever their underlying topology. This inadequacy of the Euler characteristic to reliably distinguish between different topologies in higher dimensions led to the discovery of the more sophisticated Betti numbers.
Similarly, the notion of orientability of a polyhedron is insufficient to characterise the surface twistings of toroidal polytopes, and this led to the use of torsion coefficients.
Classification
6-polytopes may be classified by properties like "convexity" and "symmetry".
A 6-polytope is convex if its boundary (including its 5-faces, 4-faces, cells, faces and edges) does not intersect itself and the line segment joining any two points of the 6-polytope is contained in the 6-polytope or its interior; otherwise, it is non-convex. Self-intersecting 6-polytope are also known as star 6-polytopes, from analogy with the star-like shapes of the non-convex Kepler-Poinsot polyhedra.
A regular 6-polytope has all identical regular 5-polytope facets. All regular 6-polytope are convex.
A semi-regular 6-polytope contains two or more types of regular 4-polytope facets. There is only one such figure, called 221.
A uniform 6-polytope has a symmetry group under which all vertices are equivalent, and its facets are uniform 5-polytopes. The faces of a uniform polytope must be regular.
A prismatic 6-polytope is constructed by the Cartesian product of two lower-dimensional polytopes. A prismatic 6-polytope is uniform if its factors are uniform. The 6-cube is prismatic (product of a squares and a cube), but is considered separately because it has symmetries other than those inherited from its factors.
A 5-space tessellation is the division of five-dimensional Euclidean space into a regular grid of 5-polytope facets. Strictly speaking, tessellations are not 6-polytopes as they do not bound a "6D" volume, but we include them here for the sake of completeness because they are similar in many ways to 6-polytope. A uniform 5-space tessellation is one whose vertices are rel |
https://en.wikipedia.org/wiki/Asymptotic%20theory%20%28statistics%29 | In statistics, asymptotic theory, or large sample theory, is a framework for assessing properties of estimators and statistical tests. Within this framework, it is often assumed that the sample size may grow indefinitely; the properties of estimators and tests are then evaluated under the limit of . In practice, a limit evaluation is considered to be approximately valid for large finite sample sizes too.
Overview
Most statistical problems begin with a dataset of size . The asymptotic theory proceeds by assuming that it is possible (in principle) to keep collecting additional data, thus that the sample size grows infinitely, i.e. . Under the assumption, many results can be obtained that are unavailable for samples of finite size. An example is the weak law of large numbers. The law states that for a sequence of independent and identically distributed (IID) random variables , if one value is drawn from each random variable and the average of the first values is computed as , then the converge in probability to the population mean as .
In asymptotic theory, the standard approach is . For some statistical models, slightly different approaches of asymptotics may be used. For example, with panel data, it is commonly assumed that one dimension in the data remains fixed, whereas the other dimension grows: and , or vice versa.
Besides the standard approach to asymptotics, other alternative approaches exist:
Within the local asymptotic normality framework, it is assumed that the value of the "true parameter" in the model varies slightly with , such that the -th model corresponds to . This approach lets us study the regularity of estimators.
When statistical tests are studied for their power to distinguish against the alternatives that are close to the null hypothesis, it is done within the so-called "local alternatives" framework: the null hypothesis is and the alternative is . This approach is especially popular for the unit root tests.
There are models where the dimension of the parameter space slowly expands with , reflecting the fact that the more observations there are, the more structural effects can be feasibly incorporated in the model.
In kernel density estimation and kernel regression, an additional parameter is assumed—the bandwidth . In those models, it is typically taken that as . The rate of convergence must be chosen carefully, though, usually .
In many cases, highly accurate results for finite samples can be obtained via numerical methods (i.e. computers); even in such cases, though, asymptotic analysis can be useful. This point was made by , as follows.
Modes of convergence of random variables
Asymptotic properties
Estimators
Consistency
A sequence of estimates is said to be consistent, if it converges in probability to the true value of the parameter being estimated:
That is, roughly speaking with an infinite amount of data the estimator (the formula for generating the estimates) would almost surely give the correct |
https://en.wikipedia.org/wiki/Tukey%27s%20test%20of%20additivity | In statistics, Tukey's test of additivity, named for John Tukey, is an approach used in two-way ANOVA (regression analysis involving two qualitative factors) to assess whether the factor variables (categorical variables) are additively related to the expected value of the response variable. It can be applied when there are no replicated values in the data set, a situation in which it is impossible to directly estimate a fully general non-additive regression structure and still have information left to estimate the error variance. The test statistic proposed by Tukey has one degree of freedom under the null hypothesis, hence this is often called "Tukey's one-degree-of-freedom test."
Introduction
The most common setting for Tukey's test of additivity is a two-way factorial analysis of variance (ANOVA) with one observation per cell. The response variable Yij is observed in a table of cells with the rows indexed by i = 1,..., m and the columns indexed by j = 1,..., n. The rows and columns typically correspond to various types and levels of treatment that are applied in combination.
The additive model states that the expected response can be expressed EYij = μ + αi + βj, where the αi and βj are unknown constant values. The unknown model parameters are usually estimated as
where Yi• is the mean of the ith row of the data table, Y•j is the mean of the jth column of the data table, and Y•• is the overall mean of the data table.
The additive model can be generalized to allow for arbitrary interaction effects by setting EYij = μ + αi + βj + γij. However, after fitting the natural estimator of γij,
the fitted values
fit the data exactly. Thus there are no remaining degrees of freedom to estimate the variance σ2, and no hypothesis tests about the γij can performed.
Tukey therefore proposed a more constrained interaction model of the form
By testing the null hypothesis that λ = 0, we are able to detect some departures from additivity based only on the single parameter λ.
Method
To carry out Tukey's test, set
Then use the following test statistic
Under the null hypothesis, the test statistic has an F distribution with 1, q degrees of freedom, where q = mn − (m + n) is the degrees of freedom for estimating the error variance.
See also
Tukey's range test for multiple comparisons
References
Analysis of variance
Statistical tests |
https://en.wikipedia.org/wiki/Regular%20extension | In field theory, a branch of algebra, a field extension is said to be regular if k is algebraically closed in L (i.e., where is the set of elements in L algebraic over k) and L is separable over k, or equivalently, is an integral domain when is the algebraic closure of (that is, to say, are linearly disjoint over k).
Properties
Regularity is transitive: if F/E and E/K are regular then so is F/K.
If F/K is regular then so is E/K for any E between F and K.
The extension L/k is regular if and only if every subfield of L finitely generated over k is regular over k.
Any extension of an algebraically closed field is regular.
An extension is regular if and only if it is separable and primary.
A purely transcendental extension of a field is regular.
Self-regular extension
There is also a similar notion: a field extension is said to be self-regular if is an integral domain. A self-regular extension is relatively algebraically closed in k. However, a self-regular extension is not necessarily regular.
References
M. Nagata (1985). Commutative field theory: new edition, Shokado. (Japanese)
A. Weil, Foundations of algebraic geometry.
Field (mathematics) |
https://en.wikipedia.org/wiki/Crime%20in%20Arizona | In 2014, 242,156 crimes were reported in the U.S. state of Arizona.
Statistics
Capital punishment laws
Capital punishment is applied in Arizona. In most circumstances, the method used is lethal injection. Inmates sentenced to death for murders committed prior to November 23, 1992 may choose lethal gas.
See also
List of people executed in Arizona
References |
https://en.wikipedia.org/wiki/Crime%20in%20California | Crime in California refers to crime occurring within the U.S. state of California.
State statistics
In 2019, there were 1,096,668 crimes reported in California including 1,679 murders, 14,720 rapes and 915,197 property crimes. In 2019, there were 1,012,441 arrests of adults and 43,181 arrests of juveniles in California.
In 2014, 1,697 people were victims of homicides. 30% of homicides were gang-related, 28% were due to an unspecified argument, 9% were domestic, and 7% were robbery related. The rest were unknown. In 2017 the violent crime rate in California rose 1.5% and was 14th highest of the 50 states.
By location
Los Angeles
In 2010, Los Angeles reported 293 homicides. The 2010 number corresponds to a rate of 7.6 per 100,000 population. Murders in Los Angeles have decreased since the peak year of 1993, when the homicide rate was 21.1 (per 100,000 population).
Legal procedure
As one of the fifty states of the United States, California follows common law criminal procedure. The principal source of law for California criminal procedure is the California Penal Code, Part 2, "Of Criminal Procedure."
Every year in California, approximately 150 thousand violent crimes and 1 million property crimes are committed. With a population of about 40 million people, approximately 1.2 million arrests are made every year in California. The California superior courts hear about 270,000 felony cases, 900,000 misdemeanor cases, and 5 million infraction cases every year. There are currently 130,000 people in state prisons and 70,000 people in county jails. Of these, there are 746 people who have been sentenced to death.
Policing
In 2018, California had 531 state and local law enforcement agencies. Those agencies employed a total of 130,451 staff. Of the total staff, 79,038 were sworn officers (defined as those with general arrest powers).
Police ratio
In 2018, California had 200 police officers per 100,000 residents.
Capital punishment laws
The death penalty (also known as capital punishment) is legal in California, although Governor Gavin Newsom issued a moratorium on the use on March 13, 2019. The last execution was issued for Clarence Ray Allen on January 17, 2006, through lethal injection.
Organized crime
Organized crime in California involves the criminal activities of organized crime groups, street gangs, criminal extremists, and terrorists in California. Traditional organized crime are in the form of Cosa Nostra (LCN), Sicilian Mafia, and Camorra. Eurasian criminal networks specialize in white-collar crime, fraud, prostitution and human trafficking. Crime cells from Southeast Asia, Latin America, and Eastern Europe impact public safety and the state's economy.
Gangs
Gangs in California are classified into three categories: criminal street gangs, prison gangs, and outlaw motorcycle gangs. Gang operations usually include "assault, auto theft, drive-by shooting, illegal drug and narcotic manufacturing, drug and
narcotic trafficking, forge |
https://en.wikipedia.org/wiki/Kontsevich%20quantization%20formula | In mathematics, the Kontsevich quantization formula describes how to construct a generalized ★-product operator algebra from a given arbitrary finite-dimensional Poisson manifold. This operator algebra amounts to the deformation quantization of the corresponding Poisson algebra. It is due to Maxim Kontsevich.
Deformation quantization of a Poisson algebra
Given a Poisson algebra , a deformation quantization is an associative unital product on the algebra of formal power series in , subject to the following two axioms,
If one were given a Poisson manifold , one could ask, in addition, that
where the are linear bidifferential operators of degree at most .
Two deformations are said to be equivalent iff they are related by a gauge transformation of the type,
where are differential operators of order at most . The corresponding induced -product, , is then
For the archetypal example, one may well consider Groenewold's original "Moyal–Weyl" -product.
Kontsevich graphs
A Kontsevich graph is a simple directed graph without loops on 2 external vertices, labeled f and g; and internal vertices, labeled . From each internal vertex originate two edges. All (equivalence classes of) graphs with internal vertices are accumulated in the set .
An example on two internal vertices is the following graph,
Associated bidifferential operator
Associated to each graph , there is a bidifferential operator defined as follows. For each edge there is a partial derivative on the symbol of the target vertex. It is contracted with the corresponding index from the source symbol. The term for the graph is the product of all its symbols together with their partial derivatives. Here f and g stand for smooth functions on the manifold, and is the Poisson bivector of the Poisson manifold.
The term for the example graph is
Associated weight
For adding up these bidifferential operators there are the weights of the graph . First of all, to each graph there is a multiplicity which counts how many equivalent configurations there are for one graph. The rule is that the sum of the multiplicities for all graphs with internal vertices is . The sample graph above has the multiplicity . For this, it is helpful to enumerate the internal vertices from 1 to .
In order to compute the weight we have to integrate products of the angle in the upper half-plane, H, as follows. The upper half-plane is , endowed with a metric
and, for two points with , we measure the angle between the geodesic from to and from to counterclockwise. This is
The integration domain is Cn(H) the space
The formula amounts
,
where t1(j) and t2(j) are the first and second target vertex of the internal vertex . The vertices f and g are at the fixed positions 0 and 1 in .
The formula
Given the above three definitions, the Kontsevich formula for a star product is now
Explicit formula up to second order
Enforcing associativity of the -product, it is straightforward to check directly that the Kontsevi |
https://en.wikipedia.org/wiki/SymbolicC%2B%2B | SymbolicC++ is a general purpose computer algebra system written in the programming language C++. It is free software released under the terms of the GNU General Public License. SymbolicC++ is used by including a C++ header file or by linking against a library.
Examples
#include <iostream>
#include "symbolicc++.h"
using namespace std;
int main(void)
{
Symbolic x("x");
cout << integrate(x+1, x); // => 1/2*x^(2)+x
Symbolic y("y");
cout << df(y, x); // => 0
cout << df(y[x], x); // => df(y[x],x)
cout << df(exp(cos(y[x])), x); // => -sin(y[x])*df(y[x],x)*e^cos(y[x])
return 0;
}
The following program fragment inverts the matrix
symbolically.
Symbolic theta("theta");
Symbolic R = ( ( cos(theta), sin(theta) ),
( -sin(theta), cos(theta) ) );
cout << R(0,1); // sin(theta)
Symbolic RI = R.inverse();
cout << RI[ (cos(theta)^2) == 1 - (sin(theta)^2) ];
The output is
[ cos(theta) −sin(theta) ]
[ sin(theta) cos(theta) ]
The next program illustrates non-commutative symbols in SymbolicC++. Here b is a Bose annihilation operator and bd is a Bose creation operator. The variable vs denotes the vacuum state . The ~ operator toggles the commutativity of a variable, i.e. if b is commutative that ~b is non-commutative and if b is non-commutative ~b is commutative.
