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https://en.wikipedia.org/wiki/Natural%20mapping | Natural mapping may refer to:
Canonical map
Natural transformation in category theory, a branch of abstract mathematics
Natural mapping (interface design) |
https://en.wikipedia.org/wiki/Ivan%20Lovri%C4%87%20%28footballer%29 | Ivan Lovrić (born 11 July 1985) is a Croatian football player. He plays as a centre-back for Hungarian club Budapest Honvéd.
Club statistics
{| class="wikitable" style="font-size:90%; text-align: center;"
|-
!rowspan="2"|Club
!rowspan="2"|Season
!colspan="2"|League
!colspan="2"|Cup
!colspan="2"|League Cup
!colspan="2"|Europe
!colspan="2"|Total
|-
!Apps
!Goals
!Apps
!Goals
!Apps
!Goals
!Apps
!Goals
!Apps
!Goals
|-||-||-||-|-
|rowspan="5" valign="center"|Kecskemét
|-
|2007–08
|21||4||0||0||0||0||0||0||21||4
|-
|2008–09
|5||0||1||0||0||0||0||0||6||0
|-
|2014–15
|27||0||1||0||2||0||0||0||30||0
|-
|- style="font-weight:bold; background-color:#eeeeee;"
|Total||52||4||2||0||2||0||0||0||56||4
|-
|rowspan="3" valign="center"|Baja
|-
|2008–09
|14||1||0||0||0||0||0||0||14||1
|-
|- style="font-weight:bold; background-color:#eeeeee;"
|Total||14||1||0||0||0||0||0||0||14||1
|-
|rowspan="3" valign="center"|Warriors
|-
|2010
|31||2||1||0||1||1||7||1||40||4
|-
|- style="font-weight:bold; background-color:#eeeeee;"
|Total||31||2||1||0||1||1||7||1||40||4
|-
|rowspan="13" valign="center"|Honvéd
|-
|2010–11
|13||4||2||0||0||0||0||0||15||4
|-
|2011–12
|29||2||1||0||4||0||0||0||34||2
|-
|2012–13
|26||1||3||0||5||0||4||0||38||1
|-
|2013–14
|24||3||2||0||5||0||4||0||35||3
|-
|2015–16
|21||0||5||1||–||–||–||–||26||1
|-
|2016–17
|26||0||5||0||–||–||–||–||31||0
|-
|2017–18
|20||0||5||0||–||–||2||0||27||0
|-
|2018–19
|11||2||9||2||–||–||0||0||20||4
|-
|2019–20
|29||0||6||1||–||–||4||0||39||1
|-
|2020–21
|26||1||3||0||–||–||1||0||30||1
|-
|2021–22
|25||0||4||0||–||–||–||–||29||0
|-
|- style="font-weight:bold; background-color:#eeeeee;"
|Total||252||13||45||4||14||0||15||0||326||17
|-
|- style="font-weight:bold; background-color:#eeeeee;"
|rowspan="2" valign="top"|Career Total
|
|350||20||48||4||17||1||22||1||437||26'''
|}Updated to games played as of 15 May 2022.''
References
HLSZ
External links
1985 births
Living people
Footballers from Split, Croatia
Men's association football defenders
Men's association football midfielders
Croatian men's footballers
Kecskeméti TE players
Bajai LSE footballers
Warriors FC players
Budapest Honvéd FC players
Nemzeti Bajnokság I players
Singapore Premier League players
Croatian expatriate men's footballers
Expatriate men's footballers in Hungary
Croatian expatriate sportspeople in Hungary
Expatriate men's footballers in Singapore
Croatian expatriate sportspeople in Singapore |
https://en.wikipedia.org/wiki/Symplectic%20frame%20bundle | In symplectic geometry, the symplectic frame bundle of a given symplectic manifold is the canonical principal -subbundle of the tangent frame bundle consisting of linear frames which are symplectic with respect to . In other words, an element of the symplectic frame bundle is a linear frame at point i.e. an ordered basis of tangent vectors at of the tangent vector space , satisfying
and
for . For , each fiber of the principal -bundle is the set of all symplectic bases of .
The symplectic frame bundle , a subbundle of the tangent frame bundle , is an example of reductive G-structure on the manifold .
See also
Metaplectic group
Metaplectic structure
Symplectic basis
Symplectic structure
Symplectic geometry
Symplectic group
Symplectic spinor bundle
Notes
Books
da Silva, A.C., Lectures on Symplectic Geometry, Springer (2001). .
Maurice de Gosson: Symplectic Geometry and Quantum Mechanics (2006) Birkhäuser Verlag, Basel .
Symplectic geometry
Structures on manifolds
Algebraic topology |
https://en.wikipedia.org/wiki/Lambda%20distribution | The lambda distribution is either of two probability distributions used in statistics:
Tukey's lambda distribution is a shape-conformable distribution used to identify an appropriate common distribution family to fit a collection of data to.
Wilks' lambda distribution is an extension of Snedecor's F-distribution for matricies used in multivariate hypothesis testing, especially with regard to the likelihood-ratio test and multivariate analysis of variance. |
https://en.wikipedia.org/wiki/Plethysm | In algebra, plethysm is an operation on symmetric functions introduced by Dudley E. Littlewood, who denoted it by {λ} ⊗ {μ}. The word "plethysm" for this operation (after the Greek word πληθυσμός meaning "multiplication") was introduced later by , who said that the name was suggested by M. L. Clark.
If symmetric functions are identified with operations in lambda rings, then plethysm corresponds to composition of operations.
In representation theory
Let V be a vector space over the complex numbers, considered as a representation of the general linear group GL(V). Each Young diagram λ corresponds to a Schur functor Lλ(-) on the category of GL(V)-representations. Given two Young diagrams λ and μ, consider the decomposition of Lλ(Lμ(V)) into a direct sum of irreducible representations of the group. By the representation theory of the general linear group we know that each summand is isomorphic to for a Young diagram . So for some nonnegative multiplicities there is an isomorphism
The problem of (outer) plethysm is to find an expression for the multiplicities .
This formulation is closely related to the classical question. The character of the GL(V)-representation Lλ(V) is a symmetric function in dim(V) variables, known as the Schur polynomial sλ corresponding to the Young diagram λ. Schur polynomials form a basis in the space of symmetric functions. Hence to understand the plethysm of two symmetric functions it would be enough to know their expressions in that basis and an expression for a plethysm of two arbitrary Schur polynomials {sλ}⊗{sμ} . The second piece of data is precisely the character of Lλ(Lμ(V)).
References
Symmetric functions |
https://en.wikipedia.org/wiki/Federal%20Office%20of%20Statistics | Federal Office of Statistics may refer to:
Federal Statistical Office (Switzerland)
Federal Office of Statistics (Bosnia and Herzegovina)
Federal Office of Statistics (Nigeria), now National Bureau of Statistics of Nigeria
See also
National Bureau of Statistics (disambiguation) |
https://en.wikipedia.org/wiki/%CE%9B-ring | In algebra, a λ-ring or lambda ring is a commutative ring together with some operations λn on it that behave like the exterior powers of vector spaces. Many rings considered in K-theory carry a natural λ-ring structure. λ-rings also provide a powerful formalism for studying an action of the symmetric functions on the ring of polynomials, recovering and extending many classical results ().
λ-rings were introduced by . For more about λ-rings see , , and .
Motivation
If V and W are finite-dimensional vector spaces over a field k, then we can form the direct sum V ⊕ W, the tensor product V ⊗ W, and the n-th exterior power of V, Λn(V). All of these are again finite-dimensional vector spaces over k. The same three operations of direct sum, tensor product and exterior power are also available when working with k-linear representations of a finite group, when working with vector bundles over some topological space, and in more general situations.
λ-rings are designed to abstract the common algebraic properties of these three operations, where we also allow for formal inverses with respect to the direct sum operation. (These formal inverses also appear in Grothendieck groups, which is why the underlying additive groups of most λ-rings are Grothendieck groups.) The addition in the ring corresponds to the direct sum, the multiplication in the ring corresponds to the tensor product, and the λ-operations to the exterior powers. For example, the isomorphism
corresponds to the formula
valid in all λ-rings, and the isomorphism
corresponds to the formula
valid in all λ-rings. Analogous but (much) more complicated formulas govern the higher order λ-operators.
Motivation with Vector Bundles
If we have a short exact sequence of vector bundles over a smooth scheme then locally, for a small enough open neighborhood we have the isomorphism
Now, in the Grothendieck group of (which is actually a ring), we get this local equation globally for free, from the defining equivalence relations. So
demonstrating the basic relation in a λ-ring, that
Definition
A λ-ring is a commutative ring R together with operations λn : R → R for every non-negative integer n. These operations are required to have the following properties valid for all x, y in R and all n, m ≥ 0:
λ0(x) = 1
λ1(x) = x
λn(1) = 0 if n ≥ 2
λn(x + y) = Σi+j=n λi(x) λj(y)
λn(xy) = Pn(λ1(x), ..., λn(x), λ1(y), ..., λn(y))
λn(λm(x)) = Pn,m(λ1(x), ..., λmn(x))
where Pn and Pn,m are certain universal polynomials with integer coefficients that describe the behavior of exterior powers on tensor products and under composition. These polynomials can be defined as follows.
Let e1, ..., emn be the elementary symmetric polynomials in the variables X1, ..., Xmn. Then Pn,m is the unique polynomial in nm variables with integer coefficients such that Pn,m(e1, ..., emn) is the coefficient of tn in the expression
(Such a polynomial exists, because the expression is symmetric in the Xi and the elementary symmetr |
https://en.wikipedia.org/wiki/Yair%20Censor | Yair Censor (Hebrew: יאיר צנזור, born November 29, 1943) is an Israeli mathematician and a professor at the University of Haifa, specializing in computational mathematics and optimization, as well as applications of these fields, in particular to medical imaging and radiation therapy treatment planning.
Biography
Yair Censor was born in Rishon LeZion. After serving in the IDF, he studied at the Technion in Haifa, where he earned his D.Sc. in 1975 under the supervision of Professor Adi Ben-Israel.
Academic career
Censor joined the department of mathematics at the University of Haifa in 1979, and became full professor in 1989. His research focuses on mathematical aspects of Intensity-Modulated Radiation Therapy (IMRT). In 2002, he founded the Center for Computational Mathematics and Scientific Computation at the University of Haifa. In recent years he is involved with research about the Superiorization Methodology.
Together with S.A. Zenios, he co-authored the book Parallel Optimization: Theory, Algorithms, and Applications (Oxford University Press, New York, NY, USA, 1997), for which he received the 1999 ICS (INFORMS Computing Society) Prize for Research Excellence in the Interface Between Operations Research and Computer Science.
Active in the struggle to preserve the academic freedom of the research universities in Israel, Censor was one of the founders of the Inter-Senate Committee (ISC) of the Universities for the Protection of Academic Independence.
See also
Education in Israel
References
External links
Yair Censor website
Convergence and Perturbation Resilience of Dynamic String-Averaging Projection Methods
An iterative approach to plan combination in radiotherapy
1943 births
Academic staff of the University of Haifa
Israeli mathematicians
Living people |
https://en.wikipedia.org/wiki/Tata%20Subba%20Rao | Tata Subba Rao (1942 – 13 April 2018] was a professor of statistics in the School of Mathematics, University of Manchester. He gained his MA at Karnatak, his PhD from Gauhati University in 1966 under the guidance of Jyotiprasad Medhi, and DSc from Manchester in 1988. He was a specialist in time series analysis especially non-stationary and non-linear time series analysis, higher order spectral analysis, theory of random fields, time series methods for analysis of environmental variables (detection of climatic changes etc.) and multivariate nonlinear models. The Subba Rao–Liporace models, and Subba Rao Gabr window for bivariate spectra are named after him.
Subba Rao retired, but was given an emeritus chair, in 2009 having worked at UMIST and the University of Manchester for 42 years. He published over 70 papers and supervised 15 PhD students. Even after retirement he remained active.
References
External links
Home page at Manchester Mathematics Staff Homepage (Archived version 2017)
1942 births
2018 deaths
British statisticians
Indian emigrants to the United Kingdom
Academics of the University of Manchester |
https://en.wikipedia.org/wiki/Springer%20resolution | In mathematics, the Springer resolution is a resolution of the variety of nilpotent elements in a semisimple Lie algebra, or the unipotent elements of a reductive algebraic group, introduced by Tonny Albert Springer in 1969. The fibers of this resolution are called Springer fibers.
If U is the variety of unipotent elements in a reductive group G, and X the variety of Borel subgroups B, then the Springer resolution of U is the variety of pairs (u,B) of U×X such that u is in the Borel subgroup B. The map to U is the projection to the first factor. The Springer resolution for Lie algebras is similar, except that U is replaced by the nilpotent elements of the Lie algebra of G and X replaced by the variety of Borel subalgebras.
The Grothendieck–Springer resolution is defined similarly, except that U is replaced by the whole group G (or the whole Lie algebra of G). When restricted to the unipotent elements of G it becomes the Springer resolution.
Examples
When G=SL(2), the Lie algebra Springer resolution is T*P1 → n, where n are the nilpotent elements of sl(2). In this example, n are the matrices x with tr(x2)=0, which is a two dimensional conical subvariety of sl(2). n has a unique singular point 0, the fibre above which in the Springer resolution is the zero section P1 .
References
Lie algebras
Singularity theory
Algebraic groups |
https://en.wikipedia.org/wiki/Homeschooling%20international%20status%20and%20statistics | Homeschooling is legal in many countries. Countries with the most prevalent homeschooling movements include Australia, Canada, New Zealand, the United Kingdom, and the United States. Some countries have highly regulated homeschooling programs as an extension of the compulsory school system; few others, such as Germany, have outlawed it entirely. In some other countries, while not restricted by law, homeschooling is not socially acceptable, or is considered undesirable, and is virtually non-existent.
Status of homeschooling across continents
Africa
North America
Latin America and the Caribbean
Asia
Europe
Oceania
Legality by country or region
Africa
Kenya
Status: Contentious
Homeschooling is currently permitted in Kenya.
The freedom of homeschooling is however under threat in Kenya, because a new education law has been proposed that does not make any allowance for homeschooling.
South Africa
Status: Legal
During apartheid, home education was illegal in South Africa. The parents Andre and Bokkie Meintjies were jailed in 1994– (this was the year Mandela was elected as President of South Africa), and their children were placed in separate orphanages while the parents were jailed at correctional facilities very far from each other and the children to prevent family contact, because they educated their children at home. However, a few years later, the Mandela government legalised home education with the publication of the South African School Act in 1996. Since it was legalised, homeschooling has the fastest growing education model in the country.
Homeschooling is legal according to South African national law, but individual provinces have the authority to set their own restrictions. The SA Schools Act (art. 51) requires parents to register their children for education at home. In practice however, most provincial departments do not have the administrative capability to register children for home education. Some of the larger provincial departments have limited administrative capabilities to register children for home education as well as a lack of follow up capacity, resulting in a serious miscommunication between government and citizens. As a result of this situation, more than 90% of homeschooling parents do not register with the department.
Since home education was legalised, it has grown exponentially. According to the census count of 2011, there were about 57,000 home learners in the country, putting South Africa in the top five countries in terms of number of home learners.
Americas
Argentina
Status: Permission required
Article 129 of the National Education Law in Argentina states that parents must ensure that their children attend school. Children can be homeschooled, but parents need to apply for permission from the Provincial Council of Education.
The law pertaining to homeschooling permissions is Article 26 of the Regulatory Decree 572/62, where the home is mentioned as one of the possible ways to ensure compulsory edu |
https://en.wikipedia.org/wiki/Free%20boundary%20problem | In mathematics, a free boundary problem (FB problem) is a partial differential equation to be solved for both an unknown function and an unknown domain . The segment of the boundary of which is not known at the outset of the problem is the free boundary.
