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https://en.wikipedia.org/wiki/Attributional%20calculus | Attributional calculus is a logic and representation system defined by Ryszard S. Michalski. It combines elements of predicate logic, propositional calculus, and multi-valued logic. Attributional calculus provides a formal language for natural induction, which is an inductive learning process whose outcomes are in human-readable forms.
References
Michalski, R.S., "ATTRIBUTIONAL CALCULUS: A Logic and Representation Language for Natural Induction," Reports of the Machine Learning and Inference Laboratory, MLI 04-2, George Mason University, Fairfax, VA, April, 2004.
Artificial intelligence
Systems of formal logic |
https://en.wikipedia.org/wiki/Ruskovce%2C%20Sobrance%20District | Ruskovce () is a village and municipality in the Sobrance District in the Košice Region of east Slovakia.
References
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Sobrance District |
https://en.wikipedia.org/wiki/Symplectization | In mathematics, the symplectization of a contact manifold is a symplectic manifold which naturally corresponds to it.
Definition
Let be a contact manifold, and let . Consider the set
of all nonzero 1-forms at , which have the contact plane as their kernel. The union
is a symplectic submanifold of the cotangent bundle of , and thus possesses a natural symplectic structure.
The projection supplies the symplectization with the structure of a principal bundle over with structure group .
The coorientable case
When the contact structure is cooriented by means of a contact form , there is another version of symplectization, in which only forms giving the same coorientation to as are considered:
Note that is coorientable if and only if the bundle is trivial. Any section of this bundle is a coorienting form for the contact structure.
Differential topology
Structures on manifolds
Symplectic geometry |
https://en.wikipedia.org/wiki/1996%E2%80%9397%20Mexican%20Primera%20Divisi%C3%B3n%20season | The following are statistics of Mexico's Primera División for the 1996–97 season.
Overview
Teams
Torneo Invierno 1996
Primera División de México (Mexican First Division) Invierno 1996 is a Mexican football tournament - one of two short tournaments that take up the entire year to determine the champion(s) of Mexican football. It began on Friday, August 9, 1996, and ran until November 24, when the regular season ended. In the final Santos defeated Necaxa and became champions for the 1st time.
Final standings (groups)
League table
Results
Top goalscorers
Players sorted first by goals scored, then by last name. Only regular season goals listed.
Source: MedioTiempo
Playoffs
Repechage
Toros Neza won 4–2 on aggregate.
Atlas won 6–3 on aggregate.
Bracket
Quarterfinals
Toros Neza won 9–2 on aggregate.
Santos Laguna won 4–2 on aggregate.
Necaxa won 3–2 on aggregate.
Puebla won 2–1 on aggregate.
Semifinals
Santos Laguna won 5–2 on aggregate.
Necaxa won 7–3 on aggregate.
Finals
First leg
Second leg
Santos Laguna won 4–3 on aggregate.
Torneo Verano 1997
Primera División de México (Mexican First Division) Verano 1997 is a Mexican football tournament - one of two short tournaments that take up the entire year to determine the champion(s) of Mexican football. It began on Saturday, January 11, 1997, and ran until May 4, when the regular season ended. In the final Guadalajara defeated Toros Neza and became champions for the 10th time.
Final standings (groups)
League table
Results
Top goalscorers
Players sorted first by goals scored, then by last name. Only regular season goals listed.
Source: MedioTiempo
Playoffs
Repechage
Morelia won 4–2 on aggregate.
Bracket
Quarterfinals
Guadalajara won 6–1 on aggregate.
Atlético Morelia won 4–1 on aggregate.
Toros Neza won 4–3 on aggregate.
3–3 on aggregate. Necaxa advanced for being the higher seeded team.
Semifinals
1–1 on aggregate. Guadalajara advanced for being the higher seeded team.
Toros Neza won 4–3 on aggregate.
Finals
First leg
Second leg
Guadalajara won 7–2 on aggregate.
Relegation table
References
External links
Mediotiempo.com (where information was obtained)
Liga MX seasons
Mex
1996–97 in Mexican football |
https://en.wikipedia.org/wiki/Giuseppe%20Bianchi%20%28astronomer%29 | Giuseppe Bianchi (13 October 1791 in Modena – 25 December 1866) was an Italian astronomer.
After studying mathematics, physics, and astronomy at Padua, Bianchi taught astronomy at the University of Modena starting in 1819, and under his direction the Observatory of Modena was built in 1826. In 1859 he was transferred to the marchese Montecuccoli's private observatory at Modena for political reasons.
He took part in the Commissione dei Pesi e Misure (Commission for Weights and Measures) and numerous scientific academies, and was secretary of the "Società italiana delle Scienze" (Italian Society of Science).
In 1834 he published "Acts of the Reale Osservatorio of Modena", describing the Observatory and all the work and observations he had carried out up to that year.
1791 births
1866 deaths
19th-century Italian astronomers |
https://en.wikipedia.org/wiki/Bramble%E2%80%93Hilbert%20lemma | In mathematics, particularly numerical analysis, the Bramble–Hilbert lemma, named after James H. Bramble and Stephen Hilbert, bounds the error of an approximation of a function by a polynomial of order at most in terms of derivatives of of order . Both the error of the approximation and the derivatives of are measured by norms on a bounded domain in . This is similar to classical numerical analysis, where, for example, the error of linear interpolation can be bounded using the second derivative of . However, the Bramble–Hilbert lemma applies in any number of dimensions, not just one dimension, and the approximation error and the derivatives of are measured by more general norms involving averages, not just the maximum norm.
Additional assumptions on the domain are needed for the Bramble–Hilbert lemma to hold. Essentially, the boundary of the domain must be "reasonable". For example, domains that have a spike or a slit with zero angle at the tip are excluded. Lipschitz domains are reasonable enough, which includes convex domains and domains with continuously differentiable boundary.
The main use of the Bramble–Hilbert lemma is to prove bounds on the error of interpolation of function by an operator that preserves polynomials of order up to , in terms of the derivatives of of order . This is an essential step in error estimates for the finite element method. The Bramble–Hilbert lemma is applied there on the domain consisting of one element (or, in some superconvergence results, a small number of elements).
The one-dimensional case
Before stating the lemma in full generality, it is useful to look at some simple special cases. In one dimension and for a function that has derivatives on interval , the lemma reduces to
where is the space of all polynomials of degree at most and indicates
the th derivative of a function .
In the case when , , , and is twice differentiable, this means that there exists a polynomial of degree one such that for all ,
This inequality also follows from the well-known error estimate for linear interpolation by choosing as the linear interpolant of .
Statement of the lemma
Suppose is a bounded domain in , , with boundary and diameter . is the Sobolev space of all function on with weak derivatives of order up to in . Here, is a multiindex, and denotes the derivative times with respect to , times with respect to , and so on. The Sobolev seminorm on consists of the norms of the highest order derivatives,
and
is the space of all polynomials of order up to on . Note that for all and , so has the same value for any .
Lemma (Bramble and Hilbert) Under additional assumptions on the domain , specified below, there exists a constant independent of and such that for any there exists a polynomial such that for all
The original result
The lemma was proved by Bramble and Hilbert under the assumption that satisfies the strong cone property; that is, there exists a finite open coveri |
https://en.wikipedia.org/wiki/Algebraist | Algebraist may refer to:
a specialist in algebra.
The Algebraist, a science fiction novel by Iain M. Banks. |
https://en.wikipedia.org/wiki/Bob%20Hale%20%28philosopher%29 | Bob Hale, FRSE (1945 – 12 December 2017) was a British philosopher, known for his contributions to the development of the neo-Fregean (neo-logicist) philosophy of mathematics in collaboration with Crispin Wright, and for his works in modality and philosophy of language.
Career
Hale obtained a BPhil in Philosophy in 1967 from Linacre College, University of Oxford. From 2006 until his death, he was a professor of philosophy in the department of philosophy at the University of Sheffield. Prior to that, he taught in the University of Glasgow, the University of St. Andrews and the University of Lancaster.
Hale produced the first published neo-Fregean construction of the real numbers. In his book (Necessary Beings), he argues for an essentialist theory of necessity and possibility.
Notable positions
British Academy Research Reader (1997–9)
Fellow of the Royal Society of Edinburgh (from 2000)
President of the Aristotelian Society (2002–3)
Leverhulme Senior Research Fellow (2009–11)
Selected works
(1987) Abstract Objects. Oxford: Blackwell Publishing.
(1997) Co-editor with Crispin Wright. The Blackwell Companion to the Philosophy of Language. Oxford: Blackwell Publishing.
(2001) With Crispin Wright. The Reason's Proper Study: Essays towards a Neo-Fregean Philosophy of Mathematics. Oxford: Oxford University Press.
(2013) "Necessary Beings – An Essay on Ontology, Modality, and the Relations Between Them" Oxford: Oxford University Press.
References
External links
Interview at 3:AM Magazine
1945 births
2017 deaths
20th-century British philosophers
21st-century British philosophers
Philosophers of language
Philosophers of mathematics
Academics of the University of Sheffield
Fellows of the Royal Society of Edinburgh
Presidents of the Aristotelian Society
Alumni of the University of Oxford |
https://en.wikipedia.org/wiki/External%20%28mathematics%29 | The term external is useful for describing certain algebraic structures. The term comes from the concept of an external binary operation which is a binary operation that draws from some external set. To be more specific, a left external binary operation on S over R is a function and a right external binary operation on S over R is a function where S is the set the operation is defined on, and R is the external set (the set the operation is defined over).
Generalizations
The external concept is a generalization rather than a specialization, and as such, it is different from many terms in mathematics. A similar but opposite concept is that of an internal binary function from R to S, defined as a function . Internal binary functions are like binary functions, but are a form of specialization, so they only accept a subset of the domains of binary functions. Here we list these terms with the function signatures they imply, along with some examples:
(binary function)
Example: exponentiation ( as in )
Examples: matrix multiplication, the tensor product, and the Cartesian product
(internal binary function)
Example: internal binary relations ()
Example: set membership ( where is the category of sets)
Examples: the dot product, the inner product, and metrics
(external binary operation)
Examples: dynamical system flows, group actions, projection maps, and scalar multiplication
(binary operation)
Examples: addition, multiplication, permutations, and the cross product
External monoids
Since monoids are defined in terms of binary operations, we can define an external monoid in terms of external binary operations. For the sake of simplicity, unless otherwise specified, a left external binary operation is implied. Using the term external, we can make the generalizations:
An external magma over R is a set S with an external binary operation. This satisfies for all (external closure).
An external semigroup over is an external magma that satisfies for all (externally associative).
An external monoid over is an external semigroup in which there exists such that for all (has external identity element).
Modules as external rings
Much of the machinery of modules and vector spaces are fairly straightforward, or discussed above. The only thing not covered yet is their distribution axioms. The external ring multiplication is externally distributive in over the ring iff:
for all and:
for all
Using these terminology we can make the following local generalizations:
An external semiring over the semiring is a commutative monoid and an external monoid where is externally distributive in over the semiring .
An external ring over the ring is an abelian group and an external monoid where is externally distributive in over the ring .
Other examples
Now that we have all the terminology we need, we can make simple connections between various structures:
Complex exponentiation forms an external monoid over |
https://en.wikipedia.org/wiki/Japanese%20theorem%20for%20cyclic%20quadrilaterals | In geometry, the Japanese theorem states that the centers of the incircles of certain triangles inside a cyclic quadrilateral are vertices of a rectangle.
Triangulating an arbitrary cyclic quadrilateral by its diagonals yields four overlapping triangles (each diagonal creates two triangles). The centers of the incircles of those triangles form a rectangle.
Specifically, let be an arbitrary cyclic quadrilateral and let , , , be the incenters of the triangles , , , . Then the quadrilateral formed by , , , is a rectangle.
Note that this theorem is easily extended to prove the Japanese theorem for cyclic polygons. To prove the quadrilateral case, simply construct the parallelogram tangent to the corners of the constructed rectangle, with sides parallel to the diagonals of the quadrilateral. The construction shows that the parallelogram is a rhombus, which is equivalent to showing that the sums of the radii of the incircles tangent to each diagonal are equal.
The quadrilateral case immediately proves the general case by induction on the set of triangulating partitions of a general polygon.
See also
Carnot's theorem
Sangaku
Japanese mathematics
References
Mangho Ahuja, Wataru Uegaki, Kayo Matsushita: In Search of the Japanese Theorem (postscript file)
Theorem at Cut-the-Knot
Wataru Uegaki: "" (On the Origin and History of the Japanese Theorem). Departmental Bulletin Paper, Mie University Scholarly E-Collections, 2001-03-01
Wilfred Reyes: An Application of Thebault’s Theorem. Forum Geometricorum, Volume 2, 2002, pp. 183–185
External links
Japanese theorem, interactive proof with animation
Euclidean plane geometry
Japanese mathematics
Theorems about quadrilaterals and circles |
https://en.wikipedia.org/wiki/Kravany%2C%20Trebi%C5%A1ov%20District | Kravany () is a village and municipality in the Trebišov District in the Košice Region of eastern Slovakia.
External links
https://web.archive.org/web/20100202015957/http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Trebišov District
Zemplín (region) |
https://en.wikipedia.org/wiki/Rhode%20Island%20Math%20League | The Rhode Island Mathematics League (RIML) competition consists of four meets spanning the entire year. It culminates at the state championship held at Bishop Hendricken High School. Top schools from the state championship are invited to the New England Association of Math Leagues (NEAML) championship.
Format
Each meet consists of five rounds and a team round. Each team consists of five students, and each school may have as many as six teams. However, each team may have a maximum of two seniors and four sophomores/juniors. At least one sophomore or freshman must be on each team (or the team may compete with an empty slot). Three students from each team participate in a round. Therefore, each student participates in three rounds and the team round. The first five rounds consist of three questions each. Beginning in 2007, one of the five rounds is designated as "calculator-free", in 2008, this number was increased to two, and in 2018, calculators were banned from all meets. The first question in each round is worth one point, the second two points, and the third three points. Each student works on the questions independently in the ten minutes allotted. All answers must be presented in simplified and rationalized form unless specified otherwise. After the completion of the first five rounds, there is a team round. All five players from each team collaborate on five questions worth two points each. The maximum score for one team is 100 points, and the maximum score for one student is 18 points.
Rounds
At the first meet the rounds are as follows:
Round 1: Arithmetic, Number Theory, and Matrices
Round 2: Algebra I
Round 3: Geometry
Round 4: Algebra II
Round 5: Miscellaneous Math
Team Round
At the second meet the rounds are as follows:
Round 1: Arithmetic, Number Theory, and Matrices
Round 2: Algebra I
Round 3: Geometry
Round 4: Algebra II
Round 5: Miscellaneous Math
Team Round
At the third meet the rounds are as follows:
Round 1: Statistics and Probability
Round 2: Algebra I
Round 3: Geometry
Round 4: Algebra II
Round 5: Miscellaneous Math
Team Round
At the fourth meet the rounds are as follows:
Round 1: Statistics and Probability
Round 2: Algebra I
Round 3: Geometry
Round 4: Algebra II
Round 5: Miscellaneous Math
Team Round
At the playoff meet the rounds are as follows:
Round 1: Arithmetic, Number Theory, and Matrices
Round 2: Statistics and Probability
Round 3: Algebra I
Round 4: Geometry
Round 5: Algebra II
Round 6: Miscellaneous Math
At the end of the six rounds, a relay round will occur, where four people from a team of six will participate. In this round, four questions are given, and each student after the first must use the answer given to them from the previous question to answer the next one.
At the end of the relay round, there will be a team round, where four people from the team will compete to answer five questions together.
Miscellaneous Math
As of the 2019-2020 year, certai |
https://en.wikipedia.org/wiki/Studentized%20range | In statistics, the studentized range, denoted q, is the difference between the largest and smallest data in a sample normalized by the sample standard deviation.
It is named after William Sealy Gosset (who wrote under the pseudonym "Student"), and was introduced by him in 1927.
The concept was later discussed by Newman (1939), Keuls (1952), and John Tukey in some unpublished notes.
Its statistical distribution is the studentized range distribution, which is used for multiple comparison procedures, such as the single step procedure Tukey's range test, the Newman–Keuls method, and the Duncan's step down procedure, and establishing confidence intervals that are still valid after data snooping has occurred.
Description
The value of the studentized range, most often represented by the variable q, can be defined based on a random sample x1, ..., xn from the N(0, 1) distribution of numbers, and another random variable s that is independent of all the xi, and νs2 has a χ2 distribution with ν degrees of freedom. Then
has the Studentized range distribution for n groups and ν degrees of freedom. In applications, the xi are typically the means of samples each of size m, s2 is the pooled variance, and the degrees of freedom are ν = n(m − 1).
The critical value of q is based on three factors:
α (the probability of rejecting a true null hypothesis)
n (the number of observations or groups)
ν (the degrees of freedom used to estimate the sample variance)
Distribution
If X1, ..., Xn are independent identically distributed random variables that are normally distributed, the probability distribution of their studentized range is what is usually called the studentized range distribution. Note that the definition of q does not depend on the expected value or the standard deviation of the distribution from which the sample is drawn, and therefore its probability distribution is the same regardless of those parameters.
Studentization
Generally, the term studentized means that the variable's scale was adjusted by dividing by an estimate of a population standard deviation (see also studentized residual). The fact that the standard deviation is a sample standard deviation rather than the population standard deviation, and thus something that differs from one random sample to the next, is essential to the definition and the distribution of the Studentized data. The variability in the value of the sample standard deviation contributes additional uncertainty into the values calculated. This complicates the problem of finding the probability distribution of any statistic that is studentized.
See also
Studentized range distribution
Tukey's range test
References
Further reading
Pearson, E.S.; Hartley, H.O. (1970) Biometrika Tables for Statisticians, Volume 1, 3rd Edition, Cambridge University Press.
John Neter, Michael H. Kutner, Christopher J. Nachtsheim, William Wasserman (1996) Applied Linear Statistical Models, fourth edition, McGraw-Hill, page 726.
Joh |
https://en.wikipedia.org/wiki/Lazard%27s%20universal%20ring | In mathematics, Lazard's universal ring is a ring introduced by Michel Lazard in over which the universal commutative one-dimensional formal group law is defined.
There is a universal commutative one-dimensional formal group law over a universal commutative ring defined as follows. We let
be
for indeterminates , and we define the universal ring R to be the commutative ring generated by the elements , with the relations that are forced by the associativity and commutativity laws for formal group laws. More or less by definition, the ring R has the following universal property:
For every commutative ring S, one-dimensional formal group laws over S correspond to ring homomorphisms from R to S.
The commutative ring R constructed above is known as Lazard's universal ring. At first sight it seems to be incredibly complicated: the relations between its generators are very messy. However Lazard proved that it has a very simple structure: it is just a polynomial ring (over the integers) on generators of degree 1, 2, 3, ..., where has degree . proved that the coefficient ring of complex cobordism is naturally isomorphic as a graded ring to Lazard's universal ring. Hence, topologists commonly regrade the Lazard ring so that has degree , because the coefficient ring of complex cobordism is evenly graded.
References
Algebraic topology
Algebraic groups
Algebraic number theory |
https://en.wikipedia.org/wiki/John%20Farrar%20%28disambiguation%29 | John Farrar (born 1945) is an Australian-born musician.
John Farrar may also refer to:
John Farrar (scientist) (1779–1853), professor of mathematics and natural philosophy at Harvard
John Farrar (minister) (1802–1884), British Methodist minister
John C. Farrar (1896–1974), American editor, writer and publisher
John Percy Farrar (1857–1929), English soldier and mountaineer
John Nutting Farrar (1839–1913), American dentist |
https://en.wikipedia.org/wiki/Bicone | In geometry, a bicone or dicone (from , and Greek: di-, both meaning "two") is the three-dimensional surface of revolution of a rhombus around one of its axes of symmetry. Equivalently, a bicone is the surface created by joining two congruent, right, circular cones at their bases.
A bicone has circular symmetry and orthogonal bilateral symmetry.
Geometry
For a circular bicone with radius R and height center-to-top H, the formula for volume becomes
For a right circular cone, the surface area is
where is the slant height.