#include <iostream>
#include "symbolicc++.h"
using namespace std;
int main(void)
{
// The operator b is the annihilation operator and bd is the creation operator
Symbolic b("b"), bd("bd"), vs("vs");
b = ~b; bd = ~bd; vs = ~vs;
Equations rules = (b*bd == bd*b + 1, b*vs == 0);
// Example 1
Symbolic result1 = b*bd*b*bd;
cout << "result1 = " << result1.subst_all(rules) << endl;
cout << "result1*vs = " << (result1*vs).subst_all(rules) << endl;
// Example 2
Symbolic result2 = (b+bd)^4;
cout << "result2 = " << result2.subst_all(rules) << endl;
cout << "result2*vs = " << (result2*vs).subst_all(rules) << endl;
return 0;
}
Further examples can be found in the books listed below.
History
SymbolicC++ is described in a series of books on computer algebra. The first book described the first version of SymbolicC++. In this version the main data type for symbolic computation was the Sum class. The list of available classes included
Verylong : An unbounded integer implementation
Rational : A template class for rational numbers
Quaternion : A template class for quaternions
Derive : A template class for automatic differentiation
Vector : A template class for vectors (see vector space)
Matrix : A template class for matrices (see matrix (mathematics))
Sum : A template class for symbolic expressions
Example:
#include <iostream>
#include "rational.h"
#include "msymbol.h"
using namespace std;
int main(void)
{
Sum<int> x("x",1);
Sum<Rational<int> > y("y",1);
cout << Int(y, y); // => 1/2 yˆ2
y.depend(x);
cout << df(y, x); // => df(y,x)
return 0;
}
The second version of SymbolicC++ featured new class |
https://en.wikipedia.org/wiki/2010%20Bangkok%20Glass%20F.C.%20season | The 2010 season was Bangkok Glass's 2nd season in the top division of Thai football. This article shows statistics of the club's players in the season, and also lists all matches that the club played in the season.
Team kit
Chronological list of events
10 November 2009: The Thai Premier League 2010 season first leg fixtures were announced.
7 July 2010: Bangkok Glass is kicked out of the FA Cup in the third round by Rajnavy Rayong.
24 October 2010: Bangkok Glass finished in 5th place in the Thai Premier League.
Squad
Transfers
In
Out
Out on loan
Matches
League
League table
FA Cup
Third round
League Cup
First round
1st leg
2nd leg
Second round
1st leg
2nd leg
Singapore Cup
Round of 16
Quarter-final
1st leg
2nd leg
Glass won 6–4 on aggregate.
References
2010
Bangkok Glass |
https://en.wikipedia.org/wiki/2010%20Bangkok%20United%20F.C.%20season | The 2010 season was Bangkok United's 7th season in the top division of Thai football. This article shows statistics of the club's players in the season, and also lists all matches that the club played in the season.
Team kit
Chronological list of events
10 November 2009: The Thai Premier League 2010 season first leg fixtures were announced.
1 January 2010: Worrakon Vijanarong announced as new head coach
17 October 2010: Bangkok United were relegated from the Thai Premier League after 7 seasons in the top flight.
24 October 2010: Bangkok United finished in 15th place in the Thai Premier League.
Squad
Transfers
In
Out
Matches
League
League table
FA Cup
Third round
Fourth round
League Cup
First round
1st leg
2nd leg
Second round
1st leg
2nd leg
Bangkok Charity Cup
References
2010
Bangkok United |
https://en.wikipedia.org/wiki/2010%20Thai%20Port%20F.C.%20season | The 2010 season was Thai Port's 14th season in the top division of Thai football. This article shows statistics of the club's players in the season, and also lists all matches that the club played in the season.
Team kit
Chronological list of events
10 November 2009: The Thai Premier League 2010 season first leg fixtures were announced.
23 June 2010: Thai Port were knocked out of the FA Cup by Pattaya United in the third round.
24 October 2010: Thai Port finished in 4th place in the Thai Premier League.
Pre-season
League table
Matches
League
FA Cup
Third round
League Cup
First round
1st leg
2nd leg
Second round
1st leg
2nd leg
Isuzu Cup
Kor Royal Cup
Queen's Cup
AFC Cup
Group stage
Round of 16
Quarter-finals
1st leg
2nd leg
References
2010
Thai Port |
https://en.wikipedia.org/wiki/Crime%20in%20Colorado | This article describes crime in the U.S. state of Colorado.
Statistics
In 2011, there were 151,125 crimes reported in Colorado.
In 2008, there were 158,236 crimes reported in Colorado, including 156 murders, 141,107 property crimes, and 2,094 rapes.
Capital punishment laws
Capital punishment is not applied in this state.
Notable crimes
Two of the country's largest mass shootings have occurred in Colorado: the Columbine High School massacre in 1999 and the Aurora movie theater massacre in 2012. Other notable mass shootings in Colorado include the Colorado YWAM and New Life shootings in 2007, the Colorado Springs Planned Parenthood shooting in 2015, the National Western Complex shootout in Denver in 2016, the 2021 Boulder shooting, the 2021 Colorado Springs shooting, and most recently the Colorado Springs nightclub shooting in 2022.
In 2004, Marvin Heemeyer, a business owner, used a modified bulldozer to smash into light poles, trees and buildings in Granby, Colorado.
See also
Law of Colorado
References |
https://en.wikipedia.org/wiki/Crime%20in%20Florida | Crime in Florida refers to crime occurring within the U.S. State of Florida.
Crime Statistics
Policing
In 2018, Florida had 373 state and local law enforcement agencies. Those agencies employed a total of 85,234 staff. Of the total staff, 47,177 were sworn officers (defined as those with general arrest powers). In 2018, Florida had 222 police officers per 100,000 residents.
Capital punishment laws
Capital punishment is applied in Florida. In 1995, the legislature modified Chapter 921 to provide that felons should serve at least 85% of their sentence.
See also
Incarceration in Florida
List of Florida state prisons
Law of Florida
References
Further reading |
https://en.wikipedia.org/wiki/Crime%20in%20Georgia%20%28U.S.%20state%29 | In 2008, there were 434,560 crimes reported in the U.S. state of Georgia, including 650 murders
Statistics
From 1877 to 1950, the state was the site of at least 586 lynchings of black people, the most of any state.
In 2008, there were 434,560 crimes reported in Georgia, including 650 murders, 387,009 property crimes, and 2,344 rapes.
Capital punishment laws
Capital punishment is applied in this state. Up until 2009, juvenile offenders could be charged as adults for crimes called the seven deadly sins.
See also
Crime in Atlanta
Gangs in Georgia
Law of Georgia (U.S. state)
References |
https://en.wikipedia.org/wiki/2010%20Osotspa%20Saraburi%20F.C.%20season | The 2010 season was Osotspa Saraburi's 12th season in the top division of Thai football. This article shows statistics of the club's players in the season, and also lists all matches that the club played in the season.
Team kit
Chronological list of events
10 November 2009: The Thai Premier League 2010 season first leg fixtures were announced.
11 August 2010: Osotspa Saraburi were knocked out of the FA Cup by Pattaya United in the fourth round.
24 October 2010: Osotspa Saraburi finished in 7th place in the Thai Premier League.
Current squad
As of January 25, 2010
Transfers
In
Out
League table
Results
Thai Premier League
FA Cup
Third round
Fourth round
League Cup
First round
1st leg
2nd leg
Second round
1st leg
2nd leg
Queen's Cup
References
Osotspa Saraburi F.C.
Osotspa Saraburi |
https://en.wikipedia.org/wiki/2010%20Royal%20Thai%20Army%20F.C.%20season | The 2010 season was the Royal Thai Army's 1st season back in the top division of Thai football after promotion from the 1st division. This article shows statistics of the club's players in the season, and also lists all matches that the club played in the season.
Team kit
Chronological list of events
10 November 2009: The Thai Premier League 2010 season first leg fixtures were announced.
6 October 2010: Royal Thai Army were knocked out of the Thai League Cup by Nong Khai in the second round.
13 October 2010: Royal Thai Army were relegated from the Thai Premier League on their 1st season back.
24 October 2010: Royal Thai Army finished in 20th place in the Thai Premier League.
Current squad
Results
Thai Premier League
League table
FA Cup
Third round
Fourth round
Quarter-final
League Cup
First round
1st leg
2nd leg
Second round
1st leg
2nd leg
Queen's Cup
References
2010
Royal Thai Army |
https://en.wikipedia.org/wiki/2010%20Samut%20Songkhram%20F.C.%20season | The 2010 season was Samut Songkhram's 3rd season in the top division of Thai football. This article shows statistics of the club's players in the season, and also lists all matches that the club played in the season.
Team kit
Chronological list of events
10 November 2009: The Thai Premier League 2010 season first leg fixtures were announced.
25 August 2010: Samut Songkhram were knocked out by Chonburi in the FA Cup fourth round.
24 October 2010: Samut Songkhram finished in 8th place in the Thai Premier League.
Players
Current squad
As of January 18, 2010
2010 Season transfers
In
Out
Results
Thai Premier League
League table
FA Cup
Third round
Fourth round
League Cup
First round
1st leg
2nd leg
First round
1st leg
2nd leg
Queen's Cup
References
2010
Thai football clubs 2010 season |
https://en.wikipedia.org/wiki/2010%20Sisaket%20F.C.%20season | The 2010 season was Sisaket's 1st season in the top division of Thai football after promotion from the 1st division. This article shows statistics of the club's players in the season, and also lists all matches that the club played in the season.
Team kit
Chronological list of events
10 November 2009: The Thai Premier League 2010 season first leg fixtures were announced.
20 October 2010: Sisaket were relegated from the Thai Premier League after their 1st season in the Premier League.
24 October 2010: Sisaket finished in 14th place in the Thai Premier League.
Players
Current squad
First team squad
As of July 31, 2010
2010 Season transfers
In
Out
Results
Thai Premier League
League table
FA Cup
Third round
Fourth round
Quarter-final
League Cup
First round
1st leg
2nd leg
Second round
1st leg
2nd leg
References
2010
Thai football clubs 2010 season |
https://en.wikipedia.org/wiki/2010%20Pattaya%20United%20F.C.%20season | The 2010 season was Pattaya United's 2nd season in the top division of Thai football. This article shows statistics of the club's players in the season, and also lists all matches that the club played in the season.
Team kit
Chronological list of events
10 November 2009: The Thai Premier League 2010 season first leg fixtures were announced.
15 September 2010: Pattaya United were knocked out of the Thai FA Cup by Chonburi in the quarter-final.
24 October 2010: Pattaya United finished in 6th place in the Thai Premier League.
Players
Current squad
2010 Season transfers
In
Out
Results
Thai Premier League
FA Cup
Third round
Fourth round
Quarter-finals
League Cup
First round
1st leg
2nd leg
Second round
1st leg
2nd leg
Third round
1st leg
2nd leg
League table
Queen's Cup
References
Thai football clubs 2010 season
Pattaya United F.C. seasons |
https://en.wikipedia.org/wiki/Crime%20in%20Illinois | In 2008, there were 446,135 crimes reported in the U.S. state of Illinois, including 790 murders.
State statistics
Policing
In 2019, Illinois had 846 state and local law enforcement agencies. Those agencies employed a total of 48,240 staff. Of the total staff, 38,539 were sworn officers (defined as those with general arrest powers). Illinois has 303 sworn officers per 100,000 residents.
Capital punishment laws
Capital punishment is not applied in Illinois. It was abolished by Governor Pat Quinn on March 9, 2011.
See also
Law of Illinois
Crime in Chicago
Crime in the United States
References |
https://en.wikipedia.org/wiki/Crime%20in%20Iowa | In 2019, there were 7,545 violent-crime incidents, and 8,237 offenses reported in the U.S. state of Iowa.
Statistics
In 2019, there were 7,545 violent-crime incidents, and 8,237 offenses reported in Iowa by 246 law enforcement agencies that submitted National Incident-Based Reporting System (NIBRS) data, and covers 98% of the total population.
Capital punishment laws
Capital punishment is not applied in this state.
References |
https://en.wikipedia.org/wiki/Crime%20in%20Kentucky | In 2020, there were 9,820 violent-crime incidents, and 11,349 offenses reported the U.S. state of Kentucky.
State statistics
In 2008, there were 122,960 crimes reported in Kentucky, including 198 murders.
In 2020, there were 9,820 violent-crime incidents, and 11,349 offenses reported in Kentucky by 423 law enforcement agencies that submitted National Incident-Based Reporting System data, and covers 99% of the total population.
2010
Capital punishment laws
Capital punishment is applied in this state.
References |
https://en.wikipedia.org/wiki/Crime%20in%20Michigan | In 2019, 43,686 crimes were reported in the U.S. state of Michigan. Crime statistics vary widely by location. For example, Dearborn has a murder rate of only 2.1 per 100,000 while sharing borders with Detroit (43.5 per 100,000) and Inkster (24.2 per 100,000), some of the highest rates in the state.
State statistics
By location
Detroit
Detroit had the 2nd highest violent crime rate in the nation in 2015 among cities with a population greater than 50,000. In 2013, with only 7% of the state population, the city of Detroit had 50% of all murders recorded in Michigan.
Detroit recorded 295 homicides in 2015 down from the recent high of 386 in 2012. The number of homicides peaked in 1974 at 714 and again in 1991 with 615. By the end of 2010, the homicide count fell to 308 for the year with an estimated population of just over 900,000, the lowest count and rate since 1967. According to a 2007 analysis, Detroit officials noted that about 65 to 70 percent of homicides in the city were confined to a narcotics catalyst.
