source stringlengths 31 168 | text stringlengths 51 3k |
|---|---|
https://en.wikipedia.org/wiki/Transversal%20%28geometry%29 | In geometry, a transversal is a line that passes through two lines in the same plane at two distinct points. Transversals play a role in establishing whether two or more other lines in the Euclidean plane are parallel. The intersections of a transversal with two lines create various types of pairs of angles: consecutive interior angles, consecutive exterior angles, corresponding angles, and alternate angles. As a consequence of Euclid's parallel postulate, if the two lines are parallel, consecutive interior angles are supplementary, corresponding angles are equal, and alternate angles are equal.
Angles of a transversal
A transversal produces 8 angles, as shown in the graph at the above left:
4 with each of the two lines, namely α, β, γ and δ and then α1, β1, γ1 and δ1; and
4 of which are interior (between the two lines), namely α, β, γ1 and δ1 and 4 of which are exterior, namely α1, β1, γ and δ.
A transversal that cuts two parallel lines at right angles is called a perpendicular transversal. In this case, all 8 angles are right angles
When the lines are parallel, a case that is often considered, a transversal produces several congruent supplementary angles. Some of these angle pairs have specific names and are discussed below: corresponding angles, alternate angles, and consecutive angles.
Alternate angles
Alternate angles are the four pairs of angles that:
have distinct vertex points,
lie on opposite sides of the transversal and
both angles are interior or both angles are exterior.
If the two angles of one pair are congruent (equal in measure), then the angles of each of the other pairs are also congruent.
Proposition 1.27 of Euclid's Elements, a theorem of absolute geometry (hence valid in both hyperbolic and Euclidean Geometry), proves that if the angles of a pair of alternate angles of a transversal are congruent then the two lines are parallel (non-intersecting).
It follows from Euclid's parallel postulate that if the two lines are parallel, then the angles of a pair of alternate angles of a transversal are congruent (Proposition 1.29 of Euclid's Elements).
Corresponding angles
Corresponding angles are the four pairs of angles that:
have distinct vertex points,
lie on the same side of the transversal and
one angle is interior and the other is exterior.
Two lines are parallel if and only if the two angles of any pair of corresponding angles of any transversal are congruent (equal in measure).
Proposition 1.28 of Euclid's Elements, a theorem of absolute geometry (hence valid in both hyperbolic and Euclidean Geometry), proves that if the angles of a pair of corresponding angles of a transversal are congruent then the two lines are parallel (non-intersecting).
It follows from Euclid's parallel postulate that if the two lines are parallel, then the angles of a pair of corresponding angles of a transversal are congruent (Proposition 1.29 of Euclid's Elements).
If the angles of one pair of corresponding angles are congruent, then t |
https://en.wikipedia.org/wiki/Fundamentals%20of%20Physics | Fundamentals of Physics is a calculus-based physics textbook by David Halliday, Robert Resnick, and Jearl Walker. The textbook is currently in its 12th edition (published October, 2021).
The current version is a revised version of the original 1960 textbook Physics for Students of Science and Engineering by Halliday and Resnick, which was published in two parts (Part I containing Chapters 1-25 and covering mechanics and thermodynamics; Part II containing Chapters 26-48 and covering electromagnetism, optics, and introducing quantum physics). A 1966 revision of the first edition of Part I changed the title of the textbook to Physics.
It is widely used in colleges as part of the undergraduate physics courses, and has been well known to science and engineering students for decades as "the gold standard" of freshman-level physics texts. In 2002, the American Physical Society named the work the most outstanding introductory physics text of the 20th century.
The first edition of the book to bear the title Fundamentals of Physics, first published in 1970, was revised from the original text by Farrell Edwards and John J. Merrill. (Editions for sale outside the USA have the title Principles of Physics.) Walker has been the revising author since 1990.
In the more recent editions of the textbook, beginning with the fifth edition, Walker has included "checkpoint" questions. These are conceptual ranking-task questions that help the student before embarking on numerical calculations.
The textbook covers most of the basic topics in physics:
Mechanics
Waves
Thermodynamics
Electromagnetism
Optics
Special Relativity
The extended edition also contains introductions to topics such as quantum mechanics, atomic theory, solid-state physics, nuclear physics and cosmology. A solutions manual and a study guide are also available.
In popular culture
A copy of Fundamentals of Physics (3rd edition) appears on the bookshelf in Leonard and Sheldon's apartment in The Big Bang Theory.
See also
Physics education
References
External links
Fundamentals of physics at Wikibooks:
Physics textbooks |
https://en.wikipedia.org/wiki/List%20of%20Melbourne%20Storm%20records | This article contains records and statistics for the Melbourne Storm Rugby League Club who have played in the Australian National Rugby League competition since 1998. Statistical information on this page is for NRL games only and does not take into account games against non NRL teams e.g. World Club Challenge games.
This article is current as round 27 of the 2023 NRL season.
Sources of information: Rugby League Project and Rugby League Tables
Melbourne Storm Win–loss record
Overall
Melbourne Storm Win–loss records
Club honours
NRL Premierships
NRL Runners Up
NRL Minor Premierships
NRL Under-20s Premierships
World Club Challenge Titles
Finals Appearances
1998, 1999, 2000, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020, 2021, 2022, 2023
Club Records
Winning Games
Top 10 Biggest Wins
Top 10 Highest Scores
Most Consecutive Wins
19, Round 4 (2 April 2021) — Round 23 (19 August 2021)
Biggest Comeback
Recovered from a 22-point deficit.
Trailed Cronulla-Sutherland Sharks 22–0 after 32 minutes to win 36–32 at Shark Park (16 March 2003).
Losing Games
Top 10 Biggest Losses
Top 10 Highest Scores Conceded
Most Consecutive Losses
6, Round 7 (27 April 2002) – Round 13 (8 June 2002)
Worst Collapse
Surrendered an 18-point lead.
Led Canberra Raiders 18–0 after 29 minutes to lose 22–18 at Melbourne Rectangular Stadium (17 August 2019).
Individual Records
Games for club
NRL Games only
Players that have played 150+ games for the club
Try Scoring Records
Top 10 Most Tries For Club
Most Tries In A Match
Most Tries In A Season
Current Record in Bold
Points Scoring Records
Top 10 Most Points For Club
Most Points In A Season
Current Record in Bold
Top 10 Most Points In a Game
Most Goals In A Game
11, Matt Orford – July 2, 2004 vs Penrith Panthers at Olympic Park
Age Records
Oldest Player Fielded
37 years and 129 days, Cameron Smith – October 25, 2020 vs Penrith Panthers at Stadium Australia
Youngest Player Fielded
17 years and 347 days, Israel Folau – March 16, 2007 vs Wests Tigers at Olympic Park
Relationship Records
Father/Son relationships
Notable Storm relationships
Anderson family
Ben Anderson Melbourne Storm forward
Chris Anderson Melbourne Storm inaugural coach
Bromwich brothers
Jesse Bromwich Melbourne Storm NRL player
Kenny Bromwich Melbourne Storm NRL player
Chan family
Alex Chan Melbourne Storm forward
Joe Chan Melbourne Storm forward
Cross brothers
Ben Cross Melbourne Storm forward
Matt Cross Melbourne Storm forward
Johns family
Matthew Johns Melbourne Storm Assistant Coach
Cooper Johns Melbourne Storm playmaker
Kaufusi brothers
Antonio Kaufusi Melbourne Storm forward
Felise Kaufusi Melbourne Storm forward
Patrick Kaufusi Melbourne Storm forward
MacDougall brothers
Ben MacDougall Melbourne Storm centre
Luke MacDougall Melbourne Storm winger
Walters family
Kevin Walters Melbourne Storm Assistant Coach
Billy Walters Melbourne Storm hooker
Discipline
Pla |
https://en.wikipedia.org/wiki/Mayhew%20Prize | The Mayhew Prize is a prize awarded annually by the Faculty of Mathematics, University of Cambridge to the student showing the greatest distinction in applied mathematics, primarily for courses offered by DAMTP, but also for some courses offered by the Statistical Laboratory, in the MASt examinations, also known as Part III of the Mathematical Tripos. This includes about half of all students taking the MASt examinations, since the rest are taking mainly pure mathematics courses, and so winning the Mayhew Prize is not equivalent to obtaining the highest mark on the MASt examinations. Since 2018 the Faculty have also awarded the Pure Mathematics Prize for pure mathematics, but due to an absence of funds there is no equivalent monetary reward.
The Mayhew Prize was founded in 1923 through a donation of £500 by William Loudon Mollison, Master of Clare College, in memory of his wife Ellen Mayhew (1846-1917).
List of winners
Most of this list is from The Times newspaper archive. The winners of the prize are published in the Cambridge University Reporter.
See also
List of mathematics awards
References
Faculty of Mathematics, University of Cambridge
Mathematics awards |
https://en.wikipedia.org/wiki/Slope%20%28disambiguation%29 | Slope or gradient of a line describes its steepness, incline, or grade, in mathematics.
Slope may also refer to:
Slope landform, a type of landform
Grade (slope) of a topographic feature or constructed element
Piste, a marked track for snow skiing or snowboarding
Roof pitch, a steepness of a roof
Slope (album), a 2007 album by Steve Jansen
Slope (ethnic slur), a pejorative for Asian people
See also
Park Slope, a neighborhood in Brooklyn, New York City
Slope County, North Dakota
Slope rating in golf
Slope stability
Slope stability analysis |
https://en.wikipedia.org/wiki/Pentagonal%20bipyramidal%20molecular%20geometry | In chemistry, a pentagonal bipyramid is a molecular geometry with one atom at the centre with seven ligands at the corners of a pentagonal bipyramid. A perfect pentagonal bipyramid belongs to the molecular point group D5h.
The pentagonal bipyramid is a case where bond angles surrounding an atom are not identical (see also trigonal bipyramidal molecular geometry). This is one of the three common shapes for heptacoordinate transition metal complexes, along with the capped octahedron and the capped trigonal prism.
Pentagonal bipyramids are claimed to be promising coordination geometries for lanthanide-based single-molecule magnets, since (a) they present no extradiagonal crystal field terms, therefore minimising spin mixing, and (b) all of their diagonal terms are in first approximation protected from low-energy vibrations, minimising vibronic coupling.
Examples
Iodine heptafluoride (IF7) with 7 bonding groups
Osmium heptafluoride (OsF7)
Peroxo chromium(IV) complexes, e.g. [Cr(O2)2(NH3)3] where the peroxo groups occupy four of the planar positions.
and
References
External links
– Images of IF7
3D Chem – Chemistry, Structures, and 3D Molecules
IUMSC – Indiana University Molecular Structure Center
Stereochemistry
Molecular geometry |
https://en.wikipedia.org/wiki/KK-theory | In mathematics, KK-theory is a common generalization both of K-homology and K-theory as an additive bivariant functor on separable C*-algebras. This notion was introduced by the Russian mathematician Gennadi Kasparov in 1980.
It was influenced by Atiyah's concept of Fredholm modules for the Atiyah–Singer index theorem, and the classification of extensions of C*-algebras by Lawrence G. Brown, Ronald G. Douglas, and Peter Arthur Fillmore in 1977. In turn, it has had great success in operator algebraic formalism toward the index theory and the classification of nuclear C*-algebras, as it was the key to the solutions of many problems in operator K-theory, such as, for instance, the mere calculation of K-groups. Furthermore, it was essential in the development of the Baum–Connes conjecture and plays a crucial role in noncommutative topology.
KK-theory was followed by a series of similar bifunctor constructions such as the E-theory and the bivariant periodic cyclic theory, most of them having more category-theoretic flavors, or concerning another class of algebras rather than that of the separable C*-algebras, or incorporating group actions.
Definition
The following definition is quite close to the one originally given by Kasparov. This is the form in which most KK-elements arise in applications.
Let A and B be separable C*-algebras, where B is also assumed to be σ-unital. The set of cycles is the set of triples (H, ρ, F), where H is a countably generated graded Hilbert module over B, ρ is a *-representation of A on H as even bounded operators which commute with B, and F is a bounded operator on H of degree 1 which again commutes with B. They are required to fulfill the condition that
for a in A are all B-compact operators. A cycle is said to be degenerate if all three expressions are 0 for all a.
Two cycles are said to be homologous, or homotopic, if there is a cycle between A and IB, where IB denotes the C*-algebra of continuous functions from [0,1] to B, such that there is an even unitary operator from the 0-end of the homotopy to the first cycle, and a unitary operator from the 1-end of the homotopy to the second cycle.
The KK-group KK(A, B) between A and B is then defined to be the set of cycles modulo homotopy. It becomes an abelian group under the direct sum operation of bimodules as the addition, and the class of the degenerate modules as its neutral element.
There are various, but equivalent definitions of the KK-theory, notably the one due to Joachim Cuntz which eliminates bimodule and 'Fredholm' operator F from the picture and puts the accent entirely on the homomorphism ρ. More precisely it can be defined as the set of homotopy classes
,
of *-homomorphisms from the classifying algebra qA of quasi-homomorphisms to the C*-algebra of compact operators of an infinite dimensional separable Hilbert space tensored with B. Here, qA is defined as the kernel of the map from the C*-algebraic free product A*A of A with itself to A defi |
https://en.wikipedia.org/wiki/Operator%20K-theory | In mathematics, operator K-theory is a noncommutative analogue of topological K-theory for Banach algebras with most applications used for C*-algebras.
Overview
Operator K-theory resembles topological K-theory more than algebraic K-theory. In particular, a Bott periodicity theorem holds. So there are only two K-groups, namely K0, which is equal to algebraic K0, and K1. As a consequence of the periodicity theorem, it satisfies excision. This means that it associates to an extension of C*-algebras to a long exact sequence, which, by Bott periodicity, reduces to an exact cyclic 6-term-sequence.
Operator K-theory is a generalization of topological K-theory, defined by means of vector bundles on locally compact Hausdorff spaces. Here, a vector bundle over a topological space X is associated to a projection in the C* algebra of matrix-valued—that is, -valued—continuous functions over X. Also, it is known that isomorphism of vector bundles translates to Murray-von Neumann equivalence of the associated projection in K ⊗ C(X), where K is the compact operators on a separable Hilbert space.
Hence, the K0 group of a (not necessarily commutative) C*-algebra A is defined as Grothendieck group generated by the Murray-von Neumann equivalence classes of projections in K ⊗ C(X). K0 is a functor from the category of C*-algebras and *-homomorphisms, to the category of abelian groups and group homomorphisms. The higher K-functors are defined via a C*-version of the suspension: Kn(A) = K0(Sn(A)), where
SA = C0(0,1) ⊗ A.
However, by Bott periodicity, it turns out that Kn+2(A) and Kn(A) are isomorphic for each n, and thus the only groups produced by this construction are K0 and K1.
The key reason for the introduction of K-theoretic methods into the study of C*-algebras was the Fredholm index: Given a bounded linear operator on a Hilbert space that has finite-dimensional kernel and cokernel, one can associate to it an integer, which, as it turns out, reflects the 'defect' on the operator - i.e. the extent to which it is not invertible. The Fredholm index map appears in the 6-term exact sequence given by the Calkin algebra. In the analysis on manifolds, this index and its generalizations played a crucial role in the index theory of Atiyah and Singer, where the topological index of the manifold can be expressed via the index of elliptic operators on it. Later on, Brown, Douglas and Fillmore observed that the Fredholm index was the missing ingredient in classifying essentially normal operators up to certain natural equivalence. These ideas, together with Elliott's classification of AF C*-algebras via K-theory led to a great deal of interest in adapting methods such as K-theory from algebraic topology into the study of operator algebras.
This, in turn, led to K-homology, Kasparov's bivariant KK-theory, and, more recently, Connes and Higson's E-theory.
References
K-theory
Operator algebras
C*-algebras |
https://en.wikipedia.org/wiki/Relative%20neighborhood%20graph | In computational geometry, the relative neighborhood graph (RNG) is an undirected graph defined on a set of points in the Euclidean plane by connecting two points and by an edge whenever there does not exist a third point that is closer to both and than they are to each other. This graph was proposed by Godfried Toussaint in 1980 as a way of defining a structure from a set of points that would match human perceptions of the shape of the set.
Algorithms
showed how to construct the relative neighborhood graph of points in the plane efficiently in time. It can be computed in expected time, for random set of points distributed uniformly in the unit square. The relative neighborhood graph can be computed in linear time from the Delaunay triangulation of the point set.
Generalizations
Because it is defined only in terms of the distances between points, the relative neighborhood graph can be defined for point sets in any and for non-Euclidean metrics. Computing the relative neighborhood graph, for higher-dimensional point sets, can be done in time .
Related graphs
The relative neighborhood graph is an example of a lens-based beta skeleton. It is a subgraph of the Delaunay triangulation. In turn, the Euclidean minimum spanning tree is a subgraph of it, from which it follows that it is a connected graph.
The Urquhart graph, the graph formed by removing the longest edge from every triangle in the Delaunay triangulation, was originally proposed as a fast method to compute the relative neighborhood graph. Although the Urquhart graph sometimes differs from the relative neighborhood graph it can be used as an approximation to the relative neighborhood graph.
References
Geometric graphs |
https://en.wikipedia.org/wiki/Nonstandard%20%28disambiguation%29 | Nonstandard describes a state not conforming to standards.
Nonstandard or non-standard may also refer to:
non-standard analysis, the use of infinitesimals to formulate calculus
non-standard model, in model theory, a model that is not isomorphic to the standard model, especially models of Peano arithmetic
non-standard cosmology, models which do not conform to current scientific consensus |
https://en.wikipedia.org/wiki/Journal%20of%20the%20Royal%20Statistical%20Society | The Journal of the Royal Statistical Society is a peer-reviewed scientific journal of statistics. It comprises three series and is published by Oxford University Press for the Royal Statistical Society.
History
The Statistical Society of London was founded in 1834, but would not begin producing a journal for four years. From 1834 to 1837, members of the society would read the results of their studies to the other members, and some details were recorded in the proceedings. The first study reported to the society in 1834 was a simple survey of the occupations of people in Manchester, England. Conducted by going door-to-door and inquiring, the study revealed that the most common profession was mill-hands, followed closely by weavers.
When founded, the membership of the Statistical Society of London overlapped almost completely with the statistical section of the British Association for the Advancement of Science. In 1837 a volume of Transactions of the Statistical Society of London were written, and in May 1838 the society began its journal. The first editor-in-chief of the journal was Rawson W. Rawson. In the early days of the society and the journal, there was dispute over whether or not opinions should be expressed, or merely the numbers. The symbol of the society was a wheatsheaf, representing a bundle of facts, and the motto Aliis exterendum, Latin for "to be threshed out by others." Many early members chafed under this prohibition, and in 1857 the motto was dropped.
From 1838 to 1886, the journal was published as the Journal of the Statistical Society of London (). In 1887 it was renamed the Journal of the Royal Statistical Society () when the society was granted a Royal Charter.
