source
stringlengths 31
168
| text
stringlengths 51
3k
|
|---|---|
https://en.wikipedia.org/wiki/Nandra%C5%BE
|
Nandraž () is a village and municipality in Revúca District in the Banská Bystrica Region of Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Revúca District
|
https://en.wikipedia.org/wiki/Meijer%20G-function
|
In mathematics, the G-function was introduced by as a very general function intended to include most of the known special functions as particular cases. This was not the only attempt of its kind: the generalized hypergeometric function and the MacRobert E-function had the same aim, but Meijer's G-function was able to include those as particular cases as well. The first definition was made by Meijer using a series; nowadays the accepted and more general definition is via a line integral in the complex plane, introduced in its full generality by Arthur Erdélyi in 1953.
With the modern definition, the majority of the established special functions can be represented in terms of the Meijer G-function. A notable property is the closure of the set of all G-functions not only under differentiation but also under indefinite integration. In combination with a functional equation that allows to liberate from a G-function G(z) any factor zρ that is a constant power of its argument z, the closure implies that whenever a function is expressible as a G-function of a constant multiple of some constant power of the function argument, f(x) = G(cxγ), the derivative and the antiderivative of this function are expressible so too.
The wide coverage of special functions also lends power to uses of Meijer's G-function other than the representation and manipulation of derivatives and antiderivatives. For example, the definite integral over the positive real axis of any function g(x) that can be written as a product G1(cxγ)·G2(dxδ) of two G-functions with rational γ/δ equals just another G-function, and generalizations of integral transforms like the Hankel transform and the Laplace transform and their inverses result when suitable G-function pairs are employed as transform kernels.
A still more general function, which introduces additional parameters into Meijer's G-function, is Fox's H-function and is used for Matrix transform by Ram Kishore Saxena
One application of the Meijer G-function has been the particle spectrum of radiation from an inertial horizon in the moving mirror model of the dynamical Casimir effect .
Definition of the Meijer G-function
A general definition of the Meijer G-function is given by the following line integral in the complex plane :
where Γ denotes the gamma function. This integral is of the so-called Mellin–Barnes type, and may be viewed as an inverse Mellin transform. The definition holds under the following assumptions:
0 ≤ m ≤ q and 0 ≤ n ≤ p, where m, n, p and q are integer numbers
ak − bj ≠ 1, 2, 3, ... for k = 1, 2, ..., n and j = 1, 2, ..., m, which implies that no pole of any Γ(bj − s), j = 1, 2, ..., m, coincides with any pole of any Γ(1 − ak + s), k = 1, 2, ..., n
z ≠ 0
Note that for historical reasons the first lower and second upper index refer to the top parameter row, while the second lower and first upper index refer to the bottom parameter row. One often encounters the following more synthetic notation using vectors
|
https://en.wikipedia.org/wiki/List%20of%20census%20divisions%20of%20Manitoba
|
Statistics Canada divides the province of Manitoba into 23 census divisions. Unlike in some other provinces, census divisions do not reflect the organization of local government in Manitoba. These areas exist solely for the purposes of statistical analysis and presentation; they have no government of their own.
See also
Administrative divisions of Canada
List of communities in Manitoba
List of municipalities in Manitoba
List of regions of Manitoba
External links
Government of Manitoba Community Profiles. Census Divisions Map
Census divisions
|
https://en.wikipedia.org/wiki/Spherical%20bearing
|
A spherical bearing is a bearing that permits rotation about a central point in two orthogonal directions (usually within a specified angular limit based on the bearing geometry). Typically these bearings support a rotating shaft in the bore of the inner ring that must move not only rotationally, but also at an angle. It can either be a plain bearing or roller bearing.
Self-aligning spherical bearings were first used by James Nasmyth around 1840 to support line shaft bearings in mills and machine shops. For long shafts it was impossible to accurately align bearings, even if the shaft was perfectly straight. Nasmyth used brass bearing shells between hemispherical brass cups to allow the bearings to self-align.
Construction
Spherical bearings can be of a hydrostatic or mechanical construction. A spherical bearing by itself consists of an outer ring and an inner ring and a locking feature that makes the inner ring captive within the outer ring in the axial direction only. The outer surface of the inner ring and the inner surface of the outer ring are spherical and are collectively considered the raceway and they slide against each other, either with a lubricant, a maintenance-free (typically polytetrafluoroethylene or PTFE) based liner, or they incorporate a rolling element such as a race of ball-bearings, allowing lower friction.
Applications
Spherical bearings are used in car suspensions, engines, driveshafts, heavy machinery, sewing machines, robotics and many other applications: wherever rotational motion must be allowed to change the alignment of its rotation axis. An example, although this is rarely used in practice, is the drive axle bearings of a vehicle control arm (or A-arm) suspension. The mechanics of the suspension allow the axle to move up and down (and the wheel to turn in order to steer the vehicle), and the axle bearings must allow the rotational axis of the axle to change without binding. This is a simple concept that illustrates a possible application of a spherical bearing. In fact, spherical bearings are used in smaller sub-components of this type of suspension, for example certain types of constant-velocity joints.
See also
– one of the earliest patented applications of this technology
References
External links
Spherical bearing 101 With single and double rows of rollers.
Bearings (mechanical)
|
https://en.wikipedia.org/wiki/Immanant
|
In mathematics, the immanant of a matrix was defined by Dudley E. Littlewood and Archibald Read Richardson as a generalisation of the concepts of determinant and permanent.
Let be a partition of an integer and let be the corresponding irreducible representation-theoretic character of the symmetric group . The immanant of an matrix associated with the character is defined as the expression
Examples
The determinant is a special case of the immanant, where is the alternating character , of Sn, defined by the parity of a permutation.
The permanent is the case where is the trivial character, which is identically equal to 1.
For example, for matrices, there are three irreducible representations of , as shown in the character table:
As stated above, produces the permanent and produces the determinant, but produces the operation that maps as follows:
Properties
The immanant shares several properties with determinant and permanent. In particular, the immanant is multilinear in the rows and columns of the matrix; and the immanant is invariant under simultaneous permutations of the rows or columns by the same element of the symmetric group.
Littlewood and Richardson studied the relation of the immanant to Schur functions in the representation theory of the symmetric group.
The necessary and sufficient conditions for the immanant of a Gram matrix to be are given by Gamas's Theorem.
References
Algebra
Linear algebra
Matrix theory
Permutations
|
https://en.wikipedia.org/wiki/Isaac%20Malitz
|
Isaac Richard Jay Malitz (born 1947, in Cleveland, Ohio) is a logician who introduced the subject of positive set theory in his 1976 Ph.D. Thesis at UCLA.
References
Isaac (Richard) Jay Malitz – entry in the Mathematics Genealogy Project
1947 births
Living people
American logicians
University of California, Los Angeles alumni
Scientists from Cleveland
Date of birth missing (living people)
20th-century American mathematicians
|
https://en.wikipedia.org/wiki/Groud
|
Groud may refer to:
Groud (mathematics), an algebraic structure
Gilbert G. Groud, African artist
|
https://en.wikipedia.org/wiki/De%20Gua%27s%20theorem
|
In mathematics, De Gua's theorem is a three-dimensional analog of the Pythagorean theorem named after Jean Paul de Gua de Malves. It states that if a tetrahedron has a right-angle corner (like the corner of a cube), then the square of the area of the face opposite the right-angle corner is the sum of the squares of the areas of the other three faces:
De Gua's theorem can be applied for proving a special case of Heron's formula.