The city has faced many cases of arson each year on Devil's Night, the evening before Halloween. The Angel's Night campaign, launched in the late 1990s, draws many volunteers to patrol the streets during Halloween week. The effort reduced arson: while there were 810 fires set in 1984, this was reduced to 742 in 1996. In recent years, fires on this three-night period have dropped even further. In 2009, the Detroit Fire Department reported 119 fires over this period, of which 91 were classified as suspected arsons.
Flint
The city of Flint has recorded murder rates higher than those of Detroit in some years. For example, in 2013 Flint had a murder rate of 48 per 100,000 compared to Detroit's 45. Flint's population fell below 100,000 and it is no longer tracked among the statistics of major cities.
Benton Harbor
The small city of Benton Harbor, population 10,000, had the highest total crime rate and highest property crime rate in Michigan in 2012. Its murder rate was the third highest in the state.
Grand Rapids
The second-largest city in Michigan, Grand Rapids recorded a murder rate of 13.8 per 100,000 in 2020, more than double of the United States rate of 7.8 per 100,000. The overall crime rate declined by one-third between 2003 and 2011, but Grand Rapids set a record with 38 homicides in 2020.
Policing
In 2018, Michigan had 564 state and local law enforcement agencies. Those agencies employed a total of 25,742 staff. Of the total staff, 18,193 were sworn officers (defined as those with general arrest powers).
Police ratio
In 2018, Michigan had 182 police officers per 100,000 residents.
Capital punishment laws
Capital punishment is not applied in this state. Capital punishment was banned early in state history and no executions were ever carried out by state authorities.
See also
Law of Michigan
List of homicides in Michigan
Crime in the United States
References
External links
Michigan Crime at DisasterCenter |
https://en.wikipedia.org/wiki/Crime%20in%20Nevada |
Statistics
Capital punishment laws
Capital punishment is legal in Nevada through lethal injection, and the most recent execution was issued to Daryl Mack on April 26, 2006.
References |
https://en.wikipedia.org/wiki/Crime%20in%20North%20Dakota | In 2020 there were 10,815 crimes reported in the U.S. state of North Dakota, including 32 murders.
Statistics
In 2010 there were 13,558 crimes reported in North Dakota, including 10 murders.
In 2011 there were 15,033 crimes reported, including 24 murders.
In 2012 there were 16,020 crimes reported, including 25 murders.
In 2013 there were 17,335 crimes reported, including 16 murders.
In 2014 there were 17,858 crimes reported, including 23 murders.
In 2015 there were 19,665 crimes reported, including 21 murders.
In 2016 there were 19,305 crimes reported, including 15 murders.
In 2017 there were 18,786 crimes reported, including 10 murders.
In 2018 there were 17,775 crimes reported, including 18 murders.
In 2019 there were 17,235 crimes reported, including 24 murders.
In 2020 there were 10,815 crimes reported, including 32 murders.
Capital punishment laws
Capital punishment is not applied in this state.
References |
https://en.wikipedia.org/wiki/Crime%20in%20Texas | In 2014 there were 923,348 crimes reported in the U.S. state of Texas, including 1,184 murders and 8,236 rapes.
State statistics
Policing
In 2008, Texas had 1,913 state and local law enforcement agencies. Those agencies employed a total of 96,116 staff. Of the total staff, 59,219 were sworn officers (defined as those with general arrest powers).
Police ratio
In 2008, Texas had 244 police officers per 100,000 residents.
According to the Texas Commission on Law Enforcement (TCOLE), the state average for police officers per 100,000 residents in Texas is 241 as of 2021. However, the ratio can vary among different cities and counties in Texas.
Capital punishment laws
Capital punishment is applied under Texas state law for capital murder if the perpetrator is 18 years of age and older and the prosecutor seeks the death penalty.
The federal death penalty may also be used in certain circumstances.
Incarceration
In 1974 the Texas Department of Corrections (TDC), since merged into the Texas Department of Criminal Justice (TDCJ), had about 17,000 prisoners; 44% were black, 39% were non-Hispanic white, 16% were Hispanic and Latino, and 1% were of other races. 96% were male and 4% were female. At the time all 14 prison units of the TDC were in Southeast Texas.
In 1974 the Federal Bureau of Prisons (BOP) operated four federal prisons in Texas: FCI Texarkana, FCI Seagoville, FPC Bryan (for women), and FCI La Tuna. These prisons had a combined population of about 2,300.
See also
Crime in the United States
Crime in Houston
Law of Texas
References
Further reading
Harnsberger, R. Scott. A Guide to Sources of Texas Criminal Justice Statistics [North Texas Crime and Criminal Justice Series, no.6]. Denton: University of North Texas Press, 2011. |
https://en.wikipedia.org/wiki/Crime%20in%20Vermont | The U.S. state of Vermont is the safest state in the country with a violent crime rate of 118 incidents per 100,000 state residents.
State statistics
In 2011 there were 16,011 crimes reported in Vermont, including 11 murders. In 2014 there were 10,173 crimes reported, including 10 murders.
From 2000 to 2013, the state experienced a 77% increase in treatment for all opiates. About 80% of all inmates are either addicted or in prison because of their addiction.
Capital punishment laws
Capital punishment is no longer applied in the state.
See also
Vermont Department of Corrections
References |
https://en.wikipedia.org/wiki/Crime%20in%20West%20Virginia | In 2014 there were 43,236 crimes reported n the U.S. state of West Virginia, including 74 murders.
State statistics
In 2008 there were 51,376 crimes reported in West Virginia, including 67 murders. West Virginia's ten worst cities statistically to live in are Fairmont, South Charleston, Martinsburg, Morgantown, Clarksburg, Parkersburg, Beckley, Wheeling, Charleston and Huntington with an annual crime rate of 394.
Capital punishment laws
Capital punishment is not applied in this state.
References |
https://en.wikipedia.org/wiki/Crime%20in%20Wyoming | This article discusses crime in the U.S. state of Wyoming.
Statistics
Capital punishment laws
Capital punishment is legal in Wyoming, although no one has been executed since January 22, 1992. On Feb 15, 2019, the Wyoming Senate rejected a bill to repeal the death penalty.
Footnotes
References |
https://en.wikipedia.org/wiki/Kenneth%20Stroud | Kenneth Stroud may refer to:
Ken Stroud, author of mathematics textbooks
Kenny Stroud, English footballer |
https://en.wikipedia.org/wiki/Enneagonal%20antiprism | In geometry, the enneagonal antiprism (or nonagonal antiprism) is one in an infinite set of convex antiprisms formed by triangle sides and two regular polygon caps, in this case two enneagons.
Antiprisms are similar to prisms except the bases are twisted relative to each other, and that the side faces are triangles, rather than quadrilaterals.
In the case of a regular 9-sided base, one usually considers the case where its copy is twisted by an angle 180°/n. Extra regularity is obtained by the line connecting the base centers being perpendicular to the base planes, making it a right antiprism. As faces, it has the two n-gonal bases and, connecting those bases, 2n isosceles triangles.
If faces are all regular, it is a semiregular polyhedron.
See also
External links
Virtual Reality Polyhedra www.georgehart.com: The Encyclopedia of Polyhedra
VRML model
polyhedronisme A9
Prismatoid polyhedra |
https://en.wikipedia.org/wiki/TenMarks%20Education%2C%20Inc. | TenMarks Education, Inc. was an American company that provided personalized online math practice and enrichment programs for K-Algebra/Geometry using a structured approach of practice, on-demand hints, video lessons and real-time results.
Founded in 2009, TenMarks Education had offices in San Francisco, CA and Boston, MA (formerly Newton, MA)
Amazon acquired TenMarks in 2013 and discontinued TenMarks apps in 2019.
Milestones
• June 2009: Launched Beta Program for Middle School Math (6th, 7th and 8th Grade), pilot at a Middle School in Boston
• July 2009: Launched the "Step Up" Math Summer Program for Middle Schoolers
• September 2009: Raised First Round of Angel Investment
• October 2009: TenMarks goes Live with Middle School Curriculum in Math
• December 2009: TenMarks Live for Elementary School - Math for Grades 3,4,5 added
• January 2010: TenMarks Rolled Out at two Schools to Measure Efficacy
• February 2010: TenMarks for High School Math Launched
• March 2010: TenMarks Placement Assessment Released to Help Automate Personalization of TenMarks Curriculum
• April 2010: TenMarks launches TeacherZone - A free library of math videos for teachers
• May 2010: TenMarks launches Summer Programs – Offering 19 different, customized step up and foundation programs for students to combat Summer Learning Loss
• May 2010: TenMarks wins Dr. Toy The Best Vacation Product Award
• October 2013: Amazon acquires TenMarks to assist in pushing education through the kindle market.
• April 2018 - Amazon announced plans to stop offering TenMarks’ learning apps after June 30, 2019.
References
External links
TenMarks Home Page
Amazon (company) acquisitions
Education companies of the United States
2009 establishments in California
2019 disestablishments in California |
https://en.wikipedia.org/wiki/Lebesgue%20integration | In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the axis. The Lebesgue integral, named after French mathematician Henri Lebesgue, extends the integral to a larger class of functions. It also extends the domains on which these functions can be defined.
Long before the 20th century, mathematicians already understood that for non-negative functions with a smooth enough graph—such as continuous functions on closed bounded intervals—the area under the curve could be defined as the integral, and computed using approximation techniques on the region by polygons. However, as the need to consider more irregular functions arose—e.g., as a result of the limiting processes of mathematical analysis and the mathematical theory of probability—it became clear that more careful approximation techniques were needed to define a suitable integral. Also, one might wish to integrate on spaces more general than the real line. The Lebesgue integral provides the necessary abstractions for this.
The Lebesgue integral plays an important role in probability theory, real analysis, and many other fields in mathematics. It is named after Henri Lebesgue (1875–1941), who introduced the integral . It is also a pivotal part of the axiomatic theory of probability.
The term Lebesgue integration can mean either the general theory of integration of a function with respect to a general measure, as introduced by Lebesgue, or the specific case of integration of a function defined on a sub-domain of the real line with respect to the Lebesgue measure.
Introduction
The integral of a positive function between limits and can be interpreted as the area under the graph of . This is straightforward for functions such as polynomials, but what does it mean for more exotic functions? In general, for which class of functions does "area under the curve" make sense? The answer to this question has great theoretical and practical importance.
As part of a general movement toward rigor in mathematics in the nineteenth century, mathematicians attempted to put integral calculus on a firm foundation. The Riemann integral—proposed by Bernhard Riemann (1826–1866)—is a broadly successful attempt to provide such a foundation. Riemann's definition starts with the construction of a sequence of easily calculated areas that converge to the integral of a given function. This definition is successful in the sense that it gives the expected answer for many already-solved problems, and gives useful results for many other problems.
However, Riemann integration does not interact well with taking limits of sequences of functions, making such limiting processes difficult to analyze. This is important, for instance, in the study of Fourier series, Fourier transforms, and other topics. The Lebesgue integral is better able to describe how and when it is possible to take limits under the integral |
https://en.wikipedia.org/wiki/Schur%E2%80%93Horn%20theorem | In mathematics, particularly linear algebra, the Schur–Horn theorem, named after Issai Schur and Alfred Horn, characterizes the diagonal of a Hermitian matrix with given eigenvalues. It has inspired investigations and substantial generalizations in the setting of symplectic geometry. A few important generalizations are Kostant's convexity theorem, Atiyah–Guillemin–Sternberg convexity theorem, Kirwan convexity theorem.
Statement
The inequalities above may alternatively be written:
The Schur–Horn theorem may thus be restated more succinctly and in plain English:
Schur–Horn theorem: Given any non-increasing real sequences of desired diagonal elements and desired eigenvalues there exists a Hermitian matrix with these eigenvalues and diagonal elements if and only if these two sequences have the same sum and for every possible integer the sum of the first desired diagonal elements never exceeds the sum of the first desired eigenvalues.
Reformation allowing unordered diagonals and eigenvalues
Although this theorem requires that and be non-increasing, it is possible to reformulate this theorem without these assumptions.
We start with the assumption
The left hand side of the theorem's characterization (that is, "there exists a Hermitian matrix with these eigenvalues and diagonal elements") depends on the order of the desired diagonal elements (because changing their order would change the Hermitian matrix whose existence is in question) but it does depend on the order of the desired eigenvalues
On the right hand right hand side of the characterization, only the values of depend on the assumption
Notice that this assumption means that the expression is just notation for the sum of the largest desired eigenvalues.
Replacing the expression with this written equivalent makes the assumption completely unnecessary:
Schur–Horn theorem: Given any desired real eigenvalues and a non-increasing real sequence of desired diagonal elements there exists a Hermitian matrix with these eigenvalues and diagonal elements if and only if these two sequences have the same sum and for every possible integer the sum of the first desired diagonal elements never exceeds the sum of the desired eigenvalues.
Permutation polytope generated by a vector
The permutation polytope generated by denoted by is defined as the convex hull of the set Here denotes the symmetric group on
In other words, the permutation polytope generated by is the convex hull of the set of all points in that can be obtained by rearranging the coordinates of The permutation polytope of for instance, is the convex hull of the set which in this case is the solid (filled) triangle whose vertices are the three points in this set.
Notice, in particular, that rearranging the coordinates of does not change the resulting permutation polytope; in other words, if a point can be obtained from by rearranging its coordinates, then
The following lemma characterizes the perm |
https://en.wikipedia.org/wiki/G-module | In mathematics, given a group G, a G-module is an abelian group M on which G acts compatibly with the abelian group structure on M. This widely applicable notion generalizes that of a representation of G. Group (co)homology provides an important set of tools for studying general G-modules.
The term G-module is also used for the more general notion of an R-module on which G acts linearly (i.e. as a group of R-module automorphisms).