See also
Sphericon
Biconical antenna
References
External links
Elementary geometry
Surfaces |
https://en.wikipedia.org/wiki/Current%20Index%20to%20Statistics | The Current Index to Statistics is an online database published by the Institute of Mathematical Statistics and the American Statistical Association that contains bibliographic data of articles in statistics, probability, and related fields. It was shut down at the end of 2019.
See also
Web of Science
IEEE Xplore
References
External links
Official website
American Statistical Association
Institute of Mathematical Statistics
Bibliographic databases and indexes
Online databases |
https://en.wikipedia.org/wiki/Jan%20Dibbets | Jan Dibbets (born 9 May 1941, in Weert) is an Amsterdam-based Dutch conceptual artist. His work is influenced by mathematics and works mainly with photography.
Life and career
In the late 1950s and early 1960s, he started as an art teacher at the Tilburg Academy and studied painting with Jan Gregoor in Eindhoven. He had his first solo exhibition in 1965 at Amsterdam's Galerie 845 and subsequently abandoned painting in 1967. At that same period, he visited London and met Richard Long and other artists working with land art. He returned to Amsterdam, incorporated land-art based theories into his work and began to use photography as a "dialogue between nature and cool geometrical design by rotating the camera on its axis" with his "perspective corrections". His work in the Dutch pavilion at the Venice Biennale in 1972 gave him an international reputation.
In 1994, he was commissioned by the Arago Association to create a memorial to the French astronomer François Arago, known as Hommage à Arago. Dibbets set 135 bronze medallions into the ground along the Paris Meridian between the north and south limits of Paris.
Dibbets's works are included in museums around the world, including the Stedelijk Museum, Amsterdam, Solomon R. Guggenheim Museum, New York, De Pont Museum of Contemporary Art in Tilburg, and the Van Abbemuseum in Eindhoven.
Further reading
Jeffery Kastner and Brian Wallis (editors): Land and Environmental Art. Phaidon Press, 1998.
Rudi Fuchs and Gloria Moure: Jan Dibbets, Interior Light. New York: Rizzoli, 1991.
Books
Robin Redbreast's Territory/Sculpture 1969. Zédélé éditions, Reprint collection, 2014 [1970].
With foreword by David Cleaton-Roberts, Director of the Alan Cristea Gallery.
Articles
Rutger Pontzen, Like before, the sea levels everything, Observatoire du Land Art, 4 October 2009. Read online through Latitudes
References
External links
UBU Web
Zédélé éditions
1941 births
Living people
Dutch conceptual artists
20th-century Dutch photographers
People from Weert
Dutch contemporary artists
21st-century Dutch photographers |
https://en.wikipedia.org/wiki/Zariski%27s%20main%20theorem | In algebraic geometry, Zariski's main theorem, proved by , is a statement about the structure of birational morphisms stating roughly that there is only one branch at any normal point of a variety. It is the special case of Zariski's connectedness theorem when the two varieties are birational.
Zariski's main theorem can be stated in several ways which at first sight seem to be quite different, but are in fact deeply related. Some of the variations that have been called Zariski's main theorem are as follows:
The total transform of a normal fundamental point of a birational map has positive dimension. This is essentially Zariski's original form of his main theorem.
A birational morphism with finite fibers to a normal variety is an isomorphism to an open subset.
The total transform of a normal point under a proper birational morphism is connected.
A closely related theorem of Grothendieck describes the structure of quasi-finite morphisms of schemes, which implies Zariski's original main theorem.
Several results in commutative algebra that imply the geometric form of Zariski's main theorem.
A normal local ring is unibranch, which is a variation of the statement that the transform of a normal point is connected.
The local ring of a normal point of a variety is analytically normal. This is a strong form of the statement that it is unibranch.
The name "Zariski's main theorem" comes from the fact that Zariski labelled it as the "MAIN THEOREM" in .
Zariski's main theorem for birational morphisms
Let f be a birational mapping of algebraic varieties V and W. Recall that f is defined by a closed subvariety (a "graph" of f) such that the projection on the first factor induces an isomorphism between an open and , and such that is an isomorphism on U too. The complement of U in V is called a fundamental variety or indeterminacy locus, and the image of a subset of V under is called a total transform of it.
The original statement of the theorem in reads:
MAIN THEOREM: If W is an irreducible fundamental variety on V of a birational correspondence T between V and V′ and if T has no fundamental elements on V′ then — under the assumption that V is locally normal at W — each irreducible component of the transform T[W] is of higher dimension than W.
Here T is essentially a morphism from V′ to V that is birational, W is a subvariety of the set where the inverse of T is not defined whose local ring is normal, and the transform T[W] means the inverse image of W under the morphism from V′ to V.
Here are some variants of this theorem stated using more recent terminology. calls the following connectedness statement "Zariski's Main theorem":
If f:X→Y is a birational projective morphism between noetherian integral schemes, then the inverse image of every normal point of Y is connected.
The following consequence of it (Theorem V.5.2,loc.cit.) also goes under this name:
If f:X→Y is a birational transformation of projective varieties with Y normal, then the to |
https://en.wikipedia.org/wiki/Separating%20set | In mathematics, a set of functions with domain is called a and is said to (or just ) if for any two distinct elements and of there exists a function such that
Separating sets can be used to formulate a version of the Stone–Weierstrass theorem for real-valued functions on a compact Hausdorff space with the topology of uniform convergence. It states that any subalgebra of this space of functions is dense if and only if it separates points. This is the version of the theorem originally proved by Marshall H. Stone.
Examples
The singleton set consisting of the identity function on separates the points of
If is a T1 normal topological space, then Urysohn's lemma states that the set of continuous functions on with real (or complex) values separates points on
If is a locally convex Hausdorff topological vector space over or then the Hahn–Banach separation theorem implies that continuous linear functionals on separate points.
See also
References
Set theory |
https://en.wikipedia.org/wiki/Connectedness%20theorem | In mathematics, the connectedness theorem may be one of
Deligne's connectedness theorem
Fulton–Hansen connectedness theorem
Grothendieck's connectedness theorem
Hartshorne's connectedness theorem
Zariski's connectedness theorem, a generalization of Zariski's main theorem |
https://en.wikipedia.org/wiki/%CE%95-quadratic%20form | In mathematics, specifically the theory of quadratic forms, an ε-quadratic form is a generalization of quadratic forms to skew-symmetric settings and to *-rings; , accordingly for symmetric or skew-symmetric. They are also called -quadratic forms, particularly in the context of surgery theory.
There is the related notion of ε-symmetric forms, which generalizes symmetric forms, skew-symmetric forms (= symplectic forms), Hermitian forms, and skew-Hermitian forms. More briefly, one may refer to quadratic, skew-quadratic, symmetric, and skew-symmetric forms, where "skew" means (−) and the * (involution) is implied.
The theory is 2-local: away from 2, ε-quadratic forms are equivalent to ε-symmetric forms: half the symmetrization map (below) gives an explicit isomorphism.
Definition
ε-symmetric forms and ε-quadratic forms are defined as follows.
Given a module M over a *-ring R, let B(M) be the space of bilinear forms on M, and let be the "conjugate transpose" involution . Since multiplication by −1 is also an involution and commutes with linear maps, −T is also an involution. Thus we can write and εT is an involution, either T or −T (ε can be more general than ±1; see below). Define the ε-symmetric forms as the invariants of εT, and the ε-quadratic forms are the coinvariants.
As an exact sequence,
As kernel and cokernel,
The notation Qε(M), Qε(M) follows the standard notation MG, MG for the invariants and coinvariants for a group action, here of the order 2 group (an involution).
Composition of the inclusion and quotient maps (but not ) as yields a map Qε(M) → Qε(M): every ε-symmetric form determines an ε-quadratic form.
Symmetrization
Conversely, one can define a reverse homomorphism , called the symmetrization map (since it yields a symmetric form) by taking any lift of a quadratic form and multiplying it by . This is a symmetric form because , so it is in the kernel. More precisely, . The map is well-defined by the same equation: choosing a different lift corresponds to adding a multiple of , but this vanishes after multiplying by . Thus every ε-quadratic form determines an ε-symmetric form.
Composing these two maps either way: or yields multiplication by 2, and thus these maps are bijective if 2 is invertible in R, with the inverse given by multiplication with 1/2.
An ε-quadratic form is called non-degenerate if the associated ε-symmetric form is non-degenerate.
Generalization from *
If the * is trivial, then , and "away from 2" means that 2 is invertible: .
More generally, one can take for any element such that . always satisfy this, but so does any element of norm 1, such as complex numbers of unit norm.
Similarly, in the presence of a non-trivial *, ε-symmetric forms are equivalent to ε-quadratic forms if there is an element such that . If * is trivial, this is equivalent to or , while if * is non-trivial there can be multiple possible λ; for example, over the complex numbers any number with real part 1/2 is such a λ.
|
https://en.wikipedia.org/wiki/Johan%20Frederik%20Steffensen | Johan Frederik Steffensen (28 February 1873, in Copenhagen – 20 December 1961) was a Danish mathematician, statistician, and actuary who did research in the fields of calculus of finite differences and interpolation. He was professor of actuarial science at the University of Copenhagen from 1923 to 1943. Steffensen's inequality and Steffensen's method (an iterative numerical method) are named after him. He was an Invited Speaker at the 1912 International Congress of Mathematicians (ICM) in Cambridge, England and at the 1924 ICM in Toronto.
Publications
Obituary
References
External links
Johan Frederik Steffensen papirer (Danish)
1873 births
1961 deaths
Danish mathematicians
Danish statisticians
Danish actuaries |
https://en.wikipedia.org/wiki/Zariski%27s%20connectedness%20theorem | In algebraic geometry, Zariski's connectedness theorem (due to Oscar Zariski) says that under certain conditions the fibers of a morphism of varieties are connected. It is an extension of Zariski's main theorem to the case when the morphism of varieties need not be birational.
Zariski's connectedness theorem gives a rigorous version of the "principle of degeneration" introduced by Federigo Enriques, which says roughly that a limit of absolutely irreducible cycles is absolutely connected.
Statement
Suppose that f is a proper surjective morphism of varieties from X to Y such that the function field of Y is separably closed in that of X. Then Zariski's connectedness theorem says that the inverse image of any normal point of Y is connected. An alternative version says that if f is proper and f* OX = OY, then f is surjective and the inverse image of any point of Y is connected.
References
Theorems in algebraic geometry |
https://en.wikipedia.org/wiki/Jacobi%E2%80%93Anger%20expansion | In mathematics, the Jacobi–Anger expansion (or Jacobi–Anger identity) is an expansion of exponentials of trigonometric functions in the basis of their harmonics. It is useful in physics (for example, to convert between plane waves and cylindrical waves), and in signal processing (to describe FM signals). This identity is named after the 19th-century mathematicians Carl Jacobi and Carl Theodor Anger.
The most general identity is given by:
where is the -th Bessel function of the first kind and is the imaginary unit,
Substituting by , we also get:
Using the relation valid for integer , the expansion becomes:
Real-valued expressions
The following real-valued variations are often useful as well:
See also
Plane wave expansion
Notes
References
External links
Special functions
Mathematical identities |
https://en.wikipedia.org/wiki/Surif | Surif () is a Palestinian City in the Hebron Governorate located 25 km northwest of the city of Hebron. According to the Palestinian Central Bureau of Statistics census, Surif had a population of 17,287 in 2011. The population is entirely Muslim.
Most of the town's 15,000 dunams is used for agriculture, in particular, olives, wheat and barley. There are seven mosques and four schools located in its vicinity.
Ahmad Lafi is the mayor.
History
In 1838 Surif was noted as a Muslim village, located between Hebron and Gaza, but subjected to the government of Hebron.
In 1863 Victor Guérin found Surif to be a village with 700 inhabitants. He further noted that beside a birket in the rock, a few cisterns and an ancient column shaft which was placed near a small mosque, all of Surif's constructions seemed more or less modern.
An official Ottoman village list from about 1870 showed 87 houses and a population of 265, counting men only.
In 1883, the PEF's Survey of Western Palestine described Surif as "A small village on a low hill, with olives to the south."
In 1896 the population of Surif was estimated to be about 1164 persons.
British Mandate Era
According to the 1922 census of Palestine conducted by the British Mandate authorities, Surif had a population of 1,265 inhabitants, all Muslims, increasing in the 1931 census to 1,640, in 344 inhabited houses.
In the 1945 statistics the population of Surif was 2,190, all Muslims, with a total of 38,876 dunams of land according to an official land and population survey. Of this, 3,493 dunams were plantations and irrigable land, 11.325 for cereals, while 54 dunams were built-up (urban) land.
Jordanian era
In the wake of the 1948 Arab–Israeli War and the 1949 Armistice Agreements, Surif came under Jordanian rule.
The Jordanian census of 1961 found 2,827 inhabitants in Surif.
Post-1967
Since the Six-Day War in 1967, Surif has been under Israeli occupation.
Israel has confiscated approximately 1,213 dunams of land from Surif since 2000, and approximately 1,300 dunums of Surif lands will be behind the Israeli West Bank barrier, when it is finished.
References
Bibliography
External links
Welcome To Surif
Surif, Welcome to Palestine
Survey of Western Palestine, Map 21: IAA, Wikimedia commons
Surif Town (Fact Sheet), Applied Research Institute–Jerusalem, ARIJ
Surif Town Profile, ARIJ
Surif aerial photo, ARIJ
The priorities and needs for development in Surif town based on the community and local authorities’ assessment, ARIJ
Towns in the West Bank
Hebron Governorate
Municipalities of the State of Palestine |
https://en.wikipedia.org/wiki/Grothendieck%27s%20connectedness%20theorem | In mathematics, Grothendieck's connectedness theorem , states that if A is a complete Noetherian local ring whose spectrum is k-connected and f is in the maximal ideal, then Spec(A/fA) is (k − 1)-connected. Here a Noetherian scheme is called k-connected if its dimension is greater than k and the complement of every closed subset of dimension less than k is connected.
It is a local analogue of Bertini's theorem.
See also
Zariski connectedness theorem
Fulton–Hansen connectedness theorem
References
Bibliography
Theorems in algebraic geometry |
https://en.wikipedia.org/wiki/Peter%20Winkler | Peter Mann Winkler is a research mathematician, author of more than 125 research papers in mathematics and patent holder in a broad range of applications, ranging from cryptography to marine navigation. His research areas include discrete mathematics, theory of computation and probability theory.
He is currently a professor of mathematics and computer science at Dartmouth College.
Peter Winkler studied mathematics at Harvard University and later received his PhD in 1975 from Yale University under the supervision of Angus McIntyre. He has also served as an assistant professor at Stanford, full professor and chair at Emory and as a mathematics research director at Bell Labs and Lucent Technologies. He was visiting professor at the Technische Universität Darmstadt.
He has published three books on mathematical puzzles: Mathematical Puzzles: A connoisseur's collection (A K Peters, 2004, ), Mathematical Mind-Benders (A K Peters, 2007, ), and Mathematical Puzzles (A K Peters, 2021, ). And he is widely considered to be a pre eminent scholar in this domain. He was the Visiting Distinguished Chair for Public Dissemination of Mathematics at the National Museum of Mathematics (MoMath), gave topical talks at the Gathering 4 Gardner conferences, and wrote novel papers related to some of these puzzles.
Winkler's book Bridge at the Enigma Club was a runner up for the 2011 Master Point Press Book Of The Year award.
Also in 2011, Winkler received the David P. Robbins Prize of the Mathematical Association of America as coauthor of one of two papers in the American Mathematical Monthly.
Paul Erdős anecdote
According to a story included in Chapter One of "The Man Who Loved Only Numbers / The Story of Paul Erdös and the Search for Mathematical Truth", Paul Erdős attended the bar mitzvah celebration for Peter Winkler's twins, and Winkler's mother-in-law tried to throw Erdős out. [Quote:]"Erdös came to my twins' bar mitzvah, notebook in hand," said Peter Winkler, a colleague of Graham's at AT&T. "He also brought gifts for my children--he loved kids--and behaved himself very well. But my mother-in-law tried to throw him out. She thought he was some guy who wandered in off the street, in a rumpled suit, carrying a pad under his arm. It is entirely possible that he proved a theorem or two during the ceremony."
References
External links
Year of birth missing (living people)
Living people
20th-century American mathematicians
Mathematics popularizers
Harvard College alumni
Yale Graduate School of Arts and Sciences alumni
Dartmouth College faculty
Academic staff of Technische Universität Darmstadt
Fellows of the American Physical Society |
https://en.wikipedia.org/wiki/Carl%20Hierholzer | Carl Hierholzer (2 October 1840 – 13 September 1871) was a German mathematician.
Biography
Hierholzer studied mathematics in Karlsruhe, and he got his Ph.D. from Ruprecht-Karls-Universität Heidelberg in 1865. His Ph.D. advisor was Ludwig Otto Hesse (1811–1874). In 1870 Hierholzer wrote his habilitation about conic sections (title: Ueber Kegelschnitte im Raum) in Karlsruhe, where he later became a Privatdozent.
Hierholzer proved that a connected graph has an Eulerian trail if and only if exactly zero or two of its vertices have an odd degree. This result had been given, with no proof of the 'if' part, by Leonhard Euler in 1736. Hierholzer apparently presented his work to a circle of fellow mathematicians not long before his premature death in 1871. A colleague then arranged for its posthumous publication in a paper that appeared in 1873.
References
C. Hierholzer: Ueber Kegelschnitte im Raume. (Habilitation in Karlsruhe.) Mathematische Annalen II (1870), 564–586.
C. Hierholzer: Ueber eine Fläche der vierten Ordnung. Mathematische Annalen IV (1871), 172–180.
C. Hierholzer: Über die Möglichkeit, einen Linienzug ohne Wiederholung und ohne Unterbrechung zu umfahren. Mathematische Annalen VI (1873), 30–32.
Barnett, Janet Heine Early Writings on Graph Theory: Euler Circuits and The Königsberg Bridge Problem
1840 births
1871 deaths
Scientists from Freiburg im Breisgau
People from the Grand Duchy of Baden
19th-century German mathematicians |
https://en.wikipedia.org/wiki/Rational%20homotopy%20theory | In mathematics and specifically in topology, rational homotopy theory is a simplified version of homotopy theory for topological spaces, in which all torsion in the homotopy groups is ignored. It was founded by and . This simplification of homotopy theory makes certain calculations much easier.
Rational homotopy types of simply connected spaces can be identified with (isomorphism classes of) certain algebraic objects called Sullivan minimal models, which are commutative differential graded algebras over the rational numbers satisfying certain conditions.
A geometric application was the theorem of Sullivan and Micheline Vigué-Poirrier (1976): every simply connected closed Riemannian manifold X whose rational cohomology ring is not generated by one element has infinitely many geometrically distinct closed geodesics. The proof used rational homotopy theory to show that the Betti numbers of the free loop space of X are unbounded. The theorem then follows from a 1969 result of Detlef Gromoll and Wolfgang Meyer.
Rational spaces
A continuous map of simply connected topological spaces is called a rational homotopy equivalence if it induces an isomorphism on homotopy groups tensored with the rational numbers . Equivalently: f is a rational homotopy equivalence if and only if it induces an isomorphism on singular homology groups with rational coefficients. The rational homotopy category (of simply connected spaces) is defined to be the localization of the category of simply connected spaces with respect to rational homotopy equivalences. The goal of rational homotopy theory is to understand this category (i.e. to determine the information that can be recovered from rational homotopy equivalences).
One basic result is that the rational homotopy category is equivalent to a full subcategory of the homotopy category of topological spaces, the subcategory of rational spaces. By definition, a rational space is a simply connected CW complex all of whose homotopy groups are vector spaces over the rational numbers. For any simply connected CW complex , there is a rational space , unique up to homotopy equivalence, with a map that induces an isomorphism on homotopy groups tensored with the rational numbers. The space is called the rationalization of . This is a special case of Sullivan's construction of the localization of a space at a given set of prime numbers.
One obtains equivalent definitions using homology rather than homotopy groups. Namely, a simply connected CW complex is a rational space if and only if its homology groups are rational vector spaces for all . The rationalization of a simply connected CW complex is the unique rational space (up to homotopy equivalence) with a map that induces an isomorphism on rational homology. Thus, one has
and
for all .