Generalizations
The Pythagorean theorem and de Gua's theorem are special cases () of a general theorem about n-simplices with a right-angle corner, proved by P. S. Donchian and H. S. M. Coxeter in 1935. This, in turn, is a special case of a yet more general theorem by Donald R. Conant and William A. Beyer (1974), which can be stated as follows.
Let U be a measurable subset of a k-dimensional affine subspace of (so ). For any subset with exactly k elements, let be the orthogonal projection of U onto the linear span of , where and is the standard basis for . Then
where is the k-dimensional volume of U and the sum is over all subsets with exactly k elements.
De Gua's theorem and its generalisation (above) to n-simplices with right-angle corners correspond to the special case where k = n−1 and U is an (n−1)-simplex in with vertices on the co-ordinate axes. For example, suppose , and U is the triangle in with vertices A, B and C lying on the -, - and -axes, respectively. The subsets of with exactly 2 elements are , and . By definition, is the orthogonal projection of onto the -plane, so is the triangle with vertices O, B and C, where O is the origin of . Similarly, and , so the Conant–Beyer theorem says
which is de Gua's theorem.
The generalisation of de Gua's theorem to n-simplices with right-angle corners can also be obtained as a special case from the Cayley–Menger determinant formula.
De Gua's theorem can also be generalized to arbitrary tetrahedra and to pyramids.
History
Jean Paul de Gua de Malves (1713–85) published the theorem in 1783, but around the same time a slightly more general version was published by another French mathematician, Charles de Tinseau d'Amondans (1746–1818), as well. However the theorem had also been known much earlier to Johann Faulhaber (1580–1635) and René Descartes (1596–1650).
See also
Vector area and projected area
Bivector
Notes
References
Sergio A. Alvarez: Note on an n-dimensional Pythagorean theorem, Carnegie Mellon University.
Theorems in geometry
Euclidean geometry
|
https://en.wikipedia.org/wiki/Lipovany
|
Lipovany () is a village and municipality in the Lučenec District in the Banská Bystrica Region of Slovakia.
External links
https://web.archive.org/web/20080111223415/http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Lučenec District
|
https://en.wikipedia.org/wiki/%C4%BDubore%C4%8D
|
Ľuboreč () is a village and municipality in the Lučenec District in the Banská Bystrica Region of Slovakia.
References
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Lučenec District
|
https://en.wikipedia.org/wiki/Outline%20of%20arithmetic
|
Arithmetic is an elementary branch of mathematics that is widely used for tasks ranging from simple day-to-day counting to advanced science and business calculations.
Essence of arithmetic
Elementary arithmetic
Decimal arithmetic
Decimal point
Numeral
Place value
Face value
History of arithmetic
Arithmetic operations and related concepts
Order of Operations
Addition
Summation – Answer after adding a sequence of numbers
Additive Inverse
Subtraction – Taking away numbers
Multiplication – Repeated addition
Multiple – Product of Multiplication
Least Common Multiple
Multiplicative Inverse
Division – Repeated Subtraction
Modulo – The remainder of division
Quotient – Result of Division
Quotition and Partition – How many parts are there, and what is the size of each part
Fraction – A number that isn't whole, often shown as a divsion equation
Decimal Fraction – Representation of a Fraction in the form of a number
Proper Fraction – Fraction with a Numerator that is less than the Denominator
Improper Fraction – Fractions with a Numerator that is any number
Ratio – Showing how much one number can go into another
Least Common Denominator – Least Common Multiple of 2 or more fractions' denominators
Factoring – Breaking a number down into its products
Fundamental theorem of arithmetic
Prime number – Number divisable by only 1 or itself
Prime number theorem
Distribution of primes
Composite number – Number made of 2 smaller integers
Factor – A number that can be divided from it's original number to get a whole number
Greatest Common Factor – Greatest Factor that is common between 2 numbers
Euclid's algorithm for finding greatest common divisors
Exponentiation (power) – Repreated Multiplication
Square root – Reversal of a power of 2 (exponent of 1/2)
Cube root – Reversal of a power of 3 (exponent of 1/3)
Properties of Operations
Associative property
Distributive property
Commutative property
Factorial – Multiplication of numbers from the current number to 0
Types of numbers
Real number
Rational number
Integer
Natural number
Composite number
Irrational number
Odd number
Even number
Positive number
Negative number
Prime number
List of prime numbers
Highly composite number
Perfect number
Algebraic number
Transcendental number
Hypercomplex number
Transfinite number
Indefinite and fictitious numbers
Elementary statistics
Mean
Weighted mean
Median
Mode
Range
Other basic concepts
Combinations
Percentage
Permutations
Proportion
Rounding
Scientific notation
Modern arithmetic
Outline of number theory
Riemann zeta function
L-functions
Multiplicative functions
Modular forms
See also
Elementary mathematics
Table of mathematical symbols
External links
MathWorld article about arithmetic
The New Student's Reference Work/Arithmetic (historical)
Maximus Planudes' the Great Calculation an early western work on arithmetic at Convergence
Arithmetic
Arithmetic
Arithmetic
|
https://en.wikipedia.org/wiki/Ma%C5%A1kov%C3%A1
|
Mašková () is a village and municipality in the Lučenec District in the Banská Bystrica Region of Slovakia.
External links
https://web.archive.org/web/20070513023228/http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Lučenec District
|
https://en.wikipedia.org/wiki/Miku%C5%A1ovce%2C%20Lu%C4%8Denec%20District
|
Mikušovce () is a village and municipality in the Lučenec District in the Banská Bystrica Region of Slovakia.
External links
https://web.archive.org/web/20071116010355/http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Lučenec District
|
https://en.wikipedia.org/wiki/Mu%C4%8D%C3%ADn
|
Mučín () is a village and municipality in the Lučenec District in the Banská Bystrica Region of Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Lučenec District
|
https://en.wikipedia.org/wiki/Polichno
|
Polichno () is a village and municipality in the Lučenec District in the Banská Bystrica Region of Slovakia.
External links
https://web.archive.org/web/20071027094149/http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Lučenec District
|
https://en.wikipedia.org/wiki/Rapovce
|
Rapovce () is a village and municipality in the Lučenec District in the Banská Bystrica Region of Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Lučenec District
|
https://en.wikipedia.org/wiki/P%C3%ADla%2C%20Lu%C4%8Denec%20District
|
Píla () is a village and municipality in the Lučenec District in the Banská Bystrica Region of Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Lučenec District
|
https://en.wikipedia.org/wiki/%C5%A0iatorsk%C3%A1%20Bukovinka
|
Šiatorská Bukovinka () is a village and municipality in the Lučenec District in the Banská Bystrica Region of Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Lučenec District
|
https://en.wikipedia.org/wiki/Ple%C5%A1%2C%20Slovakia
|
Pleš () is a village and municipality in the Lučenec District in the Banská Bystrica Region of Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Lučenec District
|
https://en.wikipedia.org/wiki/Ratka
|
Ratka () is a village and municipality in the Lučenec District in the Banská Bystrica Region of Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Lučenec District
|
https://en.wikipedia.org/wiki/Pr%C5%A1a
|
Prša () is a village and municipality in the Lučenec District in the Banská Bystrica Region of Slovakia.