Definition and basics
Let be a group. A left -module consists of an abelian group together with a left group action such that
g·(a1 + a2) = g·a1 + g·a2
where g·a denotes ρ(g,a). A right G-module is defined similarly. Given a left G-module M, it can be turned into a right G-module by defining a·g = g−1·a.
A function f : M → N is called a morphism of G-modules (or a G-linear map, or a G-homomorphism) if f is both a group homomorphism and G-equivariant.
The collection of left (respectively right) G-modules and their morphisms form an abelian category G-Mod (resp. Mod-G). The category G-Mod (resp. Mod-G) can be identified with the category of left (resp. right) ZG-modules, i.e. with the modules over the group ring Z[G].
A submodule of a G-module M is a subgroup A ⊆ M that is stable under the action of G, i.e. g·a ∈ A for all g ∈ G and a ∈ A. Given a submodule A of M, the quotient module M/A is the quotient group with action g·(m + A) = g·m + A.
Examples
Given a group G, the abelian group Z is a G-module with the trivial action g·a = a.
Let M be the set of binary quadratic forms f(x, y) = ax2 + 2bxy + cy2 with a, b, c integers, and let G = SL(2, Z) (the 2×2 special linear group over Z). Define
where
and (x, y)g is matrix multiplication. Then M is a G-module studied by Gauss. Indeed, we have
If V is a representation of G over a field K, then V is a G-module (it is an abelian group under addition).
Topological groups
If G is a topological group and M is an abelian topological group, then a topological G-module is a G-module where the action map G×M → M is continuous (where the product topology is taken on G×M).
In other words, a topological G-module is an abelian topological group M together with a continuous map G×M → M satisfying the usual relations g(a + a′) = ga + ga′, (gg′)a = g(g′a), and 1a = a.
Notes
References
Chapter 6 of
Group theory
Representation theory of groups |
https://en.wikipedia.org/wiki/Dyadic%20distribution | A dyadic (or 2-adic) distribution is a specific type of discrete probability distribution that is of some theoretical importance in data compression.
Definition
A dyadic distribution is a probability distribution whose probability mass function is
where is some whole number.
It is possible to find a binary code defined on this distribution, which has an average code length that is equal to the entropy.
References
Cover, T.M., Joy A. Thomas, J.A. (2006) Elements of information theory, Wiley.
Types of probability distributions
Data compression
Discrete distributions |
https://en.wikipedia.org/wiki/Talagrand%27s%20concentration%20inequality | In the probability theory field of mathematics , Talagrand's concentration inequality is an isoperimetric-type inequality for product probability spaces. It was first proved by the French mathematician Michel Talagrand. The inequality is one of the manifestations of the concentration of measure phenomenon.
Statement
The inequality states that if is a product space endowed with a product probability measure and
is a subset in this space, then for any
where is the complement of where this is defined by
and where is Talagrand's convex distance defined as
where , are -dimensional vectors with entries
respectively and is the -norm. That is,
References
Probabilistic inequalities
Measure theory |
https://en.wikipedia.org/wiki/Yuri%20Tarasov | Yuri Ivanovich Tarasov (; born 5 April 1960; died in 1999) was a Soviet Ukrainian professional football player.
Statistics for Metalist
Other – USSR Super Cup
The statistics in USSR Cups and Europe is made under the scheme "Autumn-Spring" and enlisted in a year of start of tournaments
Honours
Club records for most games played (214) and most goals scored (61) for FC Metalist Kharkiv in the Soviet Top League.
Soviet Cup winner: 1988.
4 games and 2 goals for FC Metalist Kharkiv in the 1988–89 European Cup Winners' Cup.
References
External links
Career summary by KLISF
1960 births
2000 deaths
Soviet men's footballers
Soviet expatriate men's footballers
Ukrainian men's footballers
Expatriate men's footballers in Israel
FC Metalist Kharkiv players
FC Mayak Valky players
Beitar Jerusalem F.C. players
FC Nyva Vinnytsia players
Soviet Top League players
Ukrainian Premier League players
Men's association football forwards
Soviet expatriate sportspeople in Israel
Footballers from Kharkiv Oblast |
https://en.wikipedia.org/wiki/P-adic%20Hodge%20theory | In mathematics, p-adic Hodge theory is a theory that provides a way to classify and study p-adic Galois representations of characteristic 0 local fields with residual characteristic p (such as Qp). The theory has its beginnings in Jean-Pierre Serre and John Tate's study of Tate modules of abelian varieties and the notion of Hodge–Tate representation. Hodge–Tate representations are related to certain decompositions of p-adic cohomology theories analogous to the Hodge decomposition, hence the name p-adic Hodge theory. Further developments were inspired by properties of p-adic Galois representations arising from the étale cohomology of varieties. Jean-Marc Fontaine introduced many of the basic concepts of the field.
General classification of p-adic representations
Let K be a local field with residue field k of characteristic p. In this article, a p-adic representation of K (or of GK, the absolute Galois group of K) will be a continuous representation ρ : GK→ GL(V), where V is a finite-dimensional vector space over Qp. The collection of all p-adic representations of K form an abelian category denoted in this article. p-adic Hodge theory provides subcollections of p-adic representations based on how nice they are, and also provides faithful functors to categories of linear algebraic objects that are easier to study. The basic classification is as follows:
where each collection is a full subcategory properly contained in the next. In order, these are the categories of crystalline representations, semistable representations, de Rham representations, Hodge–Tate representations, and all p-adic representations. In addition, two other categories of representations can be introduced, the potentially crystalline representations Reppcris(K) and the potentially semistable representations Reppst(K). The latter strictly contains the former which in turn generally strictly contains Repcris(K); additionally, Reppst(K) generally strictly contains Repst(K), and is contained in RepdR(K) (with equality when the residue field of K is finite, a statement called the p-adic monodromy theorem).
Period rings and comparison isomorphisms in arithmetic geometry
The general strategy of p-adic Hodge theory, introduced by Fontaine, is to construct certain so-called period rings such as BdR, Bst, Bcris, and BHT which have both an action by GK and some linear algebraic structure and to consider so-called Dieudonné modules
(where B is a period ring, and V is a p-adic representation) which no longer have a GK-action, but are endowed with linear algebraic structures inherited from the ring B. In particular, they are vector spaces over the fixed field . This construction fits into the formalism of B-admissible representations introduced by Fontaine. For a period ring like the aforementioned ones B∗ (for ∗ = HT, dR, st, cris), the category of p-adic representations Rep∗(K) mentioned above is the category of B∗-admissible ones, i.e. those p-adic representations V for which
or, equiv |
https://en.wikipedia.org/wiki/Lehmer%20code | In mathematics and in particular in combinatorics, the Lehmer code is a particular way to encode each possible permutation of a sequence of n numbers. It is an instance of a scheme for numbering permutations and is an example of an inversion table.
The Lehmer code is named in reference to Derrick Henry Lehmer, but the code had been known since 1888 at least.
The code
The Lehmer code makes use of the fact that there are
permutations of a sequence of n numbers. If a permutation σ is specified by the sequence (σ1, ..., σn) of its images of 1, ..., n, then it is encoded by a sequence of n numbers, but not all such sequences are valid since every number must be used only once. By contrast the encodings considered here choose the first number from a set of n values, the next number from a fixed set of values, and so forth decreasing the number of possibilities until the last number for which only a single fixed value is allowed; every sequence of numbers chosen from these sets encodes a single permutation. While several encodings can be defined, the Lehmer code has several additional useful properties; it is the sequence
in other words the term L(σ)i counts the number of terms in (σ1, ..., σn) to the right of σi that are smaller than it, a number between 0 and , allowing for different values.
A pair of indices (i,j) with and is called an inversion of σ, and L(σ)i counts the number of inversions (i,j) with i fixed and varying j. It follows that is the total number of inversions of σ, which is also the number of adjacent transpositions that are needed to transform the permutation into the identity permutation. Other properties of the Lehmer code include that the lexicographical order of the encodings of two permutations is the same as that of their sequences (σ1, ..., σn), that any value 0 in the code represents a right-to-left minimum in the permutation (i.e., a smaller than any to its right), and a value
at position i similarly signifies a right-to-left maximum, and that the Lehmer code of σ coincides with the factorial number system representation of its position in the list of permutations of n in lexicographical order (numbering the positions starting from 0).
Variations of this encoding can be obtained by counting inversions (i,j) for fixed j rather than fixed i, by counting inversions with a fixed smaller value rather than smaller index i, or by counting non-inversions rather than inversions; while this does not produce a fundamentally different type of encoding, some properties of the encoding will change correspondingly. In particular counting inversions with a fixed smaller value gives the inversion table of σ, which can be seen to be the Lehmer code of the inverse permutation.
Encoding and decoding
The usual way to prove that there are n! different permutations of n objects is to observe that the first object can be chosen in different ways, the next object in different ways (because choosing the same number as the first is |
https://en.wikipedia.org/wiki/1537%20in%20science | The year 1537 in science and technology included many events, some of which are listed here.
Mathematics
Niccolò Fontana Tartaglia publishes La Nova Scientia in Venice, applying mathematics to the study of ballistics.
Pedro Nunes publishes several treatises on navigation: Tratado em defensam da carta de marear, Tratado sobre certas dúvidas da navegação (including discussion of a rhumb line course) and Tratado da sphera com a Theorica do Sol e da Lua.
The first known complete English language arithmetic book, An Introduction for to Lerne to Recken with the Pen and with the Counters after the True Cast of Arsmetyke or Awgrym, an anonymous translation of Luca Pacioli's Summa de arithmetica, geometria, proportioni et proportionalità (Venice, 1494), is published at St Albans.
Births
Deaths
References
16th century in science
1530s in science |
https://en.wikipedia.org/wiki/Quadratic%20Lie%20algebra | A quadratic Lie algebra is a Lie algebra together with a compatible symmetric bilinear form. Compatibility means that it is invariant under the adjoint representation. Examples of such are semisimple Lie algebras, such as su(n) and sl(n,R).
Definition
A quadratic Lie algebra is a Lie algebra (g,[.,.]) together with a non-degenerate symmetric bilinear form that is invariant under the adjoint action, i.e.
([X,Y],Z)+(Y,[X,Z])=0
where X,Y,Z are elements of the Lie algebra g.
A localization/ generalization is the concept of Courant algebroid where the vector space g is replaced by (sections of) a vector bundle.
Examples
As a first example, consider Rn with zero-bracket and standard inner product
.
Since the bracket is trivial the invariance is trivially fulfilled.
As a more elaborate example consider so(3), i.e. R3 with base X,Y,Z, standard inner product, and Lie bracket
.
Straightforward computation shows that the inner product is indeed preserved. A generalization is the following.
Semisimple Lie algebras
A big group of examples fits into the category of semisimple Lie algebras, i.e. Lie algebras whose adjoint representation is faithful. Examples are sl(n,R) and su(n), as well as direct sums of them. Let thus g be a semi-simple Lie algebra with adjoint representation ad, i.e.
.
Define now the Killing form
.
Due to the Cartan criterion, the Killing form is non-degenerate if and only if the Lie algebra is semisimple.
If g is in addition a simple Lie algebra, then the Killing form is up to rescaling the only invariant symmetric bilinear form.
References
Lie algebras
Theoretical physics |
https://en.wikipedia.org/wiki/William%20C.%20Waterhouse | William Charles Waterhouse (December 31, 1941 – June 26, 2016) was an American mathematician. He was a professor emeritus of Mathematics at Pennsylvania State University, after having taught there for over 35 years. The early part of his career was at Cornell University. His research interests included abstract algebra, number theory, group schemes, and the history of mathematics.
Early life and education
Waterhouse was born in Galveston, Texas, on December 31, 1941, the son of William T. Waterhouse and Grace D. Waterhouse, but grew up in Denver, Colorado. His father was an engineer who was employed with the United States Bureau of Reclamation.
He attended East High School in Denver. In a high school mathematics competition spanning the states of Colorado, South Dakota, and Wyoming, he received the highest score in the competition's history and helped his school gain the top mark. As a senior, he took the Scholastic Aptitude Test and received a near-perfect 797 on the verbal portion and a perfect 800 on the math portion. He then received perfect 800 scores on three different college board Achievement Tests, those for English Composition, for Chemistry, and for advanced Mathematics, a feat that the Associated Press filed a story about and that ran in a number of newspapers around the country. Time magazine ran a profile of him as well. Waterhouse received the National Merit Scholarship and the General Motors Scholarship; he graduated from East High in 1959.
Waterhouse attended Harvard College. There he was a standout in the Putnam Competition: As a sophomore in the 1960 competition, he was not part of Harvard's three-person team that finished second overall, but he did achieve a top-ten individual mark; as a junior in the 1961 competitition, he attained the highest individual level – a top-five score – while helping his Harvard team to a fourth-place finish overall; and as a senior in the 1962 competition, he again was a Putnam Fellow with a top-five score and helped his Harvard team to a third-place finish overall.
After graduating from Harvard College with a bachelor's degree summa cum laude and being elected to Phi Beta Kappa, Waterhouse continued at Harvard Graduate School of Arts and Sciences where he received a master's degree. While in the graduate school, he was awarded a National Science Foundation Fellowship. He then received his Ph.D. in 1968 from Harvard for his thesis Abelian Varieties over Finite Fields under the supervision of John Tate.
Career
Waterhouse began teaching at Cornell University in 1968.
He had a career-long interest in the history of mathematics, and while at Cornell wrote a history of the early years of that university's Oliver Club, a discussion forum begun by pioneering Cornell mathematican James Edward Oliver in the 1890s.
Waterhouse remained an assistant professor at Cornell until 1975, at which point he was appointed an associate professor at Penn State. At Penn State, he subsequently became a |
https://en.wikipedia.org/wiki/Calcutta%20Mathematical%20Society | The Calcutta Mathematical Society (CalMathSoc) is an association of professional mathematicians dedicated to the interests of mathematical research and education in India. The Society has its head office located at Kolkata, India.