On its centenary in 1934, the society inaugurated a Supplement to the Journal of the Royal Statistical Society to publish work on industrial and agricultural applications. In 1948 the society reorganised its journals and the main journal became the Journal of the Royal Statistical Society, Series A (General) () and the supplement became Series B (Statistical Methodology). In 1988, Series A changed its name to Series A (Statistics in Society).
In 1952, the society founded Applied Statistics of the Journal of the Royal Statistical Society which became Series C (Applied Statistics). After merging with the Institute of Statisticians in 1993, the society published Series D (The Statistician) (), but this journal was closed in 2003, to be replaced by Significance.
Discussion papers
Traditionally papers were presented at ordinary meetings of the society and those present, whether fellows or not, were invited to comment on the presentation. The paper and subsequent discussion would then be published in the journal. This followed a format used by other scientific societies of the time, such as the Royal Society. This practice continues although papers are selected for reading and go through peer review before being presented. It is considered a significant recognitio |
https://en.wikipedia.org/wiki/Chicago%20school%20%28mathematical%20analysis%29 | The Chicago school of mathematical analysis is a school of thought in mathematics that emphasizes the applications of Fourier analysis to the study of partial differential equations. Mathematician Antoni Zygmund co-founded the school with his doctoral student Alberto Calderón at the University of Chicago in the 1950s. Over the years, Zygmund mentored over 40 doctoral students at the University of Chicago.
Key people
Antoni Zygmund
Alberto Calderón
Paul Cohen, Fields Medal winner (1966)
Charles Fefferman, Fields Medal winner (1978)
Eli Stein
Comments
The Chicago school of analysis is considered to be one of the strongest schools of mathematical analysis in the 20th century, which was responsible for some of the most important developments in analysis.
Awards
In 1986, Antoni Zygmund received the National Medal of Science, in part for his "creation and leadership of the strongest school of analytical research in the contemporary mathematical world."
See also
Joseph Fourier
Mathematical analysis
References
University of Chicago
Philosophical schools and traditions |
https://en.wikipedia.org/wiki/Semialgebraic%20set | In mathematics, a basic semialgebraic set is a set defined by polynomial equalities and polynomial inequalities, and a semialgebraic set is a finite union of basic semialgebraic sets. A semialgebraic function is a function with a semialgebraic graph. Such sets and functions are mainly studied in real algebraic geometry which is the appropriate framework for algebraic geometry over the real numbers.
Definition
Let be a real closed field. (For example could be the field of real numbers .)
A subset of is a semialgebraic set if it is a finite union of sets defined by polynomial equalities of the form and of sets defined by polynomial inequalities of the form
Properties
Similarly to algebraic subvarieties, finite unions and intersections of semialgebraic sets are still semialgebraic sets. Furthermore, unlike subvarieties, the complement of a semialgebraic set is again semialgebraic. Finally, and most importantly, the Tarski–Seidenberg theorem says that they are also closed under the projection operation: in other words a semialgebraic set projected onto a linear subspace yields another semialgebraic set (as is the case for quantifier elimination). These properties together mean that semialgebraic sets form an o-minimal structure on R.
A semialgebraic set (or function) is said to be defined over a subring A of R if there is some description as in the definition, where the polynomials can be chosen to have coefficients in A.
On a dense open subset of the semialgebraic set S, it is (locally) a submanifold. One can define the dimension of S to be the largest dimension at points at which it is a submanifold. It is not hard to see that a semialgebraic set lies inside an algebraic subvariety of the same dimension.
See also
Łojasiewicz inequality
Existential theory of the reals
Subanalytic set
piecewise algebraic space
References
.
.
.
External links
PlanetMath page
Real algebraic geometry |
https://en.wikipedia.org/wiki/Serghaya | Serghaya or Sirghaya () is a small town located in the Damascus countryside in south west Syria. According to the Syria Central Bureau of Statistics (CBS), Serghaya had a population of 7,501 in the 2004 census. Its inhabitants are predominantly Sunni Muslims.
Geography
It is from Damascus and above sea level. It is at the foot of the Anti-Lebanon Mountain.
Serghaya has a moderate climate with a temperature that varies from 25 to 32 degrees Celsius all summer season and cold climate with temperature varies from -5 up to 10 all winter when snow covers the land and mountain.
It is connected to Damascus via Al-Zabadani and also has old rail reaches to Beirut via Riyaq (or Rayak), Bekaa.
Nearby Towns
West :`Utayb (3.5 nm)
North: Yahfufah (3.4 nm), Al Khuraybah (4.0 nm), Ma`rabun (3.4 nm)
East: Al `Uwayni (0.4 nm)
South: `Ayn al Hawr (2.2 nm)
Climate
In Serghaya, there is a cool summer Mediterranean climate. Rainfall is higher in winter than in summer. The Köppen-Geiger climate classification is Csc. The average annual temperature in Serghaya is . About of precipitation falls annually.
Economy
Economy for this town is based on agricultural activities and the main crops are (Apple, Cherry, Pear, Peach and Apricot).
References
Bibliography
Populated places in Al-Zabadani District
Towns in Syria |
https://en.wikipedia.org/wiki/Topology%20%28musical%20ensemble%29 | Topology is an indie classical quintet from Australia, formed in 1997 and they eading Australian new music ensemble. They perform throughout Australia and abroad and have to date released 14 albums, including one with rock/electronica band Full Fathom Five and one with contemporary ensemble Loops. They were formerly the resident ensembles at the University of Western Sydney and Brisbane Powerhouse. The group works with composers including Tim Brady, Andrew Poppy, Michael Nyman, Jeremy Peyton Jones, Terry Riley, Steve Reich, Philip Glass, Carl Stone, Pand aul Dresher, as well as with many Australian composers.
In 2009, Topology won the Outstanding Contribution by an Organisation award at the Australasian Performing Right Association (APRA) Classical Music Awards for their work on the 2008 Brisbane Powerhouse Series.
Members
Bernard Hoey (viola)
Christa Powell (violin)
John Babbage (saxophone)
Kylie Davidson (piano)
Therese Milanovic (piano)
Robert Davidson (bass)
Discography
Albums
Awards and nominations
APRA Awards
APRA Awards of 2009: Outstanding Contribution by an Organisation win for the 2008 Brisbane Powerhouse Series by Topology.
ARIA Music Awards
The ARIA Music Awards are presented annually from 1987 by the Australian Recording Industry Association (ARIA).
!
|-
| 2014
| Share House
| Best Classical Album
|
|
|-
References
External links
Official website
Official Facebook Page
Official YouTube Page
Official Twitter Page
APRA Award winners
Contemporary classical music ensembles
New South Wales musical groups
Musical groups from Brisbane |
https://en.wikipedia.org/wiki/Asymmetric%20norm | In mathematics, an asymmetric norm on a vector space is a generalization of the concept of a norm.
Definition
An asymmetric norm on a real vector space is a function that has the following properties:
Subadditivity, or the triangle inequality:
Nonnegative homogeneity: and every non-negative real number
Positive definiteness:
Asymmetric norms differ from norms in that they need not satisfy the equality
If the condition of positive definiteness is omitted, then is an asymmetric seminorm. A weaker condition than positive definiteness is non-degeneracy: that for at least one of the two numbers and is not zero.
Examples
On the real line the function given by
is an asymmetric norm but not a norm.
In a real vector space the of a convex subset that contains the origin is defined by the formula
for
This functional is an asymmetric seminorm if is an absorbing set, which means that and ensures that is finite for each
Corresponce between asymmetric seminorms and convex subsets of the dual space
If is a convex set that contains the origin, then an asymmetric seminorm can be defined on by the formula
For instance, if is the square with vertices then is the taxicab norm Different convex sets yield different seminorms, and every asymmetric seminorm on can be obtained from some convex set, called its dual unit ball. Therefore, asymmetric seminorms are in one-to-one correspondence with convex sets that contain the origin. The seminorm is
positive definite if and only if contains the origin in its topological interior,
degenerate if and only if is contained in a linear subspace of dimension less than and
symmetric if and only if
More generally, if is a finite-dimensional real vector space and is a compact convex subset of the dual space that contains the origin, then is an asymmetric seminorm on
See also
References
S. Cobzas, Functional Analysis in Asymmetric Normed Spaces, Frontiers in Mathematics, Basel: Birkhäuser, 2013; .
Linear algebra
Norms (mathematics) |
https://en.wikipedia.org/wiki/Robbins%20lemma | In statistics, the Robbins lemma, named after Herbert Robbins, states that if X is a random variable having a Poisson distribution with parameter λ, and f is any function for which the expected value E(f(X)) exists, then
Robbins introduced this proposition while developing empirical Bayes methods.
References
Theorems in statistics
Lemmas
Poisson distribution |
https://en.wikipedia.org/wiki/Jihoz%C3%A1pad | Jihozápad (Southwest) is statistical area of the Nomenclature of Territorial Units for Statistics, level NUTS 2. It includes the Plzeň Region and the South Bohemian Region.
It covers an area of 17 617 km2 and 1,214,450 inhabitants (population density 67 inhabitants/km2).
Economy
The Gross domestic product (GDP) of the region was 20.6 billion € in 2018, accounting for 9.9% of Czech economic output. GDP per capita adjusted for purchasing power was 23,600 € or 78% of the EU27 average in the same year. The GDP per employee was also 75% of the EU average.
References
See also
NUTS of the Czech Republic
NUTS 2 statistical regions of the European Union
Subdivisions of the Czech Republic |
https://en.wikipedia.org/wiki/Schilder%27s%20theorem | In mathematics, Schilder's theorem is a generalization of the Laplace method from integrals on to functional Wiener integration. The theorem is used in the large deviations theory of stochastic processes. Roughly speaking, out of Schilder's theorem one gets an estimate for the probability that a (scaled-down) sample path of Brownian motion will stray far from the mean path (which is constant with value 0). This statement is made precise using rate functions. Schilder's theorem is generalized by the Freidlin–Wentzell theorem for Itō diffusions.
Statement of the theorem
Let C0 = C0([0, T]; Rd) be the Banach space of continuous functions such that , equipped with the supremum norm ||·||∞ and be the subspace of absolutely continuous functions whose derivative is in (the so-called Cameron-Martin space). Define the rate function
on and let be two given functions, such that (the "action") has a unique minimum .
Then under some differentiability and growth assumptions on which are detailed in Schilder 1966, one has
where denotes expectation with respect to the Wiener measure on and is the Hessian of at the minimum ; is meant in the sense of an inner product.
Application to large deviations on the Wiener measure
Let B be a standard Brownian motion in d-dimensional Euclidean space Rd starting at the origin, 0 ∈ Rd; let W denote the law of B, i.e. classical Wiener measure. For ε > 0, let Wε denote the law of the rescaled process B. Then, on the Banach space C0 = C0([0, T]; Rd) of continuous functions such that , equipped with the supremum norm ||·||∞, the probability measures Wε satisfy the large deviations principle with good rate function I : C0 → R ∪ {+∞} given by
if ω is absolutely continuous, and I(ω) = +∞ otherwise. In other words, for every open set G ⊆ C0 and every closed set F ⊆ C0,
and
Example
Taking ε = 1/c2, one can use Schilder's theorem to obtain estimates for the probability that a standard Brownian motion B strays further than c from its starting point over the time interval [0, T], i.e. the probability
as c tends to infinity. Here Bc(0; ||·||∞) denotes the open ball of radius c about the zero function in C0, taken with respect to the supremum norm. First note that
Since the rate function is continuous on A, Schilder's theorem yields
making use of the fact that the infimum over paths in the collection A is attained for . This result can be heuristically interpreted as saying that, for large and/or large
In fact, the above probability can be estimated more precisely: for a standard Brownian motion in , and any and , we have:
References
(See theorem 5.2)
Asymptotic analysis
Theorems regarding stochastic processes
Large deviations theory |
https://en.wikipedia.org/wiki/Freidlin%E2%80%93Wentzell%20theorem | In mathematics, the Freidlin–Wentzell theorem (due to Mark Freidlin and Alexander D. Wentzell) is a result in the large deviations theory of stochastic processes. Roughly speaking, the Freidlin–Wentzell theorem gives an estimate for the probability that a (scaled-down) sample path of an Itō diffusion will stray far from the mean path. This statement is made precise using rate functions. The Freidlin–Wentzell theorem generalizes Schilder's theorem for standard Brownian motion.
Statement
Let B be a standard Brownian motion on Rd starting at the origin, 0 ∈ Rd, and let Xε be an Rd-valued Itō diffusion solving an Itō stochastic differential equation of the form
where the drift vector field b : Rd → Rd is uniformly Lipschitz continuous. Then, on the Banach space C0 = C0([0, T]; Rd) equipped with the supremum norm ||·||∞, the family of processes (Xε)ε>0 satisfies the large deviations principle with good rate function I : C0 → R ∪ {+∞} given by
if ω lies in the Sobolev space H1([0, T]; Rd), and I(ω) = +∞ otherwise. In other words, for every open set G ⊆ C0 and every closed set F ⊆ C0,
and
References
(See chapter 5.6)
Asymptotic analysis
Stochastic differential equations
Theorems in statistics
Large deviations theory
Probability theorems |
https://en.wikipedia.org/wiki/Dawson%E2%80%93G%C3%A4rtner%20theorem | In mathematics, the Dawson–Gärtner theorem is a result in large deviations theory. Heuristically speaking, the Dawson–Gärtner theorem allows one to transport a large deviation principle on a “smaller” topological space to a “larger” one.
Statement of the theorem
Let (Yj)j∈J be a projective system of Hausdorff topological spaces with maps pij : Yj → Yi. Let X be the projective limit (also known as the inverse limit) of the system (Yj, pij)i,j∈J, i.e.
Let (με)ε>0 be a family of probability measures on X. Assume that, for each j ∈ J, the push-forward measures (pj∗με)ε>0 on Yj satisfy the large deviation principle with good rate function Ij : Yj → R ∪ {+∞}. Then the family (με)ε>0 satisfies the large deviation principle on X with good rate function I : X → R ∪ {+∞} given by
References
(See theorem 4.6.1)
Asymptotic analysis
Large deviations theory
Probability theorems |
https://en.wikipedia.org/wiki/Varadhan%27s%20lemma | In mathematics, Varadhan's lemma is a result from the large deviations theory named after S. R. Srinivasa Varadhan. The result gives information on the asymptotic distribution of a statistic φ(Zε) of a family of random variables Zε as ε becomes small in terms of a rate function for the variables.
Statement of the lemma
Let X be a regular topological space; let (Zε)ε>0 be a family of random variables taking values in X; let με be the law (probability measure) of Zε. Suppose that (με)ε>0 satisfies the large deviation principle with good rate function I : X → [0, +∞]. Let ϕ : X → R be any continuous function. Suppose that at least one of the following two conditions holds true: either the tail condition
where 1(E) denotes the indicator function of the event E; or, for some γ > 1, the moment condition
Then
See also
Laplace principle (large deviations theory)
References
(See theorem 4.3.1)
Asymptotic analysis
Lemmas
Probability theorems
Theorems in statistics
Large deviations theory |
https://en.wikipedia.org/wiki/Severoz%C3%A1pad | Severozápad (Northwest) is a statistical area of the Nomenclature of Territorial Units for Statistics, level NUTS 2. It includes the Karlovy Vary Region and Ústí nad Labem Region.
It covers an area of 8 649 km2 and 1,120,654 inhabitants (population density 130 inhabitants/km2).
Economy
The Gross domestic product (GDP) of the region was 15.2 billion € in 2018, accounting for 7.3% of Czech economic output. GDP per capita adjusted for purchasing power was 19,200 € or 64% of the EU27 average in the same year. The GDP per employee was also 64% of the EU average.
See also
NUTS of the Czech Republic
References
NUTS 2 statistical regions of the European Union
Subdivisions of the Czech Republic |
https://en.wikipedia.org/wiki/Smith%E2%80%93Minkowski%E2%80%93Siegel%20mass%20formula | In mathematics, the Smith–Minkowski–Siegel mass formula (or Minkowski–Siegel mass formula) is a formula for the sum of the weights of the lattices (quadratic forms) in a genus, weighted by the reciprocals of the orders of their automorphism groups. The mass formula is often given for integral quadratic forms, though it can be generalized to quadratic forms over any algebraic number field.
In 0 and 1 dimensions the mass formula is trivial, in 2 dimensions it is essentially equivalent to Dirichlet's class number formulas for imaginary quadratic fields, and in 3 dimensions some partial results were given by Gotthold Eisenstein.
The mass formula in higher dimensions was first given by , though his results were forgotten for many years.
It was rediscovered by , and an error in Minkowski's paper was found and corrected by .
Many published versions of the mass formula have errors; in particular the 2-adic densities are difficult to get right, and it is sometimes forgotten that the trivial cases of dimensions 0 and 1 are different from the cases of dimension at least 2.
give an expository account and precise statement of the mass formula for integral quadratic forms, which is reliable because they check it on a large number of explicit cases.
For recent proofs of the mass formula see and .
The Smith–Minkowski–Siegel mass formula is essentially the constant term of the Weil–Siegel formula.
Statement of the mass formula
If f is an n-dimensional positive definite integral quadratic form (or lattice) then the mass
of its genus is defined to be
where the sum is over all integrally inequivalent forms in the same genus as f, and Aut(Λ) is the automorphism group of Λ.
The form of the mass formula given by states that for n ≥ 2 the mass is given by
where mp(f) is the p-mass of f, given by
for sufficiently large r, where ps is the highest power of p dividing the determinant of f. The number N(pr) is the number of n by n matrices
X with coefficients that are integers mod p r such that
where A is the Gram matrix of f, or in other words the order of the automorphism group of the form reduced mod p r.
Some authors state the mass formula in terms of the p-adic density
instead of the p-mass. The p-mass is invariant under rescaling f but the p-density is not.
In the (trivial) cases of dimension 0 or 1 the mass formula needs some modifications. The factor of 2 in front represents the Tamagawa number of the special orthogonal group, which is only 1 in dimensions 0 and 1. Also the factor of 2 in front of mp(f) represents the index of the special orthogonal group in the orthogonal group, which is only 1 in 0 dimensions.
Evaluation of the mass
The mass formula gives the mass as an infinite product over all primes. This can be rewritten as a finite product as follows. For all but a finite number of primes (those not dividing 2 det(ƒ)) the p-mass mp(ƒ) is equal to the standard p-mass stdp(ƒ), given by
(for n = dim(ƒ) even)
(for n = dim(ƒ) odd)
where |
https://en.wikipedia.org/wiki/Smooth%20coarea%20formula | In Riemannian geometry, the smooth coarea formulas relate integrals over the domain of certain mappings with integrals over their codomains.
Let be smooth Riemannian manifolds of respective dimensions . Let be a smooth surjection such that the pushforward (differential) of is surjective almost everywhere. Let a measurable function. Then, the following two equalities hold:
where is the normal Jacobian of , i.e. the determinant of the derivative restricted to the orthogonal complement of its kernel.
Note that from Sard's lemma almost every point is a regular point of and hence the set is a Riemannian submanifold of , so the integrals in the right-hand side of the formulas above make sense.