External links
https://web.archive.org/web/20071116010355/http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Lučenec District
|
https://en.wikipedia.org/wiki/Outline%20of%20algebra
|
The following outline is provided as an overview of and topical guide to algebra:
Algebra is one of the main branches of mathematics, covering the study of structure, relation and quantity. Algebra studies the effects of adding and multiplying numbers, variables, and polynomials, along with their factorization and determining their roots. In addition to working directly with numbers, algebra also covers symbols, variables, and set elements. Addition and multiplication are general operations, but their precise definitions lead to structures such as groups, rings, and fields.
Branches
Pre-algebra
Elementary algebra
Boolean algebra
Abstract algebra
Linear algebra
Universal algebra
Algebraic equations
An algebraic equation is an equation involving only algebraic expressions in the unknowns. These are further classified by degree.
Linear equation – algebraic equation of degree one.
Polynomial equation – equation in which a polynomial is set equal to another polynomial.
Transcendental equation – equation involving a transcendental function of one of its variables.
Functional equation – equation in which the unknowns are functions rather than simple quantities.
Differential equation – equation involving derivatives.
Integral equation – equation involving integrals.
Diophantine equation – equation where the only solutions of interest of the unknowns are the integer ones.
History
History of algebra
General algebra concepts
Fundamental theorem of algebra – states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with an imaginary part equal to zero.
Equations – equality of two mathematical expressions
Linear equation – an algebraic equation with a degree of one
Quadratic equation – an algebraic equation with a degree of two
Cubic equation – an algebraic equation with a degree of three
Quartic equation – an algebraic equation with a degree of four
Quintic equation – an algebraic equation with a degree of five
Polynomial – an algebraic expression consisting of variables and coefficients
Inequalities – a comparison between values
Functions – mapping that associates a single output value with each input value
Sequences – ordered list of elements either finite or infinite
Systems of equations – finite set of equations
Vectors – element of a vector space
Matrix – two dimensional array of numbers
Vector space – basic algebraic structure of linear algebra
Field – algebraic structure with addition, multiplication and division
Groups – algebraic structure with a single binary operation
Rings – algebraic structure with addition and multiplication
See also
Table of mathematical symbols
External links
'4000 Years of Algebra', lecture by Robin Wilson, at Gresham College, 17 October 2007 (available for MP3 and MP4 download, as well as a text file).
ExampleProblems.com Example problems an
|
https://en.wikipedia.org/wiki/Outline%20of%20geometry
|
Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. Geometry is one of the oldest mathematical sciences.
Classical branches
Geometry
Analytic geometry
Differential geometry
Euclidean geometry
Non-Euclidean geometry
Projective geometry
Riemannian geometry
Contemporary branches
Absolute geometry
Affine geometry
Archimedes' use of infinitesimals
Birational geometry
Complex geometry
Combinatorial geometry
Computational geometry
Conformal geometry
Constructive solid geometry
Contact geometry
Convex geometry
Descriptive geometry
Digital geometry
Discrete geometry
Distance geometry
Elliptic geometry
Enumerative geometry
Epipolar geometry
Finite geometry
Geometry of numbers
Hyperbolic geometry
Incidence geometry
Information geometry
Integral geometry
Inversive geometry
Klein geometry
Lie sphere geometry
Numerical geometry
Ordered geometry
Parabolic geometry
Plane geometry
Quantum geometry
Ruppeiner geometry
Spherical geometry
Symplectic geometry
Synthetic geometry
Systolic geometry
Taxicab geometry
Toric geometry
Transformation geometry
Tropical geometry
History of geometry
History of geometry
Timeline of geometry
Babylonian geometry
Egyptian geometry
Ancient Greek geometry
Euclidean geometry
Pythagorean theorem
Euclid's Elements
Measurement of a Circle
Indian mathematics
Bakhshali manuscript
Modern geometry
History of analytic geometry
History of the Cartesian coordinate system
History of non-Euclidean geometry
History of topology
History of algebraic geometry
General geometry concepts
General concepts
Geometric progression — Geometric shape — Geometry — Pi — angular velocity — linear velocity — De Moivre's theorem — parallelogram rule — Pythagorean theorem — similar triangles — trigonometric identity — unit circle — Trapezoid — Triangle — Theorem — point — ray — plane — line — line segment
Measurements
Bearing
Angle
Degree
Minute
Radian
Circumference
Diameter
Trigonometric functions
Trigonometric function
Asymptotes
Circular functions
Periodic functions
Law of cosines
Law of sines
Vectors
Amplitude
Dot product
Norm (mathematics) (also known as magnitude)
Position vector
Scalar multiplication
Vector addition
Zero vector
Vector spaces and complex dimensions
Complex plane
Imaginary axis
Linear interpolation
One-to-one
Orthogonal
Polar coordinate system
Pole
Real axis
Secant line
CIrcular sector or "sector"
Semiperimeter
Lists
List of mathematical shapes
List of geometers
List of curves
List of curves topics
See also
List of basic mathematics topics
List of mathematics articles
Table of mathematical symbols
Further reading
External links
Geometry
Geometry
Geometry
|
https://en.wikipedia.org/wiki/Casio%20BASIC
|
Casio BASIC is a programming language used in the Casio calculators such as the Classpad, PRIZM Series, fx-9860G Series, fx-5800P, Algebra FX and CFX graphing calculators.
It is also known as "BasicLike" in some models.
This programming language has nothing to do with the more or less standard BASIC, which incorporated from the beginning of the 80s, the so-called "Pocket computers" or "Pocket PC" from Casio, among which the FX series can be found. -702P, Series 100 (PB-100), Series 700 (PB-100), and many others. The version of BASIC of these machines is called Casio POCKETPC BASIC
The language is a linear structured, BASIC-based programming language. It was devised to allow users to program in commonly performed calculations, such as the Pythagorean theorem and complex trigonometric calculations.
Output from the program can be in the form of scrolling or located text, graphs, or by writing data to lists and matrices in the calculator memory. Casio also makes label printers which can be used with rolls of paper for the Casio BASIC calculators. Programs, variables, data, and other items can be exchanged from one calculator to another (via SB-62 cable) and to and from a computer (via USB cable). All new models of Casio graphing calculators have both ports and include both cables.
The Casio calculators, as with those of many of the other big three manufacturers' machines, can acquire data from instruments via a data logger to which probes for temperature, light intensity, pH, sound intensity (dBA), voltage and other electrical parameters, as well as other readings, and custom probes to attach to the data logger can be built and configured for use with the data logger and calculator. Existing instruments can also be modified to interface with the calculator-data logger, in order to collect such data including such things as weather instruments and means of collecting data such as pulse, blood pressure, galvanic skin resistance, EKG and so on.
Like Tiny BASIC, the BASIC interpreter for Casio BASIC restricts variable names to the letters A-Z with just one predefined array (in Casio BASIC, Z, as compared to A in Level I BASIC and @ in Palo Alto Tiny BASIC). For Casio's graphical calculators, italic x, y, r and θ are also used as variable names for certain calculations. Therefore, extending the array size of predefined variable names from 26 to 30.
Numerical data can be stored in the lists and matrices available on Casio calculators. This data can be used to create sprites for non-text programs. In this way, the language can also be used to create games, such as Pong, Monopoly and role-playing games.
Additionally, characters can be stored as strings in the string memory.
Examples
Hello world program in Casio BASIC:
"Hello, world!"