These results for simply connected spaces extend with little change to nilpotent spaces (spaces whose fundamental group is nilpotent and acts nilpotently on the higher homotopy groups).
C |
https://en.wikipedia.org/wiki/Rudolf%20Wille | Rudolf Wille (2 November 1937 – 22 January 2017) was a German mathematician and was professor of General Algebra from 1970 to 2003 at Technische Universität Darmstadt (TU Darmstadt). His most celebrated work is the invention of formal concept analysis, an unsupervised machine learning technique that applies mathematical lattice theory to organize data based on objects and their shared attributes.
An accomplished musician and has also made contributions to Mathematics in Music, Mathematical Pedagogy and the Philosophy of Science, Wille played an active leadership role in the concept lattice research community.
Wille was a member of the Board of Directors of the Institute for Philosophy at TU Darmstadt from 1976. From 1983, was leader of the research group on Formal concept analysis and from 1993 Chairman of the "Ernst Schröder Center for Conceptual Knowledge Engineering". Wille was also a founding member of the Center for Inter-Disciplinary Research in Darmstadt and maintained a footprint in other research groups around the world as a visiting consulting/scholar.
Wille's research interests included algebra, order and lattice theory, foundations of geometry, discrete mathematics, measurement theory, mathematics in music, philosophy of science, conceptual knowledge engineering and contextual logic.
A significant international community of researchers follow Wille's work on Formal concept analysis, the main forums being the International Conferences on Formal Concept Analysis (ICFCA), Conceptual Structures (see also Conceptual graphs) (ICCS) and Concept Lattices and their Application (CLA) conferences. The first two are published in the Lecture Notes in Computer Science and the latter is a multi-stage conference that produces journal papers.
A leader, inter-disciplinarian, peace activist and prolific mentor, Wille oversaw more than 100 German "Diplom- und Staatsexamenarbeiten" in Mathematics, 51 PhD dissertations, and 8 Postdoctoral "habilitation" qualifications.
Work
Wille authored more than 250 scientific publications and co-authored the highly cited and influential textbook on Formal concept analysis with his longtime collaborator (and former PhD student) Bernhard Ganter who is now Professor Emeritus of Mathematics at TU Dresden:
B. Ganter, R. Wille (1999) Formal Concept Analysis: Mathematical Foundations, Springer-Verlag, .
References
External links
Ernst Schröder Zentrum
Rudolf Wille's page at The Mathematics Genealogy Project
ICCS
ICFCA
CLA
1937 births
2017 deaths
20th-century German mathematicians
Lattice theorists
Academic staff of Technische Universität Darmstadt
21st-century German mathematicians
People from Bremen (city) |
https://en.wikipedia.org/wiki/George%20Chrystal | George Chrystal FRSE FRS (8 March 1851 – 3 November 1911) was a Scottish mathematician. He is primarily known for his books on algebra and his studies of seiches (wave patterns in large inland bodies of water) which earned him a Gold Medal from the Royal Society of London that was confirmed shortly after his death.
Life
He was born in Old Meldrum on 8 March 1851, the son of Margaret (née Burr) and William Chrystal, a wealthy farmer and grain merchant.
He was educated at Aberdeen Grammar School and the University of Aberdeen. In 1872, he moved to study under James Clerk Maxwell at Peterhouse, Cambridge. He graduated Second Wrangler in 1875, joint with William Burnside, and was elected a fellow of Corpus Christi. He was appointed to the Regius Chair of Mathematics at the University of St Andrews in 1877, and then in 1879 to the Chair in Mathematics at the University of Edinburgh. In 1911, he was awarded the Royal Medal of the Royal Society for his researches into the surface oscillations of Scottish lochs.
He was a contributor to the drafting of the Universities (Scotland) Act 1889, and was one of the founders of the Edinburgh Mathematical Society.
He was elected a Fellow of the Royal Society of Edinburgh in 1880, his proposers including James Clerk Maxwell. He was awarded the Society's Keith Medal for 1879-81 and their Gunning Victoria Jubilee Prize for the period 1904–8. He served as Vice President of the Society from 1895-1901 and General Secretary from 1901–1911. He is credited with instigating the move of the Society from the Mound to George Street.
He was awarded honorary doctorates (LLD) from the University of Aberdeen in 1887 and the University of Glasgow in 1911.
In later life he is listed as living at 5 Belgrave Crescent in western Edinburgh.
The mathematician Alexander G. Burgess trained under him.
He grew ill in 1909 and this worsened early in 1911, leading the University to grant him leave of absence from April of that year. A work-free summer did not improve him. He died on 3 November 1911 at 5 Belgrave Crescent in Edinburgh. He is buried in Foveran Churchyard in Aberdeenshire.
Family
He married Margaret Anne Balfour (1851-1903) in 1879. She died before him and is buried in the northern Victorian extension to Dean Cemetery with their son Walter MacDonald Chrystal who died in infancy. They had four sons and two daughters.
Publications
(Out of copyright: 1900 and subsequent editions are available in reprint or online.)
Three articles within the 1911 edition of Encyclopædia Britannica: Pascal, Blaise; Perpetual Motion; and Riemann, Georg Friedrich Bernhard.
References
External links
Algebra: An Elementary Text-Book for the Higher Classes of Secondary Schools and for Colleges. Volume I, Volume II
1851 births
1911 deaths
Alumni of the University of Aberdeen
Alumni of Peterhouse, Cambridge
Fellows of Corpus Christi College, Cambridge
Academics of the University of Edinburgh
Scottish mathematicians
Royal Medal winners
S |
https://en.wikipedia.org/wiki/List%20of%20Egyptian%20pyramids | This list presents the vital statistics of the pyramids listed in chronological order, when available.
See also
Egyptian pyramids
Great Sphinx of Giza
Lepsius list of pyramids
List of Egyptian pyramidia
List of finds in Egyptian pyramids
List of the oldest buildings in the world
Umm El Qa'ab
References and notes
Bibliography
Pyramids, Egyptian
Pyramids
Pyramids in Egypt |
https://en.wikipedia.org/wiki/List%20of%20Norwich%20City%20F.C.%20records%20and%20statistics | This is a list of the most notable Norwich City F.C. club records.
Players
Appearances
Kevin Keelan holds the record for Norwich City appearances, having played 673 first-team matches between 1963 and 1980.
Goals
Ralph Hunt holds the record for the most League goals scored in a season, 31 in the 1955–56 season in Division Three (South).
Johnny Gavin the top scorer over a career - 122 between 1948 and 1955.
Transfers
The highest transfer fee received for a Norwich City player is approximately £33 million for Emiliano Buendia (to Aston Villa) in June 2021,
Most spent by the club on a player was £9.1 million for Steven Naismith from Everton in 2016.
Matches
The club's widest victory margin in the league was their 10–2 win against Coventry City in the Division Three (South) in 1930. Their heaviest defeat in the league was 10–2 against Swindon Town in 1908 in the Southern Football League.
Norwich's record home attendance is 43,984 for a sixth round FA Cup match against Leicester City on 30 March 1963. With the introduction of regulations enforcing all-seater stadiums, it is unlikely that this record will be beaten in the foreseeable future, as Carrow Road's capacity is currently 27,224.
Seasons
The club's highest league finish was third in the FA Premiership in 1992–93. The club has won the League Cup twice (most recently in 1985) and also reached the FA Cup semi-final three times, most recently in 1992. Norwich have taken part in European competition just once, reaching the third round of the UEFA Cup in 1993–94.
References
Records
English football club statistics |
https://en.wikipedia.org/wiki/Calculus%20Made%20Easy | Calculus Made Easy is a book on infinitesimal calculus originally published in 1910 by Silvanus P. Thompson, considered a classic and elegant introduction to the subject. The original text continues to be available as of 2008 from Macmillan and Co., but a 1998 update by Martin Gardner is available from St. Martin's Press which provides an introduction; three preliminary chapters explaining functions, limits, and derivatives; an appendix of recreational calculus problems; and notes for modern readers. Gardner changes "fifth form boys" to the more American sounding (and gender neutral) "high school students," updates many now obsolescent mathematical notations or terms, and uses American decimal dollars and cents in currency examples.
Calculus Made Easy ignores the use of limits with its epsilon-delta definition, replacing it with a method of approximating (to arbitrary precision) directly to the correct answer in the infinitesimal spirit of Leibniz, now formally justified in modern nonstandard analysis and smooth infinitesimal analysis.
The original text is now in the public domain under US copyright law (although Macmillan's copyright under UK law is reproduced in the 2008 edition from St. Martin's Press). It can be freely accessed on Project Gutenberg.
Further reading
Nature, Vol. 86, No. 2158 (March 9, 1911), p. 41. . A review of the first edition. Internet Archive; Google Books.
Carl Linderholm, College Mathematics Journal, Vol. 31, No. 1 (January, 2000), pp. 77-79. A review of Silvanus P. Thompson, revised by Martin Gardner, Calculus Made Easy (St. Martin's Press, 1998).
External links
Silvanus P. Thompson, Calculus Made Easy: Being a Very-Simplest Introduction to Those Beautiful Methods of Reckoning which Are Generally Called by the Terrifying Names of the Differential Calculus and the Integral Calculus (New York: MacMillan Company, 2nd Ed., 1914). Also available as the (London: MacMillan and Co., Limited, 2nd Ed., 1914) printing, which isn't published under Thompson's name, but instead has the byline of "by F.R.S." (i.e., Fellow of the Royal Society).
(Re-typeset in LaTeX)
Calculus Made Easy online
Calculus Made Easy on YouTube
1910 non-fiction books
Works by Martin Gardner
Mathematics textbooks |
https://en.wikipedia.org/wiki/Edwin%20Spanier | Edwin Henry Spanier (August 8, 1921 – October 11, 1996) was an American mathematician at the University of California at Berkeley, working in algebraic topology. He co-invented Spanier–Whitehead duality and Alexander–Spanier cohomology, and wrote what was for a long time the standard textbook on algebraic topology .
Spanier attended the University of Minnesota, graduating in 1941. During World War II, he served in the United States Army Signal Corps. He received his Ph.D. degree from the University of Michigan in 1947 for the thesis Cohomology Theory for General Spaces written under the direction of Norman Steenrod. After spending a year as a research fellow at the Institute for Advanced Study in Princeton, New Jersey, in 1948 he was appointed to the faculty of the University of Chicago, and then a professor at UC Berkeley in 1959. He had 17 doctoral students, including Morris Hirsch and Elon Lages Lima.
Publications
References
Retrieved on 2008-01-17
Retrieved on 2008-01-17
Obituary, at the Notices of the American Mathematical Society
Photos, at the Mathematical Research Institute of Oberwolfach
20th-century American mathematicians
Topologists
University of California, Berkeley faculty
University of Chicago faculty
University of Minnesota alumni
University of Michigan alumni
United States Army personnel of World War II
Academics from Washington, D.C.
1921 births
1996 deaths
Fair division researchers |
https://en.wikipedia.org/wiki/CGOL | CGOL (pronounced "see goll") is an alternative syntax featuring an extensible algebraic notation for the Lisp programming language. It was designed for MACLISP by Vaughan Pratt and subsequently ported to Common Lisp.
The notation of CGOL is a traditional infix notation, in the style of ALGOL, rather than Lisp's traditional, uniformly-parenthesized prefix notation syntax. The CGOL parser is based on Pratt's design for top-down operator precedence parsing, sometimes informally referred to as a "Pratt parser".
Semantically, CGOL is essentially just Common Lisp, with some additional reader and printer support.
CGOL may be regarded as a more successful incarnation of some of the essential ideas behind the earlier LISP 2 project. Lisp 2 was a successor to LISP 1.5 that aimed to provide ALGOL syntax. LISP 2 was abandoned, whereas it is possible to use the CGOL codebase today. This is because unlike LISP 2, CGOL is implemented as portable functions and macros written in Lisp, requiring no alterations to the host Lisp implementation.
Syntax
Special notations are available for many commonly used Common Lisp operations. For example, one can write a matrix multiply routine as:
CGOL has an infix . operation (referring to Common Lisp's cons function) and the infix @ operation (referring to Common Lisp's append function):
a.(b@c) = (a.b)@c
The preceding example corresponds to this text in native Common Lisp:
(EQUAL (CONS A (APPEND B C)) (APPEND (CONS A B) C))
CGOL uses of to read and set properties:
The preceding example corresponds to this text in native Common Lisp:
(PUTPROP X (GET (GET Y RELATIVE) 'BROTHER) 'FATHER)
This illustrates how CGOL notates a function of two arguments:
\x,y; 1/sqrt(x**2 + y**2)
The preceding example corresponds to this text in native Common Lisp:
(LAMBDA (X Y) (QUOTIENT 1 (SQRT (PLUS (EXPT X 2) (EXPT Y 2)))))
The syntax of CGOL is data-driven and so both modifiable and extensible.
Status and source code
CGOL is known to work on Armed Bear Common Lisp.
The CGOL source code and some text files containing discussions of it are available as freeware from Carnegie-Mellon University's Artificial Intelligence Repository.
References
Lisp programming language family |
https://en.wikipedia.org/wiki/Juan%20Antonio | Juan Ignacio Antonio (born 5 January 1988) is an Argentine former professional football who played as a forward.
External links
Argentine Primera statistics
Player profile on the River Plate website
1988 births
Living people
People from Trelew
Footballers from Chubut Province
Argentine men's footballers
Argentine expatriate men's footballers
Men's association football forwards
Club Atlético River Plate footballers
Brescia Calcio players
Ascoli Calcio 1898 FC players
UC Sampdoria players
SSD Varese Calcio players
Parma Calcio 1913 players
Feralpisalò players
Serie A players
Serie B players
Serie C players
Expatriate men's footballers in Italy |
https://en.wikipedia.org/wiki/Rook%20polynomial | In combinatorial mathematics, a rook polynomial is a generating polynomial of the number of ways to place non-attacking rooks on a board that looks like a checkerboard; that is, no two rooks may be in the same row or column. The board is any subset of the squares of a rectangular board with m rows and n columns; we think of it as the squares in which one is allowed to put a rook. The board is the ordinary chessboard if all squares are allowed and m = n = 8 and a chessboard of any size if all squares are allowed and m = n. The coefficient of x k in the rook polynomial RB(x) is the number of ways k rooks, none of which attacks another, can be arranged in the squares of B. The rooks are arranged in such a way that there is no pair of rooks in the same row or column. In this sense, an arrangement is the positioning of rooks on a static, immovable board; the arrangement will not be different if the board is rotated or reflected while keeping the squares stationary. The polynomial also remains the same if rows are interchanged or columns are interchanged.
The term "rook polynomial" was coined by John Riordan.
Despite the name's derivation from chess, the impetus for studying rook polynomials is their connection with counting permutations (or partial permutations) with restricted positions. A board B that is a subset of the n × n chessboard corresponds to permutations of n objects, which we may take to be the numbers 1, 2, ..., n, such that the number aj in the j-th position in the permutation must be the column number of an allowed square in row j of B. Famous examples include the number of ways to place n non-attacking rooks on:
an entire n × n chessboard, which is an elementary combinatorial problem;
the same board with its diagonal squares forbidden; this is the derangement or "hat-check" problem (this is a particular case of the problème des rencontres);
the same board without the squares on its diagonal and immediately above its diagonal (and without the bottom left square), which is essential in the solution of the problème des ménages.
Interest in rook placements arises in pure and applied combinatorics, group theory, number theory, and statistical physics. The particular value of rook polynomials comes from the utility of the generating function approach, and also from the fact that the zeroes of the rook polynomial of a board provide valuable information about its coefficients, i.e., the number of non-attacking placements of k rooks.
Definition
The rook polynomial RB(x) of a board B is the generating function for the numbers of arrangements of non-attacking rooks:
where is the number of ways to place k non-attacking rooks on the board B. There is a maximum number of non-attacking rooks the board can hold; indeed, there cannot be more rooks than the number of rows or number of columns in the board (hence the limit ).
Complete boards
For rectangular m × n boards Bm,n, we write Rm,n := RBm,n, and if m=n, Rn := Rm,n.
The firs |
https://en.wikipedia.org/wiki/Nseluka | Nseluka is a small town in northern Zambia. It is on the M1 road, which heads to Kasama in the south and Mbala/Mpulungu in the north.
Statistics
elevation –
Transport
It has a station on the TAZARA railway. It is the proposed junction for a branch railway to Mpulungu on the shores of Lake Tanganyika.
See also
Transport in Zambia
References
Populated places in Northern Province, Zambia |
https://en.wikipedia.org/wiki/M/M/1%20queue | In queueing theory, a discipline within the mathematical theory of probability, an M/M/1 queue represents the queue length in a system having a single server, where arrivals are determined by a Poisson process and job service times have an exponential distribution. The model name is written in Kendall's notation. The model is the most elementary of queueing models and an attractive object of study as closed-form expressions can be obtained for many metrics of interest in this model. An extension of this model with more than one server is the M/M/c queue.
Model definition
An M/M/1 queue is a stochastic process whose state space is the set {0,1,2,3,...} where the value corresponds to the number of customers in the system, including any currently in service.
Arrivals occur at rate λ according to a Poisson process and move the process from state i to i + 1.
Service times have an exponential distribution with rate parameter μ in the M/M/1 queue, where 1/μ is the mean service time.
All arrival times and services times are (usually) assumed to be independent of one another.
A single server serves customers one at a time from the front of the queue, according to a first-come, first-served discipline. When the service is complete the customer leaves the queue and the number of customers in the system reduces by one.
The buffer is of infinite size, so there is no limit on the number of customers it can contain.
The model can be described as a continuous time Markov chain with transition rate matrix
on the state space {0,1,2,3,...}. This is the same continuous time Markov chain as in a birth–death process. The state space diagram for this chain is as below.
Stationary analysis
The model is considered stable only if λ < μ. If, on average, arrivals happen faster than service completions the queue will grow indefinitely long and the system will not have a stationary distribution. The stationary distribution is the limiting distribution for large values of t.
Various performance measures can be computed explicitly for the M/M/1 queue. We write ρ = λ/μ for the utilization of the buffer and require ρ < 1 for the queue to be stable. ρ represents the average proportion of time which the server is occupied.
The probability that the stationary process is in state i (contains i customers, including those in service) is
Average number of customers in the system
We see that the number of customers in the system is geometrically distributed with parameter 1 − ρ. Thus the average number of customers in the system is ρ/(1 − ρ) and the variance of number of customers in the system is ρ/(1 − ρ)2. This result holds for any work conserving service regime, such as processor sharing.
Busy period of server
The busy period is the time period measured between the instant a customer arrives to an empty system until the instant a customer departs leaving behind an empty system. The busy period has probability density function
where I1 is a modified Bessel function |
https://en.wikipedia.org/wiki/PIN%20Group%20%28disambiguation%29 | PIN Group was a German courier and postal services company. "Pin group" may also refer to:
Pin group, subgroup of the Clifford algebra associated to a quadratic space
The Pin Group, a New Zealand band founded by Roy Montgomery in 1980 |
https://en.wikipedia.org/wiki/Almost%20integer | In recreational mathematics, an almost integer (or near-integer) is any number that is not an integer but is very close to one. Almost integers are considered interesting when they arise in some context in which they are unexpected.
Almost integers relating to the golden ratio and Fibonacci numbers
Well-known examples of almost integers are high powers of the golden ratio , for example:
The fact that these powers approach integers is non-coincidental, because the golden ratio is a Pisot–Vijayaraghavan number.
The ratios of Fibonacci or Lucas numbers can also make almost integers, for instance:
The above examples can be generalized by the following sequences, which generate near-integers approaching Lucas numbers with increasing precision:
As n increases, the number of consecutive nines or zeros beginning at the tenths place of a(n) approaches infinity.
Almost integers relating to e and
Other occurrences of non-coincidental near-integers involve the three largest Heegner numbers:
where the non-coincidence can be better appreciated when expressed in the common simple form:
where
and the reason for the squares is due to certain Eisenstein series. The constant
is sometimes referred to as Ramanujan's constant.