References
Chavel, Isaac (2006) Riemannian Geometry. A Modern Introduction. Second Edition.
Riemannian geometry |
https://en.wikipedia.org/wiki/Severov%C3%BDchod | Severovýchod (Northeast) is a statistical area of the Nomenclature of Territorial Units for Statistics, level 2 NUTS. It is composed of the Liberec Region, Hradec Králové Region and Pardubice Region of the Czech Republic. It covers an area of 12,440 km2, with 1,507,209 inhabitants and a (population density of 119 inhabitants/km2).
Economy
The Gross domestic product (GDP) of the region was 24.7 billion € in 2018, accounting for 11.9% of Czech economic output. GDP per capita adjusted for purchasing power was 23,000 € or 76% of the EU27 average in the same year. The GDP per employee was also 74% of the EU average.
See also
NUTS of the Czech Republic
References
NUTS 2 statistical regions of the European Union
Subdivisions of the Czech Republic |
https://en.wikipedia.org/wiki/Cube-connected%20cycles | In graph theory, the cube-connected cycles is an undirected cubic graph, formed by replacing each vertex of a hypercube graph by a cycle. It was introduced by for use as a network topology in parallel computing.
Definition
The cube-connected cycles of order n (denoted CCCn) can be defined as a graph formed from a set of n2n nodes, indexed by pairs of numbers (x, y) where 0 ≤ x < 2n and 0 ≤ y < n. Each such node is connected to three neighbors: , , and , where "⊕" denotes the bitwise exclusive or operation on binary numbers.
This graph can also be interpreted as the result of replacing each vertex of an n-dimensional hypercube graph by an n-vertex cycle. The hypercube graph vertices are indexed by the numbers x, and the positions within each cycle by the numbers y.
Properties
The cube-connected cycles of order n is the Cayley graph of a
group that acts on binary words of length n by rotation and flipping bits of the word. The generators used to form this Cayley graph from the group are the group elements that act by rotating the word one position left, rotating it one position right, or flipping its first bit. Because it is a Cayley graph, it is vertex-transitive: there is a symmetry of the graph mapping any vertex to any other vertex.
The diameter of the cube-connected cycles of order n is for any n ≥ 4; the farthest point from (x, y) is (2n − x − 1, (y + n/2) mod n). showed that the crossing number of CCCn is ((1/20) + o(1)) 4n.
According to the Lovász conjecture, the cube-connected cycle graph should always contain a Hamiltonian cycle, and this is now known to be true. More generally, although these graphs are not pancyclic, they contain cycles of all but a bounded number of possible even lengths, and when n is odd they also contain many of the possible odd lengths of cycles.
Parallel processing application
Cube-connected cycles were investigated by , who applied these graphs as the interconnection pattern of a network connecting the processors in a parallel computer. In this application, cube-connected cycles have the connectivity advantages of hypercubes while only requiring three connections per processor. Preparata and Vuillemin showed that a planar layout based on this network has optimal area × time2 complexity for many parallel processing tasks.
Notes
References
.
.
.
.
.
.
Network topology
Parametric families of graphs
Regular graphs |
https://en.wikipedia.org/wiki/Reversible%20diffusion | In mathematics, a reversible diffusion is a specific example of a reversible stochastic process. Reversible diffusions have an elegant characterization due to the Russian mathematician Andrey Nikolaevich Kolmogorov.
Kolmogorov's characterization of reversible diffusions
Let B denote a d-dimensional standard Brownian motion; let b : Rd → Rd be a Lipschitz continuous vector field. Let X : [0, +∞) × Ω → Rd be an Itō diffusion defined on a probability space (Ω, Σ, P) and solving the Itō stochastic differential equation
with square-integrable initial condition, i.e. X0 ∈ L2(Ω, Σ, P; Rd). Then the following are equivalent:
The process X is reversible with stationary distribution μ on Rd.
There exists a scalar potential Φ : Rd → R such that b = −∇Φ, μ has Radon–Nikodym derivative and
(Of course, the condition that b be the negative of the gradient of Φ only determines Φ up to an additive constant; this constant may be chosen so that exp(−2Φ(·)) is a probability density function with integral 1.)
References
(See theorem 1.4)
Stochastic differential equations
Probability theorems |
https://en.wikipedia.org/wiki/D.%20J.%20Finney | David John Finney (3 January 1917 – 12 November 2018), was a British statistician
and Professor Emeritus of Statistics at the University of Edinburgh. He was Director of the Agricultural Research Council's Unit of Statistics from 1954 to 1984 and a former President of the Royal Statistical Society and of the Biometric Society. He was a pioneer in the development of systematic monitoring of drugs for detection of adverse reactions. He turned 100 in January 2017 and died on 12 November 2018 at the age of 101 following a short illness.
Childhood and education
Finney was born in Latchford, Cheshire, Warrington. In his interview with MacNeill, Finney describes his background: "My family were never wealthy but never in want". His paternal grandfather was a schoolmaster, and his father was an accountant in the steel industry. David was the eldest child; he had no sisters. In the Preface to his "Probit Analysis" book, Finney thanks his father Robt. G. S. Finney for assistance.
Finney was educated at the coeducational Lymm Grammar School and Manchester Grammar School, where he won a Cambridge scholarship. He read mathematics and statistics at Clare College, Cambridge from 1934 to 1938. He was awarded a postgraduate scholarship for statistical work in agriculture under Ronald Fisher at the Galton Laboratory, University College London, where he worked on statistical estimation for human genetics.
Career
He became assistant to Frank Yates at Rothamsted Experimental Station in 1939, where there was great emphasis on increasing productivity of agriculture and he was involved in the design of field experiments and the interpretation of their results.
In 1945, he joined the University of Oxford as the first holder of the post of Lecturer in the Design and Analysis of Scientific Experiment. He married in 1950 and with his wife and 9-month-old daughter, left Oxford in 1952 for New Delhi where, for a year, he acted as a consultant to the United Nations Food and Agriculture Organisation on the development of the Indian Agricultural Statistics Research Institute in New Delhi. In 1951 he was elected as a Fellow of the American Statistical Association. He became a Fellow of The Royal Society of Edinburgh in 1955.
After returning from India, he moved to the University of Aberdeen where he became Reader in Statistics and also established a Unit of Statistics funded by the Agricultural Research Council, which was to provide a service for Scotland modelled on that provided by Rothamsted for England. The Agricultural Research Council moved the Unit of Statistics to the University of Edinburgh in 1966 and Finney, who moved to Edinburgh with it, became the first Professor of Statistics at the university and well as being the Director of the Unit of Statistics. He served as president of the Royal Statistical Society in 1973–4. He retired from his position at Edinburgh in 1984.
During the 1960s he became involved in the field of drug safety, providing important advice bo |
https://en.wikipedia.org/wiki/Bessel%27s%20correction | In statistics, Bessel's correction is the use of n − 1 instead of n in the formula for the sample variance and sample standard deviation, where n is the number of observations in a sample. This method corrects the bias in the estimation of the population variance. It also partially corrects the bias in the estimation of the population standard deviation. However, the correction often increases the mean squared error in these estimations. This technique is named after Friedrich Bessel.
Formulation
In estimating the population variance from a sample when the population mean is unknown, the uncorrected sample variance is the mean of the squares of deviations of sample values from the sample mean (i.e. using a multiplicative factor 1/n). In this case, the sample variance is a biased estimator of the population variance.
Multiplying the uncorrected sample variance by the factor
gives an unbiased estimator of the population variance. In some literature, the above factor is called Bessel's correction.
One can understand Bessel's correction as the degrees of freedom in the residuals vector (residuals, not errors, because the population mean is unknown):
where is the sample mean. While there are n independent observations in the sample, there are only n − 1 independent residuals, as they sum to 0. For a more intuitive explanation of the need for Bessel's correction, see .
Generally Bessel's correction is an approach to reduce the bias due to finite sample size. Such finite-sample bias correction is also needed for other estimates like skew and kurtosis, but in these the inaccuracies are often significantly larger. To fully remove such bias it is necessary to do a more complex multi-parameter estimation. For instance a correct correction for the standard deviation depends on the kurtosis (normalized central 4th moment), but this again has a finite sample bias and it depends on the standard deviation, i.e. both estimations have to be merged.
Caveats
There are three caveats to consider regarding Bessel's correction:
It does not yield an unbiased estimator of standard deviation.
The corrected estimator often has a higher mean squared error (MSE) than the uncorrected estimator. Furthermore, there is no population distribution for which it has the minimum MSE because a different scale factor can always be chosen to minimize MSE.
It is only necessary when the population mean is unknown (and estimated as the sample mean). In practice, this generally happens.
Firstly, while the sample variance (using Bessel's correction) is an unbiased estimator of the population variance, its square root, the sample standard deviation, is a biased estimate of the population standard deviation; because the square root is a concave function, the bias is downward, by Jensen's inequality. There is no general formula for an unbiased estimator of the population standard deviation, though there are correction factors for particular distributions, such as the normal; s |
https://en.wikipedia.org/wiki/Jihov%C3%BDchod | Jihovýchod (Southeast) is a statistical area of the Nomenclature of Territorial Units for Statistics, level NUTS 2. It comprises Vysočina Region and South Moravian Region.
It covers an area of 13 990 km2 and has 1,684,500 inhabitants (population density 117 inhabitants/km2).
Economy
The Gross domestic product (GDP) of the region was 30.5 billion € in 2018, accounting for 14.7% of Czech economic output. GDP per capita adjusted for purchasing power was 25,300 € or 84% of the EU27 average in the same year. The GDP per employee was also 77% of the EU average.
See also
NUTS of the Czech Republic
References
NUTS 2 statistical regions of the European Union
Subdivisions of the Czech Republic |
https://en.wikipedia.org/wiki/Isaac%20Habrecht | Isaac Habrecht is the name of:
Isaac Habrecht I (1544–1622), horologist
Isaac Habrecht II (1589–1633), doctor of medicine and philosophy / professor of astronomy and mathematics |
https://en.wikipedia.org/wiki/Albrecht%20Beutelspacher | Albrecht Beutelspacher (born 5 June 1950) is a German mathematician and founder of the Mathematikum. He is a professor emeritus at the University of Giessen, where he held the chair for geometry and discrete mathematics from 1988 to 2018.
Biography
Beutelspacher studied from 1969 to 1973 math, physics and philosophy at the University of Tübingen and received his PhD 1976 from the University of Mainz. His PhD advisor was Judita Cofman. From 1982 to 1985 he was an associate professor at the University of Mainz and from 1985 to 1988 he worked at a research department of Siemens. From 1988 to 2018 he was a tenured professor for geometry and discrete mathematics at the University of Giessen. He became a well-known popularizer of mathematics in Germany by authoring several books in the field of popular science and recreational math and by founding Germany's first math museum, the Mathematikum. He received several awards for his contributions to popularizing mathematics. He had a math column in the German popular science magazine Bild der Wissenschaft and moderated a popular math series for the TV Channel BR- (educational TV).
Awards
2004: IQ Award
2008: Hessian Cultural Prize
2016 Hessian Order of Merit
Books
regular
Einführung in die endliche Geometrie. Band 1: Blockpläne. Bibliographisches Institut Wissenschaftsverlag, Mannheim u. a. 1982, ISBN 3-411-01632-9.
Einführung in die endliche Geometrie. Band 2: Projektive Räume. Bibliographisches Institut Wissenschaftsverlag, Mannheim u. a. 1983, ISBN 3-411-01648-5.
Cryptology. Spectrum
with Lynn Batten: The Theory of finite linear Spaces. Cambridge University Press, 1993
with Uta Rosenbaum: Projective Geometry: From Foundations to Applications. Cambridge University Press
„Das ist o.B.d.A. trivial!“ Eine Gebrauchsanleitung zur Formulierung mathematischer Gedanken mit vielen praktischen Tips für Studenten der Mathematik und Informatik. Vieweg, Braunschweig u. a. 1991, ISBN 3-528-06442-0 (9th edition, Vieweg + Teubner, Wiesbaden 2009, ISBN 978-3-8348-0771-7).
Lineare Algebra. Eine Einführung in die Wissenschaft der Vektoren, Abbildungen und Matrizen. 8th edition, Springer Spektrum, Wiesbaden 2014, ISBN 978-3-658-02412-3, doi:10.1007/978-3-658-02413-0.
with Bernhard Petri: Der goldene Schnitt. 2. Auflage. Spektrum, Heidelberg, Berlin, Oxford 1996,
mit Jörg Schwenk und Klaus-Dieter Wolfenstetter: Moderne Verfahren der Kryptographie. Von RSA zu Zero-Knowledge. Vieweg, Braunschweig u. a. 1995, (7., überarbeitete Auflage. Vieweg + Teubner, Wiesbaden 2010, ).
with Marc-Alexander Zschiegner: Diskrete Mathematik für Einsteiger. Mit Anwendungen in Technik und Informatik. Vieweg, Braunschweig u. a. 2002, (4., aktualisierte Auflage. Vieweg + Teubner, Wiesbaden 2011, ).
with Heike B. Neumann und Thomas Schwarzpaul: Kryptographie in Theorie und Praxis. Mathematische Grundlagen für elektronisches Geld, Internetsicherheit und Mobilfunk. Vieweg + Teubner, Braunschweig und Wiesbaden 2005, .
Survival-Kit M |
https://en.wikipedia.org/wiki/M%C3%B6bius%E2%80%93Kantor%20configuration | In geometry, the Möbius–Kantor configuration is a configuration consisting of eight points and eight lines, with three points on each line and three lines through each point. It is not possible to draw points and lines having this pattern of incidences in the Euclidean plane, but it is possible in the complex projective plane.
Coordinates
asked whether there exists a pair of polygons with p sides each, having the property that the vertices of one polygon lie on the lines through the edges of the other polygon, and vice versa. If so, the vertices and edges of these polygons would form a projective configuration. For there is no solution in the Euclidean plane, but found pairs of polygons of this type, for a generalization of the problem in which the points and edges belong to the complex projective plane. That is, in Kantor's solution, the coordinates of the polygon vertices are complex numbers. Kantor's solution for , a pair of mutually-inscribed quadrilaterals in the complex projective plane, is called the Möbius–Kantor configuration.
supplies the following simple complex projective coordinates for the eight points of the Möbius–Kantor configuration:
(1,0,0), (0,0,1), (ω, −1, 1), (−1, 0, 1),
(−1,ω2,1), (1,ω,0), (0,1,0), (0,−1,1),
where ω denotes a complex cube root of 1.
The eight points and eight lines of the Möbius–Kantor configuration, with these coordinates, form the eight vertices and eight 3-edges of the complex polygon 3{3}3. Coxeter named it a Möbius–Kantor polygon.
Abstract incidence pattern
More abstractly, the Möbius–Kantor configuration can be described as a system of eight points and eight triples of points such that each point belongs to exactly three of the triples. With the additional conditions (natural to points and lines) that no pair of points belong to more than one triple and that no two triples have more than one point in their intersection, any two systems of this type are equivalent under some permutation of the points. That is, the Möbius–Kantor configuration is the unique projective configuration of type (8383).
The Möbius–Kantor graph derives its name from being the Levi graph of the Möbius–Kantor configuration. It has one vertex per point and one vertex per triple, with an edge connecting two vertices if they correspond to a point and to a triple that contains that point.
The points and lines of the Möbius–Kantor configuration can be described as a matroid, whose elements are the points of the configuration and whose nontrivial flats are the lines of the configuration. In this matroid, a set S of points is independent if and only if either or S consists of three non-collinear points. As a matroid, it has been called the MacLane matroid, after the work of proving that it cannot be oriented; it is one of several known minor-minimal non-orientable matroids.
Related configurations
The solution to Möbius' problem of mutually inscribed polygons for values of p greater than four is also of interest. In particu |
https://en.wikipedia.org/wiki/Dininho | Irondino Ferreira Neto or simply Dininho (born July 23, 1975), is a Brazilian central defender. He currently plays for Grêmio Catanduvense de Futebol.
Club statistics
Flamengo career statistics
(Correct October 19, 2008)
according to combined sources on the.
Honours
São Caetano
São Paulo State Championship: 2004
Palmeiras
São Paulo State Championship: 2008
References
External links
dininho official site
sambafoot
Guardian Stats Centre
zerozero.pt
globoesporte
1975 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
CR Flamengo footballers
Mirassol Futebol Clube players
Mogi Mirim Esporte Clube players
América Futebol Clube (SP) players
Associação Desportiva São Caetano players
São Paulo state football team players
Sanfrecce Hiroshima players
Sociedade Esportiva Palmeiras players
Esporte Clube Santo André players
Expatriate men's footballers in Japan
J1 League players
Men's association football defenders |
https://en.wikipedia.org/wiki/Institutiones%20calculi%20integralis | Institutiones calculi integralis (Foundations of integral calculus) is a three-volume textbook written by Leonhard Euler and published in 1768. It was on the subject of integral calculus and contained many of Euler's discoveries about differential equations.
See also
Institutiones calculi differentialis
External links
Full text available from Archive.org.
Full text (1768) available from books.google.com.
provides a complete English translation of Euler's Institutiones calculi integralis by Ian Bruce.
German translation Vollständige Anleitung zur Integralrechnung (1828) available from e-rara.ch.
1768 non-fiction books
18th-century Latin books
Mathematics textbooks
Leonhard Euler |
https://en.wikipedia.org/wiki/Marginal%20model | In statistics, marginal models (Heagerty & Zeger, 2000) are a technique for obtaining regression estimates in multilevel modeling, also called hierarchical linear models.
People often want to know the effect of a predictor/explanatory variable X, on a response variable Y. One way to get an estimate for such effects is through regression analysis.
Why the name marginal model?
In a typical multilevel model, there are level 1 & 2 residuals (R and U variables). The two variables form a joint distribution for the response variable (). In a marginal model, we collapse over the level 1 & 2 residuals and thus marginalize (see also conditional probability) the joint distribution into a univariate normal distribution. We then fit the marginal model to data.
For example, for the following hierarchical model,
level 1: , the residual is , and
level 2: , the residual is , and
Thus, the marginal model is,
This model is what is used to fit to data in order to get regression estimates.
References
Heagerty, P. J., & Zeger, S. L. (2000). Marginalized multilevel models and likelihood inference. Statistical Science, 15(1), 1-26.
Regression models |
https://en.wikipedia.org/wiki/Sullivan%20conjecture | In mathematics, Sullivan conjecture or Sullivan's conjecture on maps from classifying spaces can refer to any of several results and conjectures prompted by homotopy theory work of Dennis Sullivan. A basic theme and motivation concerns the fixed point set in group actions of a finite group . The most elementary formulation, however, is in terms of the classifying space of such a group. Roughly speaking, it is difficult to map such a space continuously into a finite CW complex in a non-trivial manner. Such a version of the Sullivan conjecture was first proved by Haynes Miller. Specifically, in 1984, Miller proved that the function space, carrying the compact-open topology, of base point-preserving mappings from to is weakly contractible.