FBs arise in various mathematical models encompassing applications that ranges from physical to economical, financial and biological phenomena, where there is an extra effect of the medium. This effect is in general a qualitative change of the medium and hence an appearance of a phase transition: ice to water, liquid to crystal, buying to selling (assets), active to inactive (biology), blue to red (coloring games), disorganized to organized (self-organizing criticality). An interesting aspect of such a criticality is the so-called sandpile dynamic (or Internal DLA).
The most classical example is the melting of ice: Given a block of ice, one can solve the heat equation given appropriate initial and boundary conditions to determine its temperature. But, if in any region the temperature is greater than the melting point of ice, this domain will be occupied by liquid water instead. The boundary formed from the ice/liquid interface is controlled dynamically by the solution of the PDE.
Two-phase Stefan problems
The melting of ice is a Stefan problem for the temperature field , which is formulated as follows. Consider a medium occupying a region consisting of two phases, phase 1 which is present when and phase 2 which is present when . Let the two phases have thermal diffusivities and . For example, the thermal diffusivity of water is 1.4×10−7 m2/s, while the diffusivity of ice is 1.335×10−6 m2/s.
In the regions consisting solely of one phase, the temperature is determined by the heat equation: in the region ,
while in the region ,
This is subject to appropriate conditions on the (known) boundary of ; represents sources or sinks of heat.
Let be the surface where at time ; this surface is the interface between the two phases. Let denote the unit outward normal vector to the second (solid) phase. The Stefan condition determines the evolution of the surface by giving an equation governing the velocity of the free surface in the direction , specifically
where is the latent heat of melting. By we mean the limit of the gradient as approaches from the region , and for we mean the limit of the gradient as approaches from the region .
In this problem, we know beforehand the whole region but we only know the ice-liquid interface at time . To solve the Stefan problem we not only have to solve the heat equation in each region, but we must also track the free boundary .
The one-phase Stefan problem corresponds to taking either or to be zero; it is a special case of the two-phase problem. In the direction of greater complexity we could also consider problems with an arbitrary number of phases.
Obstacle problems
Another famous free-boundary problem is the obstacle problem, which |
https://en.wikipedia.org/wiki/Jantzen%20filtration | In representation theory, a Jantzen filtration is a filtration of a Verma module of a semisimple Lie algebra, or a Weyl module of a reductive algebraic group of positive characteristic. Jantzen filtrations were introduced by .
Jantzen filtration for Verma modules
If M(λ) is a Verma module of a semisimple Lie algebra with highest weight λ, then the Janzen filtration is a decreasing filtration
It has the following properties:
M(λ)1=N(λ), the unique maximal proper submodule of M(λ)
The quotients M(λ)i/M(λ)i+1 have non-degenerate contravariant bilinear forms.
The Jantzen sum formula holds:
where denotes the formal character.
References
Lie algebras
Representation theory |
https://en.wikipedia.org/wiki/Weyl%20module | In algebra, a Weyl module is a representation of a reductive algebraic group, introduced by and named after Hermann Weyl. In characteristic 0 these representations are irreducible, but in positive characteristic they can be reducible, and their decomposition into irreducible components can be hard to determine.
See also
Borel–Weil–Bott theorem
Garnir relations
Further reading
Representation theory
Algebraic groups |
https://en.wikipedia.org/wiki/Ian%20Agol | Ian Agol (born May 13, 1970) is an American mathematician who deals primarily with the topology of three-dimensional manifolds.
Education and career
Agol graduated with B.S. in mathematics from the California Institute of Technology in 1992 and obtained his Ph.D. in 1998 from the University of California, San Diego. At UCSD, his advisor was Michael Freedman and his thesis was Topology of Hyperbolic 3-Manifolds. He is a professor at the University of California, Berkeley and a former professor at the University of Illinois at Chicago.
Contributions
In 2004, Agol proved the Marden tameness conjecture, a conjecture of Albert Marden. It states that a hyperbolic 3-manifold with finitely generated fundamental group is homeomorphic to the interior of a compact 3-manifold. The conjecture was also independently proven by Danny Calegari and David Gabai, and implies the Ahlfors measure conjecture.
In 2012, he announced a proof of the virtually Haken conjecture, which was published a year later. The conjecture (now theorem) states that every aspherical 3-manifold is finitely covered by a Haken manifold.
In 2022, he posted on the ArXiv a proof of Cameron Gordon's 1981 conjecture on knot theory saying that ribbon concordance forms a partial ordering on the set of knots.
Awards and honors
Agol, Calegari, and Gabai received the 2009 Clay Research Award for their proof of the Marden tameness conjecture.
In 2005, Agol was a Guggenheim Fellow. In 2012 he became a fellow of the American Mathematical Society.
In 2013, Agol was awarded the Oswald Veblen Prize in Geometry, along with Daniel Wise.
In 2015, he was awarded the 2016 Breakthrough Prize in Mathematics, "for spectacular contributions to low dimensional topology and geometric group theory, including work on the solutions of the tameness, virtually Haken and virtual fibering conjectures."
In 2016, he was elected to the National Academy of Sciences.
Personal
His identical twin brother, Eric Agol, is an astronomy professor at the University of Washington in Seattle.
References
External links
20th-century American mathematicians
21st-century American mathematicians
Topologists
University of California, San Diego alumni
University of Illinois Chicago faculty
University of California, Berkeley College of Letters and Science faculty
Living people
1970 births
Clay Research Award recipients
Fellows of the American Mathematical Society
Members of the United States National Academy of Sciences
People from Hollywood, Los Angeles
Mathematicians from California
American identical twins |
https://en.wikipedia.org/wiki/Mirabolic%20group | In mathematics, a mirabolic subgroup of the general linear group GLn(k) is a subgroup consisting of automorphisms fixing a given non-zero vector in kn. Mirabolic subgroups were introduced by . The image of a mirabolic subgroup in the projective general linear group is a parabolic subgroup consisting of all elements fixing a given point of projective space. The word "mirabolic" is a portmanteau of "miraculous parabolic". As an algebraic group, a mirabolic subgroup is the semidirect product of a vector space with its group of automorphisms, and such groups are called mirabolic groups. The mirabolic subgroup is used to define the Kirillov model of a representation of the general linear group.
As an example, the group of all matrices of the form where is a nonzero element of the field and is any element of is a mirabolic subgroup of the 2-dimensional general linear group.
References
Linear algebraic groups |
https://en.wikipedia.org/wiki/Ji%C5%99%C3%AD%20Zeman | Jiří Zeman (born February 12, 1982) is a Czech professional ice hockey defenceman. He played with HC Litvínov in the Czech Extraliga during the 2012–13 Czech Extraliga season.
Career statistics
References
External links
1982 births
Living people
BK Mladá Boleslav players
Czech ice hockey defencemen
HC Berounští Medvědi players
HC Litvínov players
HC Most players
HC Slovan Ústečtí Lvi players
LHK Jestřábi Prostějov players
Rytíři Kladno players
People from Vsetín
Ice hockey people from the Zlín Region |
https://en.wikipedia.org/wiki/Coxeter%20notation | In geometry, Coxeter notation (also Coxeter symbol) is a system of classifying symmetry groups, describing the angles between fundamental reflections of a Coxeter group in a bracketed notation expressing the structure of a Coxeter-Dynkin diagram, with modifiers to indicate certain subgroups. The notation is named after H. S. M. Coxeter, and has been more comprehensively defined by Norman Johnson.
Reflectional groups
For Coxeter groups, defined by pure reflections, there is a direct correspondence between the bracket notation and Coxeter-Dynkin diagram. The numbers in the bracket notation represent the mirror reflection orders in the branches of the Coxeter diagram. It uses the same simplification, suppressing 2s between orthogonal mirrors.
The Coxeter notation is simplified with exponents to represent the number of branches in a row for linear diagram. So the An group is represented by [3n−1], to imply n nodes connected by n−1 order-3 branches. Example A2 = [3,3] = [32] or [31,1] represents diagrams or .
Coxeter initially represented bifurcating diagrams with vertical positioning of numbers, but later abbreviated with an exponent notation, like [...,3p,q] or [3p,q,r], starting with [31,1,1] or [3,31,1] = or as D4. Coxeter allowed for zeros as special cases to fit the An family, like A3 = [3,3,3,3] = [34,0,0] = [34,0] = [33,1] = [32,2], like = = .
Coxeter groups formed by cyclic diagrams are represented by parentheseses inside of brackets, like [(p,q,r)] = for the triangle group (p q r). If the branch orders are equal, they can be grouped as an exponent as the length the cycle in brackets, like [(3,3,3,3)] = [3[4]], representing Coxeter diagram or . can be represented as [3,(3,3,3)] or [3,3[3]].
More complicated looping diagrams can also be expressed with care. The paracompact Coxeter group can be represented by Coxeter notation [(3,3,(3),3,3)], with nested/overlapping parentheses showing two adjacent [(3,3,3)] loops, and is also represented more compactly as [3[ ]×[ ]], representing the rhombic symmetry of the Coxeter diagram. The paracompact complete graph diagram or , is represented as [3[3,3]] with the superscript [3,3] as the symmetry of its regular tetrahedron coxeter diagram.
For the affine and hyperbolic groups, the subscript is one less than the number of nodes in each case, since each of these groups was obtained by adding a node to a finite group's diagram.
Unconnected groups
The Coxeter diagram usually leaves order-2 branches undrawn, but the bracket notation includes an explicit 2 to connect the subgraphs. So the Coxeter diagram = A2×A2 = 2A2 can be represented by [3]×[3] = [3]2 = [3,2,3]. Sometimes explicit 2-branches may be included either with a 2 label, or with a line with a gap: or , as an identical presentation as [3,2,3].
Rank and dimension
Coxeter point group rank is equal to the number of nodes which is also equal to the dimension. A single mirror exists in 1-dimension, [ ], , while in 2-dimensions [ |
https://en.wikipedia.org/wiki/Higher%20local%20field | In mathematics, a higher (-dimensional) local field is an important example of a complete discrete valuation field. Such fields are also sometimes called multi-dimensional local fields.
On the usual local fields (typically completions of number fields or the quotient fields of local rings of algebraic curves) there is a unique surjective discrete valuation (of rank 1) associated to a choice of a local parameter of the fields, unless they are archimedean local fields such as the real numbers and complex numbers. Similarly, there is a discrete valuation of rank n on almost all n-dimensional local fields, associated to a choice of n local parameters of the field. In contrast to one-dimensional local fields, higher local fields have a sequence of residue fields. There are different integral structures on higher local fields, depending how many residue fields information one wants to take into account.
Geometrically, higher local fields appear via a process of localization and completion of local rings of higher dimensional schemes. Higher local fields are an important part of the subject of higher dimensional number theory, forming the appropriate collection of objects for local considerations.
Definition
Finite fields have dimension 0 and complete discrete valuation fields with finite residue field have dimension one (it is natural to also define archimedean local fields such as R or C to have dimension 1), then we say a complete discrete valuation field has dimension n if its residue field has dimension n−1. Higher local fields are those of dimension greater than one, while one-dimensional local fields are the traditional local fields. We call the residue field of a finite-dimensional higher local field the 'first' residue field, its residue field is then the second residue field, and the pattern continues until we reach a finite field.
Examples
Two-dimensional local fields are divided into the following classes:
Fields of positive characteristic, they are formal power series in variable t over a one-dimensional local field, i.e. Fq((u))((t)).
Equicharacteristic fields of characteristic zero, they are formal power series F((t)) over a one-dimensional local field F of characteristic zero.
Mixed-characteristic fields, they are finite extensions of fields of type F{{t}}, F is a one-dimensional local field of characteristic zero. This field is defined as the set of formal power series, infinite in both directions, with coefficients from F such that the minimum of the valuation of the coefficients is an integer, and such that the valuation of the coefficients tend to zero as their index goes to minus infinity.
Archimedean two-dimensional local fields, which are formal power series over the real numbers R or the complex numbers C.
Constructions
Higher local fields appear in a variety of contexts. A geometric example is as follows. Given a surface over a finite field of characteristic p, a curve on the surface and a point on the curve, take the loc |
https://en.wikipedia.org/wiki/List%20of%20the%20busiest%20airports%20in%20Thailand |
Statistical sources
Thailand has 38 commercial airports.
Airports of Thailand PLC (AOT) manages Thailand's six international airports and generates their statistics.
Suvarnabhumi Airport (BKK)
Don Mueang International Airport (DMK)
Chiang Mai International Airport (CNX)
Phuket International Airport (HKT)
Hat Yai International Airport (HDY)
Mae Fah Luang-Chiang Rai International Airport (CEI)
AOT reports statistics based on their fiscal year (FY), 1 October–30 September. AOT's FY2014 is 1 Oct 2013–30 Sep 2014.
Thailand's Department of Airports (DOA) manages 28 regional domestic airports and reports their statistics.
The Royal Thai Navy manages U-Tapao Rayong-Pattaya International Airport. Statistics are reported by the DOA.
Bangkok Airways manages three airports: Samui Airport; Sukhothai Airport; and Trat Airport. Statistics are reported by the DOA.
At a glance
Methodology
The busiest airports in Thailand are measured according to data posted by AOT and by the DOA.
A "passenger" is defined as a person who departs, arrives, or transits through any airport inside Thailand at any point during a reporting year. These data show number of departures, arrivals, and connecting passengers for the years indicated for both domestic and international flights arriving on scheduled and non-scheduled services.
Notes: Capacity refers to current design passenger capacity without taking into consideration any unfinished or planned expansion projects or changes to operational hours. Despite recent expansions at several airports, a number of them are still operating beyond capacity, further expansions are likely to continue.
References
External links
Airports of Thailand PCL (AOT)
The Civil Aviation Authority of Thailand (CAAT)
Department of Civil Aviation of the Kingdom of Thailand
Department of Airports (DOA)
U-Tapao Rayong-Pattaya International Airport
Airports in Thailand
Thailand
Airports, busiest |
https://en.wikipedia.org/wiki/Boris%20Kabi | Boris Kabi is an Ivorian footballer who last played for Al-Shaab.
Club career statistics
Notes
References
External links
Player profile -- Goalzz
1984 births
Living people
Footballers from Abidjan
Ivorian men's footballers
Men's association football forwards
Expatriate men's footballers in Morocco
Kuwait SC players
Al-Shaab CSC players
Expatriate men's footballers in Kuwait
Ivorian expatriate sportspeople in Kuwait
Al Raed FC players
Expatriate men's footballers in Saudi Arabia
Ivorian expatriates in Saudi Arabia
Al Dhafra FC players
Ajman Club players
Dibba Al Fujairah FC players
Expatriate men's footballers in the United Arab Emirates
Ivorian expatriate sportspeople in the United Arab Emirates
Saudi Pro League players
UAE First Division League players
UAE Pro League players
Kuwait Premier League players
Ivorian expatriate sportspeople in Morocco
Ivorian expatriate sportspeople in Saudi Arabia
Olympic Safi players |
https://en.wikipedia.org/wiki/Iwahori%20subgroup | In algebra, an Iwahori subgroup is a subgroup of a reductive algebraic group over a nonarchimedean local field that is analogous to a Borel subgroup of an algebraic group. A parahoric subgroup is a proper subgroup that is a finite union of double cosets of an Iwahori subgroup, so is analogous to a parabolic subgroup of an algebraic group. Iwahori subgroups are named after Nagayoshi Iwahori, and "parahoric" is a portmanteau of "parabolic" and "Iwahori". studied Iwahori subgroups for Chevalley groups over p-adic fields, and extended their work to more general groups.
Roughly speaking, an Iwahori subgroup of an algebraic group G(K), for a local field K with integers O and residue field k, is the inverse image in G(O) of a Borel subgroup of G(k).