Almost integers that involve the mathematical constants and e have often puzzled mathematicians. An example is:
To date, no explanation has been given for why Gelfond's constant () is nearly identical to , which is therefore considered a mathematical coincidence.
See also
Schizophrenic number
References
External links
J.S. Markovitch Coincidence, data compression, and Mach's concept of economy of thought
Integers
Recreational mathematics |
https://en.wikipedia.org/wiki/R%C4%83zvan%20Neagu | Răzvan Neagu (born 25 May 1987) is a Romanian former football player.
Statistics
Statistics accurate as of 1 November 2011
Career honours
SC Vaslui
Cupa României
Runner-up: 2010
UEFA Intertoto Cup
Winner: 2008
External links
1987 births
Living people
Footballers from Bacău
Romanian men's footballers
CS Sporting Vaslui players
FCM Bacău players
ACF Gloria Bistrița players
FC Petrolul Ploiești players
Liga I players
CS Turnu Severin players
Men's association football forwards |
https://en.wikipedia.org/wiki/James%20Inman | James Inman (1776–1859), an English mathematician and astronomer, was professor of mathematics at the Royal Naval College, Portsmouth, and author of Inman's Nautical Tables.
Early years
Inman was born at Tod Hole in Garsdale, then in the West Riding of Yorkshire, the younger son of Richard Inman and Jane Hutchinson. He was educated at Sedbergh and St John's College, Cambridge, graduating as first Smith's prizeman and Senior Wrangler for 1800. Among his close college friends was Henry Martyn.
After graduating with first class honours in 1800, Inman intended to undertake missionary work in the Middle East, in Syria, but due to a declaration of war could travel no further than Malta, where he continued to study Arabic.
Astronomer for Matthew Flinders, 1803–04
Returning to England, the Board of Longitude appointed him as replacement astronomer (the original astronomer, suffering from severe seasickness, was discharged en route to Australia) on the expedition of under Matthew Flinders charting Australian waters in 1803–1804. Arriving at Sydney too late to join in Flinders' circumnavigation of Australia, he assisted in concluding the expedition. At this time he became a firm friend of Flinders' nephew, John Franklin, then midshipman. He also befriended the Investigator's artist, William Westall, for whom he later wrote letters of introduction. While on board the East Indiaman Warley for his return to Britain, he participated in the Battle of Pulo Auro. Here he temporarily commanded a party of lascar pikemen.
Professor, Royal Naval College
He was ordained into the Anglican ministry in 1805 when he gained his MA. Three years later he received an appointment as Professor of Nautical Mathematics at the Royal Naval College. In 1821 he published Navigation and Nautical Astronomy for Seamen; these nautical mathematical tables, known as Inman's Nautical Tables, remained in use for many years. In the third edition (1835) he introduced a new table of haversines (the term was his coinage) to simplify the calculation of distances between two points on the surface of the earth using spherical trigonometry. (For details of the calculation, see Haversine formula.)
At his suggestion, in 1810 the Admiralty established a School of Naval Architecture; the Admiralty also appointed Inman its first principal. In 1812 he conducted experiments with Flinders which led to the invention of the Flinders Bar, used for marine compass correction. At the same time as teaching in the school and publishing mathematical texts for the use of his pupils, he translated a French text on the architecture of shipbuilding, and continued his own studies, gaining his doctorate in Divinity in 1820. In recognition of his work in nautical astronomy he was elected a Fellow of the Royal Astronomical Society.
He also directed the design and construction of no less than ten British warships, of which he was proud to state that none ever had the slightest mishap due to an error of design or |
https://en.wikipedia.org/wiki/Ronald%20G.%20Douglas | Ronald George Douglas (December 10, 1938 – February 27, 2018) was an American mathematician, best known for his work on operator theory and operator algebras.
Education and career
Douglas was born in Osgood, Indiana. He was an undergraduate at the Illinois Institute of Technology, and received his Ph.D. in 1962 from Louisiana State University as a student of Pasquale Porcelli. He was at the University of Michigan until 1969, when he moved to the State University of New York at Stony Brook. Beginning in 1986 he moved into university administration, eventually becoming Vice Provost at Stony Brook in 1990, and Provost at Texas A&M University from 1996 until 2002. At the time of his death, he was Distinguished Professor in the Department of Mathematics at Texas A&M. He is survived by three children, including Michael R. Douglas, a noted string theorist.
Research
Among his best-known contributions to science is a 1977 paper with Lawrence G. Brown and Peter A. Fillmore (BDF theory), which introduced techniques from algebraic topology into the theory of operator algebras. This work was an important precursor to noncommutative geometry as later developed by Alain Connes among others. In addition to BDF theory, two other influential theories bear his names: Douglas algebra and Cowen-Douglas operators. In recent decades, he was a prominent advocator of multivariable operator theory. His coauthored book with Vern Paulsen "Hilbert modules over function algebras" introduced an analytic framework for studying commuting operator tuples. Douglas-Arveson conjecture is a well-known unsolved problem in this field.
Douglas directed 23 Ph.D students, some of whom became renowned mathematicians, and his book Banach Algebra Techniques in Operator Theory in the series Graduate Texts in Mathematics is one of the classics in operator theory.
Honors and awards
In 2012, he became a fellow of the American Mathematical Society.
See also
Douglas' lemma
References
Brown, L. G.; Douglas, R. G.; Fillmore, P. A., "Extensions of C*-algebras and K-homology", Annals of Mathematics (2) 105 (1977), no. 2, 265–324.
External links
His page at the Mathematics Genealogy Project
1938 births
2018 deaths
20th-century American mathematicians
21st-century American mathematicians
Texas A&M University faculty
University of Michigan faculty
Stony Brook University faculty
Louisiana State University alumni
Fellows of the American Mathematical Society
Illinois Institute of Technology alumni
People from Ripley County, Indiana
Operator theorists |
https://en.wikipedia.org/wiki/Marius%20Matei | Marius Matei (born 1 February 1984) is a Romanian footballer who plays as a forward for Avântul Valea Mărului.
Statistics
Career honours
FC Vaslui
UEFA Intertoto Cup
Winner: 2008
References
External links
1984 births
Living people
Romanian men's footballers
Men's association football forwards
Liga I players
Liga II players
AFC Dacia Unirea Brăila players
CS Sporting Vaslui players
ASC Oțelul Galați players
FC Botoșani players
FCV Farul Constanța players
FC Voluntari players
FC Brașov (1936) players
ACS Foresta Suceava players
CS Luceafărul Oradea players
Footballers from Galați |
https://en.wikipedia.org/wiki/Gordon%20Edwards%20%28scientist%29 | Gordon Edwards is a Canadian scientist and nuclear consultant. Edwards was born in Canada in 1940, and graduated from the University of Toronto in 1961 with a gold medal in Mathematics and Physics and a Woodrow Wilson Fellowship. At the University of Chicago he obtained two master's degrees, one in Mathematics (1962) and one in English Literature (1964). In 1972, he obtained a Ph.D. in Mathematics from Queen's University.
From 1970 to 1974, he was the editor of Survival magazine. In 1975 he co-founded the Canadian Coalition for Nuclear Responsibility, and has been its president since 1978. Edwards has worked widely as a consultant on nuclear issues and has been qualified as a nuclear expert by courts in Canada and elsewhere.
In 1972–73, Edwards was the assistant director of a nationwide study of the Mathematical Sciences in Canada conducted under the auspices of the Science Council of Canada.
Edwards has written articles and reports on radiation standards, radioactive wastes, uranium mining, nuclear proliferation, the economics of nuclear power, non-nuclear energy strategies. He has been featured on radio and television programs including David Suzuki's The Nature of Things, Pierre Berton's The Great Debate, and many others. He has worked as consultant for governmental bodies such as the Auditor General of Canada, the Select Committee on Ontario Hydro Affairs, and the Ontario Royal Commission on Electric Power Planning. In 2006, Edwards received the Nuclear-Free Future Award. He has also been awarded the Rosalie Bertell Lifetime Achievement Award and the YMCA Peacemaker Medallion.
He is a retired teacher of mathematics at Vanier College in Montreal.
See also
Anti-nuclear movement in Canada
References
External links
The Great Debate: Gordon Edwards vs Edward Teller
How I Became a Nuclear Skeptic
Canada and the Bomb: Past and Future
Canadian Coalition for Nuclear Responsibility
1940 births
Living people
Canadian anti–nuclear power activists
University of Toronto alumni
University of Chicago alumni
Canadian expatriates in the United States |
https://en.wikipedia.org/wiki/CEILIDH | CEILIDH is a public key cryptosystem based on the discrete logarithm problem in algebraic torus. This idea was first introduced by Alice Silverberg and Karl Rubin in 2003; Silverberg named CEILIDH after her cat. The main advantage of the system is the reduced size of the keys for the same security over basic schemes.
Algorithms
Parameters
Let be a prime power.
An integer is chosen such that :
The torus has an explicit rational parametrization.
is divisible by a big prime where is the Cyclotomic polynomial.
Let where is the Euler function.
Let a birational map and its inverse .
Choose of order and let .
Key agreement scheme
This Scheme is based on the Diffie-Hellman key agreement.
Alice chooses a random number .
She computes and sends it to Bob.
Bob chooses a random number .
He computes and sends it to Alice.
Alice computes
Bob computes
is the identity, thus we have :
which is the shared secret of Alice and Bob.
Encryption scheme
This scheme is based on the ElGamal encryption.
Key Generation
Alice chooses a random number as her private key.
The resulting public key is .
Encryption
The message is an element of .
Bob chooses a random integer in the range .
Bob computes and .
Bob sends the ciphertext to Alice.
Decryption
Alice computes .
Security
The CEILIDH scheme is based on the ElGamal scheme and thus has similar security properties.
If the computational Diffie-Hellman assumption holds the underlying cyclic group , then the encryption function is one-way. If the decisional Diffie-Hellman assumption (DDH) holds in , then CEILIDH achieves semantic security. Semantic security is not implied by the computational Diffie-Hellman assumption alone. See decisional Diffie-Hellman assumption for a discussion of groups where the assumption is believed to hold.
CEILIDH encryption is unconditionally malleable, and therefore is not secure under chosen ciphertext attack. For example, given an encryption of some (possibly unknown) message , one can easily construct a valid encryption of the message .
References
External links
Torus-Based Cryptography: the paper introducing the concept (in PDF from Silverberg's university web page).
Public-key encryption schemes
Key-agreement protocols |
https://en.wikipedia.org/wiki/Hanfried%20Lenz | Hanfried Lenz (22 April 1916 in Munich1 June 2013 in Berlin) was a German mathematician, who is mainly known for his work in geometry and combinatorics.
Hanfried Lenz was the eldest son of Fritz Lenz an influential German geneticist, who is associated with Eugenics and hence also with the Nazi racial policies during the Third Reich. He was also the older brother of Widukind Lenz, a geneticist. He started to study mathematics and physics at the University of Tübingen, but interrupted his studies from 1935 to 1937 to do a ( at this time, in Weimar Republic voluntary ) military service. After that he continued to study in Munich, Berlin and Leipzig. In 1939 when World War II broke out in Europe, he became a soldier in the western front and during a vacation he passed the exams for his teacher certification. He married Helene Ranke in 1943 and 1943–45 he worked on radar technology in a laboratory near Berlin.
After World War II Hanfried Lenz was classified as a "follower" by the denazification process. He started to work as a math and physics teacher in Munich and in 1949 he became an assistant at the Technical University of Munich. He received his PhD in 1951 and his Habilitation in 1953. He worked as a lecturer until he became an associate professor in 1959. In 1969 he finally became a full professor at the Free University of Berlin and worked there until his retirement in 1984.
He was also politically active and in connection with his opposition to the rebuilding of the German army in the early 50s, he became a member of the Social Democratic Party (SPD) in 1954. Later, partially due to being alienated by the student movement of the '60s, his leanings became more conservative again and in 1972 he left the SPD to join the Christian Democratic Union.
Hanfried Lenz is known for his work on the classification of projective planes and in 1954 he showed how one can introduce affine spaces axiomatically without constructing them from projective spaces or vector spaces. This result is now known as the theorem of Lenz. During his later years he also worked in the area of combinatorics and published a book on design theory (together with Dieter Jungnickel and Thomas Beth).
In 1995 the Institute of Combinatorics and its Applications awarded the Euler Medal to Hanfried Lenz.
Notes
References
Christoph Kaiser: Lernen heißt irren dürfem. Berliner Zeitung, 2002-4-15
Prof. Dr. Hanfried Lenz ist am 1. Juni 2013 gestorben - news at the math department of the Free University of Berlin (German)
Walter Benz: "Zum mathematischen Werk von Hanfried Lenz", Journal of Geometry 43, 1992 (German)
Hanfried Lenz: Mehr Glück als Verstand, Books on Demand 2002, Autobiography (German)
"Ich habe halt Schwein gehabt". FU-Nachrichten, number 5,2005 (German)
External links
Wikipedia userpage of Hanfried Lenz in the German Wikipedia
1916 births
2013 deaths
Scientists from Munich
People from the Kingdom of Bavaria
20th-century German mathematicians
All-German People's |
https://en.wikipedia.org/wiki/Surface%20gradient | In vector calculus, the surface gradient is a vector differential operator that is similar to the conventional gradient. The distinction is that the surface gradient takes effect along a surface.
For a surface in a scalar field , the surface gradient is defined and notated as
where is a unit normal to the surface. Examining the definition shows that the surface gradient is the (conventional) gradient with the component normal to the surface removed (subtracted), hence this gradient is tangent to the surface. In other words, the surface gradient is the orthographic projection of the gradient onto the surface.
The surface gradient arises whenever the gradient of a quantity over a surface is important. In the study of capillary surfaces for example, the gradient of spatially varying surface tension doesn't make much sense, however the surface gradient does and serves certain purposes.
See also
Aspect (geography)
Geomorphometry#Surface gradient Derivatives
Grade (slope)
Spatial gradient
References
Vector calculus
Surfaces
Vector physical quantities |
https://en.wikipedia.org/wiki/Jessica%20Sklar | Jessica Katherine Sklar (born 1973) is a mathematician interested in abstract algebra, recreational mathematics, mathematics and art, and mathematics and popular culture. She is a professor of mathematics at Pacific Lutheran University, and former head of the mathematics department at Pacific Lutheran.
Education and career
As a high school student, Sklar studied poetry at the Interlochen Arts Academy. She did her undergraduate studies at Swarthmore College, where her mother Elizabeth S. had earned a degree in English (later becoming an English professor at Wayne State University) and her father Lawrence Sklar had taught philosophy. Jessica completed a double major in English and mathematics in 1995.
Next, Sklar moved to the University of Oregon for graduate study in mathematics, earning a master's degree in 1997 and completing her Ph.D. there in 2001. Her dissertation, Binomial Rings and Algebras, was supervised by Frank Wylie Anderson.
She has been a faculty member in the mathematics department at Pacific Lutheran since 2001.
Combining her interests in mathematics and art she is one of 24 mathematicians and artists who make up the Mathemalchemy Team.
Selected publications
“‘Bok bok’: exploring the game of Chicken in film,” with Jennifer F. Nordstrom. In: Handbook of the Mathematics of the Arts and Sciences. Ed. Bharath Sriraman. Springer International Publishing, Cham, 2020.
“‘Elegance in design’: mathematics and the works of Ted Chiang.” In: Handbook of the Mathematics of the Arts and Sciences. Ed. Bharath Sriraman. Springer International Publishing, Cham, 2020.
“Disciple” (poem). Journal of Humanistic Mathematics 7(2) (July 2017), 418.
First-Semester Abstract Algebra: A Structural Approach. GNU Free Documentation License, 2017.
“A confused electrician uses Smith normal form,” with Tom Edgar. Mathematics Magazine 89(1) (2016), 3–13.
Mathematics in Popular Culture: Essays on Appearances in Film, Literature, Games, Television and Other Media. Jefferson, NC: McFarland & Co., 2012. Editor, with Elizabeth S. Sklar.
“The graph menagerie: abstract algebra and the Mad Veterinarian,” with G. Abrams. Mathematics Magazine 83(3) (2010), 168–179.
“Dials and levers and glyphs, oh my! Linear algebra solutions to computer game puzzles.”Mathematics Magazine 79(5) (2006), 360–367.
"Binomial rings.” Communications in Algebra 32(4) (2004), 1385–1399.
“Binomial algebras.” Communications in Algebra 30(4) (2002), 1961–1978.
Recognition
Sklar was a winner of the Carl B. Allendoerfer Award of the Mathematical Association of America in 2011 for her paper with Gene Abrams, The Graph Menagerie: Abstract Algebra and the Mad Veterinarian.
The paper provides a general solution to a class of lattice reduction puzzles exemplified by the following one:
She was the July 2012 Author of the Month at Ada's Technical Books in Seattle, Washgington.
References
External links
Home page
1973 births
Living people
21st-century American mathematicians
American women mathemat |
https://en.wikipedia.org/wiki/Capillary%20surface | In fluid mechanics and mathematics, a capillary surface is a surface that represents the interface between two different fluids. As a consequence of being a surface, a capillary surface has no thickness in slight contrast with most real fluid interfaces.
Capillary surfaces are of interest in mathematics because the problems involved are very nonlinear and have interesting properties, such as discontinuous dependence on boundary data at isolated points. In particular, static capillary surfaces with gravity absent have constant mean curvature, so that a minimal surface is a special case of static capillary surface.
They are also of practical interest for fluid management in space (or other environments free of body forces), where both flow and static configuration are often dominated by capillary effects.
The stress balance equation
The defining equation for a capillary surface is called the stress balance equation, which can be derived by considering the forces and stresses acting on a small volume that is partly bounded by a capillary surface. For a fluid meeting another fluid (the "other" fluid notated with bars) at a surface , the equation reads
where is the unit normal pointing toward the "other" fluid (the one whose quantities are notated with bars), is the stress tensor (note that on the left is a tensor-vector product), is the surface tension associated with the interface, and is the surface gradient. Note that the quantity is twice the mean curvature of the surface.
In fluid mechanics, this equation serves as a boundary condition for interfacial flows, typically complementing the Navier–Stokes equations. It describes the discontinuity in stress that is balanced by forces at the surface. As a boundary condition, it is somewhat unusual in that it introduces a new variable: the surface that defines the interface. It's not too surprising then that the stress balance equation normally mandates its own boundary conditions.
For best use, this vector equation is normally turned into 3 scalar equations via dot product with the unit normal and two selected unit tangents:
Note that the products lacking dots are tensor products of tensors with vectors (resulting in vectors similar to a matrix-vector product), those with dots are dot products. The first equation is called the normal stress equation, or the normal stress boundary condition. The second two equations are called tangential stress equations.
The stress tensor
The stress tensor is related to velocity and pressure. Its actual form will depend on the specific fluid being dealt with, for the common case of incompressible Newtonian flow the stress tensor is given by
where is the pressure in the fluid, is the velocity, and is the viscosity.
Static interfaces
In the absence of motion, the stress tensors yield only hydrostatic pressure so that , regardless of fluid type or compressibility. Considering the normal and tangential equations,
The first equation establishes that cur |
https://en.wikipedia.org/wiki/Centre%20of%20Mathematics%20and%20Design | Centre of Mathematics and Design (MAyDI) () was created at the Faculty of Architecture, Design and Urbanism of the University of Buenos Aires, in 1995, under the direction of Vera W. de Spinadel.
This Centre received several research and development grants from the Secretary of Science and Technology of the University of Buenos Aires. At the Scientific Renewable Programming 2004–2007, they received a stipendium for the creation of a Mathematics & Design Laboratory MyD_Lab. This Laboratory was officially inaugurated on April 15, 2005 and its main aim is to act as a technological pole of support, assessor ship and training in the subjects referred to the application of mathematical and informatical methodologies so as to state, develop and solve in an optimal way applied problems. The research lines in development are the following
Optimization of the Buenos Aires University Campus Habitat
Modeling in Inference Statistics
Planning, administration and control of projects
Study of urban morphologies using fractal geometry and complexity theory
Analysis of the interdisciplinarity of mathematics in relation to Design, Art and Science
Transference at the graduate and post-graduate level of multimedia material in the form of books, videos, web, seminar, “on line” courses, etc.