Program to calculate the Fibonacci sequence:
References
External links
A tutorial for making games with Casio BASIC
A source for games and programs written in Casio BASIC. (Not tested/verified)
A complete explan
|
https://en.wikipedia.org/wiki/Matt%20Visser
|
Matt Visser () is a mathematics Professor at Victoria University of Wellington, in New Zealand.
Career
Visser completed a PhD at the University of California, Berkeley, supervised by Mary K. Gaillard.
Visser's research interests include general relativity, quantum field theory and cosmology.
Visser has produced a large number of research papers on the subject of wormholes, gravitational horizons and notably the emerging subject of acoustic metrics.
He is the author of the reference book on the current state of wormhole theory, Lorentzian Wormholes — from Einstein to Hawking (1996) and co-editor of Artificial Black Holes (2002).
Awards
In 2013 Visser was awarded the Dan Walls Medal by the New Zealand Institute of Physics.
Books
David L Wiltshire, Matt Visser & Susan Scott, The Kerr Spacetime: Rotating black holes in general relativity (2009)
M Novello, Matt Visser & G E Volovik, Artificial Black Holes (2002)
Matt Visser, Lorentzian Wormholes: From Einstein To Hawking (1995)
See also
Roman ring
References
External links
Matt Visser's personalized homepage
research papers by Matt Visser on arXiv
New Zealand mathematicians
Fellows of the Royal Society of New Zealand
Academic staff of Victoria University of Wellington
Living people
New Zealand people of Dutch descent
Year of birth missing (living people)
People educated at St Bernard's College, Lower Hutt
Fellows of the American Physical Society
James Cook Research Fellows
|
https://en.wikipedia.org/wiki/Dan%20Sullivan%20%28ice%20hockey%2C%20born%201981%29
|
Dan Sullivan (born March 20, 1981) is a Canadian ice hockey player.
Career statistics
External links
1981 births
Living people
Augusta Lynx players
Baton Rouge Kingfish players
Canadian ice hockey right wingers
London Knights players
Louisiana IceGators (ECHL) players
Lowell Lock Monsters players
Mississauga IceDogs players
Owen Sound Attack players
Pensacola Ice Pilots players
People from York, Toronto
Reading Royals players
Roanoke Express players
Ice hockey people from Toronto
|
https://en.wikipedia.org/wiki/EF%20hand
|
The EF hand is a helix–loop–helix structural domain or motif found in a large family of calcium-binding proteins.
The EF-hand motif contains a helix–loop–helix topology, much like the spread thumb and forefinger of the human hand, in which the Ca2+ ions are coordinated by ligands within the loop. The motif takes its name from traditional nomenclature used in describing the protein parvalbumin, which contains three such motifs and is probably involved in muscle relaxation via its calcium-binding activity.
The EF-hand consists of two alpha helices linked by a short loop region (usually about 12 amino acids) that usually binds calcium ions. EF-hands also appear in each structural domain of the signaling protein calmodulin and in the muscle protein troponin-C.
Calcium ion binding site
The calcium ion is coordinated in a pentagonal bipyramidal configuration. The six residues involved in the binding are in positions 1, 3, 5, 7, 9 and 12; these residues are denoted by X, Y, Z, -Y, -X and -Z. The invariant Glu or Asp at position 12 provides two oxygens for liganding calcium (bidentate ligand).
The calcium ion is bound by both protein backbone atoms and by amino acid side chains, specifically those of the anionic amino acid residues aspartate and glutamate. These residues are negatively charged and will make a charge-interaction with the positively charged calcium ion. The EF hand motif was among the first structural motifs whose sequence requirements were analyzed in detail. Five of the loop residues bind calcium and thus have a strong preference for oxygen-containing side chains, especially aspartate and glutamate. The sixth residue in the loop is necessarily glycine due to the conformational requirements of the backbone. The remaining residues are typically hydrophobic and form a hydrophobic core that binds and stabilizes the two helices.
Upon binding to Ca2+, this motif may undergo conformational changes that enable Ca2+-regulated functions as seen in Ca2+ effectors such as calmodulin (CaM) and troponin C (TnC) and Ca2+ buffers such as calreticulin and calbindin D9k. While the majority of the known EF-hand calcium-binding proteins (CaBPs) contain paired EF-hand motifs, CaBPs with single EF hands have also been discovered in both bacteria and eukaryotes. In addition, "EF-hand-like motifs" have been found in a number of bacteria. Although the coordination properties remain similar with the canonical 29-residue helix–loop–helix EF-hand motif, the EF-hand-like motifs differ from EF-hands in that they contain deviations in the secondary structure of the flanking sequences and/or variation in the length of the Ca2+-coordinating loop.
EF hands have very high selectivity for calcium. For example, the dissociation constant of alpha parvalbumin for Ca2+ is ~1000 times lower than that for the similar ion Mg2+. This high selectivity is due to the relatively rigid coordination geometry, the presence of multiple charged amino acid side chains in the binding
|
https://en.wikipedia.org/wiki/Geometry%20%28disambiguation%29
|
Geometry is a branch of mathematics dealing with spatial relationships.
Geometry or geometric may also refer to:
Geometric distribution of probability theory and statistics
Geometric series, a mathematical series with a constant ratio between successive terms
Music
Geometry (Robert Rich album), a 1991 album by American musician Robert Rich
Geometry (Ivo Perelman album), a 1997 album by Brazilian saxophonist Ivo Perelman
Geometry (Jega album), a 2000 album by English musician Jega
Other uses
Geometric (typeface classification), a class of sans-serif typeface styles
Geometry (car marque) Chinese car brand manufactured by Geely
"Geometry", a Series G episode of the television series QI (2010)
Gia Metric, Canadian drag queen
See also
|
https://en.wikipedia.org/wiki/Borel%E2%80%93Moore%20homology
|
In topology, Borel−Moore homology or homology with closed support is a homology theory for locally compact spaces, introduced by Armand Borel and John Moore in 1960.
For reasonable compact spaces, Borel−Moore homology coincides with the usual singular homology. For non-compact spaces, each theory has its own advantages. In particular, a closed oriented submanifold defines a class in Borel–Moore homology, but not in ordinary homology unless the submanifold is compact.
Note: Borel equivariant cohomology is an invariant of spaces with an action of a group G; it is defined as That is not related to the subject of this article.
Definition
There are several ways to define Borel−Moore homology. They all coincide for reasonable spaces such as manifolds and locally finite CW complexes.
Definition via sheaf cohomology
For any locally compact space X, Borel–Moore homology with integral coefficients is defined as the cohomology of the dual of the chain complex which computes sheaf cohomology with compact support. As a result, there is a short exact sequence analogous to the universal coefficient theorem:
In what follows, the coefficients are not written.
Definition via locally finite chains
The singular homology of a topological space X is defined as the homology of the chain complex of singular chains, that is, finite linear combinations of continuous maps from the simplex to X. The Borel−Moore homology of a reasonable locally compact space X, on the other hand, is isomorphic to the homology of the chain complex of locally finite singular chains. Here "reasonable" means X is locally contractible, σ-compact, and of finite dimension.
In more detail, let be the abelian group of formal (infinite) sums
where σ runs over the set of all continuous maps from the standard i-simplex Δi to X and each aσ is an integer, such that for each compact subset S of X, only finitely many maps σ whose image meets S have nonzero coefficient in u. Then the usual definition of the boundary ∂ of a singular chain makes these abelian groups into a chain complex:
The Borel−Moore homology groups are the homology groups of this chain complex. That is,
If X is compact, then every locally finite chain is in fact finite. So, given that X is "reasonable" in the sense above, Borel−Moore homology coincides with the usual singular homology for X compact.