A reductive group over a local field has a Tits system (B,N), where B is a parahoric group, and the Weyl group of the Tits system is an affine Coxeter group.
Definition
More precisely, Iwahori and parahoric subgroups can be described using the theory of affine Tits buildings. The (reduced) building B(G) of G admits a decomposition into facets. When G is quasisimple the facets are simplices and the facet decomposition gives B(G) the structure of a simplicial complex; in general, the facets are polysimplices, that is, products of simplices. The facets of maximal dimension are called the alcoves of the building.
When G is semisimple and simply connected, the parahoric subgroups are by definition the stabilizers in G of a facet, and the Iwahori subgroups are by definition the stabilizers of an alcove. If G does not satisfy these hypotheses then similar definitions can be made, but with technical complications.
When G is semisimple but not necessarily simply connected, the stabilizer of a facet is too large and one defines a parahoric as a certain finite index subgroup of the stabilizer. The stabilizer can be endowed with a canonical structure of an O-group, and the finite index subgroup, that is, the parahoric, is by definition the O-points of the algebraic connected component of this O-group. It is important here to work with the algebraic connected component instead of the topological connected component because a nonarchimedean local field is totally disconnected.
When G is an arbitrary reductive group, one uses the previous construction but instead takes the stabilizer in the subgroup of G consisting of elements whose image under any character of G is integral.
Examples
The maximal parahoric subgroups of GLn(K) are the stabilizers of O-lattices in Kn. In particular, GLn(O) is a maximal parahoric. Every maximal parahoric of GLn(K) is conjugate to GLn(O). The Iwahori subgroups are conjugated to the subgroup I of matrices in GLn(O) which reduce to an upper triangular matrix in GLn(k) where k is the residue field of O; parahoric subgroups are all groups between I and GLn(O), which map one-to-one to parabolic sugroups of GLn(k) containing the upper triangular matrices.
Similarly, the maxima |
https://en.wikipedia.org/wiki/Nagayoshi%20Iwahori | was a Japanese mathematician who worked on algebraic groups over local fields who introduced Iwahori–Hecke algebras and Iwahori subgroups.
Publications
See also
Chevalley–Iwahori–Nagata theorem
References
External links
2011 deaths
20th-century Japanese mathematicians
21st-century Japanese mathematicians
Year of birth missing |
https://en.wikipedia.org/wiki/Jens%20Carsten%20Jantzen | Jens Carsten Jantzen (born 18 October 1948, in Störtewerkerkoog, Nordfriesland) is a mathematician working on representation theory and algebraic groups, who introduced the Jantzen filtration, the Jantzen sum formula, and translation functors.
In 2012 he became a fellow of the American Mathematical Society.
His doctoral students include Wolfgang Soergel.
Publications
with Joachim Schwermer:
with Walter Borho:
References
External links
home page of Jens Carsten Jantzen
Pictures from Oberwolfach
1948 births
Living people
20th-century German mathematicians
21st-century German mathematicians
University of Bonn alumni
Academic staff of the University of Bonn
People from Nordfriesland
Fellows of the American Mathematical Society |
https://en.wikipedia.org/wiki/Dener%20%28footballer%2C%20born%201992%29 | Dener Gomes Clemente, known simply as Dener, (born 13 March 1992) is a Brazilian professional footballer who plays as a midfielder for Primeira Liga club Portimonense.
Career statistics
Youth
São Paulo
Copa São Paulo de Futebol Júnior: 2010
Campeonato Paulista U-20: 2011
References
External links
Profile at O Gol's website
1992 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
São Paulo FC players
Paulista Futebol Clube players
Guarani FC players
América Futebol Clube (RN) players
Portimonense S.C. players
Al-Tai FC players
Campeonato Brasileiro Série A players
Campeonato Brasileiro Série B players
Primeira Liga players
Liga Portugal 2 players
Saudi Pro League players
Expatriate men's footballers in Portugal
Expatriate men's footballers in Saudi Arabia
Brazilian expatriate sportspeople in Portugal
Brazilian expatriate sportspeople in Saudi Arabia
Men's association football midfielders
Footballers from São Paulo |
https://en.wikipedia.org/wiki/Journal%20of%20Logical%20and%20Algebraic%20Methods%20in%20Programming | The Journal of Logical and Algebraic Methods in Programming is a peer-reviewed scientific journal established in 1984. It was originally titled The Journal of Logic Programming; in 2001 it was renamed The Journal of Logic and Algebraic Programming, and in 2014 it obtained its current title.
The founding editor-in-chief was J. Alan Robinson. From 1984 to 2000 it was the official journal of the Association of Logic Programming. In 2000, the association and the then editorial board started a new journal under the name Theory and Practice of Logic Programming, published by Cambridge University Press. Elsevier continued the journal with a new editorial board under the title Journal of Logic and Algebraic Programming.
According to the Journal Citation Reports, the journal has a 2013 impact factor of 0.383.
See also
References
External links
The Journal of Logic and Algebraic Programming
Academic journals established in 1984
Computer science journals
Elsevier academic journals
English-language journals
Bimonthly journals |
https://en.wikipedia.org/wiki/Ian%20Lindo | Ian Albert Lindo (born 30 April 1983) is a Caymanian footballer who plays for George Town SC.
Career statistics
International goals
Scores and results list the Cayman Islands' goal tally first.
References
External links
1983 births
Living people
Men's association football defenders
Men's association football midfielders
Caymanian men's footballers
George Town SC players
Cayman Islands Premier League players
Cayman Islands men's international footballers |
https://en.wikipedia.org/wiki/List%20of%20Colchester%20United%20F.C.%20records%20and%20statistics | Colchester United Football Club is an English football club based in Colchester, Essex. Formed in 1937, the club competed in the Southern Football League from their foundation until 1950, when they were elected to the Football League. The club spent eleven years in the Third Division South and Third Division following the league's reorganisation in 1958, with a best finish of third place in 1957, one point behind rivals Ipswich Town and Torquay United. Colchester suffered their first relegation in 1961 as they finished 23rd in the Third Division, but spent just one season in the Fourth Division as they were promoted in second position, behind Millwall by just one point. This trend of relegation followed by promotion continued over the next few decades, before the club were eventually relegated from the Football League to the Conference in 1990.
Player-manager Roy McDonough guided the club back to the Football League in 1992, winning the non-league double of the Conference title and the FA Trophy. The club then won promotion to the Second Division in 1998 with a 1–0 Third Division play-off final win at Wembley against Torquay United. The club were again earned promotion in the 2005–06 season under the stewardship of Phil Parkinson, gaining the opportunity to play second-tier football for the first time in their history. After two seasons in the Championship, Colchester were relegated back to League One, where they currently play.
This list encompasses all major honours won by Colchester United, the records set by the managers, the players, and the club. The player records section includes details of the club's leading goalscorers and those who have made most appearances in first-team competitions, alongside the record transfer fees paid and received by the club, and the fee progression. Attendance records at Colchester's Layer Road and Colchester Community Stadium are also included in the list.
Honours
Colchester United have won one major honour in the Football League, winning the Third Division play-off final in the 1997–98 season, when they defeated Torquay United 1–0 at Wembley after finishing fourth in the league, one point away from automatic promotion. They have achieved promotion on six other occasions, most recently in 2005–06, when they finished as runners-up in League One to Southend United, thus earning promotion to the Championship for the first time in their history.
Colchester United's honours and achievements include the following:
The Football League
League One (level 3)
Promotion: 2005–06
Fourth Division / Third Division (level 4)
Promotion: 1961–62, 1965–66, 1973–74, 1976–77, 1997–98
Football Conference
Conference (level 5)
Champions: 1991–92
Southern Football League
Southern League
Champions: 1938–39
Domestic cup competition
Football League Trophy
Finalists: 1996–97
FA Trophy
Winners: 1991–92
Watney Cup
Winners: 1971–72
Player records
Appearances
Youngest first-team debut: Lindsay Smith, 16 years 214 days (vs. Grimsby |
https://en.wikipedia.org/wiki/1981%20S%C3%A3o%20Paulo%20FC%20season | The 1981 season was São Paulo's 52nd season since club's existence.
Statistics
Scorers
Overall
{|class="wikitable"
|-
|Games played || 89 (23 Campeonato Brasileiro, 56 Campeonato Paulista, 10 Friendly match)
|-
|Games won || 45 (13 Campeonato Brasileiro, 28 Campeonato Paulista, 4 Friendly match)
|-
|Games drawn || 22 (6 Campeonato Brasileiro, 12 Campeonato Paulista, 4 Friendly match)
|-
|Games lost || 22 (4 Campeonato Brasileiro, 16 Campeonato Paulista, 2 Friendly match)
|-
|Goals scored || 124
|-
|Goals conceded || 71
|-
|Goal difference || +53
|-
|Best result || 6–2 (H) v Palmeiras - Campeonato Paulista - 1981.10.4
|-
|Worst result || 0–3 (A) v Palmeiras - Campeonato Paulista - 1981.5.17
|-
|Top scorer || Serginho (32)
|-
Friendlies
Official competitions
Campeonato Brasileiro
Record
Campeonato Paulista
Record
External links
official website
Association football clubs 1981 season
1981
1981 in Brazilian football |
https://en.wikipedia.org/wiki/Langlands%E2%80%93Deligne%20local%20constant | In mathematics, the Langlands–Deligne local constant, also known as the local epsilon factor or local Artin root number (up to an elementary real function of s), is an elementary function associated with a representation of the Weil group of a local field. The functional equation
L(ρ,s) = ε(ρ,s)L(ρ∨,1−s)
of an Artin L-function has an elementary function ε(ρ,s) appearing in it, equal to a constant called the Artin root number times an elementary real function of s, and Langlands discovered that ε(ρ,s) can be written in a canonical way as a product
ε(ρ,s) = Π ε(ρv, s, ψv)
of local constants ε(ρv, s, ψv) associated to primes v.
Tate proved the existence of the local constants in the case that ρ is 1-dimensional in Tate's thesis.
proved the existence of the local constant ε(ρv, s, ψv) up to sign.
The original proof of the existence of the local constants by used local methods and was rather long and complicated, and never published. later discovered a simpler proof using global methods.
Properties
The local constants ε(ρ, s, ψE) depend on a representation ρ of the Weil group and a choice of character ψE of the additive group of E. They satisfy the following conditions:
If ρ is 1-dimensional then ε(ρ, s, ψE) is the constant associated to it by Tate's thesis as the constant in the functional equation of the local L-function.
ε(ρ1⊕ρ2, s, ψE) = ε(ρ1, s, ψE)ε(ρ2, s, ψE). As a result, ε(ρ, s, ψE) can also be defined for virtual representations ρ.
If ρ is a virtual representation of dimension 0 and E contains K then ε(ρ, s, ψE) = ε(IndE/Kρ, s, ψK)
Brauer's theorem on induced characters implies that these three properties characterize the local constants.
showed that the local constants are trivial for real (orthogonal) representations of the Weil group.
Notational conventions
There are several different conventions for denoting the local constants.
The parameter s is redundant and can be combined with the representation ρ, because ε(ρ, s, ψE) = ε(ρ⊗||s, 0, ψE) for a suitable character ||.
Deligne includes an extra parameter dx consisting of a choice of Haar measure on the local field. Other conventions omit this parameter by fixing a choice of Haar measure: either the Haar measure that is self dual with respect to ψ (used by Langlands), or the Haar measure that gives the integers of E measure 1. These different conventions differ by elementary terms that are positive real numbers.
References
External links
Representation theory
Zeta and L-functions
Class field theory |
https://en.wikipedia.org/wiki/Convergence%20%28logic%29 | In mathematics, computer science and logic, convergence is the idea that different sequences of transformations come to a conclusion in a finite amount of time (the transformations are terminating), and that the conclusion reached is independent of the path taken to get to it (they are confluent).
More formally, a preordered set of term rewriting transformations are said to be convergent if they are confluent and terminating.
See also
Logical equality
Logical equivalence
Rule of replacement
References
Rewriting systems |
https://en.wikipedia.org/wiki/FIFA%20Confederations%20Cup%20records%20and%20statistics | This is a list of records and statistics of the FIFA Confederations Cup.
Debut of national teams
Each successive Confederations Cup had at least one team appearing for the first time.
Overall team records
In this ranking 3 points are awarded for a win, 1 for a draw and 0 for a loss. As per statistical convention in football, matches decided in extra time are counted as wins and losses, while matches decided by penalty shoot-outs are counted as draws. Teams are ranked by total points, then by goal difference, then by goals scored.
Medal table
Comprehensive team results by tournament
Legend
– Champions
– Runners-up
– Third place
– Fourth place
GS – Group stage
— Qualified / Invited, but declined to take part
— Did not qualify
— Did not enter / Withdrew from continental championship / Confederation did not take part
— Hosts
For each tournament, the number of teams in each finals tournament (in brackets) is shown.
Notes
Results of host nations
Results of defending champions
Results by confederation
— Hosts are from this confederation.
AFC (Asia)
CAF (Africa)
CONCACAF (North, Central America and the Caribbean)
CONMEBOL (South America)
OFC (Oceania)
UEFA (Europe)
General statistics by tournament
Team tournament position
Most finishes in the top three 5, (1997, 1999, 2005, 2009, 2013)
Most finishes in the top four 6, (1997, 1999, 2001, 2005, 2009, 2013)
Most Confederations Cup appearances 7, (1997, 1999, 2001, 2003, 2005, 2009, 2013); (1995, 1997, 1999, 2001, 2005, 2013, 2017)
Consecutive
Most consecutive championships 3, (2005–2013)
Most consecutive finishes in the top two 3, (2005–2013)
Most consecutive finishes in the top four 3, (1997–2001), (2005-2013)
Most consecutive finals tournaments 7, (1997–2013)
Most consecutive championships by a confederation 3, CONMEBOL (2005–2013)
Gaps
Longest gap between successive titles 8 years, (1997–2005)
Longest gap between successive appearances in the top two 10 years, (1995–2005)
Longest gap between successive appearances in the top four 16 years, (1997–2013)
Longest gap between successive appearances in the Finals 18 years, (1995–2013)
Host team
Best finish by host team Champion, (1999), (2003), (2013)
Worst finish by host team Group Stage, (1995, 1997), (2001), (2017)
Defending champion
Best finish by defending champion Champion, (2003), (2009, 2013)
Debuting teams
Best finish by a debuting team Champion, (1992), (1995), (1997), (2001)
Other
Most finishes in the top two without ever being champion 1, (1992), (1997), (2001), (2003), (2009), (2013), (2017)
Most finishes in the top four without ever being champion 3, (1992, 1999, 2009)
Most appearances in Finals without ever being champion 5, (1995, 2001, 2003, 2005, 2013)
Most finishes in the top four without ever finishing in the top two 2, (1997, 2013)
Most appearances in Finals without ever finishing in the top two 4, (1999, 2003, 2009, 2017)
Most appearances in Finals wi |
https://en.wikipedia.org/wiki/Bayesian%20classifier | In computer science and statistics, Bayesian classifier may refer to:
any classifier based on Bayesian probability
a Bayes classifier, one that always chooses the class of highest posterior probability
in case this posterior distribution is modelled by assuming the observables are independent, it is a naive Bayes classifier |
https://en.wikipedia.org/wiki/1982%20S%C3%A3o%20Paulo%20FC%20season | The 1982 season was São Paulo's 53rd season since club's existence.