External links
Centre of Mathematics & Design (MAyDI)
Fractal Geometry
University of Buenos Aires |
https://en.wikipedia.org/wiki/Classification%20of%20manifolds | In mathematics, specifically geometry and topology, the classification of manifolds is a basic question, about which much is known, and many open questions remain.
Main themes
Overview
Low-dimensional manifolds are classified by geometric structure; high-dimensional manifolds are classified algebraically, by surgery theory.
"Low dimensions" means dimensions up to 4; "high dimensions" means 5 or more dimensions. The case of dimension 4 is somehow a boundary case, as it manifests "low dimensional" behaviour smoothly (but not topologically); see discussion of "low" versus "high" dimension.
Different categories of manifolds yield different classifications; these are related by the notion of "structure", and more general categories have neater theories.
Positive curvature is constrained, negative curvature is generic.
The abstract classification of high-dimensional manifolds is ineffective: given two manifolds (presented as CW complexes, for instance), there is no algorithm to determine if they are isomorphic.
Different categories and additional structure
Formally, classifying manifolds is classifying objects up to isomorphism.
There are many different notions of "manifold", and corresponding notions of
"map between manifolds", each of which yields a different category and a different classification question.
These categories are related by forgetful functors: for instance, a differentiable manifold is also a topological manifold, and a differentiable map is also continuous, so there is a functor .
These functors are in general neither one-to-one nor onto; these failures are generally referred to in terms of "structure", as follows. A topological manifold that is in the image of is said to "admit a differentiable structure", and the fiber over a given topological manifold is "the different differentiable structures on the given topological manifold".
Thus given two categories, the two natural questions are:
Which manifolds of a given type admit an additional structure?
If it admits an additional structure, how many does it admit?
More precisely, what is the structure of the set of additional structures?
In more general categories, this structure set has more structure: in Diff it is simply a set, but in Top it is a group, and functorially so.
Many of these structures are G-structures, and the question is reduction of the structure group. The most familiar example is orientability: some manifolds are orientable, some are not, and orientable manifolds admit 2 orientations.
Enumeration versus invariants
There are two usual ways to give a classification: explicitly, by an enumeration, or implicitly, in terms of invariants.
For instance, for orientable surfaces,
the classification of surfaces enumerates them as the connect sum of tori, and an invariant that classifies them is the genus or Euler characteristic.
Manifolds have a rich set of invariants, including:
Point-set topology
Compactness
Connectedness
Classic algebraic topology |
https://en.wikipedia.org/wiki/Ministry%20of%20Interior%20%28State%20of%20Palestine%29 | The Ministry of Interior and National Security is the branch of the Palestinian National Authority (PNA) cabinet in charge of the security and the statistics of the population of the Palestinian National Authority. The Palestinian Central Bureau of Statistics (PCBS) is a sub-branch of the Interior Ministry that has the responsibility for the population and economic statistics of the Palestinian territories. Since Hamas' takeover of Gaza, the position of the Interior Ministry within the Palestinian Security Services is unclear.
History
In 2006, the Israeli Defense Forces struck the office building of the Interior Ministry multiple times as a part of a bombing campaign in Gaza. The attacks were in response to the kidnapping of Gilad Shalit, an Israeli soldier.
In June 2007, the office in Gaza was taken by Said Seyam as part of the Hamas government of June 2007. Fathi Hamad took office in January 2009, following assassination of Said Seyam on 15 January 2009 during the Gaza War. In August 2012, following government reshuffle by Prime Minister Ismail Haniyye, Fathi Hamad remained in position.
Officeholders
Interior Ministers of the Palestinian Authority
Interior Ministers in the West Bank
Interior Ministers in the Gaza Strip
See also
Palestinian Security Services
Foreign Minister of the Palestinian Authority
Finance Minister of Palestinian Authority
References
External links
Official Interior Ministry Website (Gaza Strip)
Law enforcement in the State of Palestine
Organizations based in Ramallah
Internal affairs ministries
Interior |
https://en.wikipedia.org/wiki/Institute%20of%20Mathematics%20of%20National%20Academy%20of%20Sciences%20of%20Armenia | The Institute of Mathematics of National Academy of Sciences of Armenia (Armenian: ) is owned and operated by the Armenian Academy of Sciences, located in Yerevan.
History
The Institute of Mathematics of National Academy of Sciences of Armenia originated as the Section for Mathematics and Mechanics, created within the newly formed Armenian Academy of Sciences in 1944. The section later developed into an Institute of Mathematics and Mechanics of the Armenian Academy of Sciences, whose first Director was academician Artashes Shahinian, known for his results in complex analysis. The Institute of Mathematics of Armenian Academy of Sciences separated from the latter Institute in 1971. The bearer of the office of the Director of Institute has been academician Mkhitar Djrbashian (1971-1989, 1989-1994 Honorary Director).
The academicians Sergey Mergelyan, Norair Arakelian, Alexandr Talalyan, Raphayel Alexandrian, Rouben V. Ambartzumian and Anry Nersesyan also have greatly influenced the formation of the scientific profile of the Institute and largely contributed to mathematics in general. In particular Rouben V. Ambartzumian is famous for his work in Stochastic Geometry and Integral Geometry, where he created a new branch called Combinatorial Integral Geometry. He has provided solutions to a number of classical problems in particular the solution to the Buffon Sylvester problem as well as the Hilbert's fourth problem in dimensions 2 and 3.
In the early years, the investigations carried out in the Institute concentrated on Function Theory. Gradually the sphere of investigations expanded and now includes Differential and Integral Equations, Functional Analysis, Probability Theory and Mathematical Statistics.
Journals
The institute has a journal, Izvestia NAS RA Matematika. The founder and the first Editor in Chief (1971–1990) of the journal was Mkhitar Djrbashian; under Rouben V. Ambartzumian Editor in Chief (1990 - 2010) the journal obtained international recognition and obtained an English version, Journal of Contemporary Mathematical Analysis, published initially by Allerton Press, Inc. New York and later by Springer Science+Business Media. Journal covers a host of topics including: real analysis and complex analysis; approximation theory, boundary value problems; integral geometry and stochastic geometry; differential equations; probability and statistics; integral equations; algebra.
Directors, faculty and members
At present the Institute has about 30 main researchers as well as a number of associate researchers from Yerevan State University.
Norair Arakelian in 1970 (Nice) and Mkhitar Djrbashian, Rouben V. Ambartzumian in 1974 (Vancouver) were invited speakers at the International Congresses of Mathematicians.
See also
Armenian National Academy of Sciences
Byurakan Observatory
Education in Armenia
List of International Congresses of Mathematicians Plenary and Invited Speakers
Mikael Ter-Mikaelian Institute for Physical Research
Science |
https://en.wikipedia.org/wiki/Fuzzy%20mathematics | Fuzzy mathematics is the branch of mathematics including fuzzy set theory and fuzzy logic that deals with partial inclusion of elements in a set on a spectrum, as opposed to simple binary "yes" or "no" (0 or 1) inclusion. It started in 1965 after the publication of Lotfi Asker Zadeh's seminal work Fuzzy sets. Linguistics is an example of a field that utilizes fuzzy set theory.
Definition
A fuzzy subset A of a set X is a function A: X → L, where L is the interval [0, 1]. This function is also called a membership function. A membership function is a generalization of an indicator function (also called a characteristic function) of a subset defined for L = {0, 1}. More generally, one can use any complete lattice L in a definition of a fuzzy subset A.
Fuzzification
The evolution of the fuzzification of mathematical concepts can be broken down into three stages:
straightforward fuzzification during the sixties and seventies,
the explosion of the possible choices in the generalization process during the eighties,
the standardization, axiomatization, and L-fuzzification in the nineties.
Usually, a fuzzification of mathematical concepts is based on a generalization of these concepts from characteristic functions to membership functions. Let A and B be two fuzzy subsets of X.
The intersection A ∩ B and union A ∪ B are defined as follows: (A ∩ B)(x) = min(A(x), B(x)), (A ∪ B)(x) = max(A(x), B(x)) for all x in X. Instead of and one can use t-norm and t-conorm, respectively; for example, min(a, b) can be replaced by multiplication ab. A straightforward fuzzification is usually based on and operations because in this case more properties of traditional mathematics can be extended to the fuzzy case.
An important generalization principle used in fuzzification of algebraic operations is a closure property. Let * be a binary operation on X. The closure property for a fuzzy subset A of X is that for all x, y in X, A(x*y) ≥ min(A(x), A(y)). Let (G, *) be a group and A a fuzzy subset of G. Then A is a fuzzy subgroup of G if for all x, y in G, A(x*y−1) ≥ min(A(x), A(y−1)).
A similar generalization principle is used, for example, for fuzzification of the transitivity property. Let R be a fuzzy relation on X, i.e. R is a fuzzy subset of X × X. Then R is (fuzzy-)transitive if for all x, y, z in X, R(x, z) ≥ min(R(x, y), R(y, z)).
Fuzzy analogues
Fuzzy subgroupoids and fuzzy subgroups were introduced in 1971 by A. Rosenfeld.
Analogues of other mathematical subjects have been translated to fuzzy mathematics, such as fuzzy field theory and fuzzy Galois theory, fuzzy topology, fuzzy geometry, fuzzy orderings, and fuzzy graphs.
See also
Fuzzy measure theory
Fuzzy subalgebra
Monoidal t-norm logic
Possibility theory
T-norm
References
External links
Zadeh, L.A. Fuzzy Logic - article at Scholarpedia
Hajek, P. Fuzzy Logic - article at Stanford Encyclopedia of Philosophy
Navara, M. Triangular Norms and Conorms - article at Scholarpedia
Dubois, |
https://en.wikipedia.org/wiki/Relative%20dimension | In mathematics, specifically linear algebra and geometry, relative dimension is the dual notion to codimension.
In linear algebra, given a quotient map , the difference dim V − dim Q is the relative dimension; this equals the dimension of the kernel.
In fiber bundles, the relative dimension of the map is the dimension of the fiber.
More abstractly, the codimension of a map is the dimension of the cokernel, while the relative dimension of a map is the dimension of the kernel.
These are dual in that the inclusion of a subspace of codimension k dualizes to yield a quotient map of relative dimension k, and conversely.
The additivity of codimension under intersection corresponds to the additivity of relative dimension in a fiber product. Just as codimension is mostly used for injective maps, relative dimension is mostly used for surjective maps.
References
Algebraic geometry
Geometric topology
Linear algebra
Dimension |
https://en.wikipedia.org/wiki/Delaunay%20refinement | In mesh generation, Delaunay refinements are algorithms for mesh generation based on the principle of adding Steiner points to the geometry of an input to be meshed, in a way that causes the Delaunay triangulation or constrained Delaunay triangulation of the augmented input to meet the quality requirements of the meshing application. Delaunay refinement methods include methods by Chew and by Ruppert.
Chew's second algorithm
Chew's second algorithm takes a piecewise linear system (PLS) and returns a constrained Delaunay triangulation of only quality triangles where quality is defined by the minimum angle in a triangle. Developed by L. Paul Chew for meshing surfaces embedded in three-dimensional space, Chew's second algorithm has been adopted as a two-dimensional mesh generator due to practical advantages over Ruppert's algorithm in certain cases and is the default quality mesh generator implemented in the freely available Triangle package. Chew's second algorithm is guaranteed to terminate and produce a local feature size-graded meshes with minimum angle up to about 28.6 degrees.
The algorithm begins with a constrained Delaunay triangulation of the input vertices. At each step, the circumcenter of a poor-quality triangle is inserted into the triangulation with one exception: If the circumcenter lies on the opposite side of an input segment as the poor quality triangle, the midpoint of the segment is inserted. Moreover, any previously inserted circumcenters inside the diametral ball of the original segment (before it is split) are removed from the triangulation.
Circumcenter insertion is repeated until no poor-quality triangles exist.
Ruppert's algorithm
Ruppert's algorithm takes a planar straight-line graph (or in dimension higher than two a piecewise linear system) and returns a conforming Delaunay triangulation of only quality triangles. A triangle is considered poor-quality if it has a circumradius to shortest edge ratio larger than some prescribed threshold.
Discovered by Jim Ruppert in the early 1990s,
"Ruppert's algorithm for two-dimensional quality mesh generation is perhaps the first theoretically guaranteed meshing algorithm to be truly satisfactory in practice."
Motivation
When doing computer simulations such as computational fluid dynamics, one starts with a model such as a 2D outline of a wing section.
The input to a 2D finite element method needs to be in the form of triangles that fill all space, and each triangle to be filled with one kind of material – in this example, either "air" or "wing".
Long, skinny triangles cannot be simulated accurately.
The simulation time is generally proportional to the number of triangles, and so one wants to minimize the number of triangles, while still using enough triangles to give reasonably accurate results – typically by using an unstructured grid.
The computer uses Ruppert's algorithm (or some similar meshing algorithm) to convert the polygonal model into triangles suitable for the |
https://en.wikipedia.org/wiki/MENTOR%20routing%20algorithm | The MENTOR routing algorithm is an algorithm for use in routing of mesh networks, specifically pertaining to their initial topology. It was developed in 1991 by Aaron Kershenbaum, Parviz Kermani, and George A. Grove and was published by the IEEE.
Complexity
Empirical observation has shown the complexity class of this algorithm to be O(N²), or quadratic. This represents "a significant improvement over currently used algorithms, [while still yielding] solutions of a quality competitive with other, much slower procedures."
Methodology
The algorithm assumes three things are conducive to low-"cost" (that is, minimal in distance travelled and time between destinations) topology: that paths will tend to be direct, not circuitous; that links will have a "high utilization"—that is, they will be used to nearly their maximum operating capacity; and that "long, high-capacity links [will be used] whenever possible."
The overall plan is to send traffic over a direct route between the source and destination whenever the magnitude of the requirement is sufficiently large and to send it via a path within a tree in all other cases. In the former case, we are satisfying all three of our goals--we are using a direct path of high utilization and high capacity. In the latter case we are satisfying at least the last two objectives as we are aggregating traffic as much as possible.
The minimum spanning tree on which traffic flows in the latter case is heuristically defined by Dijkstra's algorithm and Prim's algorithm.
References
Aaron Kershenbaum, Parviz Kermani, George A. Grover. "MENTOR: An Algorithm for Mesh Network Topological Optimization and Routing", IEEE Transactions on Communications, April 1991. Accessed November 4, 2007.
Routing algorithms
Mesh networking |
https://en.wikipedia.org/wiki/Win%E2%80%93stay%2C%20lose%E2%80%93switch | In psychology, game theory, statistics, and machine learning, win–stay, lose–switch (also win–stay, lose–shift) is a heuristic learning strategy used to model learning in decision situations. It was first invented as an improvement over randomization in bandit problems. It was later applied to the prisoner's dilemma in order to model the evolution of altruism.
The learning rule bases its decision only on the outcome of the previous play. Outcomes are divided into successes (wins) and failures (losses). If the play on the previous round resulted in a success, then the agent plays the same strategy on the next round. Alternatively, if the play resulted in a failure the agent switches to another action.
A large-scale empirical study of players of the game rock, paper, scissors shows that a variation of this strategy is adopted by real-world players of the game, instead of the Nash equilibrium strategy of choosing entirely at random between the three options.
References
See also
Bounded rationality
Game theory
Computational learning theory
Heuristics |
https://en.wikipedia.org/wiki/List%20of%20National%20Football%20League%20career%20scoring%20leaders | The top 25 scorers in National Football League history are all placekickers. Statistics include regular season scoring only.
List
Key
Updated through the 2022 season.
Non-kickers
The top five scoring non-kickers in NFL history are listed here with their overall scoring rank. Only one non-kicker, Jerry Rice, is in the top 50 scorers of all-time.
See also
List of National Football League annual scoring leaders
List of National Football League records (individual)
References
Scoring Leaders
National Football League lists |
https://en.wikipedia.org/wiki/F-statistic | F-statistic may refer to:
a statistic used for the F-test
a concept in biogenetics, see F-statistics |
https://en.wikipedia.org/wiki/Centroidal%20Voronoi%20tessellation | In geometry, a centroidal Voronoi tessellation (CVT) is a special type of Voronoi tessellation in which the generating point of each Voronoi cell is also its centroid (center of mass). It can be viewed as an optimal partition corresponding to an optimal distribution of generators. A number of algorithms can be used to generate centroidal Voronoi tessellations, including Lloyd's algorithm for K-means clustering or Quasi-Newton methods like BFGS.
Proofs
Gersho's conjecture, proven for one and two dimensions, says that "asymptotically speaking, all cells of the optimal CVT, while forming a tessellation, are congruent to a basic cell which depends on the dimension."
In two dimensions, the basic cell for the optimal CVT is a regular hexagon as it is proven to be the most dense packing of circles in 2D Euclidean space.
Its three dimensional equivalent is the rhombic dodecahedral honeycomb, derived from the most dense packing of spheres in 3D Euclidean space.
Applications
Centroidal Voronoi tessellations are useful in data compression, optimal quadrature, optimal quantization, clustering, and optimal mesh generation.
A weighted centroidal Voronoi diagrams is a CVT in which each centroid is weighted according to a certain function. For example, a grayscale image can be used as a density function to weight the points of a CVT, as a way to create digital stippling.
Occurrence in nature
Many patterns seen in nature are closely approximated by a centroidal Voronoi tessellation. Examples of this include the Giant's Causeway, the cells of the cornea, and the breeding pits of the male tilapia.
References
Discrete geometry
Geometric algorithms
Diagrams |
https://en.wikipedia.org/wiki/Sami%20Mermer | Sami Mermer is a Turkish Canadian documentary filmmaker of Kurdish descent.
Biography
Mermer was born in Turkey. He studied mathematics at the University of Ankara from 1994 to 1996 and from 1996 to 2000, Environment Engineering at the University of Istanbul. He pursued cinema studies at the University of Mesopotamia from 1998 to 2000 followed by studies of French and Cinema Studies at the University of Quebec from 2000 to 2002.
He worked on several documentaries and fiction films, as a director and director of photography. He was co-writer and assistant director of Ax (the land), winner of several prizes and awarded best film in the 2000 Hamburg Film Festival. In Grand Rapids, Michigan, he co-directed, with Aaron B. Smith a short fiction film called Sortie, winner of Compass School for Cinematic Arts 24 Hour Film competition.
The Box of Lanzo, 102 minutes, was his first feature documentary about homeless people which he directed, edited and shot almost entirely in Grand Rapids, Michigan. In 2006, he collaborated as writer/director & DP with Benjamin Hoekstra on a feature-length fiction project, The Extra shot in both Los Angeles and Michigan, and since reduced to a short film.
From December 2006 to July 2007 Mermer lived in Morocco and launched a production company called Turtle Productions with Hind Benchekroun based in Montreal-Casablanca.
Mermer has lived in Montreal since July 2007. In 2015, he released in collaboration with Benchekroun the long feature documentary film Callshop Istanbul, about life of refugees from all over the world in Istanbul, a gateway to immigration to Europe. In 2018, Mermer and Benchekroun released their third feature documentary film, titled Xalko. The film looks at Xalko, a kurdish village in Turkey, deserted by its men who have gone to Europe or America. It explores migration from the point of view those left behind. The film was selected in many festivals, including Montreal International Documentary Festival (RIDM) and DOXA Documentary Film Festival. It won the Prix Iris for Best Documentary Film at the 22nd Quebec Cinema Awards in 2020, and Mermer received a nomination for Best Cinematography in a Documentary. In 2019, Mermer made his last film, a short fiction called (The Room), in which he gave a little rolls to his daughters, Zeliha and Louna...
Filmography
Director
1999: Ax (the land), Fiction. Turkey. (as co-writer, director assistant). 27 min.
Best film in Hamburg film festival (Germany, 2000)
Second best film in Milano (Italy, 2000)
2004-2005: The Box of Lanzo, Doc./Fiction. Grand Rapids, MI. (as writer, director, camera, producer). 14 min.
World Urban Forum, United Nations, Vancouver.
Broadcast online on Citizenshift, website of The National Film Board of Canada.
Market Clermont-Ferrand, France.
Casablanca International Short and Documentary Film Festival.
2005: Sortie, Fiction. Grand Rapids, MI. (as writer, director). 5 min.