Definition via compactifications
Suppose that X is homeomorphic to the complement of a closed subcomplex S in a finite CW complex Y. Then Borel–Moore homology is isomorphic to the relative homology Hi(Y, S). Under the same assumption on X, the one-point compactification of X is homeomorphic to a finite CW complex. As a result, Borel–Moore homology can be viewed as the relative homology of the one-point compactification with respect to the added point.
Definition via Poincaré duality
Let X be any locally compact space with a closed embedding into an oriented manifold M of dimension m. Then
where in the right hand side, relati
|
https://en.wikipedia.org/wiki/Empedrado%2C%20Chile
|
Empedrado () is a town and commune in the Talca Province of Chile's Maule Region.
Demographics
According to the 2002 census of the National Statistics Institute, Empedrado spans an area of and has 4,225 inhabitants (2,222 men and 2,003 women). Of these, 2,499 (59.1%) lived in urban areas and 1,726 (40.9%) in rural areas. The population fell by 7.2% (329 persons) between the 1992 and 2002 censuses.
Administration
As a commune, Empedrado is a third-level administrative division of Chile administered by a municipal council, headed by an alcalde who is directly elected every four years. The 2008-2012 alcalde is Gonzalo Tejos Perez.
Within the electoral divisions of Chile, Empedrado is represented in the Chamber of Deputies by Pablo Lorenzini (PDC) and Pedro Pablo Alvarez-Salamanca (UDI) as part of the 38th electoral district, together with Curepto, Constitución, Pencahue, Maule, San Clemente, Pelarco, Río Claro and San Rafael. The commune is represented in the Senate by Juan Antonio Coloma Correa (UDI) and Andrés Zaldívar Larraín (PDC) as part of the 10th senatorial constituency (Maule-North).
References
External links
Municipality of Empedrado
Populated places in Talca Province
Communes of Chile
|
https://en.wikipedia.org/wiki/Alar%20Kotli
|
Alar Kotli (27 August 1904 in Väike-Maarja – 4 October 1963 in Tallinn) was an Estonian architect. He studied sculpture at the art school Pallas in Tartu during 1922–1923 and mathematics at the University of Tartu. He graduated from the University of technology in Gdańsk (then Free City of Danzig) in 1927 as an architect.
Among the most famous and influential Estonian architects, Kotli has created several important landmarks in Tallinn. These include the Estonian Song Festival grounds (1957–1960, with Henno Sepmann & E. Paalmann), the main building of Tallinn University (1938–1940, with Erika Nõva), the Art Fund building (1949–1953) and the administrative building in Kadriorg park (currently the residence of the president of the Republic of Estonia) in conjunction with architect Olev Siinmaa (1937–1938). Kotli has also created many experimental apartment building projects, which were widely used after World War II when there was a serious need for new dwellings. Smaller buildings (for two families) were used in the 1950s as lottery jackpots.
Kotli's style has varied over the years. He has designed many functionalistic buildings in the 1930s, for example — schoolhouses in Rakvere (1935–1938) and Tapa (1936–1939). The Presidential Palace, also dating from the 1930s, can be categorised as historicism, while his 1950s and 1960s style is similar to brutalism.
Gallery
External links
1904 births
1963 deaths
People from Väike-Maarja
People from the Governorate of Estonia
Modernist architects
Soviet architects
20th-century Estonian architects
University of Tartu alumni
Gdańsk University of Technology alumni
Burials at Metsakalmistu
|
https://en.wikipedia.org/wiki/Cumberland%20High%20School%20%28Rhode%20Island%29
|
Cumberland High School is a public school located in Cumberland, Rhode Island. In its current location since 1962, the school serves approximately 1,500 students.
Statistics
History
The current building was built in 1961 and renovated in 1973. The town's mayor presented a capital improvement plan for the school in 2002, but funding was limited and residents questioned the project's first phase plans for a wellness center (athletic complex). A few months later, the New England Association of Schools and Colleges gave the school until October 2005 to improve the school's facilities and technology or have its accreditation status placed on probation.
A town referendum was held in September 2005 asking voters to approve the town borrowing $30 million to renovate Cumberland High School. The referendum's success is credited in part to a student group called Save Our Schools, which had been organized as part of the school's requirement of 15 hours of teacher-supervised service learning.
NASSP (the National Association of Secondary School Principals) designated Alan Tenreiro of Cumberland High School as 2016 National Principal of the Year.
Campus
Cumberland High school has three main buildings. The Main building houses the school cafeteria, main office, and the majority of the CHS's classrooms and resources. The Transitional building is a circular shaped building primarily houses the ninth grade class including lockers. Additionally, the Transitional building is home to the Cumberland School Department administration. The Transitional building and Main buildings are connected by an enclosed bridge. The Wellness Center houses CHS's gymnasiums and locker rooms as well as health classrooms and a weight room. It is the newest of the three buildings.
The campus finished major renovations known as CHS 2010 at the end of the 2007–2008 school year. This project was designed to create a 21st-century learning environment on the campus of Cumberland High School. These renovations included the addition of a science and technology wing, the Wellness Center, a fine arts wing, and the refurbishing of numerous other areas.
The 2008–2009 school year marks the opening of several newly installed computer labs, including the Graphic Design Lab, the World Language Lab, and the CAD Lab.
Curriculum
Cumberland High houses departments in English, Mathematics, Sciences, History, Languages, Special Education, and Fine Arts
Extracurricular activities
The school hosts approximately 28 clubs and co-curricular activities as well as over 20 sports for girls and boys.
The more successful of these clubs include: Student Government, Symphonic Wind Ensemble, Clipper Dance Team, Mock Trial, Math Team, Debate Team, The World Language Club, Game Club, Living Lessons, Future Business Leaders of America, Ecology Club and the Clef Singers, the school's choral group.
The three publications at the school are "The Scanner", the Literary Magazine, and the French Newsletter.
Athl
|
https://en.wikipedia.org/wiki/Stephen%20Schanuel
|
Stephen H. Schanuel (14 July 1933 – 21 July 2014) was an American mathematician working in the fields of abstract algebra and category theory, number theory, and measure theory.
Life
While he was a graduate student at University of Chicago, he discovered Schanuel's lemma, an essential lemma in homological algebra. Schanuel received his Ph.D. in mathematics from Columbia University in 1963, under the supervision of Serge Lang.
Work
Shortly thereafter he stated a conjecture in the field of transcendental number theory, which remains an important open problem to this day. Schanuel was a professor emeritus of mathematics at University at Buffalo.
Books
References
External links
1933 births
2014 deaths
20th-century American mathematicians
21st-century American mathematicians
Columbia Graduate School of Arts and Sciences alumni
University at Buffalo faculty
People from St. Louis
Mathematicians from Missouri
Algebraists
|
https://en.wikipedia.org/wiki/Queensland%20Academy%20for%20Science%2C%20Mathematics%20and%20Technology
|
The Queensland Academies – Science Mathematics & Technology Campus (QASMT) is a high school in Toowong, Queensland, Australia. It was developed in partnership with the University of Queensland. QASMT offers the International Baccalaureate Diploma Programme to students in Years 11 and 12, and also offers the International Baccalaureate Middle Years Programme to Years 7–10 students.