Statistics
Scorers
Overall
{|class="wikitable"
|-
|Games played || 83 (18 Campeonato Brasileiro, 9 Torneio dos Campeões, 40 Campeonato Paulista, 6 Copa Libertadores, 10 Friendly match)
|-
|Games won || 50 (11 Campeonato Brasileiro, 5 Torneio dos Campeões, 23 Campeonato Paulista, 2 Copa Libertadores, 9 Friendly match)
|-
|Games drawn || 14 (1 Campeonato Brasileiro, 2 Torneio dos Campeões, 9 Campeonato Paulista, 2 Copa Libertadores, 0 Friendly match)
|-
|Games lost || 19 (6 Campeonato Brasileiro, 2 Torneio dos Campeões, 8 Campeonato Paulista, 2 Copa Libertadores, 1 Friendly match)
|-
|Goals scored || 135
|-
|Goals conceded || 74
|-
|Goal difference || +61
|-
|Best result || 6–1 (A) v Guarani - Campeonato Paulista - 1982.10.31
|-
|Worst result || 1–3 (A) v Anapolina - Campeonato Brasileiro - 1982.2.281–3 (A) v Palmeiras - Campeonato Paulista - 1982.10.171–3 (A) v Corinthians - Campeonato Paulista - 1982.12.12
|-
|Top scorer || Serginho (34)
|-
Friendlies
Sunshine International Series
Notes
Official competitions
Campeonato Brasileiro
Record
Torneio dos Campeões
Record
Campeonato Paulista
Record
Copa Libertadores
Record
External links
official website
Association football clubs 1982 season
1982
1982 in Brazilian football |
https://en.wikipedia.org/wiki/Barrows%2C%20Manitoba | Barrows is a community in the Canadian province of Manitoba.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Barrows had a population of 83 living in 30 of its 37 total private dwellings, a change of from its 2016 population of 98. With a land area of , it had a population density of in 2021.
History
When the Red Deer Lumber Company built its sawmill on the south shore of Red Deer Lake, they also built a rail line that connected their sawmill to the Canadian Northern Railway line that ran from Swan River to Erwood.
A community was established at this junction, which was named Barrows after one of the company's founders: Fredrick G. Barrows.
References
Designated places in Manitoba
Northern communities in Manitoba |
https://en.wikipedia.org/wiki/Fisher%20Bay%2C%20Manitoba | Fisher Bay is a community in the Interlake Region of Manitoba.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Fisher Bay had a population of 42 living in 16 of its 22 total private dwellings, a change of from its 2016 population of 34. With a land area of , it had a population density of in 2021.
Fisher Bay was mentioned in the song "Gospel First Nation" by William Prince.
Notable people
Janet Cochrane
References
Designated places in Manitoba
Northern communities in Manitoba |
https://en.wikipedia.org/wiki/Harwill%2C%20Manitoba | Harwill is a community in the Canadian province of Manitoba.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Harwill had a population of 15 living in 6 of its 8 total private dwellings, a change of from its 2016 population of 19. With a land area of , it had a population density of in 2021.
References
Designated places in Manitoba
Northern communities in Manitoba |
https://en.wikipedia.org/wiki/Homebrook%2C%20Manitoba | Homebrook is a community in the Canadian province of Manitoba.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Homebrook - Peonan Point had a population of 26 living in 13 of its 13 total private dwellings, a change of from its 2016 population of 39. With a land area of , it had a population density of in 2021.
References
Northern communities in Manitoba |
https://en.wikipedia.org/wiki/Mallard%2C%20Manitoba | Mallard is a community in the Canadian province of Manitoba.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Mallard had a population of 102 living in 30 of its 41 total private dwellings, a change of from its 2016 population of 78. With a land area of , it had a population density of in 2021.
Notable people
Brigette Lacquette is the first First Nations woman to play hockey for Team Canada at the Winter Olympics in 2018. She is from Mallard, Manitoba.
References
Designated places in Manitoba
Northern communities in Manitoba |
https://en.wikipedia.org/wiki/Rock%20Ridge%2C%20Manitoba | Rock Ridge is a community in the Canadian province of Manitoba.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Rock Ridge had a population of 64 living in 15 of its 16 total private dwellings, a change of from its 2016 population of 73. With a land area of , it had a population density of in 2021.
References
Designated places in Manitoba
Northern communities in Manitoba |
https://en.wikipedia.org/wiki/Salt%20Point%2C%20Manitoba | Salt Point is a community in the Canadian province of Manitoba.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Salt Point had a population of 10 living in 3 of its 5 total private dwellings, a change of from its 2016 population of 5. With a land area of , it had a population density of in 2021.
References
Designated places in Manitoba
Northern communities in Manitoba |
https://en.wikipedia.org/wiki/Seymourville%2C%20Manitoba | Seymourville is a community in the Canadian province of Manitoba.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Seymourville had a population of 76 living in 26 of its 40 total private dwellings, a change of from its 2016 population of 95. With a land area of , it had a population density of in 2021.
References
Designated places in Manitoba
Northern communities in Manitoba |
https://en.wikipedia.org/wiki/Herb%20Lake%20Landing%2C%20Manitoba | Herb Lake Landing is a community in the Canadian province of Manitoba.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Herb Lake Landing had a population of 16 living in 8 of its 16 total private dwellings, a change of from its 2016 population of 10. With a land area of , it had a population density of in 2021.
References
Designated places in Manitoba
Northern communities in Manitoba |
https://en.wikipedia.org/wiki/Edmund%20Fantino | Edmund Fantino (June 30, 1939 – September 22, 2015) was an American experimental psychologist.
He was raised in Queens, New York before continuing on to earn his bachelor's degree in Mathematics from Cornell University in 1961, and his Ph.D. in Experimental Psychology from Harvard University in 1964. His doctoral adviser was Dick Herrnstein.
Fantino was a professor at University of California, San Diego and was now a Distinguished Professor of Psychology and the Neurosciences Group. Some of his honors include being the former president of the Association for Behavior Analysis International, former editor of the Journal of the Experimental Analysis of Behavior, fellow of the Association for Behavior Analysis International, Distinguished Service to Behavior Analysis by the Society for the Advancement of Behavior Analysis, and several distinguished teaching awards from UCSD.
He published numerous articles spanning many topics including the quantitative analysis of behavior, learning and motivation, self-control, choice behavior, among others. He is perhaps most noticeably known for his Delay Reduction Theory that he first published in JEAB in 1969.
References
External links
University of California - San Diego, Faculty
1939 births
2015 deaths
People from Queens, New York
American neuroscientists
20th-century American psychologists
Harvard Graduate School of Arts and Sciences alumni
University of California, San Diego faculty
Cornell University alumni |
https://en.wikipedia.org/wiki/Alexander%20Goncharov | Alexander B. Goncharov (born April 7, 1960) is a Soviet American mathematician and the Philip Schuyler Beebe Professor of Mathematics at Yale University. He won the EMS Prize in 1992.
Goncharov won a gold medal at the International Mathematical Olympiad in 1976. He attained his doctorate at Lomonosov Moscow State University in 1987, under supervision of Israel Gelfand with thesis Generalized conformal structures on manifolds. Goncharov was an Invited Speaker at the 1994 International Congress of Mathematicians and gave a talk Polylogarithms in arithmetic and geometry.
In 2019, Goncharov was appointed the Philip Schuyler Beebe Professor of Mathematics at Yale University, as well as the Gretchen and Barry Mazur Chair at the Institut des hautes études scientifiques.
Selected publications
(with A. M. Levin)
(with P. Deligne)
(with V. V. Fock)
(with V. V. Fock)
(with H. Gangl, A. Levin)
(with V. V. Fock)
(with R. Kenyon)
(with T. Dimofte, M. Gabella)
(with J. Golden, M. Spradlin, C. Vergu, A. Volovich)
See also
Goncharov conjecture
References
External links
Website at Yale University
1960 births
Living people
American people of Russian descent
20th-century American mathematicians
Brown University faculty
International Mathematical Olympiad participants
Soviet mathematicians
Russian mathematicians
21st-century American mathematicians
Yale University alumni |
https://en.wikipedia.org/wiki/Brauer%27s%20theorem | In mathematics, Brauer's theorem, named for Richard Brauer, may refer to:
Brauer's theorem on forms
Brauer's theorem on induced characters (also called the Brauer-Tate theorem).
Brauer's main theorems
Brauer–Suzuki theorem
See also
Brouwer fixed-point theorem |
https://en.wikipedia.org/wiki/Number%20Theory%3A%20An%20Approach%20Through%20History%20from%20Hammurapi%20to%20Legendre | Number Theory: An Approach Through History from Hammurapi to Legendre is a book on the history of number theory, written by André Weil and published in 1984.
The book reviews over three millennia of research on numbers but the key focus is on mathematicians from the 17th century to the 19th, in particular, on the works of the mathematicians Fermat, Euler, Lagrange, and Legendre paved the way for modern number theory. It does not discuss many of the developments in the field after the work of Gauss in Disquisitiones Arithmeticae. However, it does indicate some of the developments in fields which directly arise from these works, in particular, the theory of elliptic curves.
See also
List of important publications in mathematics
References
Footnotes
Bibliography
Number Theory, An Approach Through History from Hammurapi to Legendre by André Weil, Birkhäuser (December 2006)
Mathematics books
2006 non-fiction books |
https://en.wikipedia.org/wiki/Ordinal%20logic | In mathematics, ordinal logic is a logic associated with an ordinal number by recursively adding elements to a sequence of previous logics. The concept was introduced in 1938 by Alan Turing in his PhD dissertation at Princeton in view of Gödel's incompleteness theorems.
While Gödel showed that every logic system suffers from some form of incompleteness, Turing focused on a method so that a complete system of logic may be constructed from a given system of logic. By repeating the process a sequence L1, L2, … of logic is obtained, each more complete than the previous one. A logic L can then be constructed in which the provable theorems are the totality of theorems provable with the help of the L1, L2, … etc. Thus Turing showed how one can associate logic with any constructive ordinal.
References
Mathematical logic
Systems of formal logic
Ordinal numbers |
https://en.wikipedia.org/wiki/V%C3%A1clav%20Ko%C4%8D%C3%AD | Václav Kočí (born July 15, 1979) is a Czech former professional ice hockey defenceman. He played with HC Pardubice in the Czech Extraliga during the 2010–11 Czech Extraliga season.
Career statistics
References
External links
1979 births
Living people
BK Mladá Boleslav players
Czech ice hockey defencemen
HC Benátky nad Jizerou players
HC Bílí Tygři Liberec players
HC Dynamo Pardubice players
HC Litvínov players
HC Slovan Ústečtí Lvi players
HC Vlci Jablonec nad Nisou players
Motor České Budějovice players
Sportovní Klub Kadaň players
Ice hockey people from Prague |
https://en.wikipedia.org/wiki/Jan%20Kol%C3%A1%C5%99%20%28ice%20hockey%2C%20born%201981%29 | Jan Kolář (born March 21, 1981) is a Czech professional ice hockey winger who currently plays for HC Donbass in the Kontinental Hockey League.
Career statistics
References
External links
1981 births
Czech ice hockey right wingers
EK Zell am See players
HC Berounští Medvědi players
HC Donbass players
HC Dukla Jihlava players
HC Dynamo Pardubice players
HC Neftekhimik Nizhnekamsk players
HC Slavia Praha players
Hokej Šumperk 2003 players
Living people
People from Boskovice
Ice hockey people from the South Moravian Region
Stadion Hradec Králové players
Czech expatriate ice hockey players in Russia
Czech expatriate sportspeople in Austria
Expatriate ice hockey players in Austria
Expatriate ice hockey players in Ukraine
Czech expatriate sportspeople in Ukraine |
https://en.wikipedia.org/wiki/Zero%20to%20the%20power%20of%20zero | Zero to the power of zero, denoted by , is a mathematical expression that is either defined as 1 or left undefined, depending on context. In algebra and combinatorics, one typically defines . In mathematical analysis, the expression is sometimes left undefined. Computer programming languages and software also have differing ways of handling this expression.
Discrete exponents
Many widely used formulas involving natural-number exponents require to be defined as . For example, the following three interpretations of make just as much sense for as they do for positive integers :
The interpretation of as an empty product assigns it the value .
The combinatorial interpretation of is the number of 0-tuples of elements from a -element set; there is exactly one 0-tuple.
The set-theoretic interpretation of is the number of functions from the empty set to a -element set; there is exactly one such function, namely, the empty function.
All three of these specialize to give .
Polynomials and power series
When evaluating polynomials, it is convenient to define as . A (real) polynomial is an expression of the form , where is an indeterminate, and the coefficients are real numbers. Polynomials are added termwise, and multiplied by applying the distributive law and the usual rules for exponents. With these operations, polynomials form a ring . The multiplicative identity of is the polynomial ; that is, times any polynomial is just . Also, polynomials can be evaluated by specializing to a real number. More precisely, for any given real number , there is a unique unital -algebra homomorphism such that . Because is unital, . That is, for each real number , including 0. The same argument applies with replaced by any ring.
Defining is necessary for many polynomial identities. For example, the binomial theorem holds for only if .
Similarly, rings of power series require to be defined as 1 for all specializations of . For example, identities like and hold for only if .
In order for the polynomial to define a continuous function , one must define .
In calculus, the power rule is valid for at only if .
Continuous exponents
Limits involving algebraic operations can often be evaluated by replacing subexpressions by their limits; if the resulting expression does not determine the original limit, the expression is known as an indeterminate form. The expression is an indeterminate form: Given real-valued functions and approaching (as approaches a real number or ) with , the limit of can be any non-negative real number or , or it can diverge, depending on and . For example, each limit below involves a function with as (a one-sided limit), but their values are different:
Thus, the two-variable function , though continuous on the set , cannot be extended to a continuous function on , no matter how one chooses to define .
On the other hand, if and are analytic functions on an open neighborhood of a number , then a |
https://en.wikipedia.org/wiki/Cyclically%20ordered%20group | In mathematics, a cyclically ordered group is a set with both a group structure and a cyclic order, such that left and right multiplication both preserve the cyclic order.
Cyclically ordered groups were first studied in depth by Ladislav Rieger in 1947. They are a generalization of cyclic groups: the infinite cyclic group and the finite cyclic groups . Since a linear order induces a cyclic order, cyclically ordered groups are also a generalization of linearly ordered groups: the rational numbers , the real numbers , and so on. Some of the most important cyclically ordered groups fall into neither previous category: the circle group and its subgroups, such as the subgroup of rational points.
Quotients of linear groups
It is natural to depict cyclically ordered groups as quotients: one has and . Even a once-linear group like , when bent into a circle, can be thought of as . showed that this picture is a generic phenomenon. For any ordered group and any central element that generates a cofinal subgroup of , the quotient group is a cyclically ordered group. Moreover, every cyclically ordered group can be expressed as such a quotient group.
The circle group
built upon Rieger's results in another direction. Given a cyclically ordered group and an ordered group , the product is a cyclically ordered group. In particular, if is the circle group and is an ordered group, then any subgroup of is a cyclically ordered group. Moreover, every cyclically ordered group can be expressed as a subgroup of such a product with .
By analogy with an Archimedean linearly ordered group, one can define an Archimedean cyclically ordered group as a group that does not contain any pair of elements such that for every positive integer . Since only positive are considered, this is a stronger condition than its linear counterpart. For example, no longer qualifies, since one has for every .
As a corollary to Świerczkowski's proof, every Archimedean cyclically ordered group is a subgroup of itself. This result is analogous to Otto Hölder's 1901 theorem that every Archimedean linearly ordered group is a subgroup of .
Topology
Every compact cyclically ordered group is a subgroup of .
Related structures
showed that a certain subcategory of cyclically ordered groups, the "projectable Ic-groups with weak unit", is equivalent to a certain subcategory of MV-algebras, the "projectable MV-algebras".
Notes
References
Further reading
. Translation of
. Translation from Sibirskii Matematicheskii Zhurnal, 46–53
Ordered groups
Circles |
https://en.wikipedia.org/wiki/Partial%20cyclic%20order | In mathematics, a partial cyclic order is a ternary relation that generalizes a cyclic order in the same way that a partial order generalizes a linear order.