Casablanca International Short and Documentary Film Festival.
2 |
https://en.wikipedia.org/wiki/Synaptic%20augmentation | Augmentation is one of four components of short-term synaptic plasticity that increases the probability of releasing synaptic vesicles during and after repetitive stimulation such that
when all the other components of enhancement and depression are zero, where is augmentation at time and 0 refers to the baseline response to a single stimulus. The increase in the number of synaptic vesicles that release their transmitter leads to enhancement of the post synaptic response. Augmentation can be differentiated from the other components of enhancement by its kinetics of decay and by pharmacology. Augmentation selectively decays with a time constant of about 7 seconds and its magnitude is enhanced in the presence of barium. All four components are thought to be associated with or triggered by increases in internal calcium ions that build up and decay during repetitive stimulation.
During a train of impulses the enhancement of synaptic strength due to the underlying component that gives rise to augmentation can be described by
where is the unit impulse function at the time of stimulation, is the incremental increase in with each impulse, and is the rate constant for the loss of . During a stimulus train the magnitude of augmentation added by each impulse, a*, can increase during the train such that
where is the increment added by the first impulse of the train, is a constant that determines the increase in with each impulse, is the stimulation rate, and is the duration of stimulation.
Augmentation is differentiated from the three other components of enhancement by its time constant of decay. This is shown in Table 1 where the first and second components of facilitation, F1 and F2, decay with time constants of about 50 and 300 ms, and potentiation, P, decays with a time constant than ranges from tens of seconds to minutes depending on the duration of stimulation. Also included in the table are two components of depression D1 and D2, along with their associated decay time constants of recovery decay back to normal. Depression at some synapses may arise from depletion of synaptic vesicles available for release. Depression of synaptic vesicle release may mask augmentation because of overlapping time courses. Also included in the table is the fraction change in transmitter release arising from one impulse. A magnitude of 0.8 would increase transmitter release 80%.
†The magnitude of augmentation added by each impulse can increase during the train.
‡The time constant of P can increase with repetitive stimulation.
The balance between various components of enhancement and depression at the mammalian synapse is affected by temperature so that maintenance of the components of enhancement is greatly reduced at temperatures lower than physiological. During repetitive stimulation at 23 °C components of depression dominate synaptic release, whereas at 33–38 °C synaptic strength increases due to a shift towards comp |
https://en.wikipedia.org/wiki/Odlyzko%E2%80%93Sch%C3%B6nhage%20algorithm | In mathematics, the Odlyzko–Schönhage algorithm is a fast algorithm for evaluating the Riemann zeta function at many points, introduced by . The main point is the use of the fast Fourier transform to speed up the evaluation of a finite Dirichlet series of length N at O(N) equally spaced values from O(N2) to O(N1+ε) steps (at the cost of storing O(N1+ε) intermediate values). The Riemann–Siegel formula used for
calculating the Riemann zeta function with imaginary part T uses a finite Dirichlet series with about N = T1/2 terms, so when finding about N values of the Riemann zeta function it is sped up by a factor of about T1/2. This reduces the time to find the zeros of the zeta function with imaginary part at most T from
about T3/2+ε steps to about T1+ε steps.
The algorithm can be used not just for the Riemann zeta function, but also for many other functions given by Dirichlet series.
The algorithm was used by to verify the Riemann hypothesis for the first 1013 zeros of the zeta function.
References
This unpublished book describes the implementation of the algorithm and discusses the results in detail.
Analytic number theory
Computational number theory
Zeta and L-functions |
https://en.wikipedia.org/wiki/Mark%20Pinsker | Mark Semenovich Pinsker (; April 24, 1925 – December 23, 2003) or Mark Shlemovich Pinsker () was a noted Russian mathematician in the fields of information theory, probability theory, coding theory, ergodic theory, mathematical statistics, and communication networks.
Pinsker studied stochastic processes under A. N. Kolmogorov in the 1950s, and later worked at the Institute for Information Transmission Problems (IITP), Russian Academy of Sciences, Moscow.
His accomplishments included a classic paper on the entropy theory of dynamical systems which introduced the maximal partition with zero entropy, later known as Pinsker's partition. His work in mathematical statistics was devoted mostly to the applications of information theory, including asymptotically sufficient statistics for parameter estimation and nonparametric estimation; Pinsker's inequality is named after him. He also produced notable results in the theory of switching networks and complexity problems in coding theory.
Pinsker received the IEEE Claude E. Shannon Award in 1978, and the IEEE Richard W. Hamming Medal in 1996.
Selected works
"Theory of curves in Hilbert space with stationary increments of order " Izv. Akad. Nauk SSSR Ser. Mat., 19, 1955.
Information and information stability of random variables and processes, translated and edited by Amiel Feinstein, Holden-Day, San Francisco, 1964.
L. A. Bassalygo and M. S. Pinsker, "The complexity of an optimal non-blocking commutation scheme without reorganization", Problemy Peredaci Informacii, 9(1):84–87, 1973. Translated into English in Problems of Information Transmission, 9 (1974) 64-66.
M. S. Pinsker. "On the complexity of a concentrator", 7th International Teletraffic Conference, pages 318/1-318/4, 1973.
"Estimation of error-correction complexity of Gallager low-density codes", Problems of Information Transmission, 11:18—28, 1976.
"Reflections of Some Shannon Lecturers".
Notes
References
"Review of Scientific Achievements of M. S. Pinsker", Problems of Information Transmission (translation of Problemy Peredachi Informatsii), Volume 32, Number 1, January–March, 1996, pages 3–14.
External links
"Mark Semenovich Pinsker. In Memoriam", Problems of Information Transmission, MAIK Nauka/Interperiodica, Volume 40, Number 1 / January, 2004, pages 1–4. ISSN 0032-9460. English version
Ramesh Rao. " Mark Semenovich Pinsker - On his 70th Birthday", IEEE Information Theory Society Newsletter, September 1995.
Sasha Barg. "In Memoriam - Mark Semënovich Pinsker", IEEE Information Theory Society Newsletter, Volume 54, Number 3, September 2004.
Pinsker Marks Shlemovich (1925–2003) author page at Math-Net.ru
"Reflections of Some Shannon Lecturers"
Russian information theorists
Russian mathematicians
Soviet mathematicians
1925 births
2003 deaths
Mathematical statisticians |
https://en.wikipedia.org/wiki/Beurling%E2%80%93Lax%20theorem | In mathematics, the Beurling–Lax theorem is a theorem due to and which characterizes the shift-invariant subspaces of the Hardy space . It states that each such space is of the form
for some inner function .
See also
H2
References
Jonathan R. Partington, Linear Operators and Linear Systems, An Analytical Approach to Control Theory, (2004) London Mathematical Society Student Texts 60, Cambridge University Press.
Marvin Rosenblum and James Rovnyak, Hardy Classes and Operator Theory, (1985) Oxford University Press.
Hardy spaces
Theorems in analysis
Invariant subspaces |
https://en.wikipedia.org/wiki/U-quadratic%20distribution | In probability theory and statistics, the U-quadratic distribution is a continuous probability distribution defined by a unique convex quadratic function with lower limit a and upper limit b.
Parameter relations
This distribution has effectively only two parameters a, b, as the other two are explicit functions of the support defined by the former two parameters:
(gravitational balance center, offset), and
(vertical scale).
Related distributions
One can introduce a vertically inverted ()-quadratic distribution in analogous fashion.
Applications
This distribution is a useful model for symmetric bimodal processes. Other continuous distributions allow more flexibility, in terms of relaxing the symmetry and the quadratic shape of the density function, which are enforced in the U-quadratic distribution – e.g., beta distribution and gamma distribution.
Moment generating function
Characteristic function
Beamforming
The quadratic U and inverted quadratic U distribution has an application to beamforming and pattern synthesis.
References
Continuous distributions |
https://en.wikipedia.org/wiki/Occupational%20Employment%20and%20Wage%20Statistics | The Occupational Employment and Wage Statistics) (OEWS) survey is a semi-annual survey of approximately 200,000 non-farm business establishments conducted by the Bureau of Labor Statistics (BLS), headquartered in Washington, DC with six regional offices and one office in each state. Until the spring of 2021 it was officially called the Occupational Employment Statistics (OES), and it is often cited or documented with that name or abbreviation.
Purpose
The OEWS survey is designed to produce estimates of employment and wages by occupation by four-digit North American Industry Classification System (NAICS) in each State-level Metropolitan Statistical Area (MSA-“urban”) or Balance-of-State (BOS-“rural”) geographic level, and their aggregates. Semi-annually, a "current" sample is combined with the immediate five prior samples to produce a “combined” sample of approximately 1.2 million establishments (6 x 200,000) to produce OES Estimates.
Process
During the sampling process, the National frame (using data collected by the Quarterly Census of Employment and Wages) of business establishments (approximately 7 million frame business establishments in-scope) is allocated for 1.2 million sample cases, then “divided” by 6 for each geography/industry cell (1.2 million/6=200,000).
After the sample of 200,000 is selected, data from each selected business establishment are collected by each state over the next nine months and reported back to BLS. These data include employment counts by occupation and wages paid by occupation. Wages are collected on an interval scale (rather than point data). Wages may be reported as an hourly wage or an annual wage. Wages include tips, but not overtime or any other bonus pay. Job benefits data are not collected.
After data are collected from the roughly 200,000 establishments, data are then combined with five prior collections' worth of data (for a total of 1.2 million) and weighted estimates are calculated. Most estimates are occupational-based (highly detailed using the 800 occupations in the Standard Occupational Classification System manual); estimates are calculated at the MSA-level with aggregate estimates at the state and national levels. A good example is that OES estimates calculate how many primary school teachers (including their wages) work in the Los Angeles-Long Beach-Glendale, CA MSA, which can be aggregated to how many primary school teachers (including their wages) work in California, which in turn can be aggregated to how many primary school teachers (including their wages) work in the United States.
The main occupational-based estimates that are calculated are:
employment counts
mean wages
median wages
percentile wages
upper 2/3-lower 1/3 (experienced vs. entry-level) wages
associated variances
Users of the OEWS data include colleges and trade school students, who use occupational information before entering the work force, temporary employment firms, and people who start businesses and want t |
https://en.wikipedia.org/wiki/Metric%20space%20aimed%20at%20its%20subspace | In mathematics, a metric space aimed at its subspace is a categorical construction that has a direct geometric meaning. It is also a useful step toward the construction of the metric envelope, or tight span, which are basic (injective) objects of the category of metric spaces.
Following , a notion of a metric space Y aimed at its subspace X is defined.
Informal introduction
Informally, imagine terrain Y, and its part X, such that wherever in Y you place a sharpshooter, and an apple at another place in Y, and then let the sharpshooter fire, the bullet will go through the apple and will always hit a point of X, or at least it will fly arbitrarily close to points of X – then we say that Y is aimed at X.
A priori, it may seem plausible that for a given X the superspaces Y that aim at X can be arbitrarily large or at least huge. We will see that this is not the case. Among the spaces which aim at a subspace isometric to X, there is a unique (up to isometry) universal one, Aim(X), which in a sense of canonical isometric embeddings contains any other space aimed at (an isometric image of) X. And in the special case of an arbitrary compact metric space X every bounded subspace of an arbitrary metric space Y aimed at X is totally bounded (i.e. its metric completion is compact).
Definitions
Let be a metric space. Let be a subset of , so that (the set with the metric from restricted to ) is a metric subspace of . Then
Definition. Space aims at if and only if, for all points of , and for every real , there exists a point of such that
Let be the space of all real valued metric maps (non-contractive) of . Define
Then
for every is a metric on . Furthermore, , where , is an isometric embedding of into ; this is essentially a generalisation of the Kuratowski-Wojdysławski embedding of bounded metric spaces into , where we here consider arbitrary metric spaces (bounded or unbounded). It is clear that the space is aimed at .
Properties
Let be an isometric embedding. Then there exists a natural metric map such that :
for every and .
Theorem The space Y above is aimed at subspace X if and only if the natural mapping is an isometric embedding.
Thus it follows that every space aimed at X can be isometrically mapped into Aim(X), with some additional (essential) categorical requirements satisfied.
The space Aim(X) is injective (hyperconvex in the sense of Aronszajn-Panitchpakdi) – given a metric space M, which contains Aim(X) as a metric subspace, there is a canonical (and explicit) metric retraction of M onto Aim(X) .
References
Metric geometry |
https://en.wikipedia.org/wiki/John%20Papaloizou | John Christopher Baillie Papaloizou FRS (born 1947) is a British theoretical physicist. Papaloizou is a professor at the Department of Applied Mathematics and Theoretical Physics (DAMTP) at the University of Cambridge. He works on the theory of accretion disks, with particular application to the formation of planets. He received his D.Phil. in 1972 from the University of Sussex under the supervision of Roger J. Tayler. The title of his thesis is The Vibrational Instability in Massive Stars.
He discovered the Papaloizou-Pringle instability together with Jim Pringle in 1984. Papaloizou also made major contributions in various areas such as the radial-orbit instability, toroidal modes in stars and different instabilities in accretion disks.
The asteroid 17063 Papaloizou is named after John Papaloizou.
Awards
2003 Fellow of the Royal Society
References
External links
Astrophysical Fluid Dynamics and Non-linear Patterns - Group Home Page
1947 births
Astronomers at the University of Cambridge
Alumni of Queen Mary University of London
Alumni of the University of Sussex
British theoretical physicists
20th-century British astronomers
English physicists
English people of Greek descent
Living people
Cambridge mathematicians
Fellows of the Royal Society |
https://en.wikipedia.org/wiki/Archimedes%20Geo3D | Archimedes Geo3D is a software package for dynamic geometry in three dimensions. It was released in Germany in March 2006 and won a German government award for outstanding educational software in 2007 .
Advanced features
Archimedes Geo3D can trace the movement of points, lines, segments, and circles and generate locus lines and surfaces. Arbitrary objects can be intersected with lines, locus lines, and planes.
References
External links
Archimedes Geo3D home page
Building a projection in Archimedes Geo3D
Building a double cone in Archimedes Geo3D
Interactive geometry software |
https://en.wikipedia.org/wiki/Cottlesville | Cottlesville is a rural community just outside Summerford on New World Island, Newfoundland and Labrador.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Cottlesville had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
References
Populated coastal places in Canada
Towns in Newfoundland and Labrador |
https://en.wikipedia.org/wiki/Membership%20statistics%20of%20the%20Church%20of%20Jesus%20Christ%20of%20Latter-day%20Saints | The Church of Jesus Christ of Latter-day Saints (LDS Church) releases membership, congregational, and related information on a regular basis. The latest membership information LDS Church releases includes a count of membership, stakes, wards, branches, missions, temples, and family history centers for the worldwide church and for individual countries and territories where the church is recognized. The latest information released was as of December 31, 2022.
At the end of 2022, the LDS Church had 31,330 congregations and a reported membership of 17,002,461.
Membership defined
The LDS Church defines membership as a count of living individuals who:
have been baptized and confirmed.
are under age nine and have been blessed but not baptized.
are not accountable because of intellectual disabilities, regardless of age.
are unblessed children under age eight when:
two member parents request it; or
one member parent requests it and the nonmember parent gives permission.
After baptism, blessing, or parental request stated above, membership must be recorded and maintained by the church to have and keep membership.
Membership considerations
In 2005, Peggy Fletcher Stack, longtime religion columnist for The Salt Lake Tribune, estimated that about one-third of the reported LDS membership was "active" (i.e., regularly attending church services and participating in other expected meetings and obligations). In 2005, this would have amounted to approximately 4 million active members among a worldwide LDS population of 12 million. Active membership varied from a high of 40 to 50 percent in congregations in North America and the Pacific Islands, to a low of about 25 percent in Latin America. Fletcher Stack's data was compiled from several sources, including a 2001 survey of religious affiliation by scholars at City University of New York and a demographer at LDS-owned Brigham Young University.
In 2003, church leader Dallin H. Oaks, noted that among recent converts "attrition is sharpest in the two months after baptism", which he attributed in part to difficulties adapting to the church's dietary code, the Word of Wisdom, that prohibits the use of alcohol, tobacco, coffee, and tea. In 2001, sociologist Armand Mauss estimated that about 50 percent of LDS converts in the US stopped attending church within a year of baptism, while outside the US the rate was about 70 percent.
Countries
The tables on this section represents Latter-day Saint membership, as reported by the LDS Church, as of December 31, 2022. Except where indicated, general population figures are based on the latest CIA estimates (primarily for 2023). Percentages of LDS members were calculated with this information. The link under the names of each country, territory, etc. corresponds to brief LDS history and statistical information for that particular area.
Congregations
Notes
*There are several areas that cover the US and Canada. This includes North America Central, North America North |
https://en.wikipedia.org/wiki/Noise-canceling%20microphone | A noise-canceling microphone is a microphone that is designed to filter ambient noise.
Technical details
The development is a special case of the differential microphone topology most commonly used to achieve directionality. All such microphones have at least two ports through which sound enters; a front port normally oriented toward the desired sound and another port that's more distant. The microphone's diaphragm is placed between the two ports; sound arriving from an ambient sound field reaches both ports more or less equally. Sound that's much closer to the front port than to the rear will make more of a pressure gradient between the front and back of the diaphragm, causing it to move more. The microphone's proximity effect is adjusted so that flat frequency response is achieved for sound sources very close to the front of the mic – typically 1 to 3 cm. Sounds arriving from other angles are subject to steep midrange and bass rolloff. Commercially and militarily useful noise-canceling microphones have been made since at least 1935 (Amelia Earhart used one on her 1935 flight from Hawaii to California) and have been made since the 1940s by Roanwell, Electro-Voice and others.
Alternative technologies
Another technique uses two or more microphones and active or passive circuitry to reduce the noise. The primary microphone is closer to the desired source (like a person's mouth). A second mic receives ambient noise. In a noisy environment, both microphones receive noise at a similar level, but the primary mic receives the desired sounds more strongly. Thus if one signal is subtracted from the other (in the simplest sense, by connecting the microphones out of phase) much of the noise is canceled while the desired sound is retained. Other techniques may be used as well, such as using a directional primary mic, to maximize the difference between the two signals and make the cancellation easier to do.
The internal electronic circuitry of an active noise-canceling mic attempts to subtract noise signal from the primary microphone. The circuit may employ passive or active noise canceling techniques to filter out the noise, producing an output signal that has a lower noise floor and a higher signal-to-noise ratio.
Applications
Call center headsets
Helicopter pilot headsets
Race car driver headsets
Shipboard communications
Video Gaming Headsets
See also
Noise-cancelling headphones
Acoustic quieting
Active noise control
References
U.S. Patent 7,248,708
U.S. Patent 7,162,041
Microphones
Noise reduction |
https://en.wikipedia.org/wiki/Malgrange%E2%80%93Ehrenpreis%20theorem | In mathematics, the Malgrange–Ehrenpreis theorem states that every non-zero linear differential operator with constant coefficients has a Green's function. It was first proved independently by and
.
This means that the differential equation
where P is a polynomial in several variables and δ is the Dirac delta function, has a distributional solution u. It can be used to show that
has a solution for any compactly supported distribution f. The solution is not unique in general.
The analogue for differential operators whose coefficients are polynomials (rather than constants) is false: see Lewy's example.
Proofs
The original proofs of Malgrange and Ehrenpreis were non-constructive as they used the Hahn–Banach theorem. Since then several constructive proofs have been found.
There is a very short proof using the Fourier transform and the Bernstein–Sato polynomial, as follows. By taking Fourier transforms the Malgrange–Ehrenpreis theorem is equivalent to the fact that every non-zero polynomial P has a distributional inverse. By replacing P by the product with its complex conjugate, one can also assume that P is non-negative. For non-negative polynomials P the existence of a distributional inverse follows from the existence of the Bernstein–Sato polynomial, which implies that Ps can be analytically continued as a meromorphic distribution-valued function of the complex variable s; the constant term of the Laurent expansion of Ps at s = −1 is then a distributional inverse of P.
Other proofs, often giving better bounds on the growth of a solution, are given in , and .
gives a detailed discussion of the regularity properties of the fundamental solutions.