This is equivalent to the statement that the map → from X to the function space of maps → , not necessarily preserving the base point, given by sending a point of to the constant map whose image is is a weak equivalence. The mapping space is an example of a homotopy fixed point set. Specifically, is the homotopy fixed point set of the group acting by the trivial action on . In general, for a group acting on a space , the homotopy fixed points are the fixed points of the mapping space of maps from the universal cover of to under the -action on given by in acts on a map in by sending it to . The -equivariant map from to a single point induces a natural map η: → from the fixed points to the homotopy fixed points of acting on . Miller's theorem is that η is a weak equivalence for trivial -actions on finite-dimensional CW complexes. An important ingredient and motivation for his proof is a result of Gunnar Carlsson on the homology of as an unstable module over the Steenrod algebra.
Miller's theorem generalizes to a version of Sullivan's conjecture in which the action on is allowed to be non-trivial. In, Sullivan conjectured that η is a weak equivalence after a certain p-completion procedure due to A. Bousfield and D. Kan for the group . This conjecture was incorrect as stated, but a correct version was given by Miller, and proven independently by Dwyer-Miller-Neisendorfer, Carlsson, and Jean Lannes, showing that the natural map → is a weak equivalence when the order of is a power of a prime p, and where denotes the Bousfield-Kan p-completion of . Miller's proof involves an unstable Adams spectral sequence, Carlsson's proof uses his affirmative solution of the Segal conjecture and also provides information about the homotopy fixed points before completion, and Lannes's proof involves his T-functor.
References
External links
Book extract
J. Lurie's course notes
Conjectures that have been proved
Fixed points (mathematics)
Homotopy theory |
https://en.wikipedia.org/wiki/Central%20Moravia | Central Moravia (Střední Morava) is an area of the Czech Republic defined by the Republic's Nomenclature of Territorial Units for Statistics, level NUTS 2. It is formed by the Olomouc Region and Zlín Region. It covers an area of 9 231 km2 and 1,219,394 inhabitants (population density 133 inhabitants/km2).
Economy
The Gross domestic product (GDP) of the region was 19.3 billion € in 2018, accounting for 9.3% of Czech economic output. GDP per capita adjusted for purchasing power was 22,400 € or 74% of the EU27 average in the same year. The GDP per employee was 70% of the EU average.
See also
NUTS of the Czech Republic
References
NUTS 2 statistical regions of the European Union
Subdivisions of the Czech Republic |
https://en.wikipedia.org/wiki/Generalised%20metric | In mathematics, the concept of a generalised metric is a generalisation of that of a metric, in which the distance is not a real number but taken from an arbitrary ordered field.
In general, when we define metric space the distance function is taken to be a real-valued function. The real numbers form an ordered field which is Archimedean and order complete. These metric spaces have some nice properties like: in a metric space compactness, sequential compactness and countable compactness are equivalent etc. These properties may not, however, hold so easily if the distance function is taken in an arbitrary ordered field, instead of in
Preliminary definition
Let be an arbitrary ordered field, and a nonempty set; a function is called a metric on if the following conditions hold:
if and only if ;
(symmetry);
(triangle inequality).
It is not difficult to verify that the open balls form a basis for a suitable topology, the latter called the metric topology on with the metric in
In view of the fact that in its order topology is monotonically normal, we would expect to be at least regular.
Further properties
However, under axiom of choice, every general metric is monotonically normal, for, given where is open, there is an open ball such that Take Verify the conditions for Monotone Normality.
The matter of wonder is that, even without choice, general metrics are monotonically normal.
proof.
Case I: is an Archimedean field.
Now, if in open, we may take where and the trick is done without choice.
Case II: is a non-Archimedean field.
For given where is open, consider the set
The set is non-empty. For, as is open, there is an open ball within Now, as is non-Archimdedean, is not bounded above, hence there is some such that for all Putting we see that is in
Now define We would show that with respect to this mu operator, the space is monotonically normal. Note that
If is not in (open set containing ) and is not in (open set containing ), then we'd show that is empty. If not, say is in the intersection. Then
From the above, we get that which is impossible since this would imply that either belongs to or belongs to
This completes the proof.
See also
External links
Metric geometry
Norms (mathematics)
Topology |
https://en.wikipedia.org/wiki/Hofstadter%20sequence | In mathematics, a Hofstadter sequence is a member of a family of related integer sequences defined by non-linear recurrence relations.
Sequences presented in Gödel, Escher, Bach: an Eternal Golden Braid
The first Hofstadter sequences were described by Douglas Richard Hofstadter in his book Gödel, Escher, Bach. In order of their presentation in chapter III on figures and background (Figure-Figure sequence) and chapter V on recursive structures and processes (remaining sequences), these sequences are:
Hofstadter Figure-Figure sequences
The Hofstadter Figure-Figure (R and S) sequences are a pair of complementary integer sequences defined as follows
with the sequence defined as a strictly increasing series of positive integers not present in . The first few terms of these sequences are
R: 1, 3, 7, 12, 18, 26, 35, 45, 56, 69, 83, 98, 114, 131, 150, 170, 191, 213, 236, 260, ...
S: 2, 4, 5, 6, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, ...
Hofstadter G sequence
The Hofstadter G sequence is defined as follows
The first few terms of this sequence are
0, 1, 1, 2, 3, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 9, 10, 11, 11, 12, 12, ...
Hofstadter H sequence
The Hofstadter H sequence is defined as follows
The first few terms of this sequence are
0, 1, 1, 2, 3, 4, 4, 5, 5, 6, 7, 7, 8, 9, 10, 10, 11, 12, 13, 13, 14, ...
Hofstadter Female and Male sequences
The Hofstadter Female (F) and Male (M) sequences are defined as follows
The first few terms of these sequences are
F: 1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 6, 7, 8, 8, 9, 9, 10, 11, 11, 12, 13, ...
M: 0, 0, 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 7, 8, 9, 9, 10, 11, 11, 12, 12, ...
Hofstadter Q sequence
The Hofstadter Q sequence is defined as follows
The first few terms of the sequence are
1, 1, 2, 3, 3, 4, 5, 5, 6, 6, 6, 8, 8, 8, 10, 9, 10, 11, 11, 12, ...
Hofstadter named the terms of the sequence "Q numbers"; thus the Q number of 6 is 4. The presentation of the Q sequence in Hofstadter's book is actually the first known mention of a meta-Fibonacci sequence in literature.
While the terms of the Fibonacci sequence are determined by summing the two preceding terms, the two preceding terms of a Q number determine how far to go back in the Q sequence to find the two terms to be summed. The indices of the summation terms thus depend on the Q sequence itself.
Q(1), the first element of the sequence, is never one of the two terms being added to produce a later element; it is involved only within an index in the calculation of Q(3).
Although the terms of the Q sequence seem to flow chaotically, like many meta-Fibonacci sequences its terms can be grouped into blocks of successive generations. In case of the Q sequence, the k-th generation has 2k members. Furthermore, with g being the generation that a Q number belongs to, the two terms to be summed to calculate the Q number, called its parents, reside by far mostly in generation g − 1 and only a few in generation g − 2, but never in an even older genera |
https://en.wikipedia.org/wiki/Clarkson%27s%20inequalities | In mathematics, Clarkson's inequalities, named after James A. Clarkson, are results in the theory of Lp spaces. They give bounds for the Lp-norms of the sum and difference of two measurable functions in Lp in terms of the Lp-norms of those functions individually.
Statement of the inequalities
Let (X, Σ, μ) be a measure space; let f, g : X → R be measurable functions in Lp. Then, for 2 ≤ p < +∞,
For 1 < p < 2,
where
i.e., q = p ⁄ (p − 1).
The case p ≥ 2 is somewhat easier to prove, being a simple application of the triangle inequality and the convexity of
References
.
.
.
External links
Banach spaces
Inequalities
Measure theory
Lp spaces |
https://en.wikipedia.org/wiki/Mathematics%20and%20fiber%20arts | Ideas from mathematics have been used as inspiration for fiber arts including quilt making, knitting, cross-stitch, crochet, embroidery and weaving. A wide range of mathematical concepts have been used as inspiration including topology, graph theory, number theory and algebra. Some techniques such as counted-thread embroidery are naturally geometrical; other kinds of textile provide a ready means for the colorful physical expression of mathematical concepts.
Quilting
The IEEE Spectrum has organized a number of competitions on quilt block design, and several books have been published on the subject. Notable quiltmakers include Diana Venters and Elaine Ellison, who have written a book on the subject Mathematical Quilts: No Sewing Required. Examples of mathematical ideas used in the book as the basis of a quilt include the golden rectangle, conic sections, Leonardo da Vinci's Claw, the Koch curve, the Clifford torus, San Gaku, Mascheroni's cardioid, Pythagorean triples, spidrons, and the six trigonometric functions.
Knitting and crochet
Knitted mathematical objects include the Platonic solids, Klein bottles and Boy's surface.
The Lorenz manifold and the hyperbolic plane have been crafted using crochet. Knitted and crocheted tori have also been constructed depicting toroidal embeddings of the complete graph K7 and of the Heawood graph. The crocheting of hyperbolic planes has been popularized by the Institute For Figuring; a book by Daina Taimina on the subject, Crocheting Adventures with Hyperbolic Planes, won the 2009 Bookseller/Diagram Prize for Oddest Title of the Year.
Embroidery
Embroidery techniques such as counted-thread embroidery including cross-stitch and some canvas work methods such as Bargello make use of the natural pixels of the weave, lending themselves to geometric designs.
Weaving
Ada Dietz (1882 – 1950) was an American weaver best known for her 1949 monograph Algebraic Expressions in Handwoven Textiles, which defines weaving patterns based on the expansion of multivariate polynomials.
used the Rule 90 cellular automaton to design tapestries depicting both trees and abstract patterns of triangles.
Spinning
Margaret Greig was a mathematician who articulated the mathematics of worsted spinning.
Fashion design
The silk scarves from DMCK Designs' 2013 collection are all based on Douglas McKenna's space-filling curve patterns. The designs are either generalized Peano curves, or based on a new space-filling construction technique.
The Issey Miyake Fall-Winter 2010–2011 ready-to-wear collection designs from a collaboration between fashion designer Dai Fujiwara and mathematician William Thurston. The designs were inspired by Thurston's geometrization conjecture, the statement that every 3-manifold can be decomposed into pieces with one of eight different uniform geometries, a proof of which had been sketched in 2003 by Grigori Perelman as part of his proof of the Poincaré conjecture.
See also
Mathematics and art
Referenc |
https://en.wikipedia.org/wiki/Regular%20matroid | In mathematics, a regular matroid is a matroid that can be represented over all fields.
Definition
A matroid is defined to be a family of subsets of a finite set, satisfying certain axioms. The sets in the family are called "independent sets". One of the ways of constructing a matroid is to select a finite set of vectors in a vector space, and to define a subset of the vectors to be independent in the matroid when it is linearly independent in the vector space. Every family of sets constructed in this way is a matroid, but not every matroid can be constructed in this way, and the vector spaces over different fields lead to different sets of matroids that can be constructed from them.
A matroid is regular when, for every field , can be represented by a system of vectors over .
Properties
If a matroid is regular, so is its dual matroid, and so is every one of its minors. Every direct sum of regular matroids remains regular.
Every graphic matroid (and every co-graphic matroid) is regular. Conversely, every regular matroid may be constructed by combining graphic matroids, co-graphic matroids, and a certain ten-element matroid that is neither graphic nor co-graphic, using an operation for combining matroids that generalizes the clique-sum operation on graphs.
The number of bases in a regular matroid may be computed as the determinant of an associated matrix, generalizing Kirchhoff's matrix-tree theorem for graphic matroids.
Characterizations
The uniform matroid (the four-point line) is not regular: it cannot be realized over the two-element finite field GF(2), so it is not a binary matroid, although it can be realized over all other fields. The matroid of the Fano plane (a rank-three matroid in which seven of the triples of points are dependent) and its dual are also not regular: they can be realized over GF(2), and over all fields of characteristic two, but not over any other fields than those. As showed, these three examples are fundamental to the theory of regular matroids: every non-regular matroid has at least one of these three as a minor. Thus, the regular matroids are exactly the matroids that do not have one of the three forbidden minors , the Fano plane, or its dual.
If a matroid is regular, it must clearly be realizable over the two fields GF(2) and GF(3). The converse is true: every matroid that is realizable over both of these two fields is regular. The result follows from a forbidden minor characterization of the matroids realizable over these fields, part of a family of results codified by Rota's conjecture.
The regular matroids are the matroids that can be defined from a totally unimodular matrix, a matrix in which every square submatrix has determinant 0, 1, or −1. The vectors realizing the matroid may be taken as the rows of the matrix. For this reason, regular matroids are sometimes also called unimodular matroids. The equivalence of regular matroids and unimodular matrices, and their characterization by forbidden minors |
https://en.wikipedia.org/wiki/Information%20criterion | Information criterion may refer to:
Information criterion (statistics), a method to select a model in statistics
Information criteria (information technology), a component of an information technology framework which describes the intent of the objectives |
https://en.wikipedia.org/wiki/Blessed%20Thomas%20Holford%20Catholic%20College | Blessed Thomas Holford Catholic College is a secondary school based in Altrincham, Greater Manchester. The school specialises in maths and computing, and is named after Blessed Thomas Holford, a 16th-century priest from Cheshire. The college has a Catholic identity, and all pupils are required to wear uniform.
Curriculum
The college puts emphasis on maths and computing, and follows the Key Stage process. At Key Stage 3 in year 7 & 8, the pupils take the following subjects:
English (3 hours)
Maths (4 hours)
Science (3 hours)
Religious Education (2.5 hours)
Design Technology (3 hours) - Woodwork, food technology, graphic design, art, music, PSHCE
Computer Science (1 hour)
French/Spanish (2 hours)
History (2 hours)
Geography (2 hours)
Physical Education (2 hours)
Drama (1 hour)
For GCSE, the pupils take core subjects including English language, English literature, Science, Mathematics, RE, computer science or BTEC and PE although not to complete as a GCSE. Pupils choose from a variety of additional subjects, including history, geography, art, health and social care, physical education, food technology, business studies, classics, music, Italian, drama, French, German, Japanese, Spanish. BTEC courses for sport are also available. Pupils have to take at least one, but not more than three, of History, Geography, Spanish and French. Out of all the other subjects a maximum of two can be chosen.
Academic performance and Ofsted judgements
In 2019, the school was above national average for Section 48.
It was judged Outstanding by Ofsted in 2012 and 2009. In 2022 its inspection rating was 'Requires Improvement'.
Football Academy
The school has a UEFA-standard FieldTurf artificial grass football pitch which was opened in April 2007 by Bobby Charlton. The pitch, which cost £1 million to install, was used by the England national football team for training prior to an away game against Russia at the Luzhniki Stadium in Moscow, as it uses the same surface as the Russian pitch.
References
Secondary schools in Trafford
Catholic secondary schools in the Diocese of Shrewsbury
Altrincham
Voluntary aided schools in England |
https://en.wikipedia.org/wiki/Piers%20Bohl | Piers Bohl (23 October 1865 – 25 December 1921) was a Latvian mathematician, who worked in differential equations, topology and quasi-periodic functions.
He was born in 1865 in Walk, Livonia, in the family of a poor Baltic German merchant. In 1884, after graduating from a German school in Viljandi, he entered the faculty of physics and mathematics at the University of Tartu. In 1893 Bohl was awarded his Master's degree. This was for an investigation of quasi-periodic functions. The notion of quasi-periodic functions was generalised still further by Harald Bohr when he introduced almost-periodic functions. He has been the first to prove the three-dimensional case of the Brouwer fixed-point theorem, but his work was not noticed at the time.
References
External links
Bohl biography at www-history.mcs.st-and.ac.uk
http://www.mathematics.lv/lms_10_years_after.pdf
1865 births
1921 deaths
People from Valka
People from Kreis Walk
Baltic-German people
Latvian mathematicians
University of Tartu alumni
Academic staff of Riga Technical University
Mathematicians from the Russian Empire |
https://en.wikipedia.org/wiki/Gwinnett%20School%20of%20Mathematics%2C%20Science%2C%20and%20Technology | The Gwinnett School of Mathematics, Science, and Technology (GSMST) is a special public school in Lawrenceville, Georgia, United States, and a part of Gwinnett County Public Schools. Students are admitted through a county-wide lottery, whose participants, since the school dropped its charter status in 2016, must meet multiple requirements. It features a heavy focus on project-based STEM education. Its rigorous course offerings and high student graduation rate make it one of the most prestigious high schools in the state.
History
The school was chartered in March 2006 by the Gwinnett County Board of Education through SPLOST and opened in the fall of 2007. It was temporarily housed on the Duluth High School campus, but building 100 began in 2007 as a separate school from Duluth High School. Renovations updated and modernized 18 classrooms and offices, and GSMST moved to its permanent location in 2010 at the geographic center of Gwinnett County, near Sugarloaf Parkway and Old Norcross Road, the former site of Benefield Elementary School. The new building is designed for a maximum of 1,200 students. The school is adjacent to the Maxwell High School of Technology.
In 2016, GSMST was awarded the Blue Ribbon Award.
References
Educational institutions established in 2007
Public high schools in Georgia (U.S. state)
Schools in Gwinnett County, Georgia
Charter schools in Georgia (U.S. state)
2007 establishments in Georgia (U.S. state) |
https://en.wikipedia.org/wiki/Tennessee%20Governor%27s%20Academy%20for%20Math%20and%20Science | The Tennessee Governor's Academy for Mathematics and Science, commonly Tennessee Governor's Academy or TGA, was a residential high school located in Knoxville, Tennessee on the campus of The Tennessee School for the Deaf (TSD). It was founded in 2007 by Governor Phil Bredesen as part of an effort to provide challenges for students across the academic spectrum. Its inaugural class consisted of 24 high school juniors from throughout the state. The academy was closed on May 31, 2011, due to lack of state funding.
Curriculum
Pre-Calculus
Pre-Calculus served as the introductory mathematics class at TGA for those who had not taken the course elsewhere or those who are not prepared to take Calculus upon arrival at TGA. The course was similar in content to the Math 130 course offered at the University of Tennessee at Knoxville. Generally speaking, the course content serves to prepare juniors to take Calculus during the Spring semester of the junior year. Content includes a review of algebraic, logarithmic, exponential, and trigonometric functions.
Calculus
The calculus course was designed by professors at the University of Tennessee at Knoxville and based on Math 141 for the first semester and Math 142 for the second semester of the TGA calculus course.
Mathematics for the Life Sciences
Mathematics for the Life Sciences, commonly MLS, was designed by professors at the University of Tennessee at Knoxville and based on Math 151. MLS provided an introduction to a variety of mathematical topics of use in analyzing problems arising in the biological sciences.
Physics
Also designed by university professors, this course was modeled on Physics 135, an introduction to Physics for Math and Physical Science Majors at the university. The course was calculus-based, but because most students entering TGA had no prior knowledge of calculus, began with topics for which the calculus is either unnecessary, not required by the curriculum set by the state, or easily taught around.