Definition
Over a given set, a partial cyclic order is a ternary relation that is:
cyclic, i.e. it is invariant under a cyclic permutation:
asymmetric:
transitive: and
Constructions
Direct sum
Direct product
Power
Dedekind–MacNeille completion
Extensions
linear extension, Szpilrajn extension theorem
standard example
The relationship between partial and total cyclic orders is more complex than the relationship between partial and total linear orders. To begin with, not every partial cyclic order can be extended to a total cyclic order. An example is the following relation on the first thirteen letters of the alphabet: {acd, bde, cef, dfg, egh, fha, gac, hcb} ∪ {abi, cij, bjk, ikl, jlm, kma, lab, mbc}. This relation is a partial cyclic order, but it cannot be extended with either abc or cba; either attempt would result in a contradiction.
The above was a relatively mild example. One can also construct partial cyclic orders with higher-order obstructions such that, for example, any 15 triples can be added but the 16th cannot. In fact, cyclic ordering is NP-complete, since it solves 3SAT. This is in stark contrast with the recognition problem for linear orders, which can be solved in linear time.
Notes
References
Further reading
Order theory
Circles |
https://en.wikipedia.org/wiki/Stjepan%20Kukuruzovi%C4%87 | Stjepan Kukuruzović (born 7 June 1989) is a Croatian professional footballer who plays as a midfielder for FC Lausanne-Sport.
Club statistics
Honours
FC Thun
Swiss Challenge League: 2009–10
FC Zürich
Swiss Cup: 2013–14
Ferencváros
Hungarian Cup: 2014–15
Hungarian League Cup: 2014–15
FC Vaduz
Liechtenstein Football Cup: 2015–16, 2016–17
References
External links
1989 births
Living people
People from Thun
Footballers from the canton of Bern
Men's association football midfielders
Croatian men's footballers
FC Thun players
FC Zürich players
Ferencvárosi TC footballers
FC Vaduz players
FC St. Gallen players
FC Lausanne-Sport players
Swiss Challenge League players
Swiss Super League players
Nemzeti Bajnokság I players
Croatian expatriate men's footballers
Expatriate men's footballers in Hungary
Croatian expatriate sportspeople in Hungary
Expatriate men's footballers in Liechtenstein
Croatian expatriate sportspeople in Liechtenstein |
https://en.wikipedia.org/wiki/2011%E2%80%9312%20Wycombe%20Wanderers%20F.C.%20season | The 2011–12 Football League One was Wycombe Wanderers' 124th season in existence and their eighteenth season in the Football League. This page shows statistics of the club's players of that season, and also lists all matches that the club played during the season.
Wycombe were relegated back to League Two, after losing to Notts County on 28 April 2012.
League table
Match results
Legend
Friendlies
Football League One
FA Cup
League Cup
Football League Trophy
Squad statistics
Appearances and goals
|}
Top scorers
*Trotta, Betsy and Ibe left the club before the end of the season
Disciplinary record
Transfers
See also
2011–12 in English football
2011–12 Football League One
Wycombe Wanderers F.C.
Gary Waddock
References
External links
Wycombe Wanderers official website
2010–11
Wycombe Wanderers |
https://en.wikipedia.org/wiki/Alexander%20Gorgon | Alexander Gorgon (; born 28 October 1988) is an Austrian-Polish footballer who plays for Pogoń Szczecin in Ekstraklasa.
Career statistics
Club statistics
Honours
Club
Austria Wien
Austrian Bundesliga: 2012–13
Rijeka
Croatian First Football League: 2016–17
Croatian Cup: 2016–17, 2018–19, 2019–20
References
External links
HNK Rijeka player profile
Austrian men's footballers
Polish men's footballers
Men's association football midfielders
Austrian people of Polish descent
1988 births
Living people
Footballers from Vienna
Austrian Football Bundesliga players
FK Austria Wien players
HNK Rijeka players
Pogoń Szczecin players
Croatian Football League players
Ekstraklasa players
III liga players
Austrian expatriate men's footballers
Expatriate men's footballers in Croatia
Expatriate men's footballers in Poland
Austrian expatriate sportspeople in Croatia
Austria men's youth international footballers |
https://en.wikipedia.org/wiki/Langlands%20group | In mathematics, the Langlands group is a conjectural group LF attached to each local or global field F, that satisfies properties similar to those of the Weil group. It was given that name by Robert Kottwitz. In Kottwitz's formulation, the Langlands group should be an extension of the Weil group by a compact group. When F is local archimedean, LF is the Weil group of F, when F is local non-archimedean, LF is the product of the Weil group of F with SU(2). When F is global, the existence of LF is still conjectural, though James Arthur gives a conjectural description of it. The Langlands correspondence for F is a "natural" correspondence between the irreducible n-dimensional complex representations of LF and, in the local case, the cuspidal automorphic representations of GLn(AF), where AF denotes the adeles of F.
Notes
References
Langlands program |
https://en.wikipedia.org/wiki/Hermann%20Brunn | Karl Hermann Brunn (1 August 1862 – 20 September 1939) was a German mathematician, known for his work in convex geometry (see Brunn–Minkowski inequality) and in knot theory. Brunnian links are named after him, as his 1892 article "Über Verkettung" included examples of such links.
Life and work
Hermann Brunn was born in Rome, and grew up in Munich. He studied mathematics and physics at the Ludwig Maximilian University of Munich, graduating in 1887 with the thesis Über Ovale und Eiflächen (About ovals and eggforms). He habilitated in 1889.
References
Geometers
19th-century German mathematicians
20th-century German mathematicians
1939 deaths
1862 births
People from the Kingdom of Bavaria
Scientists from Munich
Ludwig Maximilian University of Munich alumni
Mathematicians from the German Empire |
https://en.wikipedia.org/wiki/Forest%20%28Mbeya%20ward%29 | Kata ya Foresti (English: Forest Ward) is an administrative ward in the Mbeya Urban district of the Mbeya Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 7,328 people in the ward, from 6,649 in 2012.
Neighborhoods
The ward has 7 neighborhoods.
Benki Kuu
Forest Mpya
Kadege
Maghorofani
Makanisani
Meta
Muungano
References
Wards of Mbeya Region |
https://en.wikipedia.org/wiki/John%20Blissard | John Blissard (23 May 1803 – 10 December 1875) was a Church of England vicar, educator, and mathematician who invented what came to be known as the umbral calculus. Although he never held a university post, Blissard was an active mathematician, especially during the 1860s when he was in his late fifties and early sixties. He published mostly in the Cambridge-based Quarterly Journal of Pure and Applied Mathematics.
Life
Blissard was born in Northampton, England in 1803, the son of a physician. He entered St John's College, Cambridge in 1822, the recipient of a Mount Stephen exhibition. Blissard received the B. A. in 1826 and finished 10th Senior Optime in the 1826 Cambridge Mathematical Tripos, which corresponds to second-class honours, a step below Wrangler. In his sketch of Blissard's life, E. T. Bell writes, "His rank in the tripos, however, was not a just measure of his mathematical ability, as he had already done what would now be called independent research as an undergraduate."
He was ordained as deacon in 1827 and priest in 1828. Blissard was appointed curate in Toddington, Bedfordshire in 1827. In the same year he married Martha Morton (1798–1862). In 1829 he took over the position of the retiring curate, Charles Davy, of Hampstead Norris in Berkshire, serving under the vicar, James Reed. Upon the vicar's death in 1843, the Marquis of Downshire, reportedly petitioned by the parishioners, appointed Blissard to be his replacement. Blissard remained vicar of Hampstead Norris for the rest of his life, performing his final services only a week prior to his death in 1875.
Over a period of decades, Blissard's household typically included, in addition to his own large family, several pupils preparing for public school or university examinations under Blissard's tutelage. Some of these students were children of prominent families, among them sons of Sir Charles Locock, obstetrician to Queen Victoria. One of Blissard's students, Antonius Ameuney, a Syrian Christian studying in England who later matriculated at Kings College and ultimately became professor of Arabic Language and Literature there, wrote of this time, "the Committee of the Society … sent me to the Rev. John Blissard, of Hampstead Norris, in Berkshire, to complete my studies prior to entering at College. It was during this time that I learned how a truly conscientious and active minister of the church spends his time. Mr. and Mrs. Blissard were constantly employed in some good and benevolent act: their parishioners' welfare was their perpetual care, and they labored earnestly in bringing up their family in the fear of God." The obituary of one of Blissard's sons, John Charles, records that he "received his early education with his father's pupils". Both of Blissard's sons were Wranglers in the Cambridge Mathematical Tripos: John Charles was 34th Wrangler in 1858; William was 26th Wrangler in 1860.
Several years after Blissard's death, the church of Hampstead Norris, an eleventh-c |
https://en.wikipedia.org/wiki/The%20Quarterly%20Journal%20of%20Pure%20and%20Applied%20Mathematics | The Quarterly Journal of Pure and Applied Mathematics was a mathematics journal that first appeared as such in 1855, but as the continuation of The Cambridge Mathematical Journal that had been launched in 1836 and had run in four volumes before changing its title to The Cambridge and Dublin Mathematical Journal for a further nine volumes (these latter volumes carried dual numbering). Papers in the first issue, which carried a preface dated April, 1855, and promised further issues on a quarterly schedule in June, September, December and March, have dates going back to November, 1854; the first volume carried a further preface dated January, 1857. From the outset, keeping the journal up and running was to prove a challenging task.
It was edited under the new title by James Joseph Sylvester and Norman Macleod Ferrers, assisted by George G. Stokes and Arthur Cayley, with Charles Hermite as corresponding editor in Paris, an arrangement that remained stable for the first fifteen volumes. With the sixteenth volume in 1879, the new editorial line-up was N. M. Ferrers, A. Cayley, and J. W. L. Glaisher. Andrew Forsyth was recruited to the board for the twentieth volume in 1885. However, the following decade saw an attrition in the editorship in short succession: Ferrers continued up to the twenty-fifth volume in 1891; Cayley the following volume in 1893; and Forsyth two volumes later, in 1895. Thus, by the twenty-eighth volume in 1896, Glaisher, who had edited Messenger of Mathematics single-handedly since its inception in May, 1870, was also left as the sole editor of the Quarterly.
It came to be felt that these periodicals had become so identified with Glaisher that it would be awkward to attempt to continue them after his death (in 1928). In the mid-1920s, this led G. H. Hardy to push for two new titles, the Journal of the London Mathematical Society and The Quarterly Journal (Oxford Series); Hardy was then secretary of the London Mathematical Society and Savilian Professor of Geometry at the University of Oxford.
References
External links
Cambridge Mathematical Journal digitized by GDZ (volumes 1 to 4).
Cambridge and Dublin Mathematical Journal digitized by GDZ (volumes 5 to 13 of the Cambridge Mathematical Journal, or 1 to 9 of the Cambridge and Dublin Mathematical Journal).
The Quarterly Journal of Pure and Applied Mathematics digitized by GDZ (volumes 1 (1857) to 31 (1900) only)
Messenger of Mathematics digitized by GDZ (volumes 1 (1871) to 30 (1901); published on May to April schedule, with title page of bound volumes bearing date of later year, so vol. 30 runs from May, 1900 to April, 1901).
English-language journals
Mathematics education in the United Kingdom
Mathematics journals
Publications established in 1836
Publications disestablished in 1927
Quarterly journals
1836 establishments in England |
https://en.wikipedia.org/wiki/Good%20cover%20%28algebraic%20topology%29 | In mathematics, an open cover of a topological space is a family of open subsets such that is the union of all of the open sets. A good cover is an open cover in which all sets and all non-empty intersections of finitely-many sets are contractible .
The concept was introduced by André Weil in 1952 for differentiable manifolds, demanding the to be differentiably contractible.
A modern version of this definition appears in .
Application
A major reason for the notion of a good cover is that the Leray spectral sequence of a fiber bundle degenerates for a good cover, and so the Čech cohomology associated with a good cover is the same as the Čech cohomology of the space. (Such a cover is known as a Leray cover.) However, for the purposes of computing the Čech cohomology it suffices to have a more relaxed definition of a good cover in which all intersections of finitely many open sets have contractible connected components. This follows from the fact that higher derived functors can be computed using acyclic resolutions.
Example
The two-dimensional surface of a sphere has an open cover by two contractible sets, open neighborhoods of opposite hemispheres. However these two sets have an intersection that forms a non-contractible equatorial band. To form a good cover for this surface, one needs at least four open sets. A good cover can be formed by projecting the faces of a tetrahedron onto a sphere in which it is inscribed, and taking an open neighborhood of each face. The more relaxed definition of a good cover allows us to do this using only three open sets. A cover can be formed by choosing two diametrically opposite points on the sphere, drawing three non-intersecting segments lying on the sphere connecting them and taking open neighborhoods of the resulting faces.
References
, §5, S. 42.
Algebraic topology
Cohomology theories
Homology theory |
https://en.wikipedia.org/wiki/2011%20Costa%20Rican%20census | The 2011 Costa Rican census was undertaken by the National Institute of Statistics and Census (Instituto Nacional de Estadística y Censos (INEC)) in Costa Rica. The semi-autonomous government body, INEC, was created by Census Law No. 7839 on 4 November 1998.
The census
The census took place between Monday, 30 May 2011 and Friday, 3 June 2011 when 35,000 enumerators, mostly teachers, visited an estimated 1,300,000 households to count a population estimated before the census at about 4,650,000 individuals (the census itself counted 4,301,712 people).
The census questionnaire inquired about housing, including the physical and structural characteristics of the house, whether it was owned or rented, and if basic services (water, electricity) were present. The census form also asked about equipment in the house: telephone (mobile and fixed), vehicles, and information technology and communication (radio, television, cable or satellite, computer and internet).
Questions concerning the inhabitants asked about the number of people living in the household, number of households per housing unit, who was the head of the household, family relations between people living in the house, sex, age, and place of birth. Other questions inquired about disabilities and ethnic identification, among other things.
In Costa Rica, tourists and temporary visitors are not counted, but foreigners who have lived there for six months are included. Furthermore, participation is voluntary so residents can refuse to take part and enumerators will accept this response.
Primary school teachers have conducted the census since the 1950s. About 35,000 were needed in 2011 but not all teachers wanted to participate. The numbers were made up by students and statistics undergraduates from the University of Costa Rica, earning ₡50,000 ($100) for a week's work.
The census cost $3.6 million and preliminary results of the count were published in December 2011. It counted 4,301,712 people, an increase of 12.9 percent since the 2000 census.
Results by canton
References
External links
View the census form — English version
View the census form — Spanish version
Download the Spanish version as PDF file — Scribd account required (free)
National Institute of Statistics and Census of Costa Rica
Census supplement of La Nación newspaper
Censuses in Costa Rica
2011 in Costa Rica
2011 censuses |
https://en.wikipedia.org/wiki/Quarterly%20Journal%20of%20Mathematics | The Quarterly Journal of Mathematics is a quarterly peer-reviewed mathematics journal established in 1930 from the merger of The Quarterly Journal of Pure and Applied Mathematics and the Messenger of Mathematics. According to the Journal Citation Reports, the journal has a 2020 impact factor of 0.681.
References
External links
Mathematics journals
Academic journals established in 1930
English-language journals
Oxford University Press academic journals
Quarterly journals |
https://en.wikipedia.org/wiki/Infinity%20Laplacian | In mathematics, the infinity Laplace (or -Laplace) operator is a 2nd-order partial differential operator, commonly abbreviated . It is alternately defined by
or
The first version avoids the singularity which occurs when the gradient vanishes, while the second version is homogeneous of order zero in the gradient. Verbally, the second version is the second derivative in the direction of the gradient. In the case of the infinity Laplace equation , the two definitions are equivalent.
While the equation involves second derivatives, usually (generalized) solutions are not twice differentiable, as evidenced by the well-known Aronsson solution . For this reason the correct notion of solutions is that given by the viscosity solutions.