History
Calcutta Mathematical Society was established on 6 September 1908 under the stewardship of Sir Asutosh Mookerjee, the then Vice-Chancellor of Calcutta University. He was the founder president of the Society, and was assisted by Sir Gurudas Banerjee, Prof. C.E. Cullis and Prof. Gauri Sankar Dey as Vice Presidents and Prof. Phanindra Lal Ganguly as the Founder Secretary of the organization. It is said that the founders were inspired by the structure and operations of the London Mathematical Society while forming this organization.
Over more than the last 100 years, the Society has fostered teaching and research of theoretical and applied mathematical sciences through several pedagogic and technical activities. It is honored to be associated with legends like Albert Einstein, Anil Kumar Gain, S. Chandrasekhar, Abdus Salam and many more eminent scientists and researchers across the globe.
Activities
The main academic activities of the Society can broadly be classified under the following three heads: Memorial Lectures, Special Lectures and Regular Seminars and Symposiums. The Memorial Lectures are organized by the Society every year in honor of great academicians who were once associates and patrons of the organization. The Special Lectures are given on request by eminent researchers and scientists who visit Kolkata from time to time. The Seminars and Symposiums are generally held on an annual basis, focusing on the Pedagogic and Technical topics as well as topics of popular interest. 'International Symposium on Mathematical Physics in memory of S. Chandrasekhar with a special session on Abdus Salam' is notable one. National seminars on 'Theory & Methodology of Mathematics Teaching', 'Contribution of René Descartes','Contribution of Gottfried Leibniz', National Seminar on 'Power generation, Environment Pollution and related Mathematical Equations' were the most notable programme. Director of all those programs was Professor N. C. Ghosh. Professor C. G. Chakraborty was Director of National seminar on 'Satyndranath Bose & his contribution', Professor B. N. Mondal was Director of the 'Workshop on Mathematics Teaching Research & Training' organised at Calcutta Mathematical Society.
Publications
In terms of publishing substantial academic work, Calcutta Mathematical Society is the 1st Mathematical Society in India and Asia, and is the 13th in the whole world. The main publication of the Society is the Bulletin of Calcutta Mathematical Society, which commenced its journey back in 1909 and has been of great repute in the global scenario of mathematics for more than 100 years. The major publications of the Society are its four journals and bulletins as follows.
Bulletin of Calcutta Mathematical Society
Journal of |
https://en.wikipedia.org/wiki/Index%20of%20a%20Lie%20algebra | In algebra, let g be a Lie algebra over a field K. Let further be a one-form on g. The stabilizer gξ of ξ is the Lie subalgebra of elements of g that annihilate ξ in the coadjoint representation. The index of the Lie algebra is
Examples
Reductive Lie algebras
If g is reductive then the index of g is also the rank of g, because the adjoint and coadjoint representation are isomorphic and rk g is the minimal dimension of a stabilizer of an element in g. This is actually the dimension of the stabilizer of any regular element in g.
Frobenius Lie algebra
If ind g = 0, then g is called Frobenius Lie algebra. This is equivalent to the fact that the Kirillov form is non-singular for some ξ in g*. Another equivalent condition when g is the Lie algebra of an algebraic group G, is that g is Frobenius if and only if G has an open orbit in g* under the coadjoint representation.
Lie algebra of an algebraic group
If g is the Lie algebra of an algebraic group G, then the index of g is the transcendence degree of the field of rational functions on g* that are invariant under the (co)adjoint action of G.
References
Lie algebras |
https://en.wikipedia.org/wiki/2010%20BEC%20Tero%20Sasana%20F.C.%20season | The 2010 season was BEC's 14th season in the top division of Thai football. This article shows statistics of the club's players in the season, and also lists all matches that the club played in the season.
Team kit
Chronological list of events
10 November 2009: The Thai Premier League 2010 season first leg fixtures were announced.
January 2010: BEC announced that they will build an extra stand at the stadium in Nong Jork. The new stand will sit opposite the existing stand and will hold 2000 fans. It is expected to be completed in April
January 2010: BEC announce that they will use the Thephasadin Stadium whilst the Nong Jork ground is expanded.
11 August 2010: BEC Tero Sasana were knocked out of the Thai FA Cup by Nakhon Pathom in the fourth round.
15 September 2010: BEC Tero Sasana were knocked out of the Thai League Cup by Sriracha in the second round due to away goals.
24 October 2010: BEC Tero Sasana finished in 9th place in the Thai Premier League.
Squad
Transfers
In
Out
Results
Thai Premier League
League table
FA Cup
Third round
Fourth round
League Cup
First round
1st leg
2nd leg
Second round
1st leg
2nd leg
Queen's Cup
References
2010
Bec Tero Sasana |
https://en.wikipedia.org/wiki/2010%20TOT-CAT%20F.C.%20season | The 2010 season was TOT's 10th season in the top division of Thai football. This article shows statistics of the club's players in the season, and also lists all matches that the club played in the season.
Team kit
Chronological list of events
10 November 2009: The Thai Premier League 2010 season first leg fixtures were announced.
30 June 2010: TOT-CAT were knocked out of the FA Cup by Osotspa Saraburi in the third round.
24 October 2010: TOT-CAT finished in 12th place in the Thai Premier League.
Squad
Current squad
* Players in bold have senior international caps.
2010 Season transfers
In
Out
Results
Thai Premier League
League table
FA Cup
League Cup
First round
1st leg
2nd leg
Second round
1st leg
2nd leg
Queen's Cup
References
2010
Tot-Cat |
https://en.wikipedia.org/wiki/2010%20TTM%20Phichit%20F.C.%20season | The 2010 season was TTM's 10th season in the top division of Thai football. This article shows statistics of the club's players in the season, and also lists all matches that the club played in the season.
Team kit
Chronological list of events
10 November 2009: The Thai Premier League 2010 season first leg fixtures were announced.
5 September 2010: TTM Phichit were knocked out of the Thai FA Cup by Sisaket in the fourth round.
6 October 2010: TTM Phichit were knocked out of the Thai League Cup by Buriram PEA in the second round.
24 October 2010: TTM Phichit finished in 13th place in the Thai Premier League.
Squad
As of February 3, 2010
2010 Season transfers
In
Out
Results
Thai Premier League
League table
FA Cup
Third round
Fourth round
League Cup
First round
1st leg
2nd leg
Second round
1st leg
2nd leg
References
2010
Thai football clubs 2010 season |
https://en.wikipedia.org/wiki/2010%20Buriram%20PEA%20F.C.%20season | The 2010 season was PEA's 6th season in the top division of Thai football. This article shows statistics of the club's players in the season, and also lists all matches that the club played in the season.
Team kit
Chronological list of events
10 November 2009: The Thai Premier League 2010 season first leg fixtures were announced.
26 August 2010: Buriram PEA were knocked out of the Thai FA Cup by Royal Thai Army in the fourth round.
24 October 2010: Buriram PEA finished in 2nd place in the Thai Premier League.
Squad
As of August 1, 2010
Current squad
2010 Season transfers
In
Out
Results
Thai Premier League
League table
FA Cup
Third round
Fourth round
League Cup
First round
1st leg
2nd leg
Second round
1st leg
2nd leg
References
2010
Buriram Pea |
https://en.wikipedia.org/wiki/B-admissible%20representation | In mathematics, the formalism of B-admissible representations provides constructions of full Tannakian subcategories of the category of representations of a group G on finite-dimensional vector spaces over a given field E. In this theory, B is chosen to be a so-called (E, G)-regular ring, i.e. an E-algebra with an E-linear action of G satisfying certain conditions given below. This theory is most prominently used in p-adic Hodge theory to define important subcategories of p-adic Galois representations of the absolute Galois group of local and global fields.
(E, G)-rings and the functor D
Let G be a group and E a field. Let Rep(G) denote a non-trivial strictly full subcategory of the Tannakian category of E-linear representations of G on finite-dimensional vector spaces over E stable under subobjects, quotient objects, direct sums, tensor products, and duals.
An (E, G)-ring is a commutative ring B that is an E-algebra with an E-linear action of G. Let F = BG be the G-invariants of B. The covariant functor DB : Rep(G) → ModF defined by
is E-linear (ModF denotes the category of F-modules). The inclusion of DB(V) in B ⊗EV induces a homomorphism
called the comparison morphism.
Regular (E, G)-rings and B-admissible representations
An (E, G)-ring B is called regular if
B is reduced;
for every V in Rep(G), αB,V is injective;
every b ∈ B for which the line bE is G-stable is invertible in B.
The third condition implies F is a field. If B is a field, it is automatically regular.
When B is regular,
with equality if, and only if, αB,V is an isomorphism.
A representation V ∈ Rep(G) is called B-admissible if αB,V is an isomorphism. The full subcategory of B-admissible representations, denoted RepB(G), is Tannakian.
If B has extra structure, such as a filtration or an E-linear endomorphism, then DB(V) inherits this structure and the functor DB can be viewed as taking values in the corresponding category.
Examples
Let K be a field of characteristic p (a prime), and Ks a separable closure of K. If E = Fp (the finite field with p elements) and G = Gal(Ks/K) (the absolute Galois group of K), then B = Ks is a regular (E, G)-ring. On Ks there is an injective Frobenius endomorphism σ : Ks → Ks sending x to xp. Given a representation G → GL(V) for some finite-dimensional Fp-vector space V, is a finite-dimensional vector space over F=(Ks)G = K which inherits from B = Ks an injective function φD : D → D which is σ-semilinear (i.e. φ(ad) = σ(a)φ(d) for all a ∈ K and all d ∈ D). The Ks-admissible representations are the continuous ones (where G has the Krull topology and V has the discrete topology). In fact, is an equivalence of categories between the Ks-admissible representations (i.e. continuous ones) and the finite-dimensional vector spaces over K equipped with an injective σ-semilinear φ.
Potentially B-admissible representations
A potentially B-admissible representation captures the idea of a representation that becomes B-admissible when restricted to som |
https://en.wikipedia.org/wiki/Nadine%20O%27Connor | Nadine O'Connor (born March 5, 1942) is a retired mathematics teacher and a world record setting, hall of fame Masters Track and Field athlete. While she specializes in the pole vault, due to her athletic training, she also holds the World Masters Athletics world records in the 100 metres, 200 metres, the Indoor 60 metres and 200 metres. She also holds the American record in the long jump, 80 metre hurdles, 300 metre hurdles and pentathlon.
Pole vault
Coached by her partner, Olympian Bud Held, O'Connor continues to improve. Her current world record of 3.19 metres, set at age 67 in the W65 pole vault is so exceptional that it is superior to the record for athletes more than ten years her junior. In the women's rankings, she is 76 cm (about feet) above the next best performer in her age group, a beamonesque margin. It would also rate her in the top 25 performers in the Men's division, a situation unheard of in any other event. Even O'Connor herself did not jump as high when she was setting records in those younger divisions. She has already jumped higher than her world record, unofficially in her private practice facility near San Diego.
Decathlon
Her world record of 10,234 points in the emerging women's decathlon, which uses an age conversion scale for scoring, is the highest recorded score for any athlete, man or woman by almost 1200 points.
Recognition
She was elected into the USATF Masters Hall of Fame in 2007. She has been named USATF Female Masters Athlete of the year in 2005 and 2006 and USATF Athlete of the Week, out of athletes of all divisions and genders, three times
References
Living people
1942 births
American masters athletes
World record holders in masters athletics
American female pole vaulters
American female decathletes
Track and field athletes from California
21st-century American women
Female decathletes |
https://en.wikipedia.org/wiki/Steven%20E.%20Shreve | Steven Eugene Shreve is a mathematician and currently the Orion Hoch Professor of Mathematical Sciences at Carnegie Mellon University and the author of several major books on the mathematics of financial derivatives.
His first degree, awarded in 1972 was in German from West Virginia University. He then studied mathematics at Georg-August-Universität Göttingen. He then took a Masters in Electrical Engineering at the University of Illinois, where he completed a PhD in mathematics in 1977.
His textbook Stochastic Calculus for Finance is used by numerous graduate programs in quantitative finance. The book was voted "Best New Book in Quantitative Finance" in 2004 by members of Wilmott website, and has been highly praised by scholars in the field.
Shreve is a Fellow of the Institute of Mathematical Statistics.
Since 2006, he has held the Orion Hoch Chair Of Mathematical Sciences at CMU.
Books
Stochastic Optimal Control: The Discrete Time Case with Dimitri P. Bertsekas, Academic Press, 1978.
Brownian Motion and Stochastic Calculus with Ioannis Karatzas Springer-Verlag, 2nd Ed. 1991.
Methods of Mathematical Finance with Ioannis Karatzas Springer-Verlag, 1998
Stochastic Calculus for Finance. Volume I: The Binomial Asset Pricing ModelVolume II: Continuous-Time Models'' Springer-Verlag, 2004
The most recent volume was awarded "New Book of the Year" by Wilmott magazine.
References
External links
Shreve's Home Page
20th-century births
Living people
20th-century American mathematicians
21st-century American mathematicians
University of Illinois alumni
West Virginia University alumni
Carnegie Mellon University faculty
Financial economists
Mathematical analysts
Probability theorists
Fellows of the Institute of Mathematical Statistics
Year of birth missing (living people)
Place of birth missing (living people) |
https://en.wikipedia.org/wiki/Mladen%20Bestvina | Mladen Bestvina (born 1959) is a Croatian-American mathematician working in the area of geometric group theory. He is a Distinguished Professor in the Department of Mathematics at the University of Utah.