Humanities
At TGA, the Humanities course was designed to fulfill the requirements for the curriculum of the State of Tennessee for American History and English III.
Student life
Cottage life
The students at the Tennessee Governor's Academy lived in cottages on the Tennessee School for the Deaf campus. The facilities were furnished by the University of Tennessee and the State. Each cottage consists of six student rooms and two hall director rooms. The cottage is divided into two wings, each consisting of three bedrooms which are shared by two students each. Each cottage has a full kitchen, dining area, classroom, and living room, and classes are held in the cottages.
House system
During the early first semester, the students were divided into four "houses," modeled after residential housing systems. TGA's houses were named Copernicus, Divinitus, Illuminati, and Renaissance, and each was headed by a hall director or assistant hall director. Though meant to be a long lasting legacy, the ho |
https://en.wikipedia.org/wiki/Hereditary%20C%2A-subalgebra | In mathematics, a hereditary C*-subalgebra of a C*-algebra is a particular type of C*-subalgebra whose structure is closely related to that of the larger C*-algebra. A C*-subalgebra B of A is a hereditary C*-subalgebra if for all a ∈ A and b ∈ B such that 0 ≤ a ≤ b, we have a ∈ B.
Properties
A hereditary C*-subalgebra of an approximately finite-dimensional C*-algebra is also AF. This is not true for subalgebras that are not hereditary. For instance, every abelian C*-algebra can be embedded into an AF C*-algebra.
A C*-subalgebra is called full if it is not contained in any proper (two-sided) closed ideal. Two C*-algebras A and B are called stably isomorphic if A ⊗ K ≅ B ⊗ K, where K is the C*-algebra of compact operators on a separable infinite-dimensional Hilbert space. C*-algebras are stably isomorphic to their full hereditary C*-subalgebras. Hence, two C*-algebras are stably isomorphic if they contain stably isomorphic full hereditary C*-subalgebras.
Also hereditary C*-subalgebras are those C*-subalgebras in which the restriction of any irreducible representation is also irreducible.
Correspondence with closed left ideals
There is a bijective correspondence between closed left ideals and hereditary C*-subalgebras of A. If L ⊂ A is a closed left ideal, let L* denote the image of L under the *-operation. The set L* is a right ideal and L* ∩ L is a C*-subalgebra. In fact, L* ∩ L is hereditary and the map L L* ∩ L is a bijection. It follows from this correspondence that every closed ideal is a hereditary C*-subalgebra. Another corollary is that a hereditary C*-subalgebra of a simple C*-algebra is also simple.
Connections with positive elements
If p is a projection of A (or a projection of the multiplier algebra of A), then pAp is a hereditary C*-subalgebra known as a corner of A. More generally, given a positive a ∈ A, the closure of the set aAa is the smallest hereditary C*-subalgebra containing a, denoted by Her(a). If A is separable, then every hereditary C*-subalgebra has this form.
These hereditary C*-subalgebras can bring some insight into the notion of Cuntz subequivalence. In particular, if a and b are positive elements of a C*-algebra A, then if b ∈ Her(a). Hence, a ~ b if Her(a) = Her(b).
If A is unital and the positive element a is invertible, then Her(a) = A. This suggests the following notion for the non-unital case: a ∈ A is said to be strictly positive if Her(a) = A. For example, in the C*-algebra K(H) of compact operators acting on Hilbert space H, a compact operator is strictly positive if and only if its range is dense in H. A commutative C*-algebra contains a strictly positive element if and only if the spectrum of the algebra is σ-compact. More generally, a C*-algebra contains a strictly positive element if and only if the algebra has a sequential approximate identity.
References
C*-algebras |
https://en.wikipedia.org/wiki/Barber%E2%80%93Johnson%20diagram | A Barber–Johnson diagram is a method of presenting hospital statistics combining four different variables in a unique graph, introduced in 1973. The method constructs a scattergram where length of stay, turnover interval, discharges, and deaths per available bed are combined. These four variables have a common relationship between them and their combination in the diagram permitted a new improved way for analyzing efficiency and performance of the hospital sector. The most complete reference about how to construct the diagram could be found in Yates. In this book, the appendix explains in detail the way for elaborating this kind of diagram.
References
External links
https://web.archive.org/web/20060908085540/http://www.publichealth.pitt.edu/supercourse/SupercoursePPT/7011-8001/7891.ppt
Statistical charts and diagrams
Medical statistics |
https://en.wikipedia.org/wiki/United%20Nations%20Security%20Council%20Resolution%20203 | United Nations Security Council Resolution 203, adopted on May 14, 1965, in the face of growing instability, a developing civil war and the probability of foreign intervention in the Dominican Republic, the Council called for a strict cease-fire and invited the Secretary-General to send a representative to the Dominican Republic to report to the Council on the present situation.
The resolution was passed unanimously.
See also
Dominican Civil War
List of United Nations Security Council Resolutions 201 to 300 (1965–1971)
Operation Power Pack
References
Text of the Resolution at undocs.org
External links
0203
1965 in the Dominican Republic
0203
May 1965 events |
https://en.wikipedia.org/wiki/Clay%20Mathematics%20Monographs | Clay Mathematics Monographs is a series of expositions in mathematics co-published by AMS and Clay Mathematics Institute. Each volume in the series offers an exposition of an active area of current research, provided by a group of mathematicians.
List of books
External links
Clay Mathematics Monographs list at ams.org
Series of mathematics books |
https://en.wikipedia.org/wiki/Contraction%20principle%20%28large%20deviations%20theory%29 | In mathematics — specifically, in large deviations theory — the contraction principle is a theorem that states how a large deviation principle on one space "pushes forward" (via the pushforward of a probability measure) to a large deviation principle on another space via a continuous function.
Statement
Let X and Y be Hausdorff topological spaces and let (με)ε>0 be a family of probability measures on X that satisfies the large deviation principle with rate function I : X → [0, +∞]. Let T : X → Y be a continuous function, and let νε = T∗(με) be the push-forward measure of με by T, i.e., for each measurable set/event E ⊆ Y, νε(E) = με(T−1(E)). Let
with the convention that the infimum of I over the empty set ∅ is +∞. Then:
J : Y → [0, +∞] is a rate function on Y,
J is a good rate function on Y if I is a good rate function on X, and
(νε)ε>0 satisfies the large deviation principle on Y with rate function J.
References
(See chapter 4.2.1)
Asymptotic analysis
Large deviations theory
Mathematical principles
Probability theorems |
https://en.wikipedia.org/wiki/Contraction%20principle | In mathematics, contraction principle may refer to:
Contraction principle (large deviations theory), a theorem that states how a large deviation principle on one space "pushes forward" to another space
Banach contraction principle, a tool in the theory of metric spaces |
https://en.wikipedia.org/wiki/Richardson%27s%20theorem | In mathematics, Richardson's theorem establishes the undecidability of the equality of real numbers defined by expressions involving integers, , and exponential and sine functions. It was proved in 1968 by mathematician and computer scientist Daniel Richardson of the University of Bath.
Specifically, the class of expressions for which the theorem holds is that generated by rational numbers, the number π, the number ln 2, the variable x, the operations of addition, subtraction, multiplication, composition, and the sin, exp, and abs functions.
For some classes of expressions (generated by other primitives than in Richardson's theorem) there exist algorithms that can determine whether an expression is zero.
Statement of the theorem
Richardson's theorem can be stated as follows:
Let E be a set of expressions that represent functions. Suppose that E includes these expressions:
x (representing the identity function)
ex (representing the exponential functions)
sin x (representing the sin function)
all rational numbers, ln 2, and π (representing constant functions that ignore their input and produce the given number as output)
Suppose E is also closed under a few standard operations. Specifically, suppose that if A and B are in E, then all of the following are also in E:
A + B (representing the pointwise addition of the functions that A and B represent)
A − B (representing pointwise subtraction)
AB (representing pointwise multiplication)
A∘B (representing the composition of the functions represented by A and B)
Then the following decision problems are unsolvable:
Deciding whether an expression A in E represents a function that is nonnegative everywhere
If E includes also the expression |x| (representing the absolute value function), deciding whether an expression A in E represents a function that is zero everywhere
If E includes an expression B representing a function whose antiderivative has no representative in E, deciding whether an expression A in E represents a function whose antiderivative can be represented in E. (Example: has an antiderivative in the elementary functions if and only if .)
Extensions
After Hilbert's tenth problem was solved in 1970, B. F. Caviness observed that the use of ex and ln 2 could be removed.
Wang later noted that under the same assumptions under which the question of whether there was x with A(x) < 0 was insolvable, the question of whether there was x with A(x) = 0 was also insolvable.
Miklós Laczkovich removed also the need for π and reduced the use of composition. In particular, given an expression A(x) in the ring generated by the integers, x, sin xn, and sin(x sin xn) (for n ranging over positive integers), both the question of whether A(x) > 0 for some x and whether A(x) = 0 for some x are unsolvable.
By contrast, the Tarski–Seidenberg theorem says that the first-order theory of the real field is decidable, so it is not possible to remove the sine function entirely.
See also
References
Furt |
https://en.wikipedia.org/wiki/Map%20of%20lattices | The concept of a lattice arises in order theory, a branch of mathematics. The Hasse diagram below depicts the inclusion relationships among some important subclasses of lattices.
Proofs of the relationships in the map
1. A boolean algebra is a complemented distributive lattice. (def)
2. A boolean algebra is a heyting algebra.
3. A boolean algebra is orthocomplemented.
4. A distributive orthocomplemented lattice is orthomodular.
5. A boolean algebra is orthomodular. (1,3,4)
6. An orthomodular lattice is orthocomplemented. (def)
7. An orthocomplemented lattice is complemented. (def)
8. A complemented lattice is bounded. (def)
9. An algebraic lattice is complete. (def)
10. A complete lattice is bounded.
11. A heyting algebra is bounded. (def)
12. A bounded lattice is a lattice. (def)
13. A heyting algebra is residuated.
14. A residuated lattice is a lattice. (def)
15. A distributive lattice is modular.
16. A modular complemented lattice is relatively complemented.
17. A boolean algebra is relatively complemented. (1,15,16)
18. A relatively complemented lattice is a lattice. (def)
19. A heyting algebra is distributive.
20. A totally ordered set is a distributive lattice.
21. A metric lattice is modular.
22. A modular lattice is semi-modular.
23. A projective lattice is modular.
24. A projective lattice is geometric. (def)
25. A geometric lattice is semi-modular.
26. A semi-modular lattice is atomic.
27. An atomic lattice is a lattice. (def)
28. A lattice is a semi-lattice. (def)
29. A semi-lattice is a partially ordered set. (def)
Notes
References
Lattice theory |
https://en.wikipedia.org/wiki/Peter%20Johnstone%20%28mathematician%29 | Peter Tennant Johnstone (born December 28, 1948) is Professor of the Foundations of Mathematics at the University of Cambridge, and a fellow of St. John's College.
He invented or developed a broad range of fundamental ideas in topos theory. His thesis, completed at the University of Cambridge in 1974, was entitled "Some Aspects of Internal Category Theory in an Elementary Topos".
Peter Johnstone is a choral singer, having sung for over thirty years with the Cambridge University Musical Society and since 2004 with the (London) Bach Choir. Following a severe bout of COVID-19 in 2020, he was invited by the Bach Choir's musical director David Hill to provide the text for a new choral work about the pandemic which the Choir commissioned from the composer Richard Blackford; the piece, `Vision of a Garden', was performed at the Bach Choir's first post-lockdown concert in October 2021 in the Royal Festival Hall, london, and again in July 2023 in King's College Chapel, Cambridge
He is a great-great-great nephew of the Reverend George Gilfillan who was eulogised in William McGonagall's first poem.
Books
.
— "[F]ar too hard to read, and not for the faint-hearted"
.
.
(v.3 in preparation)
References
External links
Johnstone's web page
Category theorists
Living people
Cambridge mathematicians
Fellows of St John's College, Cambridge
1948 births |
https://en.wikipedia.org/wiki/Tilted%20large%20deviation%20principle | In mathematics — specifically, in large deviations theory — the tilted large deviation principle is a result that allows one to generate a new large deviation principle from an old one by "tilting", i.e. integration against an exponential functional. It can be seen as an alternative formulation of Varadhan's lemma.
Statement of the theorem
Let X be a Polish space (i.e., a separable, completely metrizable topological space), and let (με)ε>0 be a family of probability measures on X that satisfies the large deviation principle with rate function I : X → [0, +∞]. Let F : X → R be a continuous function that is bounded from above. For each Borel set S ⊆ X, let
and define a new family of probability measures (νε)ε>0 on X by
Then (νε)ε>0 satisfies the large deviation principle on X with rate function IF : X → [0, +∞] given by
References
Asymptotic analysis
Mathematical principles
Probability theorems
Large deviations theory |
https://en.wikipedia.org/wiki/Differentiation%20of%20integrals | In mathematics, the problem of differentiation of integrals is that of determining under what circumstances the mean value integral of a suitable function on a small neighbourhood of a point approximates the value of the function at that point. More formally, given a space X with a measure μ and a metric d, one asks for what functions f : X → R does
for all (or at least μ-almost all) x ∈ X? (Here, as in the rest of the article, Br(x) denotes the open ball in X with d-radius r and centre x.) This is a natural question to ask, especially in view of the heuristic construction of the Riemann integral, in which it is almost implicit that f(x) is a "good representative" for the values of f near x.
Theorems on the differentiation of integrals
Lebesgue measure
One result on the differentiation of integrals is the Lebesgue differentiation theorem, as proved by Henri Lebesgue in 1910. Consider n-dimensional Lebesgue measure λn on n-dimensional Euclidean space Rn. Then, for any locally integrable function f : Rn → R, one has
for λn-almost all points x ∈ Rn. It is important to note, however, that the measure zero set of "bad" points depends on the function f.
Borel measures on Rn
The result for Lebesgue measure turns out to be a special case of the following result, which is based on the Besicovitch covering theorem: if μ is any locally finite Borel measure on Rn and f : Rn → R is locally integrable with respect to μ, then
for μ-almost all points x ∈ Rn.
Gaussian measures
The problem of the differentiation of integrals is much harder in an infinite-dimensional setting. Consider a separable Hilbert space (H, ⟨ , ⟩) equipped with a Gaussian measure γ. As stated in the article on the Vitali covering theorem, the Vitali covering theorem fails for Gaussian measures on infinite-dimensional Hilbert spaces. Two results of David Preiss (1981 and 1983) show the kind of difficulties that one can expect to encounter in this setting:
There is a Gaussian measure γ on a separable Hilbert space H and a Borel set M ⊆ H so that, for γ-almost all x ∈ H,
There is a Gaussian measure γ on a separable Hilbert space H and a function f ∈ L1(H, γ; R) such that
However, there is some hope if one has good control over the covariance of γ. Let the covariance operator of γ be S : H → H given by
or, for some countable orthonormal basis (ei)i∈N of H,
In 1981, Preiss and Jaroslav Tišer showed that if there exists a constant 0 < q < 1 such that
then, for all f ∈ L1(H, γ; R),
where the convergence is convergence in measure with respect to γ. In 1988, Tišer showed that if
for some α > 5 ⁄ 2, then
for γ-almost all x and all f ∈ Lp(H, γ; R), p > 1.
As of 2007, it is still an open question whether there exists an infinite-dimensional Gaussian measure γ on a separable Hilbert space H so that, for all f ∈ L1(H, γ; R),
for γ-almost all x ∈ H. However, it is conjectured that no such measure exists, since the σi would have to decay very rapidly.
See also
Referenc |
https://en.wikipedia.org/wiki/The%20Mathematical%20Experience | The Mathematical Experience (1981) is a book by Philip J. Davis and Reuben Hersh that discusses the practice of modern mathematics from a historical and philosophical perspective. The book discusses the psychology of mathematicians, and gives examples of famous proofs and outstanding problems. It goes on to speculate about what a proof really means, in relationship to actual truth. Other topics include mathematics in education and some of the math that occurs in computer science.
The first paperback edition won a U.S. National Book Award in Science. It is cited by some mathematicians as influential in their decision to continue their studies in graduate school; and has been hailed as a classic of mathematical literature.
On the other hand, Martin Gardner disagreed with some of the authors' philosophical opinions.
A new edition, published in 1995, includes exercises and problems, making the book more suitable for classrooms. There is also The Companion Guide to The Mathematical Experience, Study Edition. Both were co-authored with Elena Marchisotto. Davis and Hersh wrote a follow-up book, Descartes' Dream: The World According to Mathematics (Harcourt, 1986), and each has written other books with related themes, such as Mathematics And Common Sense: A Case of Creative Tension by Davis and What is Mathematics, Really? by Hersh.
Notes
References
External links
Book Review of the 1995 edition, by Kenneth C. Millett at the American Mathematical Society.
The Mathematical Experience from the Internet Archive
Books about mathematics
National Book Award-winning works
1981 non-fiction books
Collaborative non-fiction books |
https://en.wikipedia.org/wiki/Tom%20Jennings%20%28pool%20player%29 | Tom Jennings (born 1951) is an American professional pocket billiards (pool) player and mathematics professor. He won the BCA U.S. Open Straight Pool Championship in 1976 and 1977, being the first player since Steve Mizerak to win consecutive championships. He won both titles while also a full-time mathematics professor at Middlesex County College in New Jersey.
Early years
At 17 years of age, Jennings was a highly skilled player who claimed to be capable of making runs of 300.
Career
In August 1976, despite having never won a single match in four prior BCA U.S. Open Straight Pool Championships, Jennings was victorious in the straight pool (14.1 continuous) championship held in Chicago. In 1977, the tournament was held in Dayton, Ohio in September. Jennings was not positioned well for a repeat performance. He lost in the first match of the double elimination tournament to Tom Kollins by a score of 150–135. He won the next five matches easily, and earned the right to play Dick Lane in the championship match.The championship was played where the winner being the first to score 200 points. Lane was a formidable opponent, having made a run of 111 points earlier in the tournament. After 19½ innings, Lane was in a 64-point run and was leading by a score of 196–42, four points away from the championship. On the 65th shot of his run, Lane missed the break, giving Jennings an opening. Jennings returned immediately with a 71-point run of his own (his personal best for the tournament), which closed the gap to 196–113. The two mostly traded for ten more innings, during which time Lane only sank a single additional ball while Jennings inched his way closer to Lane's score. With the score at 197–171 after 30 innings, Jennings put together a final run of 29 points to seal his second consecutive US Open championship. Jennings' 158 points to Lane's 1 over the last eleven and a half innings has been called "Billiards' Biggest Comeback."