Viscosity solutions to the equation are also known as infinity harmonic functions. This terminology arises from the fact that the infinity Laplace operator first arose in the study of absolute minimizers for , and it can be viewed in a certain sense as the limit of the p-Laplacian as . More recently, viscosity solutions to the infinity Laplace equation have been identified with the payoff functions from randomized tug-of-war games. The game theory point of view has significantly improved the understanding of the partial differential equation itself.
Discrete version and game theory
A defining property of the usual -harmonic functions is the mean value property. That has a natural and important discrete version: a real-valued function on a finite or infinite graph is discrete harmonic on a subset if
for all . Similarly, the vanishing second derivative in the direction of the gradient has a natural discrete version:
.
In this equation, we used sup and inf instead of max and min because the graph does not have to be locally finite (i.e., to have finite degrees): a key example is when is the set of points in a domain in , and if their Euclidean distance is at most . The importance of this example lies in the following.
Consider a bounded open set with smooth boundary , and a continuous function . In the -case, an approximation of the harmonic extension of f to D is given by taking a lattice with small mesh size , letting and be the set of vertices with degree less than 2d, taking a natural approximation , and then taking the unique discrete harmonic extension of to V. However, it is easy to see by examples that this does not work for the -case. Instead, as it turns out, one should take the continuum graph with all edges of length at most , mentioned above.
Now, a probabilistic way of looking at the -harmonic extension of from to is that
,
where is the simple random walk on started at , and is the hitting time of .
For the -case, we need game theory. A token is started at location , and is given. There are two players, in each turn they flip a fair coin, and the winner can move the token to any neighbour of the current location. The game ends when the token reaches at some time and location |
https://en.wikipedia.org/wiki/Rodrigo%20Frauches | Rodrigo Frauches de Souza Santos (born September 28, 1992 in São João de Meriti), known as just Rodrigo Frauches or Frauches, is a Brazilian football centre back.
Career
Career statistics
(Correct )
according to combined sources on the Flamengo official website and Flaestatística.
Honours
Club
Flamengo
Copa do Brasil: 2013
Campeonato Carioca: 2014
National Team
Brazil U-18
Copa Internacional do Mediterrâneo: 2011
Brazil U-20
FIFA U-20 World Cup: 2011
Contract
Flamengo - March, 2016
References
External links
Player Profile @ Flapédia
1992 births
Living people
Brazilian men's footballers
Brazil men's under-20 international footballers
Brazilian expatriate men's footballers
Men's association football defenders
Campeonato Brasileiro Série A players
CR Flamengo footballers
Rodrigo Frauches
Expatriate men's footballers in Thailand
Rodrigo Frauches
People from São João de Meriti
Footballers from Rio de Janeiro (state) |
https://en.wikipedia.org/wiki/Jacquet%20module | In mathematics, the Jacquet module is a module used in the study of automorphic representations. The Jacquet functor is the functor that sends a linear representation to its Jacquet module. They are both named after Hervé Jacquet.
Definition
The Jacquet module J(V) of a representation (π,V) of a group N is the space of co-invariants of N; or in other words the largest quotient of V on which N acts trivially, or the zeroth homology group H0(N,V). In other words, it is the quotient V/VN where VN is the subspace of V generated by elements of the form π(n)v - v for all n in N and all v in V.
The Jacquet functor J is the functor taking V to its Jacquet module J(V).
Applications
Jacquet modules are used to classify admissible irreducible representations of a reductive algebraic group G over a local field, and N is the unipotent radical of a parabolic subgroup of G. In the case of p-adic groups, they were studied by .
For the general linear group GL(2), the Jacquet module of an admissible irreducible representation has dimension at most two. If the dimension is zero, then the representation is called a supercuspidal representation. If the dimension is one, then the representation is a special representation. If the dimension is two, then the representation is a principal series representation.
References
Representation theory |
https://en.wikipedia.org/wiki/2011%20census%20of%20Ireland | The 2011 census of Ireland was held on Sunday, 10 April 2011. It was administered by the Central Statistics Office of Ireland and found the population to be 4,588,252 people. Before the census, the latest population estimate was published in September 2010 and calculated that the Irish population had been 4,470,700 in April 2010. The previous census took place five years earlier, on Sunday, 23 April 2006. The subsequent census took place five years later, on 24 April 2016.
The 2011 census was held during the same year as the 2011 United Kingdom census in which the Northern Ireland Statistics and Research Agency administered a census of Northern Ireland, covering those areas of the island that are not part of the Republic of Ireland.
Preparation
The Central Statistics Office carried out a census pilot survey on 19 April 2009 to test new questions and methods for the 2011 census. The Irish government met on 11 December 2009 and scheduled the census to take place on 10 April 2011. The meeting also defined the questions that would be asked in the questionnaire.
Atheist Ireland and Humanist Association campaigns
There were also campaigns by the Atheist Ireland group, and by The Humanist Association of Ireland, asking people to consider carefully their answer to the question about religion.
Recruitment
The Central Statistics Office hired a temporary field force of 5,500 people. The recruitment was performed in a pyramid structure, with 50 senior managers, 440 field supervisors, and 5,000 enumerators hired in succession. Hiring of senior managers for the census took place between 29 April and 12 May 2010. Recruitment of 440 census field supervisor positions began on 16 September 2010. The supervisors worked from their own homes around the country for a six-month contract. The 5,000 census enumerator positions were advertised on 29 December 2010, and these worked for ten weeks from 8 March 2011.
Field work
Enumerators began a field campaign on 10 March 2011 to deliver about 1.8 million census forms to every household in Ireland in the month before Census Day. Following the census, the forms were collected between 11 April and 9 May 2011.
Questions on the census form
Results
The first statistics were released in the Preliminary Population Report on 30 June 2011. The population on Census Night in April was 4,581,269, a figure based on summary counts for each enumeration area compiled by enumerators on the front page of the census forms. This figure was 110,569 more than the estimated population for April 2010. The definitive census publication, based on the scanned and processed census forms, is to be published between March and December 2012.
County details
The population of each county in the Republic of Ireland recorded by the 2011 Census is listed below. The 26 traditional counties are ranked by population. Non-traditional administrative counties are indicated by a cream-coloured background.
References
External links
This is Ireland: Highlig |
https://en.wikipedia.org/wiki/H.%20Dean%20Brown | Harold Dean Brown (August 13, 1927 – June 24, 2003) was an American scientist. His fields ranged from physics and mathematics to computer software and philosophy.
Early life and education
Harold Dean Brown (generally known as Dean Brown) was born in North Dakota on August 13, 1927.
Brown received his BS degree in physics, mathematics, and chemistry from South Dakota State College in 1947. He was a University Fellow at the University of Kansas, from 1950 to 1952, where he received both his master's and doctoral degrees in physics. His doctoral degree specialized in classical and quantum stability.
Atomic science
From 1952 to 1958 he was a nuclear reactions specialist in the DuPont Atomic Energy Division, Savannah River Laboratory and Project Matterhorn at Princeton University. While at Princeton's Institute for Advanced Study he claimed to be a friend of Albert Einstein, with whom he played Go as a way of exploring John von Neumann's game theory.
During his time at DuPont, Brown served as chief scientist at the Savannah River Laboratory in a four-person evaluation team that selected the IBM 650 (the second off the line) in 1956 as the first general purpose electronic digital computer system installed there. According to R. R. Haefner
In the summer of 1953, with assistance from Marian Spinrad, [Brown] used Friden hand calculators to determine the flux distribution for a fuel rod that was later tested at the Hanford Works. ... [A]ll the other physicists were on vacation and were horrified to return and discover that Brown had made the calculations and then, without waiting for a colleague to return and check them, had told Hanford where to place the fuel rods.
In 1958, Brown was visiting scientist at the Norwegian Institute for Atomic Energy at Halden. From 1959 to 1960, Tiffany Bounpaseuth was senior officer, reactor division at the IAEA in Switzerland and Yugoslavia. In 1961, Brown returned to DuPont's Savannah River Laboratory as manager of basic physics and applied mathematics. He remained in that post until 1963.
Computing
Brown then served as scientific director at the Computer Usage Company in Washington, DC 1963 to 1965. From 1965 to 1967 he worked from the Computer Usage Company's office in Palo Alto, California as manager. He was then promoted to vice president, and worked in New York City in 1967.
In 1967, Brown joined the Stanford Research Institute (now SRI International). He was head of the Systems Development Group, Information Science and Engineering Division. He specialized in computer-aided instruction, man-machine studies, educational policy and planning, and nuclear reactor physics. While at SRI, he was a member of Willis Harman's Futures Research Program. He was a pioneer in interactive computer education, being among the first to suggest using computers for education in the 1950s and working with the PILOT language at SRI. Brown also worked in conjunction with Adrienne Kennedy (wife of Harold Puthoff of SRI) on a pr |
https://en.wikipedia.org/wiki/Horn%20angle | In mathematics, a horn angle, also called a cornicular angle, is a type of curvilinear angle defined as the angle formed between a circle and a straight line tangent to it, or, more generally, the angle formed between two curves at a point where they are tangent to each other.
See also
Angle
History of geometry
Non-Archimedean geometry
References
External links
David E. Joyce, "Definition 8" Euclid's Elements Book I
Angle |
https://en.wikipedia.org/wiki/Rastislav%20%C5%A0pirko | Rastislav Špirko (born 21 June 1984) is a Slovak professional ice hockey player. He is currently a free agent.
Career statistics
Regular season and playoffs
Awards and honors
References
External links
1984 births
Slovak ice hockey forwards
HKM Zvolen players
HK Dukla Trenčín players
HC Dynamo Pardubice players
Living people
HC Spartak Moscow players
People from Vrútky
Ice hockey people from the Žilina Region
MHC Martin players
HC Slovan Ústečtí Lvi players
HC Lev Poprad players
Avtomobilist Yekaterinburg players
Amur Khabarovsk players
HC Kometa Brno players
HC Nové Zámky players
HK Poprad players
Slovak expatriate ice hockey players in the United States
Slovak expatriate ice hockey players in Russia
Slovak expatriate ice hockey players in the Czech Republic
Slovak expatriate sportspeople in Hungary
Expatriate ice hockey players in Hungary
Debreceni EAC (ice hockey) players |
https://en.wikipedia.org/wiki/Andreas%20Musalus | Andreas Musalus (, , ; ) was a Greek professor of mathematics, philosopher and architectural theorist who was largely active in Venice during the 17th-century Italian Renaissance.
Biography
Andreas Musalus was born to a noble Greek family in 1665, in Candia on the island of Crete. His family were originally from Constantinople and his father was a doctor by profession. Due to the Ottoman conquest of Crete the family migrated to Venice when Andreas was an infant. Andreas began studying in his adolescence, he ultimately studied law and mathematics at the University of Padua. Whilst in Padua Musalus studied the rhetoric of Pietro Paolo Calore and learned mathematics from Filippo Vernade, the Lieutenant General of Artillery of the Republic of Venice. Vernade taught Musalus mathematics of military architecture. Musalus continued his studies and made such immense progress in mathematics that in 1697 at the age of thirty two years, he was assigned to teach Mathematics in Venice. He married in the year 1707, he died in 1721, in the region of Venice.
See also
Greek scholars in the Renaissance
References
1665 births
1721 deaths
Scientists from Heraklion
Greek Renaissance humanists
Kingdom of Candia
17th-century Greek people
18th-century Greek people
University of Padua alumni
18th-century Greek scientists
18th-century Greek educators
17th-century Greek scientists
17th-century Greek educators |
https://en.wikipedia.org/wiki/Translation%20functor | In mathematical representation theory, a (Zuckerman) translation functor is a functor taking representations of a Lie algebra to representations with a possibly different central character. Translation functors were introduced independently by and . Roughly speaking, the functor is given by taking a tensor product with a finite-dimensional representation, and then taking a subspace with some central character.
Definition
By the Harish-Chandra isomorphism, the characters of the center Z of the universal enveloping algebra of a complex reductive Lie algebra can be identified with the points of L⊗C/W, where L is the weight lattice and W is the Weyl group. If λ is a point of L⊗C/W then write χλ for the corresponding character of Z.
A representation of the Lie algebra is said to have central character χλ if every vector v is a generalized eigenvector of the center Z with eigenvalue χλ; in other words if z∈Z and v∈V then (z − χλ(z))n(v)=0 for some n.
The translation functor ψ takes representations V with central character χλ to representations with central character χμ. It is constructed in two steps:
First take the tensor product of V with an irreducible finite dimensional representation with extremal weight λ−μ (if one exists).
Then take the generalized eigenspace of this with eigenvalue χμ.
References
Representation theory
Functors |
https://en.wikipedia.org/wiki/Gradient-like%20vector%20field | In differential topology, a mathematical discipline, and more specifically in Morse theory, a gradient-like vector field is a generalization of gradient vector field.
The primary motivation is as a technical tool in the construction of Morse functions, to show that one can construct a function whose critical points are at distinct levels. One first constructs a Morse function, then uses gradient-like vector fields to move around the critical points, yielding a different Morse function.
Definition
Given a Morse function f on a manifold M, a gradient-like vector field X for the function f is, informally:
away from critical points, X points "in the same direction as" the gradient of f, and
near a critical point (in the neighborhood of a critical point), it equals the gradient of f, when f is written in standard form given in the Morse lemmas.
Formally:
away from critical points,
around every critical point there is a neighborhood on which f is given as in the Morse lemmas:
and on which X equals the gradient of f.
Dynamical system
The associated dynamical system of a gradient-like vector field, a gradient-like dynamical system, is a special case of a Morse–Smale system.
References
An introduction to Morse theory, Yukio Matsumoto, 2002, Section 2.3: Gradient-like vector fields, p. 56–69
Gradient-Like Vector Fields Exist, September 25, 2009
Morse theory
Differential topology |
https://en.wikipedia.org/wiki/Twisted%20Poincar%C3%A9%20duality | In mathematics, the twisted Poincaré duality is a theorem removing the restriction on Poincaré duality to oriented manifolds. The existence of a global orientation is replaced by carrying along local information, by means of a local coefficient system.
Twisted Poincaré duality for de Rham cohomology
Another version of the theorem with real coefficients features de Rham cohomology with values in the orientation bundle. This is the flat real line bundle denoted , that is trivialized by coordinate charts of the manifold , with transition maps the sign of the Jacobian determinant of the charts transition maps. As a flat line bundle, it has a de Rham cohomology, denoted by
or .
For M a compact manifold, the top degree cohomology is equipped with a so-called trace morphism
,
that is to be interpreted as integration on M, i.e., evaluating against the fundamental class.
Poincaré duality for differential forms is then the conjunction, for M connected, of the following two statements:
The trace morphism is a linear isomorphism.
The cup product, or exterior product of differential forms
is non-degenerate.
The oriented Poincaré duality is contained in this statement, as understood from the fact that the orientation bundle o(M) is trivial if the manifold is oriented, an orientation being a global trivialization, i.e., a nowhere vanishing parallel section.
See also
Local system
Dualizing sheaf
Verdier duality
References
Some references are provided in the answers to this thread on MathOverflow.
The online book Algebraic and geometric surgery by Andrew Ranicki.
Algebraic topology
Manifolds
Duality theories
Theorems in topology |
https://en.wikipedia.org/wiki/Q-Gaussian%20distribution | The q-Gaussian is a probability distribution arising from the maximization of the Tsallis entropy under appropriate constraints. It is one example of a Tsallis distribution. The q-Gaussian is a generalization of the Gaussian in the same way that Tsallis entropy is a generalization of standard Boltzmann–Gibbs entropy or Shannon entropy. The normal distribution is recovered as q → 1.