Biographical info
Mladen Bestvina is a three-time medalist at the International Mathematical Olympiad (two silver medals in 1976 and 1978 and a bronze medal in 1977). He received a B. Sc. in 1982 from the University of Zagreb. He obtained a PhD in Mathematics in 1984 at the University of Tennessee under the direction of John Walsh. He was a visiting scholar at the Institute for Advanced Study in 1987-88 and again in 1990–91. Bestvina had been a faculty member at UCLA, and joined the faculty in the Department of Mathematics at the University of Utah in 1993. He was appointed a Distinguished Professor at the University of Utah in 2008.
Bestvina received the Alfred P. Sloan Fellowship in 1988–89 and a Presidential Young Investigator Award in 1988–91.
Bestvina gave an invited address at the International Congress of Mathematicians in Beijing in 2002.
He also gave a Unni Namboodiri Lecture in Geometry and Topology at the University of Chicago.
Bestvina served as an Editorial Board member for the Transactions of the American Mathematical Society and as an associate editor of the Annals of Mathematics. Currently he is an editorial board member for Duke Mathematical Journal, Geometric and Functional Analysis, Geometry and Topology, the Journal of Topology and Analysis, Groups, Geometry and Dynamics, Michigan Mathematical Journal, Rocky Mountain Journal of Mathematics, and Glasnik Matematicki.
In 2012 he became a fellow of the American Mathematical Society.
Mathematical contributions
A 1988 monograph of Bestvina gave an abstract topological characterization of universal Menger compacta in all dimensions; previously only the cases of dimension 0 and 1 were well understood. John Walsh wrote in a review of Bestvina's monograph: 'This work, which formed the author's Ph.D. thesis at the University of Tennessee, represents a monumental step forward, having moved the status of the topological structure of higher-dimensional Menger compacta from one of "close to total ignorance" to one of "complete understanding".'
In a 1992 paper Bestvina and Feighn obtained a Combination Theorem for word-hyperbolic groups. The theorem provides a set of sufficient conditions for amalgamated free products and HNN extensions of word-hyperbolic groups to again be word-hyperbolic. The Bestvina–Feighn Combination Theorem became a standard tool in geometric group theory and has had many applications and generalizations (e.g.).
Bestvina and Feighn also gave the first published treatment of Rips' theory of stable group actions on R-trees (the Rips machine) In particular their paper gives a proof of the Morgan–Shalen conjecture that a finitely generated group G admits a free isometric action on an R-tree if and only if G is a free product of surface groups, free groups and free |
https://en.wikipedia.org/wiki/Unibranch%20local%20ring | In algebraic geometry, a local ring A is said to be unibranch if the reduced ring Ared (obtained by quotienting A by its nilradical) is an integral domain, and the integral closure B of Ared is also a local ring. A unibranch local ring is said to be geometrically unibranch if the residue field of B is a purely inseparable extension of the residue field of Ared. A complex variety X is called topologically unibranch at a point x if for all complements Y of closed algebraic subsets of X there is a fundamental system of neighborhoods (in the classical topology) of x whose intersection with Y is connected.
In particular, a normal ring is unibranch. The notions of unibranch and geometrically unibranch points are used in some theorems in algebraic geometry. For example, there is the following result:
Theorem Let X and Y be two integral locally noetherian schemes and a proper dominant morphism. Denote their function fields by K(X) and K(Y), respectively. Suppose that the algebraic closure of K(Y) in K(X) has separable degree n and that is unibranch. Then the fiber has at most n connected components. In particular, if f is birational, then the fibers of unibranch points are connected.
In EGA, the theorem is obtained as a corollary of Zariski's main theorem.
References
Algebraic geometry
Commutative algebra |
https://en.wikipedia.org/wiki/Constructible%20sheaf | In mathematics, a constructible sheaf is a sheaf of abelian groups over some topological space X, such that X is the union of a finite number of locally closed subsets on each of which the sheaf is a locally constant sheaf. It has its origins in algebraic geometry, where in étale cohomology constructible sheaves are defined in a similar way . For the derived category of constructible sheaves, see a section in ℓ-adic sheaf.
The finiteness theorem in étale cohomology states that the higher direct images of a constructible sheaf are constructible.
Definition of étale constructible sheaves on a scheme X
Here we use the definition of constructible étale sheaves from the book by Freitag and Kiehl referenced below. In what follows in this subsection, all sheaves on schemes are étale sheaves unless otherwise noted.
A sheaf is called constructible if can be written as a finite union of locally closed subschemes such that for each subscheme of the covering, the sheaf is a finite locally constant sheaf. In particular, this means for each subscheme appearing in the finite covering, there is an étale covering such that for all étale subschemes in the cover of , the sheaf is constant and represented by a finite set.
This definition allows us to derive, from Noetherian induction and the fact that an étale sheaf is constant if and only if its restriction from to is constant as well, where is the reduction of the scheme . It then follows that a representable étale sheaf is itself constructible.
Of particular interest to the theory of constructible étale sheaves is the case in which one works with constructible étale sheaves of Abelian groups. The remarkable result is that constructible étale sheaves of Abelian groups are precisely the Noetherian objects in the category of all torsion étale sheaves (cf. Proposition I.4.8 of Freitag-Kiehl).
Examples in algebraic topology
Most examples of constructible sheaves come from intersection cohomology sheaves or from the derived pushforward of a local system on a family of topological spaces parameterized by a base space.
Derived Pushforward on P1
One nice set of examples of constructible sheaves come from the derived pushforward (with or without compact support) of a local system on . Since any loop around is homotopic to a loop around we only have to describe the monodromy around and . For example, we can set the monodromy operators to be
where the stalks of our local system are isomorphic to . Then, if we take the derived pushforward or of for we get a constructible sheaf where the stalks at the points compute the cohomology of the local systems restricted to a neighborhood of them in .
Weierstrass Family of Elliptic Curves
For example, consider the family of degenerating elliptic curves
over . At this family of curves degenerates into a nodal curve. If we denote this family by then
and
where the stalks of the local system are isomorphic to . This local monodromy around of this loca |
https://en.wikipedia.org/wiki/Locally%20closed%20subset | In topology, a branch of mathematics, a subset of a topological space is said to be locally closed if any of the following equivalent conditions are satisfied:
is the intersection of an open set and a closed set in
For each point there is a neighborhood of such that is closed in
is an open subset of its closure
The set is closed in
is the difference of two closed sets in
is the difference of two open sets in
The second condition justifies the terminology locally closed and is Bourbaki's definition of locally closed. To see the second condition implies the third, use the facts that for subsets is closed in if and only if and that for a subset and an open subset
Examples
The interval is a locally closed subset of For another example, consider the relative interior of a closed disk in It is locally closed since it is an intersection of the closed disk and an open ball.
On the other hand, is not a locally closed subset of .
Recall that, by definition, a submanifold of an -manifold is a subset such that for each point in there is a chart around it such that Hence, a submanifold is locally closed.
Here is an example in algebraic geometry. Let U be an open affine chart on a projective variety X (in the Zariski topology). Then each closed subvariety Y of U is locally closed in X; namely, where denotes the closure of Y in X. (See also quasi-projective variety and quasi-affine variety.)
Properties
Finite intersections and the pre-image under a continuous map of locally closed sets are locally closed. On the other hand, a union and a complement of locally closed subsets need not be locally closed. (This motivates the notion of a constructible set.)
Especially in stratification theory, for a locally closed subset the complement is called the boundary of (not to be confused with topological boundary). If is a closed submanifold-with-boundary of a manifold then the relative interior (that is, interior as a manifold) of is locally closed in and the boundary of it as a manifold is the same as the boundary of it as a locally closed subset.
A topological space is said to be if every subset is locally closed. See Glossary of topology#S for more of this notion.
See also
Notes
References
Bourbaki, Topologie générale, 2007.
External links
General topology |
https://en.wikipedia.org/wiki/David%20Donoho | David Leigh Donoho (born March 5, 1957) is an American statistician. He is a professor of statistics at Stanford University, where he is also the Anne T. and Robert M. Bass Professor in the Humanities and Sciences. His work includes the development of effective methods for the construction of low-dimensional representations for high-dimensional data problems (multiscale geometric analysis), development of wavelets for denoising and compressed sensing. He was elected a Member of the American Philosophical Society in 2019.
Academic biography
Donoho did his undergraduate studies at Princeton University, graduating in 1978. His undergraduate thesis advisor was John W. Tukey. Donoho obtained his Ph.D. from Harvard University in 1983, under the supervision of Peter J. Huber. He was on the faculty of the University of California, Berkeley, from 1984 to 1990 before moving to Stanford.
He has been the Ph.D. advisor of at least 20 doctoral students, including Jianqing Fan and Emmanuel Candès.
Awards and honors
In 1991, Donoho was named a MacArthur Fellow. He was elected a Fellow of the American Academy of Arts and Sciences in 1992. He was the winner of the COPSS Presidents' Award in 1994. In 2001, he won the John von Neumann Prize of the Society for Industrial and Applied Mathematics. In 2002, he was appointed to the Bass professorship. He was elected a SIAM Fellow and a foreign associate of the French Académie des sciences in 2009, and in the same year received an honorary doctorate from the University of Chicago. In 2010 he won the Norbert Wiener Prize in Applied Mathematics, given jointly by SIAM and the American Mathematical Society. He is also a member of the United States National Academy of Sciences. In 2012 he became a fellow of the American Mathematical Society. In 2013 he was awarded the Shaw Prize for Mathematics. In 2016, he was awarded an honorary degree at the University of Waterloo. In 2018, he was awarded the Gauss Prize from IMU.
See also
Miriam Gasko Donoho, statistician married to Donoho
References
External links
David Donoho professional home page
Videos on International Congress of Mathematicians 2002, Beijing
1957 births
Living people
20th-century American mathematicians
21st-century American mathematicians
American statisticians
Fellows of the American Academy of Arts and Sciences
Fellows of the American Mathematical Society
Members of the American Philosophical Society
Members of the French Academy of Sciences
Members of the United States National Academy of Sciences
Princeton University alumni
Harvard University alumni
MacArthur Fellows
Stanford University Department of Statistics faculty
University of California, Berkeley College of Letters and Science faculty
Mathematical statisticians |
https://en.wikipedia.org/wiki/Rectified%206-simplexes | In six-dimensional geometry, a rectified 6-simplex is a convex uniform 6-polytope, being a rectification of the regular 6-simplex.
There are three unique degrees of rectifications, including the zeroth, the 6-simplex itself. Vertices of the rectified 6-simplex are located at the edge-centers of the 6-simplex. Vertices of the birectified 6-simplex are located in the triangular face centers of the 6-simplex.
Rectified 6-simplex
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S. It is also called 04,1 for its branching Coxeter-Dynkin diagram, shown as .
Alternate names
Rectified heptapeton (Acronym: ril) (Jonathan Bowers)
Coordinates
The vertices of the rectified 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,0,1,1). This construction is based on facets of the rectified 7-orthoplex.
Images
Birectified 6-simplex
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S. It is also called 03,2 for its branching Coxeter-Dynkin diagram, shown as .
Alternate names
Birectified heptapeton (Acronym: bril) (Jonathan Bowers)
Coordinates
The vertices of the birectified 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,1,1,1). This construction is based on facets of the birectified 7-orthoplex.
Images
Related uniform 6-polytopes
The rectified 6-simplex polytope is the vertex figure of the 7-demicube, and the edge figure of the uniform 241 polytope.
These polytopes are a part of 35 uniform 6-polytopes based on the [3,3,3,3,3] Coxeter group, all shown here in A6 Coxeter plane orthographic projections.
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.
o3x3o3o3o3o - ril, o3x3o3o3o3o - bril
External links
Polytopes of Various Dimensions
Multi-dimensional Glossary
6-polytopes |
https://en.wikipedia.org/wiki/Rectified%206-cubes | In six-dimensional geometry, a rectified 6-cube is a convex uniform 6-polytope, being a rectification of the regular 6-cube.
There are unique 6 degrees of rectifications, the zeroth being the 6-cube, and the 6th and last being the 6-orthoplex. Vertices of the rectified 6-cube are located at the edge-centers of the 6-cube. Vertices of the birectified 6-cube are located in the square face centers of the 6-cube.
Rectified 6-cube
Alternate names
Rectified hexeract (acronym: rax) (Jonathan Bowers)
Construction
The rectified 6-cube may be constructed from the 6-cube by truncating its vertices at the midpoints of its edges.
Coordinates
The Cartesian coordinates of the vertices of the rectified 6-cube with edge length are all permutations of:
Images
Birectified 6-cube
Alternate names
Birectified hexeract (acronym: brox) (Jonathan Bowers)
Rectified 6-demicube
Construction
The birectified 6-cube may be constructed from the 6-cube by truncating its vertices at the midpoints of its edges.
Coordinates
The Cartesian coordinates of the vertices of the rectified 6-cube with edge length are all permutations of:
Images
Related polytopes
These polytopes are part of a set of 63 uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex.
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.
o3x3o3o3o4o - rax, o3o3x3o3o4o - brox,
External links
Polytopes of Various Dimensions
Multi-dimensional Glossary
6-polytopes |
https://en.wikipedia.org/wiki/Rectified%205-cubes | In five-dimensional geometry, a rectified 5-cube is a convex uniform 5-polytope, being a rectification of the regular 5-cube.
There are 5 degrees of rectifications of a 5-polytope, the zeroth here being the 5-cube, and the 4th and last being the 5-orthoplex. Vertices of the rectified 5-cube are located at the edge-centers of the 5-cube. Vertices of the birectified 5-cube are located in the square face centers of the 5-cube.
Rectified 5-cube
Alternate names
Rectified penteract (acronym: rin) (Jonathan Bowers)
Construction
The rectified 5-cube may be constructed from the 5-cube by truncating its vertices at the midpoints of its edges.