In 2021, Better Education ranked Queensland Academies for Science Mathematics & Technology Campus as the top performing school in the state of Queensland.
History
Premier Peter Beattie announced the creation of the Queensland Academies on 17 April 2005 as part of the Queensland Government's Smart State Strategy – a policy designed to foster knowledge, creativity, and innovation within QLD. The Queensland Academies – Science Mathematics & Technology Campus (QASMT) subsequently opened in January 2007. The school was established in partnership with the University of Queensland with a focus on sciences and mathematics. The site occupied by QASMT was formerly Toowong College; this location was chosen "to capitalise on its close educational and geographic links with the University of Queensland."
The site was occupied by the house known as Ormlie originally and later as Easton Gray and owned by Sir Arthur Hunter Palmer, Premier of Queensland and subsequently the residence of his brother-in-law Hugh Mosman (who discovered gold at Charters Towers). Easton Gray was sold in 1944 for the construction of Toowong State High School, later Toowong College.
The first cohort of students graduated from QASMT in 2008.
Two other academies were created with close ties to QASMT. In 2007, the Queensland Academy for Creative Industries (QACI) was established in partnership with the Queensland University of Technology with a focus towards media, film, design and technology, music, theatre arts, and visual arts. In 2008, the Queensland Academy for Health Sciences (QAHS) was developed in partnership with Griffith University with a focus towards medicine, dentistry, physiotherapy, optometry, and medical research.
In 2019, QASMT introduced its Middle School Program with its new Grade 7 cohort. In 2021, QASMT became a fully complete 7–12 school. To cater for the new students, QASMT was expanded in a 2-stage approach.
Stage 1 was completed in January 2019 to accommodate the first cohort of Year 7 students. Stage 1 included the refurbishment of existing buildings and the installation of prefabricated accommodation while stage 2 was being delivered.
Stage 2 delivered new educational infrastructure and facilities to cater for the Years 8 and 9 students at the school. Stage 2 works commenced in February 2019 and includes the delivery of a new Northern Learning Centre and a new Eastern Science Technology Engineering & Mathematics (STEM) hub.
The construction of the new Eastern Science Technology Engineering & Mathematics was completed in December 2019 and is currently in use. The remaining Stage
|
https://en.wikipedia.org/wiki/Homotopical%20connectivity
|
In algebraic topology, homotopical connectivity is a property describing a topological space based on the dimension of its holes. In general, low homotopical connectivity indicates that the space has at least one low-dimensional hole. The concept of n-connectedness generalizes the concepts of path-connectedness and simple connectedness.
An equivalent definition of homotopical connectivity is based on the homotopy groups of the space. A space is n-connected (or n-simple connected) if its first n homotopy groups are trivial.
Homotopical connectivity is defined for maps, too. A map is n-connected if it is an isomorphism "up to dimension n, in homotopy".
Definition using holes
All definitions below consider a topological space X.
A hole in X is, informally, a thing that prevents some suitably-placed sphere from continuously shrinking to a point. Equivalently, it is a sphere that cannot be continuously extended to a ball. Formally,
A d-dimensional sphere in X is a continuous function .
A d-dimensional ball in X is a continuous function .
A d-dimensional-boundary hole in X is a d-dimensional sphere that is not nullhomotopic (- cannot be shrunk continuously to a point). Equivalently, it is a d-dimensional sphere that cannot be continuously extended to a (d+1)-dimensional ball. It is sometimes called a (d+1)-dimensional hole (d+1 is the dimension of the "missing ball").
X is called n-connected if it contains no holes of boundary-dimension d ≤ n.
The homotopical connectivity of X, denoted , is the largest integer n for which X is n-connected.
A slightly different definition of connectivity, which makes some computations simpler, is: the smallest integer d such that X contains a d-dimensional hole. This connectivity parameter is denoted by , and it differs from the previous parameter by 2, that is, .
Examples
A 2-dimensional hole (a hole with a 1-dimensional boundary) is a circle (S1) in X, that cannot be shrunk continuously to a point in X. An example is shown on the figure at the right. The yellow region is the topological space X; it is a pentagon with a triangle removed. The blue circle is a 1-dimensional sphere in X. It cannot be shrunk continuously to a point in X; therefore; X has a 2-dimensional hole. Another example is the punctured plane - the Euclidean plane with a single point removed, . To make a 2-dimensional hole in a 3-dimensional ball, make a tunnel through it. In general, a space contains a 1-dimensional-boundary hole if and only if it is not simply-connected. Hence, simply-connected is equivalent to 1-connected. X is 0-connected but not 1-connected, so . The lowest dimension of a hole is 2, so .
A 3-dimensional hole (a hole with a 2-dimensional boundary) is shown on the figure at the right. Here, X is a cube (yellow) with a ball removed (white). The 2-dimensional sphere (blue) cannot be continuously shrunk to a single point. X is simply-connected but not 2-connected, so . The smallest dimension of a hole is 3, so
|
https://en.wikipedia.org/wiki/NNLS
|
NNLS may refer to
Non-negative least squares, an optimization problem in mathematics
New North London Synagogue, see Sternberg Centre
|
https://en.wikipedia.org/wiki/Bessel%20polynomials
|
In mathematics, the Bessel polynomials are an orthogonal sequence of polynomials. There are a number of different but closely related definitions. The definition favored by mathematicians is given by the series
Another definition, favored by electrical engineers, is sometimes known as the reverse Bessel polynomials
The coefficients of the second definition are the same as the first but in reverse order. For example, the third-degree Bessel polynomial is
while the third-degree reverse Bessel polynomial is
The reverse Bessel polynomial is used in the design of Bessel electronic filters.
Properties
Definition in terms of Bessel functions
The Bessel polynomial may also be defined using Bessel functions from which the polynomial draws its name.
where Kn(x) is a modified Bessel function of the second kind, yn(x) is the ordinary polynomial, and θn(x) is the reverse polynomial . For example:
Definition as a hypergeometric function
The Bessel polynomial may also be defined as a confluent hypergeometric function
A similar expression holds true for the generalized Bessel polynomials (see below):
The reverse Bessel polynomial may be defined as a generalized Laguerre polynomial:
from which it follows that it may also be defined as a hypergeometric function:
where (−2n)n is the Pochhammer symbol (rising factorial).
Generating function
The Bessel polynomials, with index shifted, have the generating function
Differentiating with respect to , cancelling , yields the generating function for the polynomials
Similar generating function exists for the polynomials as well:
Upon setting , one has the following representation for the exponential function:
Recursion
The Bessel polynomial may also be defined by a recursion formula:
and
Differential equation
The Bessel polynomial obeys the following differential equation:
and
Orthogonality
The Bessel polynomials are orthogonal with respect to the weight integrated over the unit circle of the complex plane. In other words, if ,
Generalization
Explicit Form
A generalization of the Bessel polynomials have been suggested in literature, as following:
the corresponding reverse polynomials are
The explicit coefficients of the polynomials are:
Consequently, the polynomials can explicitly be written as follows:
For the weighting function
they are orthogonal, for the relation
holds for m ≠ n and c a curve surrounding the 0 point.
They specialize to the Bessel polynomials for α = β = 2, in which situation ρ(x) = exp(−2 / x).