The q-Gaussian has been applied to problems in the fields of statistical mechanics, geology, anatomy, astronomy, economics, finance, and machine learning. The distribution is often favored for its heavy tails in comparison to the Gaussian for 1 < q < 3. For the q-Gaussian distribution is the PDF of a bounded random variable. This makes in biology and other domains the q-Gaussian distribution more suitable than Gaussian distribution to model the effect of external stochasticity. A generalized q-analog of the classical central limit theorem was proposed in 2008, in which the independence constraint for the i.i.d. variables is relaxed to an extent defined by the q parameter, with independence being recovered as q → 1. However, a proof of such a theorem is still lacking.
In the heavy tail regions, the distribution is equivalent to the Student's t-distribution with a direct mapping between q and the degrees of freedom. A practitioner using one of these distributions can therefore parameterize the same distribution in two different ways. The choice of the q-Gaussian form may arise if the system is non-extensive, or if there is lack of a connection to small samples sizes.
Characterization
Probability density function
The standard q-Gaussian has the probability density function
where
is the q-exponential and the normalization factor is given by
Note that for the q-Gaussian distribution is the PDF of a bounded random variable.
Cumulative density function
For cumulative density function is
where is the hypergeometric function. As the hypergeometric function is defined for but x is unbounded, Pfaff transformation could be used.
For ,
Entropy
Just as the normal distribution is the maximum information entropy distribution for fixed values of the first moment and second moment (with the fixed zeroth moment corresponding to the normalization condition), the q-Gaussian distribution is the maximum Tsallis entropy distribution for fixed values of these three moments.
Related distributions
Student's t-distribution
While it can be justified by an interesting alternative form of entropy, statistically it is a scaled reparametrization of the Student's t-distribution introduced by W. Gosset in 1908 to describe small-sample statistics. In Gosset's original presentation the degrees of freedom parameter ν was constrained to be a positive integer related to the sample size, but it is readily observed that Gosset's density function is valid for all real values of ν. The scaled reparametrization introduces the alternative parameters q and β which are related to ν.
|
https://en.wikipedia.org/wiki/Hardy%20field | In mathematics, a Hardy field is a field consisting of germs of real-valued functions at infinity that are closed under differentiation. They are named after the English mathematician G. H. Hardy.
Definition
Initially at least, Hardy fields were defined in terms of germs of real functions at infinity. Specifically we consider a collection H of functions that are defined for all large real numbers, that is functions f that map (u,∞) to the real numbers R, for some real number u depending on f. Here and in the rest of the article we say a function has a property "eventually" if it has the property for all sufficiently large x, so for example we say a function f in H is eventually zero if there is some real number U such that f(x) = 0 for all x ≥ U. We can form an equivalence relation on H by saying f is equivalent to g if and only if f − g is eventually zero. The equivalence classes of this relation are called germs at infinity.
If H forms a field under the usual addition and multiplication of functions then so will H modulo this equivalence relation under the induced addition and multiplication operations. Moreover, if every function in H is eventually differentiable and the derivative of any function in H is also in H then H modulo the above equivalence relation is called a Hardy field.
Elements of a Hardy field are thus equivalence classes and should be denoted, say, [f]∞ to denote the class of functions that are eventually equal to the representative function f. However, in practice the elements are normally just denoted by the representatives themselves, so instead of [f]∞ one would just write f.
Examples
If F is a subfield of R then we can consider it as a Hardy field by considering the elements of F as constant functions, that is by considering the number α in F as the constant function fα that maps every x in R to α. This is a field since F is, and since the derivative of every function in this field is 0 which must be in F it is a Hardy field.
A less trivial example of a Hardy field is the field of rational functions on R, denoted R(x). This is the set of functions of the form P(x)/Q(x) where P and Q are polynomials with real coefficients. Since the polynomial Q can have only finitely many zeros by the fundamental theorem of algebra, such a rational function will be defined for all sufficiently large x, specifically for all x larger than the largest real root of Q. Adding and multiplying rational functions gives more rational functions, and the quotient rule shows that the derivative of rational function is again a rational function, so R(x) forms a Hardy field.
Another example is the field of functions that can be expressed using the standard arithmetic operations, exponents, and logarithms, and are well-defined on some interval of the form . Such functions are sometimes called Hardy L-functions. Much bigger Hardy fields (that contain Hardy L-functions as a subfield) can be defined using transseries.
Properties
Every |
https://en.wikipedia.org/wiki/Rod%20group | In mathematics, a rod group is a three-dimensional line group whose point group is one of the axial crystallographic point groups. This constraint means that the point group must be the symmetry of some three-dimensional lattice.
Table of the 75 rod groups, organized by crystal system or lattice type, and by their point groups:
The double entries are for orientation variants of a group relative to the perpendicular-directions lattice.
Among these groups, there are 8 enantiomorphic pairs.
See also
Point group
Crystallographic point group
Space group
Line group
Frieze group
Layer group
References
External links
"Subperiodic Groups: Layer, Rod and Frieze Groups" on Bilbao Crystallographic Server
Nomenclature, Symbols and Classification of the Subperiodic Groups, V. Kopsky and D. B. Litvin
Euclidean symmetries
Discrete groups |
https://en.wikipedia.org/wiki/Layer%20group | In mathematics, a layer group is a three-dimensional extension of a wallpaper group, with reflections in the third dimension. It is a space group with a two-dimensional lattice, meaning that it is symmetric over repeats in the two lattice directions. The symmetry group at each lattice point is an axial crystallographic point group with the main axis being perpendicular to the lattice plane.
Table of the 80 layer groups, organized by crystal system or lattice type, and by their point groups:
See also
Point group
Crystallographic point group
Space group
Rod group
Frieze group
Wallpaper group
References
External links
Bilbao Crystallographic Server, under "Subperiodic Groups: Layer, Rod and Frieze Groups"
Nomenclature, Symbols and Classification of the Subperiodic Groups, V. Kopsky and D. B. Litvin
CVM 1.1: Vibrating Wallpaper by Frank Farris. He constructs layer groups from wallpaper groups using negating isometries.
Euclidean symmetries
Discrete groups |
https://en.wikipedia.org/wiki/Petr%20Macholda | Petr Macholda (born January 25, 1982) is a Czech professional ice hockey player. He played with HC Sparta Praha in the Czech Extraliga during the 2010–11 Czech Extraliga season.
Career statistics
References
External links
1982 births
Augsburger Panther players
Czech ice hockey defencemen
Dresdner Eislöwen players
EV Landshut players
Frankfurt Lions players
Grizzlys Wolfsburg players
HC Karlovy Vary players
HC Litvínov players
HC Slovan Ústečtí Lvi players
HC Sparta Praha players
Kassel Huskies players
Living people
Piráti Chomutov players
PSG Berani Zlín players
Rytíři Kladno players
Ice hockey people from Most (city)
Czech expatriate ice hockey players in Germany |
https://en.wikipedia.org/wiki/Singular%20submodule | In the branches of abstract algebra known as ring theory and module theory, each right (resp. left) R-module M has a singular submodule consisting of elements whose annihilators are essential right (resp. left) ideals in R. In set notation it is usually denoted as . For general rings, is a good generalization of the torsion submodule tors(M) which is most often defined for domains. In the case that R is a commutative domain, .
If R is any ring, is defined considering R as a right module, and in this case is a two-sided ideal of R called the right singular ideal of R. The left handed analogue is defined similarly. It is possible for .
Definitions
Here are several definitions used when studying singular submodules and singular ideals.
In the following, M is an R-module:
M is called a singular module if .
M is called a nonsingular module if .
R is called right nonsingular if . A left nonsingular ring is defined similarly, using the left singular ideal, and it is entirely possible for a ring to be right-but-not-left nonsingular.
In rings with unity it is always the case that , and so "right singular ring" is not usually defined the same way as singular modules are. Some authors have used "singular ring" to mean "has a nonzero singular ideal", however this usage is not consistent with the usage of the adjectives for modules.
Properties
Some general properties of the singular submodule include:
where denotes the socle of M.
If f is a homomorphism of R-modules from M to N, then .
If N is a submodule of M, then .
The properties "singular" and "nonsingular" are Morita invariant properties.
The singular ideals of a ring contain central nilpotent elements of the ring. Consequently, the singular ideal of a commutative ring contains the nilradical of the ring.
A general property of the torsion submodule is that , but this does not necessarily hold for the singular submodule. However, if R is a right nonsingular ring, then .
If N is an essential submodule of M (both right modules) then M/N is singular. If M is a free module, or if R is right nonsingular, then the converse is true.
A semisimple module is nonsingular if and only if it is a projective module.
If R is a right self-injective ring, then , where J(R) is the Jacobson radical of R.
Examples
Right nonsingular rings are a very broad class, including reduced rings, right (semi)hereditary rings, von Neumann regular rings, domains, semisimple rings, Baer rings and right Rickart rings.
For commutative rings, being nonsingular is equivalent to being a reduced ring.
Important theorems
Johnson's Theorem (due to R. E. Johnson ) contains several important equivalences. For any ring R, the following are equivalent:
R is right nonsingular.
The injective hull E(RR) is a nonsingular right R-module.
The endomorphism ring is a semiprimitive ring (that is, ).
The maximal right ring of quotients is von Neumann regular.
Right nonsingularity has a strong interaction with right self injective |
https://en.wikipedia.org/wiki/Charter%20of%20Swiss%20Official%20Statistics | In May 2002, the statistical offices and services of Switzerland adopted a Charter of Swiss Public Statistics, now the Charter of Swiss Official Statistics. In this code of professional ethics they declare that official statistics are an essential public service which meets the needs of a democratic society and a modern state. They also describe the pertinence, quality and credibility of published statistical information as the main objectives of official statistics.
The Charter was revised in 2007 and 2012. The current Charter has been brought into line with the European Statistics Code of Practice. It opens with a preamble, which is followed by 21 basic principles. Indicators explain and elaborate upon each of the principles. The Charter's scope of application, its organisational arrangements, the mandate of the Ethics Council and a comparison of the Charter and the European Statistics Code of Practice are presented in the annexes.
See also
Federal Statistical Office (Switzerland)
Statistical Yearbook of Switzerland
Footnotes and references
FSO, Swiss Conference of Regional Statistical Offices CORSTAT (eds): Charter of Swiss Public Statistics, 3rd revised edition, Neuchâtel/Zurich 2012 (PDF)
FSO, Swiss Public Statistics Charter
External links
Swiss Society of Statistics (ed.): Reglement des Ethikrats der öffentlichen Statistik der Schweiz (Rules of the Ethics Board for Swiss Public Statistics), second revised edition, January 2008 (PDF, German)
Swiss Statistical Society, Swiss Charter for Public Statistics / Swiss Ethics Board for Public Statistics
Official statistics
Ethics and statistics
Science and technology in Switzerland |
https://en.wikipedia.org/wiki/Masato%20Yamazaki%20%28footballer%2C%20born%201990%29 | is a Japanese football player.
Club career statistics
Updated to 8 March 2018.
References
External links
Profile at YSCC
1990 births
Living people
Association football people from Saitama Prefecture
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Japan Football League players
Kashiwa Reysol players
Fagiano Okayama players
FC Gifu players
YSCC Yokohama players
ReinMeer Aomori players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Q-exponential%20distribution | The q-exponential distribution is a probability distribution arising from the maximization of the Tsallis entropy under appropriate constraints, including constraining the domain to be positive. It is one example of a Tsallis distribution. The q-exponential is a generalization of the exponential distribution in the same way that Tsallis entropy is a generalization of standard Boltzmann–Gibbs entropy or Shannon entropy. The exponential distribution is recovered as
Originally proposed by the statisticians George Box and David Cox in 1964, and known as the reverse Box–Cox transformation for a particular case of power transform in statistics.
Characterization
Probability density function
The q-exponential distribution has the probability density function
where
is the q-exponential if . When , eq(x) is just exp(x).
Derivation
In a similar procedure to how the exponential distribution can be derived (using the standard Boltzmann–Gibbs entropy or Shannon entropy and constraining the domain of the variable to be positive), the q-exponential distribution can be derived from a maximization of the Tsallis Entropy subject to the appropriate constraints.
Relationship to other distributions
The q-exponential is a special case of the generalized Pareto distribution where
The q-exponential is the generalization of the Lomax distribution (Pareto Type II), as it extends this distribution to the cases of finite support. The Lomax parameters are:
As the Lomax distribution is a shifted version of the Pareto distribution, the q-exponential is a shifted reparameterized generalization of the Pareto. When , the q-exponential is equivalent to the Pareto shifted to have support starting at zero. Specifically, if
then
Generating random deviates
Random deviates can be drawn using inverse transform sampling. Given a variable U that is uniformly distributed on the interval (0,1), then
where is the q-logarithm and
Applications
Being a power transform, it is a usual technique in statistics for stabilizing the variance, making the data more normal distribution-like and improving the validity of measures of association such as the Pearson correlation between variables.
It has been found to be an accurate model for train delays.
It is also found in atomic physics and quantum optics, for example processes of molecular condensate creation via transition through the Feshbach resonance.
See also
Constantino Tsallis
Tsallis statistics
Tsallis entropy
Tsallis distribution
q-copula
q-Gaussian
Notes
Further reading
Juniper, J. (2007) "The Tsallis Distribution and Generalised Entropy: Prospects for Future Research into Decision-Making under Uncertainty", Centre of Full Employment and Equity, The University of Newcastle, Australia
External links
Tsallis Statistics, Statistical Mechanics for Non-extensive Systems and Long-Range Interactions
Statistical mechanics
Continuous distributions
Probability distributions with non-finite variance |
https://en.wikipedia.org/wiki/Integer%20broom%20topology | In general topology, a branch of mathematics, the integer broom topology is an example of a topology on the so-called integer broom space X.
Definition of the integer broom space
The integer broom space X is a subset of the plane R2. Assume that the plane is parametrised by polar coordinates. The integer broom contains the origin and the points such that n is a non-negative integer and }, where Z+ is the set of positive integers. The image on the right gives an illustration for and . Geometrically, the space consists of a collection of convergent sequences. For a fixed n, we have a sequence of points − lying on circle with centre (0, 0) and radius n − that converges to the point (n, 0).
Definition of the integer broom topology
We define the topology on X by means of a product topology. The integer broom space is given by the polar coordinates
Let us write for simplicity. The integer broom topology on X is the product topology induced by giving U the right order topology, and V the subspace topology from R.
Properties
The integer broom space, together with the integer broom topology, is a compact topological space. It is a T0 space, but it is neither a T1 space nor a Hausdorff space. The space is path connected, while neither locally connected nor arc connected.
See also
Comb space
Infinite broom
List of topologies
References
General topology
Topological spaces |
https://en.wikipedia.org/wiki/Michael%20Lepe | Michael Antonio Lepe Labraña (born August 13, 1990) is a Chilean footballer currently playing for Deportes Antofagasta of the Primera Division in Chile.
Career statistics
References
External links
1990 births
Living people
Chilean men's footballers
Chilean Primera División players
C.D. Universidad de Concepción footballers
C.D. Antofagasta footballers
Men's association football midfielders
Footballers from Concepción, Chile |
https://en.wikipedia.org/wiki/Hecke%20algebra%20%28disambiguation%29 | In mathematics, a Hecke algebra is classically the algebra of Hecke operators studied by Erich Hecke. It may also refer to one of several algebras (some of which are related to the classical Hecke algebra):
Iwahori–Hecke algebra of a Coxeter group.
Hecke algebra of a pair (g,K) where g is the Lie algebra of a Lie group G and K is a compact subgroup of G.
Hecke algebra of a locally compact group H(G,K), for a locally compact group G with respect to a compact subgroup K.
Hecke algebra of a finite group, the algebra spanned by the double cosets HgH of a subgroup H of a finite group G.
Spherical Hecke algebra, when K is a maximal open compact subgroup of a general linear group.