Titles
1976 Maine 14.1 Championship
1976 BCA U.S. Open Straight Pool Championship
1977 BCA U.S. Open Straight Pool Championship
References
Living people
American pool players
1951 births |
https://en.wikipedia.org/wiki/Differentiation%20of%20measures | In mathematics, differentiation of measures may refer to:
the problem of differentiation of integrals, also known as the differentiation problem for measures;
the Radon–Nikodym derivative of one measure with respect to another.
the theory of differentiable measures. |
https://en.wikipedia.org/wiki/Stein%E2%80%93Str%C3%B6mberg%20theorem | In mathematics, the Stein–Strömberg theorem or Stein–Strömberg inequality is a result in measure theory concerning the Hardy–Littlewood maximal operator. The result is foundational in the study of the problem of differentiation of integrals. The result is named after the mathematicians Elias M. Stein and Jan-Olov Strömberg.
Statement of the theorem
Let λn denote n-dimensional Lebesgue measure on n-dimensional Euclidean space Rn and let M denote the Hardy–Littlewood maximal operator: for a function f : Rn → R, Mf : Rn → R is defined by
where Br(x) denotes the open ball of radius r with center x. Then, for each p > 1, there is a constant Cp > 0 such that, for all natural numbers n and functions f ∈ Lp(Rn; R),
In general, a maximal operator M is said to be of strong type (p, p) if
for all f ∈ Lp(Rn; R). Thus, the Stein–Strömberg theorem is the statement that the Hardy–Littlewood maximal operator is of strong type (p, p) uniformly with respect to the dimension n.
References
Inequalities
Theorems in measure theory
Operator theory |
https://en.wikipedia.org/wiki/Bhaskara%27s%20lemma | Bhaskara's Lemma is an identity used as a lemma during the chakravala method. It states that:
for integers and non-zero integer .
Proof
The proof follows from simple algebraic manipulations as follows: multiply both sides of the equation by , add , factor, and divide by .
So long as neither nor are zero, the implication goes in both directions. (The lemma holds for real or complex numbers as well as integers.)
References
C. O. Selenius, "Rationale of the chakravala process of Jayadeva and Bhaskara II", Historia Mathematica, 2 (1975), 167-184.
C. O. Selenius, Kettenbruch theoretische Erklarung der zyklischen Methode zur Losung der Bhaskara-Pell-Gleichung, Acta Acad. Abo. Math. Phys. 23 (10) (1963).
George Gheverghese Joseph, The Crest of the Peacock: Non-European Roots of Mathematics (1975).
External links
Introduction to chakravala
Diophantine equations
Number theoretic algorithms
Lemmas in algebra
Indian mathematics
Articles containing proofs |
https://en.wikipedia.org/wiki/Horosphere | In hyperbolic geometry, a horosphere (or parasphere) is a specific hypersurface in hyperbolic n-space. It is the boundary of a horoball, the limit of a sequence of increasing balls sharing (on one side) a tangent hyperplane and its point of tangency. For n = 2 a horosphere is called a horocycle.
A horosphere can also be described as the limit of the hyperspheres that share a tangent hyperplane at a given point, as their radii go towards infinity. In Euclidean geometry, such a "hypersphere of infinite radius" would be a hyperplane, but in hyperbolic geometry it is a horosphere (a curved surface).
History
The concept has its roots in a notion expressed by F. L. Wachter in 1816 in a letter to his teacher Gauss. Noting that in Euclidean geometry the limit of a sphere as its radius tends to infinity is a plane, Wachter affirmed that even if the fifth postulate were false, there would nevertheless be a geometry on the surface identical with that of the ordinary plane. The terms horosphere and horocycle are due to Lobachevsky, who established various results showing that the geometry of horocycles and the horosphere in hyperbolic space were equivalent to those of lines and the plane in Euclidean space. The term "horoball" is due to William Thurston, who used it in his work on hyperbolic 3-manifolds. The terms horosphere and horoball are often used in 3-dimensional hyperbolic geometry.
Models
In the conformal ball model, a horosphere is represented by a sphere tangent to the horizon sphere. In the upper half-space model, a horosphere can appear either as a sphere tangent to the horizon plane, or as a plane parallel to the horizon plane. In the hyperboloid model, a horosphere is represented by a plane whose normal lies in the asymptotic cone.
Curvature
A horosphere has a critical amount of (isotropic) curvature: if the curvature were any greater, the surface would be able to close, yielding a sphere, and if the curvature were any less, the surface would be an (N − 1)-dimensional hypercycle.
References
Appendix, the theory of space Janos Bolyai, 1987, p.143
3-manifolds
Curves
Hyperbolic geometry |
https://en.wikipedia.org/wiki/Category%20algebra | In category theory, a field of mathematics, a category algebra is an associative algebra, defined for any locally finite category and commutative ring with unity. Category algebras generalize the notions of group algebras and incidence algebras, just as categories generalize the notions of groups and partially ordered sets.
Definition
If the given category is finite (has finitely many objects and morphisms), then the following two definitions of the category algebra agree.
Group algebra-style definition
Given a group G and a commutative ring R, one can construct RG, known as the group algebra; it is an R-module equipped with a multiplication. A group is the same as a category with a single object in which all morphisms are isomorphisms (where the elements of the group correspond to the morphisms of the category), so the following construction generalizes the definition of the group algebra from groups to arbitrary categories.
Let C be a category and R be a commutative ring with unity. Define RC (or R[C]) to be the free R-module with the set of morphisms of C as its basis. In other words, RC consists of formal linear combinations (which are finite sums) of the form , where fi are morphisms of C, and ai are elements of the ring R. Define a multiplication operation on RC as follows, using the composition operation in the category:
where if their composition is not defined. This defines a binary operation on RC, and moreover makes RC into an associative algebra over the ring R. This algebra is called the category algebra of C.
From a different perspective, elements of the free module RC could also be considered as functions from the morphisms of C to R which are finitely supported. Then the multiplication is described by a convolution: if (thought of as functionals on the morphisms of C), then their product is defined as:
The latter sum is finite because the functions are finitely supported, and therefore .
Incidence algebra-style definition
The definition used for incidence algebras assumes that the category C is locally finite (see below), is dual to the above definition, and defines a different object. This isn't a useful assumption for groups, as a group that is locally finite as a category is finite.
A locally finite category is one where every morphism can be written in only finitely many ways as the composition of two non-identity morphisms (not to be confused with the "has finite Hom-sets" meaning). The category algebra (in this sense) is defined as above, but allowing all coefficients to be non-zero.
In terms of formal sums, the elements are all formal sums
where there are no restrictions on the (they can all be non-zero).
In terms of functions, the elements are any functions from the morphisms of C to R, and multiplication is defined as convolution. The sum in the convolution is always finite because of the local finiteness assumption.
Dual
The module dual of the category algebra (in the group algebra sense of the definition |
https://en.wikipedia.org/wiki/Chris%20Hall%20%28cryptographer%29 | Christopher Hall is an American cryptographer and mathematician, specializing in arithmetic geometry. He is one of the creators of the cryptosystem Twofish. He obtained a BS from the University of Colorado-Boulder Department of Computer Science and a PhD in Mathematics from Princeton University in 2003, under Nick Katz.
He is an associate professor of mathematics at the University of Western Ontario.
References
External links
Home page
American cryptographers
Living people
Year of birth missing (living people)
Princeton University alumni
Academic staff of the University of Western Ontario |
https://en.wikipedia.org/wiki/Babu%C5%A1ka%E2%80%93Lax%E2%80%93Milgram%20theorem | In mathematics, the Babuška–Lax–Milgram theorem is a generalization of the famous Lax–Milgram theorem, which gives conditions under which a bilinear form can be "inverted" to show the existence and uniqueness of a weak solution to a given boundary value problem. The result is named after the mathematicians Ivo Babuška, Peter Lax and Arthur Milgram.
Background
In the modern, functional-analytic approach to the study of partial differential equations, one does not attempt to solve a given partial differential equation directly, but by using the structure of the vector space of possible solutions, e.g. a Sobolev space W k,p. Abstractly, consider two real normed spaces U and V with their continuous dual spaces U∗ and V∗ respectively. In many applications, U is the space of possible solutions; given some partial differential operator Λ : U → V∗ and a specified element f ∈ V∗, the objective is to find a u ∈ U such that
However, in the weak formulation, this equation is only required to hold when "tested" against all other possible elements of V. This "testing" is accomplished by means of a bilinear function B : U × V → R which encodes the differential operator Λ; a weak solution to the problem is to find a u ∈ U such that
The achievement of Lax and Milgram in their 1954 result was to specify sufficient conditions for this weak formulation to have a unique solution that depends continuously upon the specified datum f ∈ V∗: it suffices that U = V is a Hilbert space, that B is continuous, and that B is strongly coercive, i.e.
for some constant c > 0 and all u ∈ U.
For example, in the solution of the Poisson equation on a bounded, open domain Ω ⊂ Rn,
the space U could be taken to be the Sobolev space H01(Ω) with dual H−1(Ω); the former is a subspace of the Lp space V = L2(Ω); the bilinear form B associated to −Δ is the L2(Ω) inner product of the derivatives:
Hence, the weak formulation of the Poisson equation, given f ∈ L2(Ω), is to find uf such that
Statement of the theorem
In 1971, Babuška provided the following generalization of Lax and Milgram's earlier result, which begins by dispensing with the requirement that U and V be the same space. Let U and V be two real Hilbert spaces and let B : U × V → R be a continuous bilinear functional. Suppose also that B is weakly coercive: for some constant c > 0 and all u ∈ U,
and, for all 0 ≠ v ∈ V,
Then, for all f ∈ V∗, there exists a unique solution u = uf ∈ U to the weak problem
Moreover, the solution depends continuously on the given data:
See also
Lions–Lax–Milgram theorem
References
External links
Theorems in analysis
Partial differential equations |
https://en.wikipedia.org/wiki/Analytic%20semigroup | In mathematics, an analytic semigroup is particular kind of strongly continuous semigroup. Analytic semigroups are used in the solution of partial differential equations; compared to strongly continuous semigroups, analytic semigroups provide better regularity of solutions to initial value problems, better results concerning perturbations of the infinitesimal generator, and a relationship between the type of the semigroup and the spectrum of the infinitesimal generator.
Definition
Let Γ(t) = exp(At) be a strongly continuous one-parameter semigroup on a Banach space (X, ||·||) with infinitesimal generator A. Γ is said to be an analytic semigroup if
for some 0 < θ < π/ 2, the continuous linear operator exp(At) : X → X can be extended to t ∈ Δθ ,
and the usual semigroup conditions hold for s, t ∈ Δθ : exp(A0) = id, exp(A(t + s)) = exp(At) exp(As), and, for each x ∈ X, exp(At)x is continuous in t;
and, for all t ∈ Δθ \ {0}, exp(At) is analytic in t in the sense of the uniform operator topology.
Characterization
The infinitesimal generators of analytic semigroups have the following characterization:
A closed, densely defined linear operator A on a Banach space X is the generator of an analytic semigroup if and only if there exists an ω ∈ R such that the half-plane Re(λ) > ω is contained in the resolvent set of A and, moreover, there is a constant C such that
for Re(λ) > ω and where is the resolvent of the operator A. Such operators are called sectorial. If this is the case, then the resolvent set actually contains a sector of the form
for some δ > 0, and an analogous resolvent estimate holds in this sector. Moreover, the semigroup is represented by
where γ is any curve from e−iθ∞ to e+iθ∞ such that γ lies entirely in the sector
with π/ 2 < θ < π/ 2 + δ.
References
Functional analysis
Partial differential equations
Semigroup theory |
https://en.wikipedia.org/wiki/European%20Congress%20of%20Mathematics | The European Congress of Mathematics (ECM) is the second largest international conference of the mathematics community, after the International Congresses of Mathematicians (ICM).
The ECM are held every four years and are timed precisely between the ICM. The ECM is held under the auspices of the European Mathematical Society (EMS), and was one of its earliest initiatives. It was founded by Max Karoubi and the first edition took place in Paris in 1992.
Its objectives are "to present various new aspects of pure and applied mathematics to a wide audience, to be a forum for discussion of the relationship between mathematics and society in Europe, and to enhance cooperation among mathematicians from all European countries."
Activities
The Congresses generally last a week and consist of plenary lectures, parallel (invited) lectures and several mini-symposia devoted to a particular subject, where participants can contribute with posters and short talks. Many editions featured also special lectures, e.g. by prize winners, and public sessions aimed at a general audience.
Other mathematics conferences and workshops organised in the same period become often satellite events of the ECM.
Prizes
Several prizes are awarded at the beginning of the Congress:
The EMS Prize (awarded since the first edition in 1992), to up to ten young mathematicians of European nationality or working in Europe
The Felix Klein Prize (awarded since 2000), to at most three young applied mathematicians
The Otto Neugebauer Prize (awarded since 2012) to a researcher in history of mathematics
List of congresses
1st edition – Paris (1992)
2nd edition – Budapest (1996)
3rd edition – Barcelona (2000)
4th edition – Stockholm (2004)
5th edition – Amsterdam (2008)
6th edition – Kraków (2012)
7th edition – Berlin (2016)
8th edition – Portorož (2021)
The 9th European Congress of Mathematics will be held in Seville in 2024.
References
Recurring events established in 1992
Mathematics conferences
Math
Quadrennial events |
https://en.wikipedia.org/wiki/Pan-African%20Congress%20of%20Mathematicians | The Pan-African Congress of Mathematicians (PACOM) is an international congress of mathematics, held under the auspices of the African Mathematical Union.
List of congresses
2008 –
2004 – Tunis, Tunisia
2000 – Cape Town, South Africa
1995 – Ifrane, Morocco
1991 – Nairobi, Kenya
1986 – Jos, Nigeria
1976 – Rabat, Morocco
External links
7th PACOM 2008
Recurring events established in 1976
Mathematics conferences |
https://en.wikipedia.org/wiki/Cram%C3%A9r%27s%20decomposition%20theorem | Cramér’s decomposition theorem for a normal distribution is a result of probability theory. It is well known that, given independent normally distributed random variables ξ1, ξ2, their sum is normally distributed as well. It turns out that the converse is also true. The latter result, initially announced by Paul Lévy, has been proved by Harald Cramér. This became a starting point for a new subfield in probability theory, decomposition theory for random variables as sums of independent variables (also known as arithmetic of probabilistic distributions).
The precise statement of the theorem
Let a random variable ξ be normally distributed and admit a decomposition as a sum ξ=ξ1+ξ2 of two independent random variables. Then the summands ξ1 and ξ2 are normally distributed as well.
A proof of Cramér's decomposition theorem uses the theory of entire functions.
See also
Raikov's theorem: Similar result for Poisson distribution.
References
Probability theorems
Theorems in statistics
Characterization of probability distributions |
https://en.wikipedia.org/wiki/Densely%20defined%20operator | In mathematics – specifically, in operator theory – a densely defined operator or partially defined operator is a type of partially defined function. In a topological sense, it is a linear operator that is defined "almost everywhere". Densely defined operators often arise in functional analysis as operations that one would like to apply to a larger class of objects than those for which they a priori "make sense".
Definition
A densely defined linear operator from one topological vector space, to another one, is a linear operator that is defined on a dense linear subspace of and takes values in written Sometimes this is abbreviated as when the context makes it clear that might not be the set-theoretic domain of
Examples
Consider the space of all real-valued, continuous functions defined on the unit interval; let denote the subspace consisting of all continuously differentiable functions. Equip with the supremum norm ; this makes into a real Banach space. The differentiation operator given by is a densely defined operator from to itself, defined on the dense subspace The operator is an example of an unbounded linear operator, since
This unboundedness causes problems if one wishes to somehow continuously extend the differentiation operator to the whole of
The Paley–Wiener integral, on the other hand, is an example of a continuous extension of a densely defined operator. In any abstract Wiener space with adjoint there is a natural continuous linear operator (in fact it is the inclusion, and is an isometry) from to under which goes to the equivalence class of in It can be shown that is dense in Since the above inclusion is continuous, there is a unique continuous linear extension of the inclusion to the whole of This extension is the Paley–Wiener map.
See also
References
Functional analysis
Hilbert spaces
Linear operators
Operator theory |
https://en.wikipedia.org/wiki/Unicyclic | Unicyclic may refer to:
Unicyclic graph, a graph in mathematics with one cycle
One-loop Feynman diagram, a type of pictorial representation in physics
A cyclic compound in chemistry with one ring |
https://en.wikipedia.org/wiki/R%C3%A9gis%20Pitbull | Régis Fernandes Silva (born September 22, 1976 in São Paulo), better known as Régis Pitbull or simply Régis, is a former Brazilian footballer who played as a forward.
Club statistics
Personal life
Régis has battled drug addiction. In April 2021, he was admitted to a rehabilitation clinic.
References
External links
1976 births
Brazilian men's footballers
Brazilian expatriate men's footballers
Men's association football forwards
Living people
Ceará Sporting Club players
C.S. Marítimo players
Kyoto Sanga FC players
Associação Atlética Ponte Preta players
CR Vasco da Gama players
Daejeon Hana Citizen players
K League 1 players
ABC Futebol Clube players
Sport Club Corinthians Paulista players
Associação Portuguesa de Desportos players
Esporte Clube Bahia players
Expatriate men's footballers in South Korea
Expatriate men's footballers in Japan
J1 League players
Expatriate men's footballers in Turkey
Brazilian expatriate sportspeople in South Korea
São Raimundo Esporte Clube (AM) footballers
Footballers from São Paulo |
https://en.wikipedia.org/wiki/Mariano%20Torresi | Luis Mariano Torresi (born 26 January 1981, in Mendoza) is an Argentine footballer who plays as a midfielder. He currently plays for Atlético Uruguay.
External links
Statistics at BDFA
1981 births
Living people
Footballers from Mendoza, Argentina
Argentine men's footballers
Argentine expatriate men's footballers
Godoy Cruz Antonio Tomba footballers
San Martín de San Juan footballers
Apollon Limassol FC players
Instituto Atlético Central Córdoba footballers
Club Libertad footballers
Argentine Primera División players
Cypriot First Division players
Men's association football midfielders
Expatriate men's footballers in Cyprus
Expatriate men's footballers in Mexico
Expatriate men's footballers in Paraguay |
https://en.wikipedia.org/wiki/Baire%20measure | In mathematics, a Baire measure is a measure on the σ-algebra of Baire sets of a topological space whose value on every compact Baire set is finite. In compact metric spaces the Borel sets and the Baire sets are the same, so Baire measures are the same as Borel measures that are finite on compact sets. In general Baire sets and Borel sets need not be the same. In spaces with non-Baire Borel sets, Baire measures are used because they connect to the properties of continuous functions more directly.
Variations
There are several inequivalent definitions of Baire sets, so correspondingly there are several inequivalent concepts of Baire measure on a topological space. These all coincide on spaces that are locally compact σ-compact Hausdorff spaces.
Relation to Borel measure
In practice Baire measures can be replaced by regular Borel measures. The relation between Baire measures and regular Borel measures is as follows:
The restriction of a finite Borel measure to the Baire sets is a Baire measure.
A finite Baire measure on a compact space is always regular.
A finite Baire measure on a compact space is the restriction of a unique regular Borel measure.
On compact (or σ-compact) metric spaces, Borel sets are the same as Baire sets and Borel measures are the same as Baire measures.