A short constructive proof was presented in :
is a fundamental solution of P(∂), i.e., P(∂)E = δ, if Pm is the principal part of P, η ∈ Rn with Pm(η) ≠ 0, the real numbers λ0, ..., λm are pairwise different, and
References
Differential equations
Theorems in analysis
Schwartz distributions |
https://en.wikipedia.org/wiki/Paul%20Kline | Paul Kline (1937 – 25 September 1999) was a British psychologist noted for his contribution to psychometrics.
Career
Kline was originally educated in classics, in education, and in statistics: he studied at the University of Reading, University College Swansea, the University of Aberdeen and the University of Manchester. When he first joined the University of Exeter, it was as a staff member in the university's then Institute of Education. However, in 1969 he joined the Department of Psychology as a Lecturer, rising eventually to become the university's first Professor of Psychometrics.
Research
Kline was interested in depth psychology, especially theories of Sigmund Freud, the founder of psychoanalysis. He was also an expert in psychometrics and carried out extensive research in the statistical analysis of personality and intelligence.
In his 1972 book Fact and Fantasy in Freudian Theory, widely translated, he brought these two interests together, examining the objective evidence for various ideas of Freudian theory, finding that some, but not all, were supported by the evidence. He also wrote introductory books to psychometrics, for example An easy guide to factor analysis (1994). He was a prolific author, writing or editing at least 14 books, and over 150 scientific papers are listed under his name in Web of Science.
Among colleagues, Kline had a reputation as an opinionated controversialist who remained a genial and supportive colleague; he was revered by students for the wit and clarity of his lectures.
Publications
Fact and fantasy in Freudian Theory (1972/1981)
Psychometrics and psychology (1979)
Personality : measurement and theory (1983)
A handbook of test construction : introduction to psychometric design (1986)
Intelligence : the psychometric view (1991)
The Handbook of Psychological Testing (1993/2000)
An easy guide to factor analysis (1994)
The new psychometrics : science, psychology, and measurement (2000/2014)
References
1937 births
1999 deaths
English psychologists
Academics of the University of Exeter
20th-century British psychologists
Alumni of the University of Reading
Alumni of the University of Aberdeen |
https://en.wikipedia.org/wiki/Oscillator%20strength | In spectroscopy, oscillator strength is a dimensionless quantity that expresses the probability of absorption or emission of electromagnetic radiation in transitions between energy levels of an atom or molecule. For example, if an emissive state has a small oscillator strength, nonradiative decay will outpace radiative decay. Conversely, "bright" transitions will have large oscillator strengths. The oscillator strength can be thought of as the ratio between the quantum mechanical transition rate and the classical absorption/emission rate of a single electron oscillator with the same frequency as the transition.
Theory
An atom or a molecule can absorb light and undergo a transition from
one quantum state to another.
The oscillator strength of a transition from a lower state
to an upper state may be defined by
where is the mass of an electron and is
the reduced Planck constant. The quantum states 1,2, are assumed to have several
degenerate sub-states, which are labeled by . "Degenerate" means
that they all have the same energy .
The operator is the sum of the x-coordinates
of all electrons in the system, etc.:
The oscillator strength is the same for each sub-state .
The definition can be recast by inserting the Rydberg energy and Bohr radius
In case the matrix elements of are the same, we can get rid of the sum and of the 1/3 factor
Thomas–Reiche–Kuhn sum rule
To make equations of the previous section applicable to the states belonging to the continuum spectrum, they should be rewritten in terms of matrix elements of the momentum . In absence of magnetic field, the Hamiltonian can be written as , and calculating a commutator in the basis of eigenfunctions of results in the relation between matrix elements
.
Next, calculating matrix elements of a commutator in the same basis and eliminating matrix elements of , we arrive at
Because , the above expression results in a sum rule
where are oscillator strengths for quantum transitions between the states and . This is the Thomas-Reiche-Kuhn sum rule, and the term with has been omitted because in confined systems such as atoms or molecules the diagonal matrix element due to the time inversion symmetry of the Hamiltonian . Excluding this term eliminates divergency because of the vanishing denominator.
Sum rule and electron effective mass in crystals
In crystals, the electronic energy spectrum has a band structure . Near the minimum of an isotropic energy band, electron energy can be expanded in powers of as where is the electron effective mass. It can be shown that it satisfies the equation
Here the sum runs over all bands with . Therefore, the ratio of the free electron mass to its effective mass in a crystal can be considered as the oscillator strength for the transition of an electron from the quantum state at the bottom of the band into the same state.
See also
Atomic spectral line
Sum rule in quantum mechanics
Electronic band structure
Einstein coefficien |
https://en.wikipedia.org/wiki/Sieve%20estimator | In statistics, sieve estimators are a class of non-parametric estimators which use progressively more complex models to estimate an unknown high-dimensional function as more data becomes available, with the aim of asymptotically reducing error towards zero as the amount of data increases. This method is generally attributed to Ulf Grenander.
Method of sieves in positron emission tomography
Sieve estimators have been used extensively for estimating density functions in high-dimensional spaces such as in Positron emission tomography (PET). The first exploitation of Sieves in PET for solving the maximum-likelihood image reconstruction problem was by Donald Snyder and Michael Miller, where they stabilized the time-of-flight PET problem originally solved by Shepp and Vardi.
Shepp and Vardi's introduction of Maximum-likelihood estimators in emission tomography exploited the use of the Expectation-Maximization algorithm, which as it ascended towards the maximum-likelihood estimator developed a series of artifacts associated to the fact that the underlying emission density was of too high a dimension for any fixed sample size of Poisson measured counts. Grenander's method of sieves was used to stabilize the estimator, so that for any fixed sample size a resolution could be set which was consistent for the number of counts. As the observe PET imaging time would go to infinity, the dimension of the sieve would increase as well in such a manner that the density was appropriate for each sample size.
See also
Nonparametric regression
References
External links
Estimator |
https://en.wikipedia.org/wiki/Sieve%20method | Sieve method, or the method of sieves, can mean:
in mathematics and computer science, the sieve of Eratosthenes, a simple method for finding prime numbers
in number theory, any of a variety of methods studied in sieve theory
in combinatorics, the set of methods dealt with in sieve theory or more specifically, the inclusion–exclusion principle
in statistics, and particularly econometrics, the use of sieve estimators |
https://en.wikipedia.org/wiki/Georg%20N%C3%B6beling | Georg August Nöbeling (12 November 1907 – 16 February 2008) was a German mathematician.
Education and career
Born and raised in Lüdenscheid, Nöbeling studied mathematics and physics at University of Göttingen between 1927 and 1929 and University of Vienna, where he was a student of Karl Menger and received his PhD in 1931 on a generalization of the embedding theorem, which for one special case can be visualized by the Menger sponge. Nöbeling worked and researched in Menger's Mathematical Colloquium with Kurt Gödel, Franz Alt, Abraham Wald, Olga Taussky-Todd and others.
In 1933, he moved to the University of Erlangen, where he habilitated in 1935 under Otto Haupt and obtained a professorship at the same place in 1940. His work focused on analysis, topology, and geometry. 1968/1969 he solved Specker's theorem on abelian groups.
As Rector (1962–1963) of the University of Erlangen he oversaw the merge with the business college in Nuremberg. He also served twice as the chairman of the German Mathematical Society and is a member of the Bavarian Academy of Sciences and Humanities. He celebrated his 100th birthday in 2007.
Publications (selected)
Georg Nöbeling: "Über eine n-dimensionale Universalmenge im (on a n-dimensional universal set for metric spaces in ." Mathematische Annalen 104 (1931), pp. 71–80.
Georg Nöbeling: "Verallgemeinerung eines Satzes von E. Specker (Generalization of a Theorem by E. Specker)". Inventiones mathematicae 6 (1968), pp. 41–55.
See also
Universal space
Notes
External links
List of deceased fellows of the Bavarian Academy of Sciences and Humanities
1907 births
2008 deaths
20th-century German mathematicians
Mathematical analysts
Topologists
German centenarians
Men centenarians
University of Göttingen alumni
University of Vienna alumni
Academic staff of the University of Erlangen-Nuremberg
People from Lüdenscheid |
https://en.wikipedia.org/wiki/Henrik%20Bundgaard | Henrik Bundgaard (born 20 March 1975) is a former Danish professional football goalkeeper.
External links
Danish Superliga player statistics at danskfodbold.com
1975 births
Living people
Danish men's footballers
Aabyhøj IF players
AC Horsens players
Aarhus Gymnastikforening players
Danish Superliga players
Brabrand IF players
Men's association football goalkeepers |
https://en.wikipedia.org/wiki/Schuette%E2%80%93Nesbitt%20formula | In mathematics, the Schuette–Nesbitt formula is a generalization of the inclusion–exclusion principle. It is named after Donald R. Schuette and Cecil J. Nesbitt.
The probabilistic version of the Schuette–Nesbitt formula has practical applications in actuarial science, where it is used to calculate the net single premium for life annuities and life insurances based on the general symmetric status.
Combinatorial versions
Consider a set and subsets . Let
denote the number of subsets to which belongs, where we use the indicator functions of the sets . Furthermore, for each , let
denote the number of intersections of exactly sets out of , to which belongs, where the intersection over the empty index set is defined as , hence . Let denote a vector space over a field such as the real or complex numbers (or more generally a module over a ring with multiplicative identity). Then, for every choice of ,
where denotes the indicator function of the set of all with , and is a binomial coefficient. Equality () says that the two -valued functions defined on are the same.
Proof of ()
We prove that () holds pointwise. Take and define .
Then the left-hand side of () equals .
Let denote the set of all those indices such that , hence contains exactly indices.
Given with elements, then belongs to the intersection if and only if is a subset of .
By the combinatorial interpretation of the binomial coefficient, there are such subsets (the binomial coefficient is zero for ).
Therefore the right-hand side of () evaluated at equals
where we used that the first binomial coefficient is zero for .
Note that the sum (*) is empty and therefore defined as zero for .
Using the factorial formula for the binomial coefficients, it follows that
Rewriting (**) with the summation index und using the binomial formula for the third equality shows that
which is the Kronecker delta. Substituting this result into the above formula and noting that choose equals for , it follows that the right-hand side of () evaluated at also reduces to .
Representation in the polynomial ring
As a special case, take for the polynomial ring with the indeterminate . Then () can be rewritten in a more compact way as
This is an identity for two polynomials whose coefficients depend on , which is implicit in the notation.
Proof of () using (): Substituting for into () and using the binomial formula shows that
which proves ().
Representation with shift and difference operators
Consider the linear shift operator and the linear difference operator , which we define here on the sequence space of by
and
Substituting in () shows that
where we used that with denoting the identity operator. Note that and equal the identity operator on the sequence space, and denote the -fold composition.
Let denote the 0th component of the -fold composition applied to , where denotes the identity. Then () can be rewritten in a more compact way as
Probabilistic versions
C |
https://en.wikipedia.org/wiki/Moving%20average%20crossover | In the statistics of time series, and in particular the stock market technical analysis, a moving-average crossover occurs when, on plotting two moving averages each based on different degrees of smoothing, the traces of these moving averages cross. It does not predict future direction but shows trends. This indicator uses two (or more) moving averages, a slower moving average and a faster moving average. The faster moving average is a short term moving average. For end-of-day stock markets, for example, it may be 5-, 10- or 25-day period while the slower moving average is medium or long term moving average (e.g. 50-, 100- or 200-day period). A short term moving average is faster because it only considers prices over short period of time and is thus more reactive to daily price changes. On the other hand, a long term moving average is deemed slower as it encapsulates prices over a longer period and is more lethargic. However, it tends to smooth out price noises which are often reflected in short term moving averages.
A moving average, as a line by itself, is often overlaid in price charts to indicate price trends. A crossover occurs when a faster moving average (i.e., a shorter period moving average) crosses a slower moving average (i.e. a longer period moving average). In other words, this is when the shorter period moving average line crosses a longer period moving average line. In stock investing, this meeting point is used either to enter (buy or sell) or exit (sell or buy) the market.
The particular case where simple equally weighted moving-averages are used is sometimes called a simple moving-average (SMA) crossover. Such a crossover can be used to signal a change in trend and can be used to trigger a trade in a black box trading system.
There are several types of moving average cross traders use in trading. Golden cross occurs when 50 days simple moving average crosses 200 days simple moving average from below. Death cross is an opposite situation, when 50 days simple moving average crosses 200 days simple moving average from above. Death cross is not a reliable indicator of future market declines.
References
Time series
Technical indicators
External links
Understanding Moving Average Crossovers and how they are used in technical analysis
Tuned, Using Moving Average Crossovers Programmatically |
https://en.wikipedia.org/wiki/Malgrange%20preparation%20theorem | In mathematics, the Malgrange preparation theorem is an analogue of the Weierstrass preparation theorem for smooth functions. It was conjectured by René Thom and proved by .
Statement of Malgrange preparation theorem
Suppose that f(t,x) is a smooth complex function of t∈R and x∈Rn near the origin, and let k be the smallest integer such that
Then one form of the preparation theorem states that near the origin f can be written as the product of a smooth function c that is nonzero at the origin and a smooth function that as a function of t is a polynomial of degree k. In other words,
where the functions c and a are smooth and c is nonzero at the origin.
A second form of the theorem, occasionally called the Mather division theorem, is a sort of "division with remainder" theorem: it says that if f and k satisfy the conditions above and g is a smooth function near the origin, then we can write
where q and r are smooth, and as a function of t, r is a polynomial of degree less than k. This means that
for some smooth functions rj(x).
The two forms of the theorem easily imply each other: the first form is the special case of the "division with remainder" form where g is tk, and the division with remainder form follows from the first form of the theorem as we may assume that f as a function of t is a polynomial of degree k.
If the functions f and g are real, then the functions c, a, q, and r can also be taken to be real. In the case of the Weierstrass preparation theorem these functions are uniquely determined by f and g, but uniqueness no longer holds for the Malgrange preparation theorem.
Proof of Malgrange preparation theorem
The Malgrange preparation theorem can be deduced from the Weierstrass preparation theorem. The obvious way of doing this does not work: although smooth functions have a formal power series expansion at the origin, and the Weierstrass preparation theorem applies to formal power series, the formal power series will not usually converge to smooth functions near the origin. Instead one can use the idea of decomposing a smooth function as a sum of analytic functions by applying a partition of unity to its Fourier transform.
For a proof along these lines see or
Algebraic version of the Malgrange preparation theorem
The Malgrange preparation theorem can be restated as a theorem about modules over rings of smooth, real-valued germs. If X is a manifold, with p∈X, let C∞p(X) denote the ring of real-valued germs of smooth functions at p on X. Let Mp(X) denote the unique maximal ideal of C∞p(X), consisting of germs which vanish at p. Let A be a C∞p(X)-module, and let f:X → Y be a smooth function between manifolds. Let q = f(p). f induces a ring homomorphism f*:C∞q(Y) → C∞p(X) by composition on the right with f. Thus we can view A as a C∞q(Y)-module. Then the Malgrange preparation theorem says that if A is a finitely-generated C∞p(X)-module, then A is a finitely-generated C∞q(Y)-module if and only if A/Mq(Y)A is a finite-dim |
https://en.wikipedia.org/wiki/Cardinal%20function | In mathematics, a cardinal function (or cardinal invariant) is a function that returns cardinal numbers.
Cardinal functions in set theory
The most frequently used cardinal function is a function that assigns to a set A its cardinality, denoted by | A |.
Aleph numbers and beth numbers can both be seen as cardinal functions defined on ordinal numbers.
Cardinal arithmetic operations are examples of functions from cardinal numbers (or pairs of them) to cardinal numbers.
Cardinal characteristics of a (proper) ideal I of subsets of X are:
The "additivity" of I is the smallest number of sets from I whose union is not in I any more. As any ideal is closed under finite unions, this number is always at least ; if I is a σ-ideal, then
The "covering number" of I is the smallest number of sets from I whose union is all of X. As X itself is not in I, we must have add(I) ≤ cov(I).
The "uniformity number" of I (sometimes also written ) is the size of the smallest set not in I. Assuming I contains all singletons, add(I) ≤ non(I).
The "cofinality" of I is the cofinality of the partial order (I, ⊆). It is easy to see that we must have non(I) ≤ cof(I) and cov(I) ≤ cof(I).
In the case that is an ideal closely related to the structure of the reals, such as the ideal of Lebesgue null sets or the ideal of meagre sets, these cardinal invariants are referred to as cardinal characteristics of the continuum.
For a preordered set the bounding number and dominating number are defined as
In PCF theory the cardinal function is used.
Cardinal functions in topology
Cardinal functions are widely used in topology as a tool for describing various topological properties. Below are some examples. (Note: some authors, arguing that "there are no finite cardinal numbers in general topology", prefer to define the cardinal functions listed below so that they never taken on finite cardinal numbers as values; this requires modifying some of the definitions given below, for example by adding "" to the right-hand side of the definitions, etc.)
Perhaps the simplest cardinal invariants of a topological space are its cardinality and the cardinality of its topology, denoted respectively by and
The weight of a topological space is the cardinality of the smallest base for When the space is said to be second countable.
The -weight of a space is the cardinality of the smallest -base for (A -base is a set of nonempty opens whose supersets includes all opens.)
The network weight of is the smallest cardinality of a network for A network is a family of sets, for which, for all points and open neighbourhoods containing there exists in for which
The character of a topological space at a point is the cardinality of the smallest local base for The character of space is When the space is said to be first countable.
The density of a space is the cardinality of the smallest dense subset of When the space is said to be separable.
The Lindelö |
https://en.wikipedia.org/wiki/National%20Bureau%20of%20Statistics%20of%20China | The National Bureau of Statistics () is a deputy-cabinet level agency directly under the State Council of China. Established in August 1952, the bureau is responsible for collection, investigation, research and publication of statistics concerning the nation's economy, population and other aspects of the society.
Kang Yi has served as the commissioner of the bureau since 3 March 2022.
Responsibilities
The bureau's authority and responsibilities are defined in Statistics Law of the People's Republic of China. It is responsible for the research of the nation's overall statistics and oversee the operations of its local counterparts.
Organizations
The bureau is led by a commissioner, with several deputy commissioners (currently four), a chief methodologist, a chief economist, and a chief information officer. It is composed of 18 departments, oversees 12 affiliated institutions and manages 32 survey organizations stationed in respective provinces. It also operates China Statistics Press (), which was founded in 1955.
The national bureau has 535 employees as authorized by the State Council.
Commissioner
Xue Muqiao (August 1952 – November 1958)
Jia Qiyun (November 1958 – June 1961)
Wang Sihua (June 1961 – December 1969)
Chen Xian (September 1974 – October 1981)
Li Chengrui (October 1981 – May 1984)
Zhang Sai (May 1984 – February 1997)
Liu Hong (February 1997 – June 2000)
Zhu Zhixin (June 2000 – March 2003)
Li Deshui (March 2003 – March 2006)
Qiu Xiaohua (March 2006 – October 2006)
Xie Fuzhan (October 2006 – September 2008)
Ma Jiantang (September 2008 – April 2015)
Wang Bao'an (April 2015 – January 2016)
Ning Jizhe (February 2016 – March 2022)
Kang Yi (March 2022 – present)
Access
Its Statistical Communiqué on the National Economic and Social Development and the China Statistical Yearbook are the bureau's most notable publications. It also runs and publishes the national census of economy, population and agriculture.
Internet
National Data (National Statistical Data Repository) is operated by the bureau, which have both Chinese and English interfaces. All publishable statistical results by the bureau are released on this website, includes monthly, quarterly, and annual data of price indices, industrial data etc.
Since December 2018, the agency began to release detailed dataset to authorized researchers and universities, which uses an application-based system for researchers resides within China. This includes dataset from the 3rd economic census, 6th population census, 3rd agricultural census, sampled 1% population survey of 2015, residents income survey and financials survey on industrial enterprises above designated size.
Archive
The bureau also operates an archive filled with almanacs ever published by the bureau since 1982, some of the files are not digitalized, hence only accessible through the archive. Citizens can access the archive with their national identification card.