Rodrigues formula for Bessel polynomials
The Rodrigues formula for the Bessel polynomials as particular solutions of the above differential equation is :
where a are normalization coefficients.
Associated Bessel polynomials
According to this generalization we have the following generalized differential equation for associated Bessel polynomials:
where . The solutions are,
Zeros
If one denotes the zeros of as , and that of the by , then the following estimates e
|
https://en.wikipedia.org/wiki/Normal%20variance-mean%20mixture
|
In probability theory and statistics, a normal variance-mean mixture with mixing probability density is the continuous probability distribution of a random variable of the form
where , and are real numbers, and random variables and are independent, is normally distributed with mean zero and variance one, and is continuously distributed on the positive half-axis with probability density function . The conditional distribution of given is thus a normal distribution with mean and variance . A normal variance-mean mixture can be thought of as the distribution of a certain quantity in an inhomogeneous population consisting of many different normal distributed subpopulations. It is the distribution of the position of a Wiener process (Brownian motion) with drift and infinitesimal variance observed at a random time point independent of the Wiener process and with probability density function . An important example of normal variance-mean mixtures is the generalised hyperbolic distribution in which the mixing distribution is the generalized inverse Gaussian distribution.
The probability density function of a normal variance-mean mixture with mixing probability density is
and its moment generating function is
where is the moment generating function of the probability distribution with density function , i.e.
See also
Normal-inverse Gaussian distribution
Variance-gamma distribution
Generalised hyperbolic distribution
References
O.E Barndorff-Nielsen, J. Kent and M. Sørensen (1982): "Normal variance-mean mixtures and z-distributions", International Statistical Review, 50, 145–159.
Continuous distributions
Compound probability distributions
|
https://en.wikipedia.org/wiki/Frattini%27s%20argument
|
In group theory, a branch of mathematics, Frattini's argument is an important lemma in the structure theory of finite groups. It is named after Giovanni Frattini, who used it in a paper from 1885 when defining the Frattini subgroup of a group. The argument was taken by Frattini, as he himself admits, from a paper of Alfredo Capelli dated 1884.
Frattini's Argument
Statement
If is a finite group with normal subgroup , and if is a Sylow p-subgroup of , then
where denotes the normalizer of in and means the product of group subsets.
Proof
The group is a Sylow -subgroup of , so every Sylow -subgroup of is an -conjugate of , that is, it is of the form , for some (see Sylow theorems). Let be any element of . Since is normal in , the subgroup is contained in . This means that is a Sylow -subgroup of . Then by the above, it must be -conjugate to : that is, for some
,
and so
.
Thus,
,
and therefore . But was arbitrary, and so
Applications
Frattini's argument can be used as part of a proof that any finite nilpotent group is a direct product of its Sylow subgroups.
By applying Frattini's argument to , it can be shown that whenever is a finite group and is a Sylow -subgroup of .
More generally, if a subgroup contains for some Sylow -subgroup of , then is self-normalizing, i.e. .
External links
Frattini's Argument on ProofWiki
References
(See Chapter 10, especially Section 10.4.)
Lemmas in group theory
Articles containing proofs
|
https://en.wikipedia.org/wiki/Normal-inverse%20Gaussian%20distribution
|
The normal-inverse Gaussian distribution (NIG, also known as the normal-Wald distribution) is a continuous probability distribution that is defined as the normal variance-mean mixture where the mixing density is the inverse Gaussian distribution. The NIG distribution was noted by Blaesild in 1977 as a subclass of the generalised hyperbolic distribution discovered by Ole Barndorff-Nielsen. In the next year Barndorff-Nielsen published the NIG in another paper. It was introduced in the mathematical finance literature in 1997.
The parameters of the normal-inverse Gaussian distribution are often used to construct a heaviness and skewness plot called the NIG-triangle.
Properties
Moments
The fact that there is a simple expression for the moment generating function implies that simple expressions for all moments are available.
Linear transformation
This class is closed under affine transformations, since it is a particular case of the Generalized hyperbolic distribution, which has the same property. If
then
Summation
This class is infinitely divisible, since it is a particular case of the Generalized hyperbolic distribution, which has the same property.
Convolution
The class of normal-inverse Gaussian distributions is closed under convolution in the following sense: if and are independent random variables that are NIG-distributed with the same values of the parameters and , but possibly different values of the location and scale parameters, , and , respectively, then is NIG-distributed with parameters and
Related distributions
The class of NIG distributions is a flexible system of distributions that includes fat-tailed and skewed distributions, and the normal distribution, arises as a special case by setting and letting .
Stochastic process
The normal-inverse Gaussian distribution can also be seen as the marginal distribution of the normal-inverse Gaussian process which provides an alternative way of explicitly constructing it. Starting with a drifting Brownian motion (Wiener process), , we can define the inverse Gaussian process Then given a second independent drifting Brownian motion, , the normal-inverse Gaussian process is the time-changed process . The process at time has the normal-inverse Gaussian distribution described above. The NIG process is a particular instance of the more general class of Lévy processes.
As a variance-mean mixture
Let denote the inverse Gaussian distribution and denote the normal distribution. Let , where ; and let , then follows the NIG distribution, with parameters, . This can be used to generate NIG variates by ancestral sampling. It can also be used to derive an EM algorithm for maximum-likelihood estimation of the NIG parameters.
References
Continuous distributions
|
https://en.wikipedia.org/wiki/Non-associative%20algebra
|
A non-associative algebra (or distributive algebra) is an algebra over a field where the binary multiplication operation is not assumed to be associative. That is, an algebraic structure A is a non-associative algebra over a field K if it is a vector space over K and is equipped with a K-bilinear binary multiplication operation A × A → A which may or may not be associative. Examples include Lie algebras, Jordan algebras, the octonions, and three-dimensional Euclidean space equipped with the cross product operation. Since it is not assumed that the multiplication is associative, using parentheses to indicate the order of multiplications is necessary. For example, the expressions (ab)(cd), (a(bc))d and a(b(cd)) may all yield different answers.
While this use of non-associative means that associativity is not assumed, it does not mean that associativity is disallowed. In other words, "non-associative" means "not necessarily associative", just as "noncommutative" means "not necessarily commutative" for noncommutative rings.
An algebra is unital or unitary if it has an identity element e with ex = x = xe for all x in the algebra. For example, the octonions are unital, but Lie algebras never are.
The nonassociative algebra structure of A may be studied by associating it with other associative algebras which are subalgebras of the full algebra of K-endomorphisms of A as a K-vector space. Two such are the derivation algebra and the (associative) enveloping algebra, the latter being in a sense "the smallest associative algebra containing A".
More generally, some authors consider the concept of a non-associative algebra over a commutative ring R: An R-module equipped with an R-bilinear binary multiplication operation. If a structure obeys all of the ring axioms apart from associativity (for example, any R-algebra), then it is naturally a -algebra, so some authors refer to non-associative -algebras as non-associative rings.
Algebras satisfying identities
Ring-like structures with two binary operations and no other restrictions are a broad class, one which is too general to study.
For this reason, the best-known kinds of non-associative algebras satisfy identities, or properties, which simplify multiplication somewhat.
These include the following ones.
Usual properties
Let , and denote arbitrary elements of the algebra over the field .
Let powers to positive (non-zero) integer be recursively defined by and either (right powers) or (left powers) depending on authors.
Unital: there exist an element so that ; in that case we can define .
Associative: .
Commutative: .
Anticommutative: .