Examples
Counting measure on the unit interval is a measure on the Baire sets that is not regular (or σ-finite).
The (left or right) Haar measure on a locally compact group is a Baire measure invariant under the left (right) action of the group on itself. In particular, if the group is an abelian group, the left and right Haar measures coincide and we say the Haar measure is translation invariant. See also Pontryagin duality.
References
Leonard Gillman and Meyer Jerison, Rings of Continuous Functions, Springer Verlag #43, 1960
Measures (measure theory) |
https://en.wikipedia.org/wiki/Shilov%20boundary | In functional analysis, a branch of mathematics, the Shilov boundary is the smallest closed subset of the structure space of a commutative Banach algebra where an analog of the maximum modulus principle holds. It is named after its discoverer, Georgii Evgen'evich Shilov.
Precise definition and existence
Let be a commutative Banach algebra and let be its structure space equipped with the relative weak*-topology of the dual . A closed (in this topology) subset of is called a boundary of if for all .
The set is called the Shilov boundary. It has been proved by Shilov that is a boundary of .
Thus one may also say that Shilov boundary is the unique set which satisfies
is a boundary of , and
whenever is a boundary of , then .
Examples
Let be the open unit disc in the complex plane and let be the disc algebra, i.e. the functions holomorphic in and continuous in the closure of with supremum norm and usual algebraic operations. Then and .
References
Notes
See also
James boundary
Furstenberg boundary
Banach algebras |
https://en.wikipedia.org/wiki/Justice%20Research%20and%20Statistics%20Association | Justice Research and Statistics Association (JRSA) is a national nonprofit organization of state Statistical Analysis Centers, researchers, and practitioners throughout government, academia, and justice organizations. Justice Research and Statistics Association's members form a network of justice professionals dedicated to policy-relevant research and practice. The association was created in 1974 to promote cooperation and the exchange of criminal justice information among the states. JRSA is a 501(c)(3) nonprofit organization.
Statistical Analysis Centers contribute to viable, effective policy development in their states through statistical services, research, evaluation, and policy analysis.
Through the United States Bureau of Justice Statistics' State Justice Statistics Program, Statistical Analysis Centers also undertake statistical research and analysis on themes selected by Bureau of Justice Statistics and JRSA that reflect issues of current concern and significance to the justice community.
JRSA collects information annually in a computerized index called the Infobase of State Activities and Research on Statistical Analysis Centers' research, analyses, and activities, as well as reports and publications.
JRSA provides access to state-based information such as the Infobase of State Activities and Research, a searchable clearinghouse of Statistical Analysis Center research, and programs, and the Statistical Analysis Center Digest, an electronic compilation of Statistical Analysis Centers' publication abstracts.
JRSA publishes The Forum, a quarterly newsletter, and Justice Research and Policy, a semiannual peer-reviewed journal.
JRSA conducts multi-state research on statewide and system-wide problems and practices.
Justice professionals share information and hear about new research, programs, and technologies at conferences convened by JRSA.
JRSA provides training and technical assistance to justice-related organizations on a wide range of topics, such as automated systems planning and management; crime analysis, including spatial analysis of crime data and analysis of incident-based data; valuation and research methods; and computer technologies for records management, data analysis, and forecasting. Justice Research and Statistics Association has also advised agencies that award and manage grants, such as the Office of Justice Programs.
See also
SEARCH, The National Consortium for Justice Information and Statistics
References
External links
Criminal justice
Legal organizations based in the United States
Non-profit organizations based in Washington, D.C.
Organizations established in 1976
Statistical organizations in the United States |
https://en.wikipedia.org/wiki/Fermat%27s%20Last%20Theorem%20in%20fiction | The problem in number theory known as "Fermat's Last Theorem" has repeatedly received attention in fiction and popular culture. It was proved by Andrew Wiles in 1994.
Prose fiction
The theorem plays a key role in the 1948 mystery novel Murder by Mathematics by Hector Hawton.
Arthur Porges' short story "The Devil and Simon Flagg" features a mathematician who bargains with the Devil that the latter cannot produce a proof of Fermat's Last Theorem within twenty-four hours. The devil is not successful and is last seen beginning a collaboration with the hero. The story was first published in 1954 in The Magazine of Fantasy and Science Fiction.
In Douglas Hofstadter's 1979 book Gödel, Escher, Bach, the statement, "I have discovered a truly remarkable proof of this theorem which this margin is too small to contain" is repeatedly rephrased and satirized, including a pun on "fermata".
In Robert Forward's 1984/1985 science fiction novel Rocheworld, Fermat's Last Theorem is unproved far enough into the future for interstellar explorers to describe it to one of the mathematically inclined natives of another star system, who finds a proof.
In the 2003 book The Oxford Murders by Guillermo Martinez, Wiles's announcement in Cambridge of his proof of Fermat's Last Theorem forms a peripheral part of the action.
In Stieg Larsson's 2006 book The Girl Who Played With Fire, the main character Lisbeth Salander is mesmerized by the theorem. Fields medalist Timothy Gowers criticized Larsson's portrayal of the theorem as muddled and confused.
In Jasper Fforde's 2007 book First Among Sequels, 9 year-old Tuesday Next, seeing the equation on the sixth-form's math classroom's chalkboard, and thinking it homework, finds a simple counterexample.
Arthur C. Clarke and Frederik Pohl's 2008 novel The Last Theorem tells of the rise to fame and world prominence of a young Sri Lankan mathematician who devises an elegant proof of the theorem.
Television
"The Royale", an episode (first aired 27 March 1989) of Star Trek: The Next Generation, begins with Picard attempting to solve the puzzle in his ready room; he remarks to Riker that the theorem had remained unproven for 800 years. The captain ends the episode with the line "Like Fermat's theorem, it is a puzzle we may never solve." Wiles' proof was released five years after the episode aired. The theorem was again mentioned in a subsequent Star Trek: Deep Space Nine episode called "Facets" in June 1995, in which Jadzia Dax comments that one of her previous hosts, Tobin Dax, had "the most original approach to the proof since Wiles over 300 years ago." Most recently it was mentioned in "Grounded", an episode of Star Trek: Lower Decks as having been solved by a little boy.
A sum, proved impossible by the theorem, appears in the 1995 episode of The Simpsons, "Treehouse of Horror VI". In the three-dimensional world in "Homer3", the equation is visible, just as the dimension begins to collapse. The joke is that the twelfth root of the su |
https://en.wikipedia.org/wiki/Multiplication%20%28disambiguation%29 | Multiplication is an elementary mathematical operation.
Multiplication or multiply may also refer to:
A generalized multiplicative function, in number theory
Multiply (website), e-commerce website based in Jakarta, Indonesia
Multiplication of money, the compounding of central bank funds by commercial lending
Multiplication (alchemy), an alchemical process
Product (mathematics), a result of multiplying
Music
Multiplication (music), methods of applying multiplication in music
Multiply Records, a record label
Multiply (Jamie Lidell album), 2005
x (Ed Sheeran album), 2014
X∞Multiplies, a 1980 album by Yellow Magic Orchestra
"Multiplication" (song), a 1961 song by Bobby Darin
"Multiply" (ASAP Rocky song), 2014
"Multiply" (Xzibit song), 2002 |
https://en.wikipedia.org/wiki/Elon%20Lages%20Lima | Elon Lages Lima (July 9, 1929 – May 7, 2017) was a Brazilian mathematician whose research concerned differential topology, algebraic topology, and differential geometry. Lima was an influential figure in the development of mathematics in Brazil.
Lima was professor emeritus at Instituto Nacional de Matemática Pura e Aplicada of which he was the director during three separate periods. Lima has been twice recipient of the Prêmio Jabuti from the Câmara Brasileira do Livro, for his textbooks Espaços Métricos and Álgebra Linear, and of the Anísio Teixeira Prize from the Ministry of Education and Sports. His mathematical style was heavily influenced by Bourbaki's.
Biography
He began his career as a high school teacher in Fortaleza, Ceará. Lima graduated with a bachelor's degree in mathematics from the Universidade do Brasil (today UFRJ) in 1953. He obtained his doctorate in 1958 from the University of Chicago with Edwin Henry Spanier as advisor. He is a former Guggenheim Fellow, and holds memberships in the Academia Brasileira de Ciências (Brazilian Academy of Sciences) and the TWAS, the Academy of Sciences for the Developing World. He is a professor Honoris Causa of the Universidade Federal do Ceará and of the University of Brasília. He was a member of the Upper Board of FAPERJ from 1987 to 1991. He was also a member of the National Board of Education.
He wrote over thirty books in mathematics, some of which were intended for secondary school teachers. Between 1990 and 1995, he coordinated the IMPA-VITAE project, which held skills improvement courses for mathematics teachers in eleven cities from eight states throughout Brazil.
He received the grã-cruz ("Great Cross") of the Ordem Nacional do Mérito Científico ("National Order of Scientific Merit") of Brazil.
Selected publications
Lima, E. L. (1964). "Common singularities of commuting vector fields on 2-manifolds". Commentarii Mathematici Helvetici. vol. 39, pp. 97–110.
Lima, E. L. (1965). "Commuting vector fields on S3". Annals of Mathematics. vol. 81, pp. 70–88.
do Carmo, M. and Lima, E. L. (1969). "Isometric immersions with semi-definite second quadratic forms". Archiv der Mathematik vol. 20, pp. 173–175.
Lima, E. L. (1987). "Orientability of smooth hypersurfaces and the Jordan-Brouwer separation theorem". Expositiones Mathematicae. vol. 5, pp. 283–286.
See also
Spectrum (topology), a notion introduced by Lima.
Notes
Silva, C. P. . "Sobre o início e consolidação da pesquisa matemática no Brasil, Parte 2", Revista Brasileira de História da Matemática (RBHM), Vol. 6, n. 12, p. 165-196, 2006
1929 births
2017 deaths
Brazilian educators
People from Maceió
University of Chicago alumni
Recipients of the Great Cross of the National Order of Scientific Merit (Brazil)
Topologists
Textbook writers
Instituto Nacional de Matemática Pura e Aplicada researchers
Members of the Brazilian Academy of Sciences
21st-century Brazilian mathematicians
20th-century Brazilian mathematicians
20th-century Braz |
https://en.wikipedia.org/wiki/Leonardo%20Devanir | Leonardo Devanir de Paula or simply Leonardo (born March 12, 1977 in Juiz de Fora), is a Brazilian central defender who currently plays for Nova Iguaçu.
Career
Flamengo career statistics
(Correct 13 July 2008)
according to combined sources on the.
Honours
Coritiba
Paraná State Championship: 1999
Palmeiras
Brazilian Série B: 2003
Goiás
Goiás State Championship: 2006
References
External links
sambafoot
Guardian Stats Centre
zerozero.pt
goiasesporteclube.com
1977 births
Living people
Footballers from Juiz de Fora
Brazilian men's footballers
Tupi Football Club players
Coritiba Foot Ball Club players
Goiás Esporte Clube players
Sociedade Esportiva Palmeiras players
CR Flamengo footballers
Vila Nova Futebol Clube players
Ipatinga Futebol Clube players
Expatriate men's footballers in Syria
Men's association football defenders
Syrian Premier League players |
https://en.wikipedia.org/wiki/Intersection%20theory%20%28disambiguation%29 | Intersection theory may refer to:
Intersection theory, especially in algebraic geometry
Intersection (set theory) |
https://en.wikipedia.org/wiki/Classification%20of%20the%20sciences%20%28Peirce%29 | The philosopher Charles Sanders Peirce (1839–1914) did considerable work over a period of years on the classification of
sciences (including mathematics). His classifications are of interest both as a map for navigating his philosophy and as an accomplished polymath's survey of research in his time. Peirce himself was well grounded and produced work in many research fields, including logic, mathematics, statistics, philosophy, spectroscopy, gravimetry, geodesy, chemistry, and experimental psychology.
Classifications
Philosophers have done little work on classification of the sciences and mathematics since Peirce's time. Noting Peirce's "important" contribution, Denmark's Birger Hjørland commented: "There is not today (2005), to my knowledge, any organized research program about the classification of the sciences in any discipline or in any country". As Miksa (1998) writes, the "interest for this question largely died in the beginning of the 20th century". It is not clear whether Hjørland includes the classification of mathematics in that characterization.
Taxa
In 1902 and 1903 Peirce elaborates classifications of the sciences in:
"A Detailed Classification of the Sciences" in Minute Logic (Feb.–Apr. 1902), Collected Papers of Charles Sanders Peirce (CP) v. 1, paragraphs 203–283
July 1902 application to the Carnegie institution (MS L75)
"An Outline Classification of the Sciences (CP 1.180-202) in his "A Syllabus of Certain Topics in Logic" (1903), wherein his classifications of the sciences take more or less their final form
However, only in the "Detailed Classification" and the Carnegie application does he discuss the taxa which he used, which were inspired by the biological taxa of Louis Agassiz.
Sciences
In 1902, he divided science into Theoretical and Practical. Theoretical Science consisted of Science of Discovery and Science of Review, the latter of which he also called "Synthetic Philosophy", a name taken from the title of the vast work, written over many years, by Herbert Spencer. Then, in 1903, he made it a three-way division: Science of Discovery, Science of Review, and Practical Science. In 1903 he characterized Science of Review as:
...arranging the results of discovery, beginning with digests, and going on to endeavor to form a philosophy of science. Such is the nature of Humboldt's Cosmos, of Comte's Philosophie positive, and of Spencer's Synthetic Philosophy. The classification of the sciences belongs to this department.
Peirce had already for a while divided the Sciences of Discovery into:
(1) Mathematics – draws necessary conclusions about hypothetical objects
(2) Cenoscopy – philosophy about positive phenomena in general, such as confront a person at every waking moment, rather than special classes, and not settling theoretical issues by special experiences or experiments
(3) Idioscopy – the special sciences, about special classes of positive phenomena, and settling theoretical issues by special experiences or experimen |
https://en.wikipedia.org/wiki/Multiplier%20algebra | In mathematics, the multiplier algebra, denoted by M(A), of a C*-algebra A is a unital C*-algebra that is the largest unital C*-algebra that contains A as an ideal in a "non-degenerate" way. It is the noncommutative generalization of Stone–Čech compactification. Multiplier algebras were introduced by .
For example, if A is the C*-algebra of compact operators on a separable Hilbert space, M(A) is B(H), the C*-algebra of all bounded operators on H.
Definition
An ideal I in a C*-algebra B is said to be essential if I ∩ J is non-trivial for every ideal J. An ideal I is essential if and only if I⊥, the "orthogonal complement" of I in the Hilbert C*-module B is {0}.
Let A be a C*-algebra. Its multiplier algebra M(A) is any C*-algebra satisfying the following universal property: for all C*-algebra D containing A as an ideal, there exists a unique *-homomorphism φ: D → M(A) such that φ extends the identity homomorphism on A and φ(A⊥) = {0}.
Uniqueness up to isomorphism is specified by the universal property. When A is unital, M(A) = A. It also follows from the definition that for any D containing A as an essential ideal, the multiplier algebra M(A) contains D as a C*-subalgebra.
The existence of M(A) can be shown in several ways.
A double centralizer of a C*-algebra A is a pair (L, R) of bounded linear maps on A such that aL(b) = R(a)b for all a and b in A. This implies that ||L|| = ||R||. The set of double centralizers of A can be given a C*-algebra structure. This C*-algebra contains A as an essential ideal and can be identified as the multiplier algebra M(A). For instance, if A is the compact operators K(H) on a separable Hilbert space, then each x ∈ B(H) defines a double centralizer of A by simply multiplication from the left and right.
Alternatively, M(A) can be obtained via representations. The following fact will be needed:
Lemma. If I is an ideal in a C*-algebra B, then any faithful nondegenerate representation π of I can be extended uniquely to B.
Now take any faithful nondegenerate representation π of A on a Hilbert space H. The above lemma, together with the universal property of the multiplier algebra, yields that M(A) is isomorphic to the idealizer of π(A) in B(H). It is immediate that M(K(H)) = B(H).
Lastly, let E be a Hilbert C*-module and B(E) (resp. K(E)) be the adjointable (resp. compact) operators on E M(A) can be identified via a *-homomorphism of A into B(E). Something similar to the above lemma is true:
Lemma. If I is an ideal in a C*-algebra B, then any faithful nondegenerate *-homomorphism π of I into B(E)can be extended uniquely to B.
Consequently, if π is a faithful nondegenerate *-homomorphism of A into B(E), then M(A) is isomorphic to the idealizer of π(A). For instance, M(K(E)) = B(E) for any Hilbert module E.
The C*-algebra A is isomorphic to the compact operators on the Hilbert module A. Therefore, M(A) is the adjointable operators on A.
Strict topology
Consider the topology on M(A) specified by the semi |
https://en.wikipedia.org/wiki/Jacobsthal%20number | In mathematics, the Jacobsthal numbers are an integer sequence named after the German mathematician Ernst Jacobsthal. Like the related Fibonacci numbers, they are a specific type of Lucas sequence for which P = 1, and Q = −2—and are defined by a similar recurrence relation: in simple terms, the sequence starts with 0 and 1, then each following number is found by adding the number before it to twice the number before that. The first Jacobsthal numbers are:
0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, 1365, 2731, 5461, 10923, 21845, 43691, 87381, 174763, 349525, …
A Jacobsthal prime is a Jacobsthal number that is also prime. The first Jacobsthal primes are:
3, 5, 11, 43, 683, 2731, 43691, 174763, 2796203, 715827883, 2932031007403, 768614336404564651, 201487636602438195784363, 845100400152152934331135470251, 56713727820156410577229101238628035243, …
Jacobsthal numbers
Jacobsthal numbers are defined by the recurrence relation:
The next Jacobsthal number is also given by the recursion formula
or by
The second recursion formula above is also satisfied by the powers of 2.
The Jacobsthal number at a specific point in the sequence may be calculated directly using the closed-form equation:
The generating function for the Jacobsthal numbers is
The sum of the reciprocals of the Jacobsthal numbers is approximately 2.7186, slightly larger than e.
The Jacobsthal numbers can be extended to negative indices using the recurrence relation or the explicit formula, giving
(see )
The following identity holds
(see )
Jacobsthal–Lucas numbers
Jacobsthal–Lucas numbers represent the complementary Lucas sequence . They satisfy the same recurrence relation as Jacobsthal numbers but have different initial values:
The following Jacobsthal–Lucas number also satisfies:
The Jacobsthal–Lucas number at a specific point in the sequence may be calculated directly using the closed-form equation:
The first Jacobsthal–Lucas numbers are:
2, 1, 5, 7, 17, 31, 65, 127, 257, 511, 1025, 2047, 4097, 8191, 16385, 32767, 65537, 131071, 262145, 524287, 1048577, … .
Jacobsthal Oblong numbers
The first Jacobsthal Oblong numbers are:
0, 1, 3, 15, 55, 231, 903, 3655, 14535, 58311, …
References
Eponymous numbers in mathematics
Integer sequences
Recurrence relations |
https://en.wikipedia.org/wiki/Forbidden%20graph%20characterization | In graph theory, a branch of mathematics, many important families of graphs can be described by a finite set of individual graphs that do not belong to the family and further exclude all graphs from the family which contain any of these forbidden graphs as (induced) subgraph or minor.