See also
Census in China
China microcensus
Economic st |
https://en.wikipedia.org/wiki/National%20Bureau%20of%20Statistics | National Bureau of Statistics may refer to:
National Bureau of Statistics of China
National Bureau of Statistics of Moldova
National Bureau of Statistics, Nigeria
National Bureau of Statistics of Tanzania
Australian Bureau of Statistics
See also
List of national and international statistical services |
https://en.wikipedia.org/wiki/Non-Hausdorff%20manifold | In geometry and topology, it is a usual axiom of a manifold to be a Hausdorff space. In general topology, this axiom is relaxed, and one studies non-Hausdorff manifolds: spaces locally homeomorphic to Euclidean space, but not necessarily Hausdorff.
Examples
Line with two origins
The most familiar non-Hausdorff manifold is the line with two origins, or bug-eyed line. This is the quotient space of two copies of the real line
and obtained by identifying points and whenever
An equivalent description of the space is to take the real line and replace the origin with two origins and The subspace retains its usual Euclidean topology. And a local base of open neighborhoods at each origin is formed by the sets with an open neighborhood of in
For each origin the subspace obtained from by replacing with is an open neighborhood of homeomorphic to Since every point has a neighborhood homeomorphic to the Euclidean line, the space is locally Euclidean. In particular, it is locally Hausdorff, in the sense that each point has a Hausdorff neighborhood. But the space is not Hausdorff, as every neighborhood of intersect every neighbourhood of It is however a T1 space.
The space is second countable.
The space exhibits several phenomena that don't happen in Hausdorff spaces:
The space is path connected but not arc connected. In particular, to get a path from one origin to the other one can first move left from to within the line through the first origin, and then move back to the right from to within the line through the second origin. But it is impossible to join the two origins with an arc, which is an injective path; intuitively, if one moves first to the left, one has to eventually backtrack and move back to the right.
The intersection of two compact sets need not be compact. For example, the sets and are compact, but their intersection is not.
The space is locally compact in the sense that every point has a local base of compact neighborhoods. But the line through one origin does not contain a closed neighborhood of that origin, as any neighborhood of one origin contains the other origin in its closure. So the space is not a regular space, and even though every point has at least one closed compact neighborhood, the origin points do not admit a local base of closed compact neighborhoods.
The space does not have the homotopy type of a CW-complex, or of any Hausdorff space.
Line with many origins
The line with many origins is similar to the line with two origins, but with an arbitrary number of origins. It is constructed by taking an arbitrary set with the discrete topology and taking the quotient space of that identifies points and whenever Equivalently, it can be obtained from by replacing the origin with many origins one for each The neighborhoods of each origin are described as in the two origin case.
If there are infinitely many origins, the space illustrates that the closure of a compact se |
https://en.wikipedia.org/wiki/Geometry%20and%20topology | In mathematics, geometry and topology is an umbrella term for the historically distinct disciplines of geometry and topology, as general frameworks allow both disciplines to be manipulated uniformly, most visibly in local to global theorems in Riemannian geometry, and results like the Gauss–Bonnet theorem and Chern–Weil theory.
Sharp distinctions between geometry and topology can be drawn, however, as discussed below.
It is also the title of a journal Geometry & Topology that covers these topics.
Scope
It is distinct from "geometric topology", which more narrowly involves applications of topology to geometry.
It includes:
Differential geometry and topology
Geometric topology (including low-dimensional topology and surgery theory)
It does not include such parts of algebraic topology as homotopy theory, but some areas of geometry and topology (such as surgery theory, particularly algebraic surgery theory) are heavily algebraic.
Distinction between geometry and topology
Geometry has local structure (or infinitesimal), while topology only has global structure. Alternatively, geometry has continuous moduli, while topology has discrete moduli.
By examples, an example of geometry is Riemannian geometry, while an example of topology is homotopy theory. The study of metric spaces is geometry, the study of topological spaces is topology.
The terms are not used completely consistently: symplectic manifolds are a boundary case, and coarse geometry is global, not local.
Local versus global structure
By definition, differentiable manifolds of a fixed dimension are all locally diffeomorphic to Euclidean space, so aside from dimension, there are no local invariants. Thus, differentiable structures on a manifold are topological in nature.
By contrast, the curvature of a Riemannian manifold is a local (indeed, infinitesimal) invariant (and is the only local invariant under isometry).
Moduli
If a structure has a discrete moduli (if it has no deformations, or if a deformation of a structure is isomorphic to the original structure), the structure is said to be rigid, and its study (if it is a geometric or topological structure) is topology. If it has non-trivial deformations, the structure is said to be flexible, and its study is geometry.
The space of homotopy classes of maps is discrete, so studying maps up to homotopy is topology.
Similarly, differentiable structures on a manifold is usually a discrete space, and hence an example of topology, but exotic R4s have continuous moduli of differentiable structures.
Algebraic varieties have continuous moduli spaces, hence their study is algebraic geometry. These are finite-dimensional moduli spaces.
The space of Riemannian metrics on a given differentiable manifold is an infinite-dimensional space.
Symplectic manifolds
Symplectic manifolds are a boundary case, and parts of their study are called symplectic topology and symplectic geometry.
By Darboux's theorem, a symplectic manifold has no local structu |
https://en.wikipedia.org/wiki/Mazur%20manifold | In differential topology, a branch of mathematics, a Mazur manifold is a contractible, compact, smooth four-dimensional manifold (with boundary) which is not diffeomorphic to the standard 4-ball. The boundary of a Mazur manifold is necessarily a homology 3-sphere.
Frequently the term Mazur manifold is restricted to a special class of the above definition: 4-manifolds that have a handle decomposition containing exactly three handles: a single 0-handle, a single 1-handle and single 2-handle. This is equivalent to saying the manifold must be of the form union a 2-handle. An observation of Mazur's shows that the double of such manifolds is diffeomorphic to with the standard smooth structure.
History
Barry Mazur and Valentin Poenaru discovered these manifolds simultaneously. Akbulut and Kirby showed that the Brieskorn homology spheres , and are boundaries of Mazur manifolds. These results were later generalized to other contractible manifolds by Casson, Harer and Stern. One of the Mazur manifolds is also an example of an Akbulut cork which can be used to construct exotic 4-manifolds.
Mazur manifolds have been used by Fintushel and Stern to construct exotic actions of a group of order 2 on the 4-sphere.
Mazur's discovery was surprising for several reasons:
Every smooth homology sphere in dimension is homeomorphic to the boundary of a compact contractible smooth manifold. This follows from the work of Kervaire and the h-cobordism theorem. Slightly more strongly, every smooth homology 4-sphere is diffeomorphic to the boundary of a compact contractible smooth 5-manifold (also by the work of Kervaire). But not every homology 3-sphere is diffeomorphic to the boundary of a contractible compact smooth 4-manifold. For example, the Poincaré homology sphere does not bound such a 4-manifold because the Rochlin invariant provides an obstruction.
The h-cobordism Theorem implies that, at least in dimensions there is a unique contractible -manifold with simply-connected boundary, where uniqueness is up to diffeomorphism. This manifold is the unit ball . It's an open problem as to whether or not admits an exotic smooth structure, but by the h-cobordism theorem, such an exotic smooth structure, if it exists, must restrict to an exotic smooth structure on . Whether or not admits an exotic smooth structure is equivalent to another open problem, the smooth Poincaré conjecture in dimension four. Whether or not admits an exotic smooth structure is another open problem, closely linked to the Schoenflies problem in dimension four.
Mazur's observation
Let be a Mazur manifold that is constructed as union a 2-handle. Here is a sketch of Mazur's argument that the double of such a Mazur manifold is . is a contractible 5-manifold constructed as union a 2-handle. The 2-handle can be unknotted since the attaching map is a framed knot in the 4-manifold . So union the 2-handle is diffeomorphic to . The boundary of is . But the boundary of is the double |
https://en.wikipedia.org/wiki/%28a%2Cb%2C0%29%20class%20of%20distributions | In probability theory, a member of the (a, b, 0) class of distributions is any distribution of a discrete random variable N whose values are nonnegative integers whose probability mass function satisfies the recurrence formula
for some real numbers a and b, where .
Only the Poisson, binomial and negative binomial distributions satisfy the full form of this relationship. These are also the three discrete distributions among the six members of the natural exponential family with quadratic variance functions (NEF–QVF).
More general distributions can be defined by fixing some initial values of pj and applying the recursion to define subsequent values. This can be of use in fitting distributions to empirical data. However, some further well-known distributions are available if the recursion above need only hold for a restricted range of values of k: for example the logarithmic distribution and the discrete uniform distribution.
The (a, b, 0) class of distributions has important applications in actuarial science in the context of loss models.
Properties
Sundt proved that only the binomial distribution, the Poisson distribution and the negative binomial distribution belong to this class of distributions, with each distribution being represented by a different sign of a. Furthermore, it was shown by Fackler that there is a universal formula for all three distributions, called the (united) Panjer distribution.
The more usual parameters of these distributions are determined by both a and b. The properties of these distributions in relation to the present class of distributions are summarised in the following table. Note that denotes the probability generating function.
Note that the Panjer distribution reduces to the Poisson distribution in the limit case ; it coincides with the negative binomial distribution for positive, finite real numbers , and it equals the binomial distribution for negative integers .
Plotting
An easy way to quickly determine whether a given sample was taken from a distribution from the (a,b,0) class is by graphing the ratio of two consecutive observed data (multiplied by a constant) against the x-axis.
By multiplying both sides of the recursive formula by , you get
which shows that the left side is obviously a linear function of . When using a sample of data, an approximation of the 's need to be done. If represents the number of observations having the value , then is an unbiased estimator of the true .
Therefore, if a linear trend is seen, then it can be assumed that the data is taken from an (a,b,0) distribution. Moreover, the slope of the function would be the parameter , while the ordinate at the origin would be .
See also
Panjer recursion
References
Discrete distributions
Systems of probability distributions
Actuarial science |
https://en.wikipedia.org/wiki/Thomas%20D.%20Baird | Thomas D. Baird (July 14, 1819 – June 9, 1873) was an educator born in Newark, Ohio, United States. Baird was the first professor of mathematics of Westminster College, and the fifth principal of Baltimore City College. He died in Baltimore, Maryland.
Biography
Thomas D. Baird, the second son of the Presbyterian minister, Reverend Thomas D. Baird, was born on July 14, 1819, in Newark, Ohio. Baird was educated at Jefferson College in Pennsylvania, where he received both his Bachelor of Arts and Master of Arts.
In 1839, he began his career as an educator by teaching mathematics in Baltimore, Maryland. During this time, he also studied law and was admitted to the bar in 1844, although he was never a practicing lawyer. In 1847, Baird became a professor of mathematics at Marshall College in Mercersburg, Pennsylvania. He held this position for three years before deciding to open a private school in Baltimore, which was devoted to the study of classics and mathematics. Baird presided over the school for four years before he was elected professor of mathematics at Westminster College in Missouri. He remained in this position for three years before returning to Baltimore to serve as the fifth principal and professor of moral and mental philosophy at the Central High School of Baltimore—later renamed Baltimore City College, the third oldest public high school in the United States. Baird held this position until his death on June 9, 1873.
Baird received a Ph.D. from Concordia College, Missouri and an LL.D. from Centre College, Kentucky.
Central High School of Baltimore
Under Baird's leadership, the school underwent a series of improvements aimed at strengthening the caliber of students admitted to the school and the quality of education. Baird increased the rigor of entrance examinations to the school as well as standards required for promotion to the next grade. This was evidenced in his first year as principal, when only about a third of the first year class was promoted to the second year. Baird also brought about changes in discipline at the school. He eliminated corporal punishment, and instead instituted a system in which a student's behavior was factored into his standing at the school.
In 1866, under the recommendation of Baird, a process was begun to elevate the high school to the status of a college. The school was renamed the Baltimore City College and a five-year course of study was created, in addition to the standard four-year course. Despite the change, the school was never elevated, and in 1869, Baird terminated the five-year course.
References
Baltimore City College faculty
Washington & Jefferson College alumni
Centre College alumni
Westminster College (Missouri) faculty
People from Newark, Ohio
1819 births
1873 deaths |
https://en.wikipedia.org/wiki/Absorbing%20element | In mathematics, an absorbing element (or annihilating element) is a special type of element of a set with respect to a binary operation on that set. The result of combining an absorbing element with any element of the set is the absorbing element itself. In semigroup theory, the absorbing element is called a zero element because there is no risk of confusion with other notions of zero, with the notable exception: under additive notation zero may, quite naturally, denote the neutral element of a monoid. In this article "zero element" and "absorbing element" are synonymous.
Definition
Formally, let be a set S with a closed binary operation • on it (known as a magma). A zero element is an element z such that for all s in S, . This notion can be refined to the notions of left zero, where one requires only that , and right zero, where .
Absorbing elements are particularly interesting for semigroups, especially the multiplicative semigroup of a semiring. In the case of a semiring with 0, the definition of an absorbing element is sometimes relaxed so that it is not required to absorb 0; otherwise, 0 would be the only absorbing element.
Properties
If a magma has both a left zero z and a right zero z′, then it has a zero, since .
A magma can have at most one zero element.
Examples
The most well known example of an absorbing element comes from elementary algebra, where any number multiplied by zero equals zero. Zero is thus an absorbing element.
The zero of any ring is also an absorbing element. For an element r of a ring R, r0=r(0+0)=r0+r0, so 0=r0, as zero is the unique element a for which r-r=a for any r in the ring R. This property holds true also in a rng since multiplicative identity isn't required.
Floating point arithmetics as defined in IEEE-754 standard contains a special value called Not-a-Number ("NaN"). It is an absorbing element for every operation; i.e., , , etc.
The set of binary relations over a set X, together with the composition of relations forms a monoid with zero, where the zero element is the empty relation (empty set).
The closed interval with is also a monoid with zero, and the zero element is 0.
More examples:
See also
Idempotent (ring theory)an element x of a ring such that x2 = x
Identity element
Null semigroup
Notes
References
M. Kilp, U. Knauer, A.V. Mikhalev, Monoids, Acts and Categories with Applications to Wreath Products and Graphs, De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000, .
External links
Absorbing element at PlanetMath
Semigroup theory
Absorbing element
Algebraic properties of elements |
https://en.wikipedia.org/wiki/Power%20of%20three | In mathematics, a power of three is a number of the form where is an integer, that is, the result of exponentiation with number three as the base and integer as the exponent.
In a context where only integers are considered, is restricted to non-negative values, so there are 1, 3, and 3 multiplied by itself a certain number of times.
The first ten powers of 3 for non-negative values of are:
1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, ...
Applications
The powers of three give the place values in the ternary numeral system.
Graph theory
In graph theory, powers of three appear in the Moon–Moser bound on the number of maximal independent sets of an -vertex graph, and in the time analysis of the Bron–Kerbosch algorithm for finding these sets. Several important strongly regular graphs also have a number of vertices that is a power of three, including the Brouwer–Haemers graph (81 vertices), Berlekamp–van Lint–Seidel graph (243 vertices), and Games graph (729 vertices).
Enumerative combinatorics
In enumerative combinatorics, there are signed subsets of a set of elements. In polyhedral combinatorics, the hypercube and all other Hanner polytopes have a number of faces (not counting the empty set as a face) that is a power of three. For example, a , or square, has 4 vertices, 4 edges and 1 face, and . Kalai's conjecture states that this is the minimum possible number of faces for a centrally symmetric polytope.
Inverse power of three lengths
In recreational mathematics and fractal geometry, inverse power-of-three lengths occur in the constructions leading to the Koch snowflake, Cantor set, Sierpinski carpet and Menger sponge, in the number of elements in the construction steps for a Sierpinski triangle, and in many formulas related to these sets. There are possible states in an -disk Tower of Hanoi puzzle or vertices in its associated Hanoi graph. In a balance puzzle with weighing steps, there are possible outcomes (sequences where the scale tilts left or right or stays balanced); powers of three often arise in the solutions to these puzzles, and it has been suggested that (for similar reasons) the powers of three would make an ideal system of coins.
Perfect totient numbers
In number theory, all powers of three are perfect totient numbers. The sums of distinct powers of three form a Stanley sequence, the lexicographically smallest sequence that does not contain an arithmetic progression of three elements. A conjecture of Paul Erdős states that this sequence contains no powers of two other than 1, 4, and 256.
Graham's number
Graham's number, an enormous number arising from a proof in Ramsey theory, is (in the version popularized by Martin Gardner) a power of three.
However, the actual publication of the proof by Ronald Graham used a different number.
Table of values
All of these numbers above represent exponents in base-3, as mentioned above.
Powers of three whose exponents are powers of three
All of these numbers above end |
https://en.wikipedia.org/wiki/Topological%20index | In the fields of chemical graph theory, molecular topology, and mathematical chemistry, a topological index, also known as a connectivity index, is a type of a molecular descriptor that is calculated based on the molecular graph of a chemical compound. Topological indices are numerical parameters of a graph which characterize its topology and are usually graph invariant. Topological indices are used for example in the development of quantitative structure-activity relationships (QSARs) in which the biological activity or other properties of molecules are correlated with their chemical structure.
Calculation
Topological descriptors are derived from hydrogen-suppressed molecular graphs, in which the atoms are represented by vertices and the bonds by edges. The connections between the atoms can be described by various types of topological matrices (e.g., distance or adjacency matrices), which can be mathematically manipulated so as to derive a single number, usually known as graph invariant, graph-theoretical index or topological index. As a result, the topological index can be defined as two-dimensional descriptors that can be easily calculated from the molecular graphs, and do not depend on the way the graph is depicted or labeled and no need of energy minimization of the chemical structure.
Types
The simplest topological indices do not recognize double bonds and atom types (C, N, O etc.) and ignore hydrogen atoms ("hydrogen suppressed") and defined for connected undirected molecular graphs only. More sophisticated topological indices also take into account the hybridization state of each of the atoms contained in the molecule. The Hosoya index is the first topological index recognized in chemical graph theory, and it is often referred to as "the" topological index. Other examples include the Wiener index, Randić's molecular connectivity index, Balaban’s J index, and the TAU descriptors. The extended topochemical atom (ETA) indices have been developed based on refinement of TAU descriptors.
Global and local indices
Hosoya index and Wiener index are global (integral) indices to describe entire molecule, Bonchev and Polansky introduced local (differential) index for every atom in a molecule. Another examples of local indices are modifications of Hosoya index.
Discrimination capability and superindices
A topological index may have the same value for a subset of different molecular graphs, i.e. the index is unable to discriminate the graphs from this subset. The discrimination capability is very important characteristic of topological index. To increase the discrimination capability a few topological indices may be combined to superindex.
Computational complexity
Computational complexity is another important characteristic of topological index. The Wiener index, Randic's molecular connectivity index, Balaban's J index may be calculated by fast algorithms, in contrast to Hosoya index and its modifications for which non-exponential a |
https://en.wikipedia.org/wiki/Haruo%20Hosoya | is a Japanese chemist and emeritus professor of Ochanomizu University, Tokyo, Japan. He is the namesake of the Hosoya index used in discrete mathematics and computational chemistry.
Hosoya was born in Kamakura, Japan to a family of an office worker. During 1955-1959 he studied at the University of Tokyo. In 1964 he wrote his Ph.D. thesis, "Study on the Structure of Reactive Intermediates and Reaction Mechanism". After postdoc work abroad (Ann Arbor, Michigan, with prof. John Platt), in 1969 he became associate professor at the Ochanomizu University, where he worked for 33 years until his retirement in 2002. After retirement he keeps working in computational chemistry.
In 1971, Hosoya defined the topological index (a graph invariant) now known as the Hosoya index as the total number of matchings of a graph plus 1. The Hosoya index is often used in computer (mathematical) chemistry investigations for organic compounds.
In 2002-2003 the Internet Electronic Journal of Molecular Design dedicated a series of issues to commemorate the 65th birthday of professor Hosoya.
Hosoya's article "The Topological Index Z Before and After 1971" describes the history of the notion and the associated inside stories and details other Hosoya's achievements.
Hosoya also introduced the triangle of numbers known as Hosoya's triangle (originally "Fibonacci triangle", but that name can be ambiguous).
Notes
Japanese chemists
1936 births
Living people
Mathematical chemistry
Computational chemists
Riken personnel
Academic staff of Ochanomizu University |
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