Coordinates
The Cartesian coordinates of the vertices of the rectified 5-cube with edge length is given by all permutations of:
Images
Birectified 5-cube
E. L. Elte identified it in 1912 as a semiregular polytope, identifying it as Cr52 as a second rectification of a 5-dimensional cross polytope.
Alternate names
Birectified 5-cube/penteract
Birectified pentacross/5-orthoplex/triacontiditeron
Penteractitriacontiditeron (acronym: nit) (Jonathan Bowers)
Rectified 5-demicube/demipenteract
Construction and coordinates
The birectified 5-cube may be constructed by birectifying the vertices of the 5-cube at of the edge length.
The Cartesian coordinates of the vertices of a birectified 5-cube having edge length 2 are all permutations of:
Images
Related polytopes
Related polytopes
These polytopes are a part of 31 uniform polytera generated from the regular 5-cube or 5-orthoplex.
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.
o3x3o3o4o - rin, o3o3x3o4o - nit
External links
Polytopes of Various Dimensions
Multi-dimensional Glossary
5-polytopes |
https://en.wikipedia.org/wiki/Cantic%206-cube | In six-dimensional geometry, a cantic 6-cube (or a truncated 6-demicube) is a uniform 6-polytope.
Alternate names
Truncated 6-demicube/demihexeract (Acronym thax) (Jonathan Bowers)
Cartesian coordinates
The Cartesian coordinates for the 480 vertices of a cantic 6-cube centered at the origin and edge length 6 are coordinate permutations:
(±1,±1,±3,±3,±3,±3)
with an odd number of plus signs.
Images
Related polytopes
There are 47 uniform polytopes with D6 symmetry, 31 are shared by the B6 symmetry, and 16 are unique:
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.
x3x3o *b3o3o3o – thax
External links
Polytopes of Various Dimensions
Multi-dimensional Glossary
6-polytopes |
https://en.wikipedia.org/wiki/Cantic%205-cube | In geometry of five dimensions or higher, a cantic 5-cube, cantihalf 5-cube, truncated 5-demicube is a uniform 5-polytope, being a truncation of the 5-demicube. It has half the vertices of a cantellated 5-cube.
Cartesian coordinates
The Cartesian coordinates for the 160 vertices of a cantic 5-cube centered at the origin and edge length 6 are coordinate permutations:
(±1,±1,±3,±3,±3)
with an odd number of plus signs.
Alternate names
Cantic penteract, truncated demipenteract
Truncated hemipenteract (thin) (Jonathan Bowers)
Images
Related polytopes
It has half the vertices of the cantellated 5-cube, as compared here in the B5 Coxeter plane projections:
This polytope is based on the 5-demicube, a part of a dimensional family of uniform polytopes called demihypercubes for being alternation of the hypercube family.
There are 23 uniform 5-polytope that can be constructed from the D5 symmetry of the 5-demicube, of which are unique to this family, and 15 are shared within the 5-cube family.
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.
External links
Polytopes of Various Dimensions
Multi-dimensional Glossary
5-polytopes |
https://en.wikipedia.org/wiki/Truncated%205-cubes | In five-dimensional geometry, a truncated 5-cube is a convex uniform 5-polytope, being a truncation of the regular 5-cube.
There are four unique truncations of the 5-cube. Vertices of the truncated 5-cube are located as pairs on the edge of the 5-cube. Vertices of the bitruncated 5-cube are located on the square faces of the 5-cube. The third and fourth truncations are more easily constructed as second and first truncations of the 5-orthoplex.
Truncated 5-cube
Alternate names
Truncated penteract (Acronym: tan) (Jonathan Bowers)
Construction and coordinates
The truncated 5-cube may be constructed by truncating the vertices of the 5-cube at of the edge length. A regular 5-cell is formed at each truncated vertex.
The Cartesian coordinates of the vertices of a truncated 5-cube having edge length 2 are all permutations of:
Images
The truncated 5-cube is constructed by a truncation applied to the 5-cube. All edges are shortened, and two new vertices are added on each original edge.
Related polytopes
The truncated 5-cube, is fourth in a sequence of truncated hypercubes:
Bitruncated 5-cube
Alternate names
Bitruncated penteract (Acronym: bittin) (Jonathan Bowers)
Construction and coordinates
The bitruncated 5-cube may be constructed by bitruncating the vertices of the 5-cube at of the edge length.
The Cartesian coordinates of the vertices of a bitruncated 5-cube having edge length 2 are all permutations of:
Images
Related polytopes
The bitruncated 5-cube is third in a sequence of bitruncated hypercubes:
Related polytopes
This polytope is one of 31 uniform 5-polytope generated from the regular 5-cube or 5-orthoplex.
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.
o3o3o3x4x - tan, o3o3x3x4o - bittin
External links
Polytopes of Various Dimensions
Multi-dimensional Glossary
5-polytopes |
https://en.wikipedia.org/wiki/Truncated%205-orthoplexes | In five-dimensional geometry, a truncated 5-orthoplex is a convex uniform 5-polytope, being a truncation of the regular 5-orthoplex.
There are 4 unique truncations of the 5-orthoplex. Vertices of the truncation 5-orthoplex are located as pairs on the edge of the 5-orthoplex. Vertices of the bitruncated 5-orthoplex are located on the triangular faces of the 5-orthoplex. The third and fourth truncations are more easily constructed as second and first truncations of the 5-cube.
Truncated 5-orthoplex
Alternate names
Truncated pentacross
Truncated triacontaditeron (Acronym: tot) (Jonathan Bowers)
Coordinates
Cartesian coordinates for the vertices of a truncated 5-orthoplex, centered at the origin, are all 80 vertices are sign (4) and coordinate (20) permutations of
(±2,±1,0,0,0)
Images
The truncated 5-orthoplex is constructed by a truncation operation applied to the 5-orthoplex. All edges are shortened, and two new vertices are added on each original edge.
Bitruncated 5-orthoplex
The bitruncated 5-orthoplex can tessellate space in the tritruncated 5-cubic honeycomb.
Alternate names
Bitruncated pentacross
Bitruncated triacontiditeron (acronym: bittit) (Jonathan Bowers)
Coordinates
Cartesian coordinates for the vertices of a truncated 5-orthoplex, centered at the origin, are all 80 vertices are sign and coordinate permutations of
(±2,±2,±1,0,0)
Images
The bitrunacted 5-orthoplex is constructed by a bitruncation operation applied to the 5-orthoplex.
Related polytopes
This polytope is one of 31 uniform 5-polytopes generated from the regular 5-cube or 5-orthoplex.
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.
x3x3o3o4o - tot, o3x3x3o4o - bittit
External links
Polytopes of Various Dimensions
Multi-dimensional Glossary
5-polytopes |
https://en.wikipedia.org/wiki/Truncated%206-cubes | In six-dimensional geometry, a truncated 6-cube (or truncated hexeract) is a convex uniform 6-polytope, being a truncation of the regular 6-cube.
There are 5 truncations for the 6-cube. Vertices of the truncated 6-cube are located as pairs on the edge of the 6-cube. Vertices of the bitruncated 6-cube are located on the square faces of the 6-cube. Vertices of the tritruncated 6-cube are located inside the cubic cells of the 6-cube.
Truncated 6-cube
Alternate names
Truncated hexeract (Acronym: tox) (Jonathan Bowers)
Construction and coordinates
The truncated 6-cube may be constructed by truncating the vertices of the 6-cube at of the edge length. A regular 5-simplex replaces each original vertex.
The Cartesian coordinates of the vertices of a truncated 6-cube having edge length 2 are the permutations of:
Images
Related polytopes
The truncated 6-cube, is fifth in a sequence of truncated hypercubes:
Bitruncated 6-cube
Alternate names
Bitruncated hexeract (Acronym: botox) (Jonathan Bowers)
Construction and coordinates
The Cartesian coordinates of the vertices of a bitruncated 6-cube having edge length 2 are the permutations of:
Images
Related polytopes
The bitruncated 6-cube is fourth in a sequence of bitruncated hypercubes:
Tritruncated 6-cube
Alternate names
Tritruncated hexeract (Acronym: xog) (Jonathan Bowers)
Construction and coordinates
The Cartesian coordinates of the vertices of a tritruncated 6-cube having edge length 2 are the permutations of:
Images
Related polytopes
Related polytopes
These polytopes are from a set of 63 Uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex.
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.
o3o3o3o3x4x - tox, o3o3o3x3x4o - botox, o3o3x3x3o4o - xog
External links
Polytopes of Various Dimensions
Multi-dimensional Glossary
6-polytopes |
https://en.wikipedia.org/wiki/Truncated%206-orthoplexes | In six-dimensional geometry, a truncated 6-orthoplex is a convex uniform 6-polytope, being a truncation of the regular 6-orthoplex.
There are 5 degrees of truncation for the 6-orthoplex. Vertices of the truncated 6-orthoplex are located as pairs on the edge of the 6-orthoplex. Vertices of the bitruncated 6-orthoplex are located on the triangular faces of the 6-orthoplex. Vertices of the tritruncated 6-orthoplex are located inside the tetrahedral cells of the 6-orthoplex.
Truncated 6-orthoplex
Alternate names
Truncated hexacross
Truncated hexacontatetrapeton (Acronym: tag) (Jonathan Bowers)
Construction
There are two Coxeter groups associated with the truncated hexacross, one with the C6 or [4,3,3,3,3] Coxeter group, and a lower symmetry with the D6 or [33,1,1] Coxeter group.
Coordinates
Cartesian coordinates for the vertices of a truncated 6-orthoplex, centered at the origin, are all 120 vertices are sign (4) and coordinate (30) permutations of
(±2,±1,0,0,0,0)
Images
Bitruncated 6-orthoplex
Alternate names
Bitruncated hexacross
Bitruncated hexacontatetrapeton (Acronym: botag) (Jonathan Bowers)
Images
Related polytopes
These polytopes are a part of a set of 63 uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex.
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.
x3x3o3o3o4o - tag, o3x3x3o3o4o - botag
External links
Polytopes of Various Dimensions
Multi-dimensional Glossary
6-polytopes |
https://en.wikipedia.org/wiki/Truncated%206-simplexes | In six-dimensional geometry, a truncated 6-simplex is a convex uniform 6-polytope, being a truncation of the regular 6-simplex.
There are unique 3 degrees of truncation. Vertices of the truncation 6-simplex are located as pairs on the edge of the 6-simplex. Vertices of the bitruncated 6-simplex are located on the triangular faces of the 6-simplex. Vertices of the tritruncated 6-simplex are located inside the tetrahedral cells of the 6-simplex.
Truncated 6-simplex
Alternate names
Truncated heptapeton (Acronym: til) (Jonathan Bowers)
Coordinates
The vertices of the truncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,0,1,2). This construction is based on facets of the truncated 7-orthoplex.
Images
Bitruncated 6-simplex
Alternate names
Bitruncated heptapeton (Acronym: batal) (Jonathan Bowers)
Coordinates
The vertices of the bitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,1,2,2). This construction is based on facets of the bitruncated 7-orthoplex.
Images
Tritruncated 6-simplex
The tritruncated 6-simplex is an isotopic uniform polytope, with 14 identical bitruncated 5-simplex facets.
The tritruncated 6-simplex is the intersection of two 6-simplexes in dual configuration: and .
Alternate names
Tetradecapeton (as a 14-facetted 6-polytope) (Acronym: fe) (Jonathan Bowers)
Coordinates
The vertices of the tritruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,2,2). This construction is based on facets of the bitruncated 7-orthoplex. Alternately it can be centered on the origin as permutations of (-1,-1,-1,0,1,1,1).
Images
Related polytopes
Related uniform 6-polytopes
The truncated 6-simplex is one of 35 uniform 6-polytopes based on the [3,3,3,3,3] Coxeter group, all shown here in A6 Coxeter plane orthographic projections.
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.
o3x3o3o3o3o - til, o3x3x3o3o3o - batal, o3o3x3x3o3o - fe
External links
Polytopes of Various Dimensions
Multi-dimensional Glossary
6-polytopes |
https://en.wikipedia.org/wiki/Mathematical%20knowledge%20management | Mathematical knowledge management (MKM) is the study of how society can effectively make use of the vast and growing literature on mathematics. It studies approaches such as databases of mathematical knowledge, automated processing of formulae and the use of semantic information, and artificial intelligence. Mathematics is particularly suited to a systematic study of automated knowledge processing due to the high degree of interconnectedness between different areas of mathematics.
See also
OMDoc
QED manifesto
Areas of mathematics
MathML
External links
www.nist.gov/mathematical-knowledge-management, NIST's MKM page
The MKM Interest Group (archived)
9th International Conference on MKM, Paris, France, 2010
Big Proof Conference , a programme at the Isaac Newton Institute directed at the challenges of bringing proof technology into mainstream mathematical practice.
Big Proof Two
Mathematics and culture
Information science |
https://en.wikipedia.org/wiki/Detlef%20Laugwitz | Detlef Laugwitz (1932–2000) was a German mathematician and historian, who worked in differential geometry, history of mathematics, functional analysis, and non-standard analysis.
Biography
He was born on 11 May 1932 in Breslau, Germany. Starting in 1949, he studied mathematics, physics, and philosophy at the Georg-August-University at Göttingen, where he received his doctorate in 1954. Until 1956 he worked in the Mathematical Research Institute of Oberwolfach. In 1958 he became a lecturer at the Technical University of Munich, where he obtained his Habilitation. In 1958 he moved to the Technical University of Darmstadt, where in 1962 he became a professor, and remained until his retirement. From 1976 to 1984 he was a visiting professor at Caltech.