A prototypical example of this phenomenon is Kuratowski's theorem, which states that a graph is planar (can be drawn without crossings in the plane) if and only if it does not contain either of two forbidden graphs, the complete graph and the complete bipartite graph . For Kuratowski's theorem, the notion of containment is that of graph homeomorphism, in which a subdivision of one graph appears as a subgraph of the other. Thus, every graph either has a planar drawing (in which case it belongs to the family of planar graphs) or it has a subdivision of at least one of these two graphs as a subgraph (in which case it does not belong to the planar graphs).
Definition
More generally, a forbidden graph characterization is a method of specifying a family of graph, or hypergraph, structures, by specifying substructures that are forbidden to exist within any graph in the family. Different families vary in the nature of what is forbidden. In general, a structure G is a member of a family if and only if a forbidden substructure is not contained in G. The forbidden substructure might be one of:
subgraphs, smaller graphs obtained from subsets of the vertices and edges of a larger graph,
induced subgraphs, smaller graphs obtained by selecting a subset of the vertices and using all edges with both endpoints in that subset,
homeomorphic subgraphs (also called topological minors), smaller graphs obtained from subgraphs by collapsing paths of degree-two vertices to single edges, or
graph minors, smaller graphs obtained from subgraphs by arbitrary edge contractions.
The set of structures that are forbidden from belonging to a given graph family can also be called an obstruction set for that family.
Forbidden graph characterizations may be used in algorithms for testing whether a graph belongs to a given family. In many cases, it is possible to test in polynomial time whether a given graph contains any of the members of the obstruction set, and therefore whether it belongs to the family defined by that obstruction set.
In order for a family to have a forbidden graph characterization, with a particular type of substructure, the family must be closed under substructures.
That is, every substructure (of a given type) of a graph in the family must be another graph in the family. Equivalently, if a graph is not part of the family, all larger graphs containing it as a substructure must also be excluded from the family. When this is true, there always exists an obstruction set (the set of graphs that are not in the family but whose smaller substructures all belong to the family). However, for some notions of what a substructure is, this obstruction set could be infinite. The Robe |
https://en.wikipedia.org/wiki/Orthogonal%20Procrustes%20problem | The orthogonal Procrustes problem is a matrix approximation problem in linear algebra. In its classical form, one is given two matrices and and asked to find an orthogonal matrix which most closely maps to . Specifically, the orthogonal Procrustes problem is an optimization problem given by
where denotes the Frobenius norm. This is a special case of Wahba's problem (with identical weights; instead of considering two matrices, in Wahba's problem the columns of the matrices are considered as individual vectors). Another difference is, that Wahba's problem tries to find a proper rotation matrix, instead of just an orthogonal one.
The name Procrustes refers to a bandit from Greek mythology who made his victims fit his bed by either stretching their limbs or cutting them off.
Solution
This problem was originally solved by Peter Schönemann in a 1964 thesis, and shortly after appeared in the journal Psychometrika.
This problem is equivalent to finding the nearest orthogonal matrix to a given matrix , i.e. solving the closest orthogonal approximation problem
.
To find matrix , one uses the singular value decomposition (for which the entries of are non-negative)
to write
Proof of Solution
One proof depends on basic properties of the Frobenius inner product that induces the Frobenius norm:
This quantity is an orthogonal matrix (as it is a product of orthogonal matrices) and thus the expression is maximised when equals the identity matrix . Thus
where is the solution for the optimal value of that minimizes the norm squared .
Generalized/constrained Procrustes problems
There are a number of related problems to the classical orthogonal Procrustes problem. One might generalize it by seeking the closest matrix in which the columns are orthogonal, but not necessarily orthonormal.
Alternately, one might constrain it by only allowing rotation matrices (i.e. orthogonal matrices with determinant 1, also known as special orthogonal matrices). In this case, one can write (using the above decomposition )
where is a modified , with the smallest singular value replaced by (+1 or -1), and the other singular values replaced by 1, so that the determinant of R is guaranteed to be positive. For more information, see the Kabsch algorithm.
The unbalanced Procrustes problem concerns minimizing the norm of , where , and , with , or alternately with complex valued matrices. This is a problem over the Stiefel manifold , and has no currently known closed form. To distinguish, the standard Procrustes problem () is referred to as the balanced problem in these contexts.
See also
Procrustes analysis
Procrustes transformation
Wahba's problem
Kabsch algorithm
Point set registration
References
Linear algebra
Matrix theory
Singular value decomposition |
https://en.wikipedia.org/wiki/Scripta%20Mathematica | Scripta Mathematica was a quarterly journal published by Yeshiva University devoted to the Philosophy, history, and expository treatment of mathematics. It was said to be, at its time, "the only mathematical magazine in the world edited by specialists for laymen."
The journal was established in 1932 under the editorship of Jekuthiel Ginsburg, a professor of mathematics at Yeshiva University, and its first issue appeared in 1933 at a subscription price of three dollars per year. It ceased publication in 1973. Notable papers published in Scripta Mathematica included work by Nobelist Percy Williams Bridgman concerning the implications for physics of set-theoretic paradoxes, and Hermann Weyl's obituary of Emmy Noether.
Some sources describe Scripta Mathematica as having been assigned but it ceased publication prior to the establishment of the ISSN system.
References
Mathematics journals
Academic journals established in 1932
Publications disestablished in 1973
Defunct journals of the United States |
https://en.wikipedia.org/wiki/Takashi%20Kondo%20%28footballer%29 | is a Japanese football player who plays for Nagano Parceiro.
Club statistics
Updated to end of 2018 season.
References
External links
Profile at Ehime FC
1992 births
Living people
Waseda University alumni
Association football people from Tokyo
Japanese men's footballers
J2 League players
J3 League players
Ehime FC players
Omiya Ardija players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Hereditary%20property | In mathematics, a hereditary property is a property of an object that is inherited by all of its subobjects, where the meaning of subobject depends on the context. These properties are particularly considered in topology and graph theory, but also in set theory.
In topology
In topology, a topological property is said to be hereditary if whenever a topological space has that property, then so does every subspace of it. If the latter is true only for closed subspaces, then the property is called weakly hereditary or
closed-hereditary.
For example, second countability and metrisability are hereditary properties. Sequentiality and Hausdorff compactness are weakly hereditary, but not hereditary. Connectivity is not weakly hereditary.
If P is a property of a topological space X and every subspace also has property P, then X is said to be "hereditarily P".
In combinatorics and graph theory
The notion of hereditary properties occurs throughout combinatorics and graph theory, although they are known by a variety of names. For example, in the context of permutation patterns, hereditary properties are typically called permutation classes.
In graph theory, a hereditary property is a property of a graph which also holds for (is "inherited" by) its induced subgraphs. Alternately, a hereditary property is preserved by the removal of vertices. A graph class is called hereditary if it is closed under induced subgraphs. Examples of hereditary graph classes are independent graphs (graphs with no edges), which is a special case (with c = 1) of being c-colorable for some number c, being forests, planar, complete, complete multipartite etc.
In some cases, the term "hereditary" has been defined with reference to graph minors, but this is more properly called a minor-hereditary property. The Robertson–Seymour theorem implies that a minor-hereditary property may be characterized in terms of a finite set of forbidden minors.
The term "hereditary" has been also used for graph properties that are closed with respect to taking subgraphs. In such a case, properties that are closed with respect to taking induced subgraphs, are called induced-hereditary. The language of hereditary properties and induced-hereditary properties provides a powerful tool for study of structural properties of various types of generalized colourings. The most important result from this area is the unique factorization theorem.
Monotone property
There is no consensus for the meaning of "monotone property" in graph theory. Examples of definitions are:
Preserved by the removal of edges.
Preserved by the removal of edges and vertices (i.e., the property should hold for all subgraphs).
Preserved by the addition of edges and vertices (i.e., the property should hold for all supergraphs).
Preserved by the addition of edges. (This meaning is used in the statement of the Aanderaa–Karp–Rosenberg conjecture.)
The complementary property of a property that is preserved by the removal of edges is |
https://en.wikipedia.org/wiki/Perpendicular%20distance | In geometry, the perpendicular distance between two objects is the distance from one to the other, measured along a line that is perpendicular to one or both.
The distance from a point to a line is the distance to the nearest point on that line. That is the point at which a segment from it to the given point is perpendicular to the line.
Likewise, the distance from a point to a curve is measured by a line segment that is perpendicular to a tangent line to the curve at the nearest point on the curve.
The distance from a point to a plane is measured as the length from the point along a segment that is perpendicular to the plane, meaning that it is perpendicular to all lines in the plane that pass through the nearest point in the plane to the given point.
Other instances include:
Point on plane closest to origin, for the perpendicular distance from the origin to a plane in three-dimensional space
Nearest distance between skew lines, for the perpendicular distance between two non-parallel lines in three-dimensional space
Perpendicular regression fits a line to data points by minimizing the sum of squared perpendicular distances from the data points to the line.
Other geometric curve fitting methods using perpendicular distance to measure the quality of a fit exist, as in total least squares.
The concept of perpendicular distance may be generalized to
orthogonal distance, between more abstract non-geometric orthogonal objects, as in linear algebra (e.g., principal components analysis);
normal distance, involving a surface normal, between an arbitrary point and its foot on the surface. It can be used for surface fitting and for defining offset surfaces.
See also
Distance between sets
Hypercycle (geometry)
Moment of inertia
Signed distance
References
Orthogonality
Distance |
https://en.wikipedia.org/wiki/Mathematics%2C%20Form%20and%20Function | Mathematics, Form and Function, a book published in 1986 by Springer-Verlag, is a survey of the whole of mathematics, including its origins and deep structure, by the American mathematician Saunders Mac Lane.
Mathematics and human activities
Throughout his book, and especially in chapter I.11, Mac Lane informally discusses how mathematics is grounded in more ordinary concrete and abstract human activities. The following table is adapted from one given on p. 35 of Mac Lane (1986). The rows are very roughly ordered from most to least fundamental. For a bullet list that can be compared and contrasted with this table, see section 3 of Where Mathematics Comes From.
Also see the related diagrams appearing on the following pages of Mac Lane (1986): 149, 184, 306, 408, 416, 422-28.
Mac Lane (1986) cites a related monograph by Lars Gårding (1977).
Mac Lane's relevance to the philosophy of mathematics
Mac Lane cofounded category theory with Samuel Eilenberg, which enables a unified treatment of mathematical structures and of the relations among them, at the cost of breaking away from their cognitive grounding. Nevertheless, his views—however informal—are a valuable contribution to the philosophy and anthropology of mathematics. His views anticipate, in some respects, the more detailed account of the cognitive basis of mathematics given by George Lakoff and Rafael E. Núñez in their Where Mathematics Comes From. Lakoff and Núñez argue that mathematics emerges via conceptual metaphors grounded in the human body, its motion through space and time, and in human sense perceptions.
See also
1986 in philosophy
Notes
References
Gårding, Lars, 1977. Encounter with Mathematics. Springer-Verlag.
Reuben Hersh, 1997. What Is Mathematics, Really? Oxford Univ. Press.
George Lakoff and Rafael E. Núñez, 2000. Where Mathematics Comes From. Basic Books.
Leslie White, 1947, "The Locus of Mathematical Reality: An Anthropological Footnote," Philosophy of Science 14: 289-303. Reprinted in Hersh, R., ed., 2006. 18 Unconventional Essays on the Nature of Mathematics. Springer: 304–19.
1986 non-fiction books
Mathematics books
Philosophy of mathematics literature
Cognitive science literature |
https://en.wikipedia.org/wiki/COROP | A COROP region is a division of the Netherlands for statistical purposes, used by Statistics Netherlands, among others. The Dutch abbreviation stands for (Coordination Commission Regional Research Programme). These divisions are also used in the EU designation as NUTS 3.
List of municipalities by COROP region
Northern Netherlands
Groningen province
Friesland province
Drenthe province
Eastern Netherlands
Overijssel province
Gelderland province
Flevoland province
Western Netherlands
Utrecht province
North Holland province
South Holland province
Zeeland province
Southern Netherlands
North Brabant province
Limburg province
See also
Indeling van Nederland in 40 COROP-gebieden per 01-01-2017 (kaart), website CBS
COROP-indeling per 01-01-2012 (kaart), website CBS
COROP-indeling per 01-01-2012 (tekst), website CBS
Subdivisions of the Netherlands
Netherlands |
https://en.wikipedia.org/wiki/Uncompensated%20risk | In investments, uncompensated risk is the level of additional risk for which no additional returns are generated and when taking systematic withdrawals make the probability of failure unacceptably high. Uncompensated risk is reduced by diversifying investment.
References
Investment |
https://en.wikipedia.org/wiki/Eigendecomposition%20of%20a%20matrix | In linear algebra, eigendecomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way. When the matrix being factorized is a normal or real symmetric matrix, the decomposition is called "spectral decomposition", derived from the spectral theorem.
Fundamental theory of matrix eigenvectors and eigenvalues
A (nonzero) vector of dimension is an eigenvector of a square matrix if it satisfies a linear equation of the form
for some scalar . Then is called the eigenvalue corresponding to . Geometrically speaking, the eigenvectors of are the vectors that merely elongates or shrinks, and the amount that they elongate/shrink by is the eigenvalue. The above equation is called the eigenvalue equation or the eigenvalue problem.
This yields an equation for the eigenvalues
We call the characteristic polynomial, and the equation, called the characteristic equation, is an th order polynomial equation in the unknown . This equation will have distinct solutions, where . The set of solutions, that is, the eigenvalues, is called the spectrum of .
If the field of scalars is algebraically closed, then we can factor as
The integer is termed the algebraic multiplicity of eigenvalue . The algebraic multiplicities sum to :
For each eigenvalue , we have a specific eigenvalue equation
There will be linearly independent solutions to each eigenvalue equation. The linear combinations of the solutions (except the one which gives the zero vector) are the eigenvectors associated with the eigenvalue . The integer is termed the geometric multiplicity of . It is important to keep in mind that the algebraic multiplicity and geometric multiplicity may or may not be equal, but we always have . The simplest case is of course when . The total number of linearly independent eigenvectors, , can be calculated by summing the geometric multiplicities
The eigenvectors can be indexed by eigenvalues, using a double index, with being the th eigenvector for the th eigenvalue. The eigenvectors can also be indexed using the simpler notation of a single index , with .
Eigendecomposition of a matrix
Let be a square matrix with linearly independent eigenvectors (where ). Then can be factorized as
where is the square matrix whose th column is the eigenvector of , and is the diagonal matrix whose diagonal elements are the corresponding eigenvalues, . Note that only diagonalizable matrices can be factorized in this way. For example, the defective matrix (which is a shear matrix) cannot be diagonalized.
The eigenvectors are usually normalized, but they need not be. A non-normalized set of eigenvectors, can also be used as the columns of . That can be understood by noting that the magnitude of the eigenvectors in gets canceled in the decomposition by the presence of . If one of the eigenvalues has more tha |
https://en.wikipedia.org/wiki/JLO%20cocycle | In noncommutative geometry, the JLO cocycle is a cocycle (and thus defines a cohomology class) in entire cyclic cohomology. It is a non-commutative version of the classic Chern character of the conventional differential geometry. In noncommutative geometry, the concept of a manifold is replaced by a noncommutative algebra of "functions" on the putative noncommutative space. The cyclic cohomology of the algebra contains the information about the topology of that noncommutative space, very much as the de Rham cohomology contains the information about the topology of a conventional manifold.
The JLO cocycle is associated with a metric structure of non-commutative differential geometry known as a -summable spectral triple (also known as a -summable Fredholm module).
-summable spectral triples
A -summable spectral triple consists of the following data:
(a) A Hilbert space such that acts on it as an algebra of bounded operators.
(b) A -grading on , . We assume that the algebra is even under the -grading, i.e. , for all .
(c) A self-adjoint (unbounded) operator , called the Dirac operator such that
(i) is odd under , i.e. .
(ii) Each maps the domain of , into itself, and the operator is bounded.
(iii) , for all .
A classic example of a -summable spectral triple arises as follows. Let be a compact spin manifold, , the algebra of smooth functions on , the Hilbert space of square integrable forms on , and the standard Dirac operator.
The cocycle
The JLO cocycle is a sequence
of functionals on the algebra , where
for . The cohomology class defined by is independent of the value of .
External links
- The original paper introducing the JLO cocycle.
- A nice set of lectures.
Noncommutative geometry |
https://en.wikipedia.org/wiki/Infinite%20divisibility%20%28probability%29 | In probability theory, a probability distribution is infinitely divisible if it can be expressed as the probability distribution of the sum of an arbitrary number of independent and identically distributed (i.i.d.) random variables. The characteristic function of any infinitely divisible distribution is then called an infinitely divisible characteristic function.
More rigorously, the probability distribution F is infinitely divisible if, for every positive integer n, there exist n i.i.d. random variables Xn1, ..., Xnn whose sum Sn = Xn1 + … + Xnn has the same distribution F.
The concept of infinite divisibility of probability distributions was introduced in 1929 by Bruno de Finetti. This type of decomposition of a distribution is used in probability and statistics to find families of probability distributions that might be natural choices for certain models or applications. Infinitely divisible distributions play an important role in probability theory in the context of limit theorems.
Examples
Examples of continuous distributions that are infinitely divisible are the normal distribution, the Cauchy distribution, the Lévy distribution, and all other members of the stable distribution family, as well as the Gamma distribution, the chi-square distribution, the Wald distribution, the Log-normal distribution and the Student's t-distribution.
Among the discrete distributions, examples are the Poisson distribution and the negative binomial distribution (and hence the geometric distribution also). The one-point distribution whose only possible outcome is 0 is also (trivially) infinitely divisible.
The uniform distribution and the binomial distribution are not infinitely divisible, nor are any other distributions with bounded support (≈ finite-sized domain), other than the one-point distribution mentioned above. The distribution of the reciprocal of a random variable having a Student's t-distribution is also not infinitely divisible.
Any compound Poisson distribution is infinitely divisible; this follows immediately from the definition.
Limit theorem
Infinitely divisible distributions appear in a broad generalization of the central limit theorem: the limit as n → +∞ of the sum Sn = Xn1 + … + Xnn of independent uniformly asymptotically negligible (u.a.n.) random variables within a triangular array
approaches — in the weak sense — an infinitely divisible distribution. The uniformly asymptotically negligible (u.a.n.) condition is given by
Thus, for example, if the uniform asymptotic negligibility (u.a.n.) condition is satisfied via an appropriate scaling of identically distributed random variables with finite variance, the weak convergence is to the normal distribution in the classical version of the central limit theorem. More generally, if the u.a.n. condition is satisfied via a scaling of identically distributed random variables (with not necessarily finite second moment), then the weak convergence is to a stable distribution. On the other h |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.