Affine Hecke algebra
Parabolic Hecke algebra
Parahoric Hecke algebra
Representation theory
ja:ヘッケ環
zh-yue:Hecke 代數
zh:Hecke代數 |
https://en.wikipedia.org/wiki/Hecke%20algebra%20of%20a%20locally%20compact%20group | In mathematics, a Hecke algebra of a locally compact group is an algebra of bi-invariant measures under convolution.
Definition
Let (G,K) be a pair consisting of a unimodular locally compact topological group G and a closed subgroup K of G. Then the space of bi-K-invariant continuous functions of compact support
C[K\G/K]
can be endowed with a structure of an associative algebra under the operation of convolution. This algebra is denoted
H(G//K)
and called the Hecke ring of the pair (G,K). If we start with a Gelfand pair then the resulting algebra turns out to be commutative.
Examples
SL(2)
In particular, this holds when
G = SLn(Qp) and K = SLn(Zp)
and the representations of the corresponding commutative Hecke ring were studied by Ian G. Macdonald.
GL(2)
On the other hand, in the case
G = GL2(Q) and K = GL2(Z)
we have the classical Hecke algebra, which is the commutative ring of Hecke operators in the theory of modular forms.
Iwahori
The case leading to the Iwahori–Hecke algebra of a finite Weyl group is when G is the finite Chevalley group over a finite field with pk elements, and B is its Borel subgroup. Iwahori showed that the Hecke ring
H(G//B)
is obtained from the generic Hecke algebra Hq of the Weyl group W of G by specializing the indeterminate q of the latter algebra to pk, the cardinality of the finite field. George Lusztig remarked in 1984 (Characters of reductive groups over a finite field, xi, footnote):
Iwahori and Matsumoto (1965) considered the case when G is a group of points of a reductive algebraic group over a non-archimedean local field F, such as Qp, and K is what is now called an Iwahori subgroup of G. The resulting Hecke ring is isomorphic to the Hecke algebra of the affine Weyl group of G, or the affine Hecke algebra, where the indeterminate q has been specialized to the cardinality of the residue field of F.
See also
Group algebra of a locally compact group
References
Representation theory |
https://en.wikipedia.org/wiki/Hecke%20algebra%20of%20a%20pair | In mathematical representation theory, the Hecke algebra of a pair (g,K) is an algebra with an approximate identity, whose approximately unital modules are the same as K-finite representations of the pairs (g,K). Here K is a compact subgroup of a Lie group with Lie algebra g.
Definition
The Hecke algebra of a pair (g,K) is the algebra of K-finite distributions on G with support in K, with the product given by convolution.
References
Representation theory |
https://en.wikipedia.org/wiki/Products%20in%20algebraic%20topology | In algebraic topology, several types of products are defined on homological and cohomological theories.
The cross product
The cap product
The slant product
The cup product
This product can be understood as induced by the exterior product of differential forms in de Rham cohomology. It makes the singular cohomology of a connected manifold into a unitary supercommutative ring.
See also
Singular homology
Differential graded algebra: the algebraic structure arising on the cochain level for the cup product
Poincaré duality: swaps some of these
Intersection theory: for a similar theory in algebraic geometry
Algebraic topology
Homology theory
Operations on structures
References
Hatcher, A., Algebraic Topology, Cambridge University Press (2002) , especially chapter 3. |
https://en.wikipedia.org/wiki/Necklace%20ring | In mathematics, the necklace ring is a ring introduced by to elucidate the multiplicative properties of necklace polynomials.
Definition
If A is a commutative ring then the necklace ring over A consists of all infinite sequences of elements of A. Addition in the necklace ring is given by pointwise addition of sequences. Multiplication is given by a sort of arithmetic convolution: the product of and has components
where is the least common multiple of and , and is their greatest common divisor.
This ring structure is isomorphic to the multiplication of formal power series written in "necklace coordinates": that is, identifying an integer sequence with the power series .
See also
Witt vector
References
Ring theory |
https://en.wikipedia.org/wiki/Binomial%20ring | In mathematics, a binomial ring is a commutative ring whose additive group is torsion-free and contains all binomial coefficients
for x in the ring and n a positive integer. Binomial rings were introduced by .
showed that binomial rings are essentially the same as λ-rings for which all Adams operations are the identity.
References
Ring theory |
https://en.wikipedia.org/wiki/Critical%20pair%20%28order%20theory%29 | In order theory, a discipline within mathematics, a critical pair is a pair of elements in a partially ordered set that are incomparable but that could be made comparable without requiring any other changes to the partial order.
Formally, let be a partially ordered set. Then a critical pair is an ordered pair of elements of with the following three properties:
and are incomparable in ,
for every in , if then , and
for every in , if then .
If is a critical pair, then the binary relation obtained from by adding the single relationship is also a partial order. The properties required of critical pairs ensure that, when the relationship is added, the addition does not cause any violations of the transitive property.
A set of linear extensions of is said to reverse a critical pair in if there exists a linear extension in for which occurs earlier than . This property may be used to characterize realizers of finite partial orders: A nonempty set of linear extensions is a realizer if and only if it reverses every critical pair.
References
.
Order theory |
https://en.wikipedia.org/wiki/Tennis%20Masters%20Series%20singles%20records%20and%20statistics | In tennis, the ATP Masters events, currently known as ATP Tour Masters 1000 series, are an annual series of nine top-level tournaments featuring the elite men's players on the ATP Tour since 1990. The Masters tournaments along with the Grand Slam tournaments and the year-end championships make up the most coveted titles on the annual ATP Tour calendar. In addition to the quadrennial Olympics, they are collectively known as the 'Big Titles'.
Twelve tournaments have been held as Masters events so far, nine each year. They have been played on three different surfaces: hard outdoors: Indian Wells, Miami, Canada, Cincinnati and Shanghai; indoors: Stockholm (1991–94), Stuttgart (1998–2001), Madrid (2002–08) and Paris; clay: Hamburg (1990–2008), Monte Carlo, Madrid and Rome; carpet indoors: Stockholm (1990) and Stuttgart (1995–97).
Champions by year
Title leaders
79 champions in 297 events as of the 2023 Shanghai Masters.
Career Golden Masters
The achievement of winning all of the nine active ATP Masters tournaments over the course of a player's career.
Career totals
Active players denoted in bold.
Season records
Season totals
Most years of success
Consecutive records
Spanning consecutive events
Spanning non-consecutive events
Most consecutive years of title success
Tournament records
Tournaments won with no sets dropped
Miscellaneous records
Youngest & oldest
Calendar Masters combinations
Triples
Doubles
Title defence
Note: Currently active tournaments in bold.
Statistics
Seeds statistics
No. 1 vs. No. 2 seeds in final
Most finals contested between two players
Top 4 seeds in semifinals
Top 8 seeds in quarterfinals
15 of Top-16 seeds in R16
Qualifiers in final
No seeds in final
Borna Ćorić is the lowest-ranked (No. 152) Masters champion.
Andrei Pavel is the lowest-ranked (No. 191) Masters finalist.
Match statistics
Age statistics
All countrymen statistics
All countrymen in final
All countrymen in semifinals
Titles won by decade
Titles by country
See also
ATP Tour
ATP Tour Masters 1000
Tennis Masters Series doubles records and statistics
Grand Prix Super Series
WTA Tour
WTA 1000
WTA 1000 Series singles records and statistics
WTA 1000 Series doubles records and statistics
References
External links
ATP Masters tournaments
ATP Masters records and statistics
records and statistics
Masters |
https://en.wikipedia.org/wiki/Flat%20vector%20bundle | In mathematics, a vector bundle is said to be flat if it is endowed with a linear connection with vanishing curvature, i.e. a flat connection.
de Rham cohomology of a flat vector bundle
Let denote a flat vector bundle, and be the covariant derivative associated to the flat connection on E.
Let denote the vector space (in fact a sheaf of modules over ) of differential forms on X with values in E. The covariant derivative defines a degree-1 endomorphism d, the differential of , and the flatness condition is equivalent to the property .
In other words, the graded vector space is a cochain complex. Its cohomology is called the de Rham cohomology of E, or de Rham cohomology with coefficients twisted by the local coefficient system E.
Flat trivializations
A trivialization of a flat vector bundle is said to be flat if the connection form vanishes in this trivialization. An equivalent definition of a flat bundle is the choice of a trivializing atlas with locally constant transition maps.
Examples
Trivial line bundles can have several flat bundle structures. An example is the trivial bundle over with the connection forms 0 and . The parallel vector fields are constant in the first case, and proportional to local determinations of the square root in the second.
The real canonical line bundle of a differential manifold M is a flat line bundle, called the orientation bundle. Its sections are volume forms.
A Riemannian manifold is flat if and only if its Levi-Civita connection gives its tangent vector bundle a flat structure.
See also
Vector-valued differential forms
Local system, the more general notion of a locally constant sheaf.
Orientation character, a characteristic form related to the orientation line bundle, useful to formulate Twisted Poincaré duality
Picard group whose connected component, the Jacobian variety, is the moduli space of algebraic flat line bundles.
Monodromy, or representations of the fundamental group by parallel transport on flat bundles.
Holonomy, the obstruction to flatness.
Vector bundles |
https://en.wikipedia.org/wiki/Partial%20integration | Partial integration may refer to:
Integration by parts, a technique in mathematics;
Partial integration (contract law), a situation that occurs when a contract contains only some of the terms to which the parties agree. |
https://en.wikipedia.org/wiki/Antoninho%20%28footballer%2C%20born%201939%29 | Benedicto Antonio Angeli (February 10, 1939 - June 3, 2021), known as Antoninho, was a former Brazilian football manager.
Managerial statistics
Honours
Fiorentina
Coppa Italia: 1960-61
UEFA Cup Winners' Cup: 1960-61
References
External links
1939 births
2021 deaths
Footballers from São Paulo (state)
Brazilian men's footballers
Brazilian football managers
Serie A players
J1 League managers
Campeonato Brasileiro Série A players
Sociedade Esportiva Palmeiras players
Botafogo Futebol Clube (SP) players
ACF Fiorentina players
América Futebol Clube (SP) players
Comercial Futebol Clube (Ribeirão Preto) players
Sertãozinho Futebol Clube players
América Futebol Clube (SP) managers
Uberaba Sport Club managers
Comercial Futebol Clube (Ribeirão Preto) managers
Associação Atlética Francana managers
Sertãozinho Futebol Clube managers
Kashiwa Reysol managers
Botafogo Futebol Clube (SP) managers
Shimizu S-Pulse managers
Men's association football forwards |
https://en.wikipedia.org/wiki/Lu%C3%ADs%20dos%20Reis | Luís dos Reis Goncalves (born 1 February 1962) is a Brazilian football manager, currently in charge of Samambaia.
Managerial statistics
References
External links
Luis dos Reis Goncalves - profile at Lamontville Golden Arrows Football Club
1962 births
Living people
Brazilian football managers
Campeonato Brasileiro Série C managers
Botafogo Futebol Clube (SP) managers
Associação Atlética Caldense managers
Clube Náutico Marcílio Dias managers
Esporte Clube Primavera managers
São José Esporte Clube managers
Camboriú Futebol Clube managers
Associação Atlética Portuguesa (Santos) managers
Associação Atlética Internacional (Limeira) managers
Marília Atlético Clube managers
Sociedade Esportiva Matonense managers
Rio Claro Futebol Clube managers
Associação Esportiva Velo Clube Rioclarense managers
Sociedade Imperatriz de Desportos managers
J2 League managers
Ventforet Kofu managers
Brazilian expatriate football managers
Brazilian expatriate sportspeople in Japan
Brazilian expatriate sportspeople in South Africa
Expatriate football managers in Japan
Brasília Futebol Clube managers |
https://en.wikipedia.org/wiki/Valmir%20Louruz | Valmir Louruz (Porto Alegre, March 13, 1944 – April 29, 2015) was a Brazilian football manager.
Managerial statistics
Honors
Player
Internacional
Campeonato Gaúcho: 1969, 1970, 1971
Manager
CSA
Campeonato Alagoano: 1981
Vitória
Campeonato Baiano: 1989
Júbilo Iwata
J. League Cup: 1998
Juventude
Copa do Brasil: 1999
References
External links
1944 births
2015 deaths
Footballers from Porto Alegre
Brazilian men's footballers
Men's association football defenders
Esporte Clube Pelotas players
Sociedade Esportiva Palmeiras players
Sport Club Internacional players
Centro Sportivo Alagoano players
Campeonato Brasileiro Série A players
Brazilian football managers
Esporte Clube Juventude managers
Esporte Clube Pelotas managers
Centro Sportivo Alagoano managers
Grêmio Esportivo Brasil managers
Londrina Esporte Clube managers
Tuna Luso Brasileira managers
Esporte Clube Vitória managers
Clube Náutico Capibaribe managers
Santa Cruz Futebol Clube managers
Paysandu Sport Club managers
Júbilo Iwata managers
Sport Club Internacional managers
Figueirense FC managers
Vila Nova Futebol Clube managers
São José Esporte Clube managers
Al-Ahli Saudi FC managers
Duque de Caxias Futebol Clube managers
Clube de Regatas Brasil managers
Campeonato Brasileiro Série A managers
Campeonato Brasileiro Série B managers
J1 League managers
Saudi Pro League managers
Brazilian expatriate football managers
Brazilian expatriate sportspeople in Japan
Brazilian expatriate sportspeople in Saudi Arabia
Brazilian expatriate sportspeople in Kuwait
Expatriate football managers in Japan
Expatriate football managers in Kuwait
Expatriate football managers in Saudi Arabia |
https://en.wikipedia.org/wiki/Johann%20Georg%20Liebknecht | Johann Georg Liebknecht (23 April 1679 in Wasungen, Thuringia – 17 September 1749 in Giessen) was a German theologian and scientist. He was professor of mathematics and theology at the Ludoviciana (University) in Giessen, Germany.
Biography
He was born the son of Michael Liebknecht, schoolmaster, of Wasungen, and his wife, Margarethe Turckin and was educated in the Gymnasium at Schleusingen and at Jena. He was awarded MA (1702), BD (1717) and DD (1719)
Liebknecht was offered a position, on the recommendation of Gottfried Wilhelm Leibniz, at the small state university in Giessen; he was versatile and could teach several subjects competently. He was both a respected evangelical theologian and a leading mathematician. Other focal points of his work lay in the application of mathematics in the military (artillery, fortresses), geology (mineral deposits), archeology (excavations of grave mounds near Giessen), fossils and astronomy. Like other Protestant theologians he avoided, even after 200 years after Copernicus, supporting the heliocentric world view. He was in contact with famous scientists such as Leibniz. From 1707 to 1737 he was Professor of Mathematics at Giessen and from 1721 until his death also Professor of Theology.
From 1715 he was a member of the Leopoldina (Deutsche Akademie der Naturforscher Leopoldina) and from 1716 a member of the Royal Prussian Society of Sciences. In 1728 he was elected a Fellow of the Royal Society.
Sidus Ludoviciana
Liebknecht was a keen astronomer. He made some of the rare aurora observations during the period 1711–1721. For another observations, however, he gained no credit. On 2 December 1722, he observed a faint star in the telescope and thought he observed relative motion (proper motion) to the neighbouring stars of the Big Dipper, Mizar and Alcor. He called the supposed new planet Sidus Ludoviciana. Liebknecht was unaware, however, that this star had already been observed in the same position in 1616 by Benedetto Castelli and could not therefore be a planet.
Family
He married twice; firstly Catharine Elisabeth, daughter of Nikolaus Caspar Elwerten, physician, of Bensheim and secondly Regina Sophie, daughter of Johann Just Hoffmann, physician, of Isenburg. He had 21 children from his two marriages.
References
1679 births
1749 deaths
People from Schmalkalden-Meiningen
18th-century German Protestant theologians
18th-century German astronomers
Fellows of the Royal Society
German male non-fiction writers
18th-century German male writers
Members of the German National Academy of Sciences Leopoldina
Members of the Prussian Academy of Sciences |
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