Work
Laugwitz worked in differential geometry of infinite dimensional vector spaces (his dissertation) and in Finsler geometry. In 1958 he and Curt Schmieden developed their own approach to infinitesimals through field extensions, independently of Abraham Robinson. They described this as "infinitesimal mathematics" and leading back to the historical roots in Leibniz. In 1996 he published the standard biography of Bernhard Riemann.
Notes
Publications
.
20th-century German mathematicians
German historians of mathematics
Differential geometers
Technical University of Munich alumni
Academic staff of the Technical University of Munich
1932 births
2000 deaths
20th-century German historians
Academic staff of Technische Universität Darmstadt |
https://en.wikipedia.org/wiki/Madhava%20series | In mathematics, a Madhava series is one of the three Taylor series expansions for the sine, cosine, and arctangent functions discovered in 14th or 15th century Kerala by the mathematician and astronomer Madhava of Sangamagrama (c. 1350 – c. 1425) or his followers in the Kerala school of astronomy and mathematics. Using modern notation, these series are:
All three series were later independently discovered in 17th century Europe. The series for sine and cosine were rediscovered by Isaac Newton in 1669, and the series for arctangent was rediscovered by James Gregory in 1671 and Gottfried Leibniz in 1673, and is conventionally called Gregory's series. The specific value can be used to calculate the circle constant , and the arctangent series for is conventionally called Leibniz's series.
In recognition of Madhava's priority, in recent literature these series are sometimes called the Madhava–Newton series, Madhava–Gregory series, or Madhava–Leibniz series (among other combinations).
No surviving works of Madhava contain explicit statements regarding the expressions which are now referred to as Madhava series. However, in the writing of later Kerala school mathematicians Nilakantha Somayaji and Jyeshthadeva one can find unambiguous attributions of these series to Madhava. These later works also include proofs and commentary which suggest how Madhava may have arrived at the series.
Madhava series in "Madhava's own words"
None of Madhava's works, containing any of the series expressions attributed to him, have survived. These series expressions are found in the writings of the followers of Madhava in the Kerala school. At many places these authors have clearly stated that these are "as told by Madhava". Thus the enunciations of the various series found in Tantrasamgraha and its commentaries can be safely assumed to be in "Madhava's own words". The translations of the relevant verses as given in the Yuktidipika commentary of Tantrasamgraha (also known as Tantrasamgraha-vyakhya) by Sankara Variar (circa. 1500 - 1560 CE) are reproduced below. These are then rendered in current mathematical notations.
Madhava's sine series
In Madhava's own words
Madhava's sine series is stated in verses 2.440 and 2.441 in Yukti-dipika commentary (Tantrasamgraha-vyakhya) by Sankara Variar. A translation of the verses follows.
Multiply the arc by the square of the arc, and take the result of repeating that (any number of times). Divide by the squares of the successive even numbers (such that current is multiplied by previous) increased by that number and multiplied by the square of the radius. Place the arc and the successive results so obtained one below the other, and subtract each from the one above. These together give the jiva [sine], as collected together in the verse beginning with "vidvan" etc.
Rendering in modern notations
Let r denote the radius of the circle and s the arc-length.
The following numerators are formed first:
These are then divided by q |
https://en.wikipedia.org/wiki/Hodge%20bundle | In mathematics, the Hodge bundle, named after W. V. D. Hodge, appears in the study of families of curves, where it provides an invariant in the moduli theory of algebraic curves. Furthermore, it has applications to the theory of modular forms on reductive algebraic groups and string theory.
Definition
Let be the moduli space of algebraic curves of genus g curves over some scheme. The Hodge bundle is a vector bundle on whose fiber at a point C in is the space of holomorphic differentials on the curve C. To define the Hodge bundle, let be the universal algebraic curve of genus g and let be its relative dualizing sheaf. The Hodge bundle is the pushforward of this sheaf, i.e.,
.
See also
ELSV formula
Notes
References
Moduli theory
Invariant theory
Algebraic curves |
https://en.wikipedia.org/wiki/Lefschetz%20theorem%20on%20%281%2C1%29-classes | In algebraic geometry, a branch of mathematics, the Lefschetz theorem on (1,1)-classes, named after Solomon Lefschetz, is a classical statement relating holomorphic line bundles on a compact Kähler manifold to classes in its integral cohomology. It is the only case of the Hodge conjecture which has been proved for all Kähler manifolds.
Statement of the theorem
Let X be a compact Kähler manifold. The first Chern class c1 gives a map from holomorphic line bundles to . By Hodge theory, the de Rham cohomology group H2(X, C) decomposes as a direct sum , and it can be proven that the image of c1 lies in H1,1(X). The theorem says that the map to is surjective.
In the special case where X is a projective variety, holomorphic line bundles are in bijection with linear equivalences class of divisors, and given a divisor D on X with associated line bundle O(D), the class c1(O(D)) is Poincaré dual to the homology class given by D. Thus, this establishes the usual formulation of the Hodge conjecture for divisors in projective varieties.
Proof using normal functions
Lefschetz's original proof worked on projective surfaces and used normal functions, which were introduced by Poincaré. Suppose that Ct is a pencil of curves on X. Each of these curves has a Jacobian variety JCt (if a curve is singular, there is an appropriate generalized Jacobian variety). These can be assembled into a family , the Jacobian of the pencil, which comes with a projection map π to the base T of the pencil. A normal function is a (holomorphic) section of π.
Fix an embedding of X in PN, and choose a pencil of curves Ct on X. For a fixed curve Γ on X, the intersection of Γ and Ct is a divisor on Ct, where d is the degree of X. Fix a base point p0 of the pencil. Then the divisor is a divisor of degree zero, and consequently it determines a class νΓ(t) in the Jacobian JCt for all t. The map from t to νΓ(t) is a normal function.
Henri Poincaré proved that for a general pencil of curves, all normal functions arose as νΓ(t) for some choice of Γ. Lefschetz proved that any normal function determined a class in H2(X, Z) and that the class of νΓ is the fundamental class of Γ. Furthermore, he proved that a class in H2(X, Z) is the class of a normal function if and only if it lies in H1,1. Together with Poincaré's existence theorem, this proves the theorem on (1,1)-classes.
Proof using sheaf cohomology
Because X is a complex manifold, it admits an exponential sheaf sequence
Taking sheaf cohomology of this exact sequence gives maps
The group of line bundles on X is isomorphic to . The first Chern class map is c1 by definition, so it suffices to show that i* is zero.
Because X is Kähler, Hodge theory implies that . However, i* factors through the map from H2(X, Z) to H2(X, C), and on H2(X, C), i* is the restriction of the projection onto H0,2(X). It follows that it is zero on , and consequently that the cycle class map is surjective.
References
Bibliography
Reprinted in
Theo |
https://en.wikipedia.org/wiki/1956%20Meistaradeildin | Statistics of Meistaradeildin in the 1956 season.
Overview
It was contested by 5 teams, and KÍ Klaksvík won the championship.
League table
Results
References
RSSSF
Meistaradeildin seasons
Faroe
Faroe |
https://en.wikipedia.org/wiki/1957%20Meistaradeildin | Statistics of Meistaradeildin in the 1957 season.
Overview
It was contested by 5 teams, and KÍ Klaksvík won the championship.
League table
Results
References
RSSSF
Meistaradeildin seasons
Faroe
Faroe |
https://en.wikipedia.org/wiki/1958%20Meistaradeildin | Statistics of Meistaradeildin in the 1958 season.
Overview
It was contested by 5 teams, and KÍ Klaksvík won the championship.
League table
Results
References
RSSSF
Meistaradeildin seasons
Faroe
Faroe |
https://en.wikipedia.org/wiki/1959%20Meistaradeildin | Statistics of Meistaradeildin in the 1959 season.
Overview
It was contested by 5 teams, and B36 Tórshavn won the championship.
League table
Results
References
RSSSF
Meistaradeildin seasons
Faroe
Faroe |
https://en.wikipedia.org/wiki/1960%20Meistaradeildin | Statistics of Meistaradeildin in the 1960 season.
Overview
It was contested by 4 teams, and Havnar Bóltfelag won the championship.
League table
Results
References
RSSSF
Meistaradeildin seasons
Faroe
Faroe |
https://en.wikipedia.org/wiki/1961%20Meistaradeildin | Statistics of Meistaradeildin in the 1961 season.
Overview
It was contested by 4 teams, and KÍ Klaksvík won the championship.
League table
Results
References
RSSSF
Meistaradeildin seasons
Faroe
Faroe |
https://en.wikipedia.org/wiki/1962%20Meistaradeildin | Statistics of Meistaradeildin in the 1962 season.
Overview
It was contested by 4 teams, and B36 Tórshavn won the championship.
League table
Results
References
RSSSF
Meistaradeildin seasons
Faroe
Faroe |
https://en.wikipedia.org/wiki/1963%20Meistaradeildin | Statistics of Meistaradeildin in the 1963 season.
Overview
It was contested by 4 teams, and Havnar Bóltfelag won the championship.
League table
Results
References
RSSSF
Meistaradeildin seasons
Faroe
Faroe |
https://en.wikipedia.org/wiki/1964%20Meistaradeildin | Statistics of Meistaradeildin in the 1964 season.
Overview
It was contested by 3 teams, and Havnar Bóltfelag won the championship.
League table
Results
References
RSSSF
Meistaradeildin seasons
Faroe
Faroe
pl:Meistaradeildin (1963) |
https://en.wikipedia.org/wiki/1965%20Meistaradeildin | Statistics of Meistaradeildin in the 1965 season.
Overview
It was contested by 4 teams, and Havnar Bóltfelag won the championship.
League table
Results
References
RSSSF
Meistaradeildin seasons
Faroe
Faroe |
https://en.wikipedia.org/wiki/1967%20Meistaradeildin | Statistics of Meistaradeildin in the 1967 season.
Overview
It was contested by 5 teams, and KÍ Klaksvík won the championship.
League table
Results
References
RSSSF
Meistaradeildin seasons
Faroe
Faroe |
https://en.wikipedia.org/wiki/1969%20Meistaradeildin | Statistics of Meistaradeildin in the 1969 season.
Overview
It was contested by 5 teams, and KÍ Klaksvík won the championship.
League table
Results
References
RSSSF
Meistaradeildin seasons
Faroe
Faroe |
https://en.wikipedia.org/wiki/1970%20Meistaradeildin | Statistics of Meistaradeildin in the 1970 season.
Overview
It was contested by 5 teams, and KÍ Klaksvík won the championship.
League table
Results
References
RSSSF
Meistaradeildin seasons
Faroe
Faroe |
https://en.wikipedia.org/wiki/1971%20Meistaradeildin | Statistics of Meistaradeildin in the 1971 season.
Overview
It was contested by 6 teams, and Havnar Bóltfelag won the championship.
League table
Results
References
RSSSF
Meistaradeildin seasons
Faroe
Faroe |
https://en.wikipedia.org/wiki/1972%20Meistaradeildin | Statistics of Meistaradeildin in the 1972 season.
Overview
It was contested by 6 teams, and KÍ Klaksvík won the championship.
League table
Results
References
RSSSF
Meistaradeildin seasons
Faroe
Faroe |
https://en.wikipedia.org/wiki/1973%20Meistaradeildin | Statistics of Meistaradeildin in the 1973 season.
Overview
It was contested by 6 teams, and Havnar Bóltfelag won the championship.
League table
Results
References
RSSSF
Meistaradeildin seasons
Faroe
Faroe |
https://en.wikipedia.org/wiki/1974%20Meistaradeildin | Statistics of Meistaradeildin in the 1974 season.
Overview
It was contested by 6 teams, and Havnar Bóltfelag won the championship.
League table
Results
References
RSSSF
Meistaradeildin seasons
Faroe
Faroe |
https://en.wikipedia.org/wiki/1975%20Meistaradeildin | Statistics of Meistaradeildin in the 1975 season.
Overview
There were 6 teams competing for the championship, and Havnar Bóltfelag won.
League table
Results
References
RSSSF
Meistaradeildin seasons
Faroe
Faroe
1975 in the Faroe Islands |
https://en.wikipedia.org/wiki/1924%20Latvian%20Football%20Championship | Statistics of Latvian Higher League in the 1924 season.
Overview
RFK won the championship.
League standings
Kaiserwald withdrew after 2 rounds because of the decision of Latvia Football Union (LFS - Latvijas Futbola Savieniba) which prohibited foreign players to participate in the championship.
2nd stage: RFK [Riga] - Cesu VB [Cesis] 5-1
References
RSSSF
1924
Lat
Lat
Football Championship |
https://en.wikipedia.org/wiki/1925%20Latvian%20Football%20Championship | Statistics of Latvian Higher League in the 1925 season.
Overview
RFK won the championship.
League standings
2nd stage: RFK [Riga] – Olimpija [Liepaja] 4–3
References
RSSSF
1925
Lat
Lat
Football Championship |
https://en.wikipedia.org/wiki/1926%20Latvian%20Football%20Championship | Statistics of the Latvian Higher League for the 1926 season- RFK were the league champions:
League standings
1st stage: Riga Group
2nd stage: Finals
References
External links
RSSSF
1926
Lat
Lat
Football Championship |
https://en.wikipedia.org/wiki/1927%20Latvian%20Higher%20League | Statistics of Latvian Higher League in the 1927 season.
Overview
It was contested by 4 teams, and Olimpija won the championship.
League standings
References
RSSSF
Latvian Higher League seasons
Lat
Lat
Football |
https://en.wikipedia.org/wiki/1928%20Latvian%20Higher%20League | Statistics of Latvian Higher League in the 1928 season.
Overview
It was contested by 5 teams, and Olimpija won the championship.
League standings
References
RSSSF
Latvian Higher League seasons
1
Latvia
Latvia |
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