Jacobi identity: or depending on authors.
Jordan identity: or depending on authors.
Alternative: (left alternative) and (right alternative).
Flexible: .
th power associative with : for all integers so that .
Third power associative: .
Fourth power associative: (compare with fourth power commutative below).
Power associative: the subalgebra
|
https://en.wikipedia.org/wiki/Variance-gamma%20distribution
|
The variance-gamma distribution, generalized Laplace distribution or Bessel function distribution is a continuous probability distribution that is defined as the normal variance-mean mixture where the mixing density is the gamma distribution. The tails of the distribution decrease more slowly than the normal distribution. It is therefore suitable to model phenomena where numerically large values are more probable than is the case for the normal distribution. Examples are returns from financial assets and turbulent wind speeds. The distribution was introduced in the financial literature by Madan and Seneta. The variance-gamma distributions form a subclass of the generalised hyperbolic distributions.
The fact that there is a simple expression for the moment generating function implies that simple expressions for all moments are available. The class of variance-gamma distributions is closed under convolution in the following sense. If and are independent random variables that are variance-gamma distributed with the same values of the parameters and , but possibly different values of the other parameters, , and , respectively, then is variance-gamma distributed with parameters , , and .
The variance-gamma distribution can also be expressed in terms of three inputs parameters (C,G,M) denoted after the initials of its founders. If the "C", here, parameter is integer then the distribution has a closed form 2-EPT distribution. See 2-EPT Probability Density Function. Under this restriction closed form option prices can be derived.
If , and , the distribution becomes a Laplace distribution with scale parameter . As long as , alternative choices of and will produce distributions related to the Laplace distribution, with skewness, scale and location depending on the other parameters.
For a symmetric variance-gamma distribution, the kurtosis can be given by .
See also Variance gamma process.
Notes
Continuous distributions
Infinitely divisible probability distributions
|
https://en.wikipedia.org/wiki/Skew%20polygon
|
In geometry, a skew polygon is a polygon whose vertices are not all coplanar. Skew polygons must have at least four vertices. The interior surface (or area) of such a polygon is not uniquely defined.
Skew infinite polygons (apeirogons) have vertices which are not all colinear.
A zig-zag skew polygon or antiprismatic polygon has vertices which alternate on two parallel planes, and thus must be even-sided.
Regular skew polygons in 3 dimensions (and regular skew apeirogons in two dimensions) are always zig-zag.
Antiprismatic skew polygon in three dimensions
A regular skew polygon is isogonal with equal edge lengths. In 3 dimensions a regular skew polygon is a zig-zag skew (or antiprismatic) polygon, with vertices alternating between two parallel planes. The side edges of an n-antiprism can define a regular skew 2n-gon.
A regular skew n-gon can be given a Schläfli symbol {p}#{ } as a blend of a regular polygon {p} and an orthogonal line segment { }. The symmetry operation between sequential vertices is glide reflection.
Examples are shown on the uniform square and pentagon antiprisms. The star antiprisms also generate regular skew polygons with different connection order of the top and bottom polygons. The filled top and bottom polygons are drawn for structural clarity, and are not part of the skew polygons.
A regular compound skew 2n-gon can be similarly constructed by adding a second skew polygon by a rotation. These share the same vertices as the prismatic compound of antiprisms.
Petrie polygons are regular skew polygons defined within regular polyhedra and polytopes. For example, the five Platonic solids have 4-, 6-, and 10-sided regular skew polygons, as seen in these orthogonal projections with red edges around their respective projective envelopes. The tetrahedron and the octahedron include all the vertices in their respective zig-zag skew polygons, and can be seen as a digonal antiprism and a triangular antiprism respectively.
Regular skew polygon as vertex figure of regular skew polyhedron
A regular skew polyhedron has regular polygon faces, and a regular skew polygon vertex figure.
Three infinite regular skew polyhedra are space-filling in 3-space; others exist in 4-space, some within the uniform 4-polytopes.
Isogonal skew polygons in three dimensions
An isogonal skew polygon is a skew polygon with one type of vertex, connected by two types of edges. Isogonal skew polygons with equal edge lengths can also be considered quasiregular. It is similar to a zig-zag skew polygon, existing on two planes, except allowing one edge to cross to the opposite plane, and the other edge to stay on the same plane.
Isogonal skew polygons can be defined on even-sided n-gonal prisms, alternatingly following an edge of one side polygon, and moving between polygons. For example, on the vertices of a cube. Vertices alternate between top and bottom squares with red edges between sides, and blue edges along each side.
Regular skew polygons in four di
|
https://en.wikipedia.org/wiki/Moni%20Nag
|
Moni Nag (1925 – 7 December 2015) was an Indian anthropologist specialising in the politics of sexuality.
Education and career
Born in India, Nag earned a master's degree in statistics from the University of Calcutta in 1946 and a PhD in anthropology from Yale University in 1961. He started his career in the Indian Statistical Institute and worked on the Anthropological Survey of India before joining the Department of Anthropology at Columbia University in New York in 1966; he was a lecturer and later an adjunct professor and headed the social demography section in the International Institute for the Study of Human Reproduction. He was also a senior associate in the Population Council in New York and a patron and vice president of the Elmhirst Institute of Community Studies at Santiniketan, and served as chair of the population commission in the International Union of Anthropological and Ethnological Sciences.
Research and publications
Nag was a pioneer of demographic anthropology. He researched and published in the fields of human sexuality, fertility, family planning, HIV prevention, and sex work, with a focus on India, and both studied and worked for the rights of prostitutes in the Kolkata red-light district of Sonagachi; he was one of several academics working with the Durbar Mahila Samanwaya Committee there.
Selected books
Factors Affecting Human Fertility in Nonindustrial Societies: A Cross-Cultural Study (Yale University, 1962)
Population and Social Organization (editor; Mouton, 1975)
Sexual Behaviour and AIDS in India (Vikas, 1996)
Sex Workers of India: Diversity in Practice of Prostitution and Ways of Life (Allied Publishers, 2006)
References
1925 births
Columbia University faculty
Indian emigrants to the United States
Indian anthropologists
University of Calcutta alumni
2015 deaths
|
https://en.wikipedia.org/wiki/Panick%C3%A9%20Dravce
|
Panické Dravce () is a village and municipality in the Lučenec District in the Banská Bystrica Region of Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Panické Dravce na stránke Novohradu
Villages and municipalities in Lučenec District
|
https://en.wikipedia.org/wiki/Podre%C4%8Dany
|
Podrečany () is a village and municipality in the Lučenec District in the Banská Bystrica Region of Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Lučenec District
|
https://en.wikipedia.org/wiki/Star%C3%A1%20Hali%C4%8D
|
Stará Halič () is a village and municipality in the Lučenec District in the Banská Bystrica Region of Slovakia.
References
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Lučenec District
|
https://en.wikipedia.org/wiki/%C5%A0urice
|
Šurice () is a village and municipality in the Lučenec District in the Banská Bystrica Region of Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Lučenec District
|
https://en.wikipedia.org/wiki/%C5%A0%C3%A1vo%C4%BE
|
Šávoľ () is a village and municipality in the Lučenec District in the Banská Bystrica Region of Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Lučenec District
|
https://en.wikipedia.org/wiki/%C5%A0%C3%ADd
|
Šíd () is a village and municipality in the Lučenec District in the Banská Bystrica Region of Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Lučenec District
|
https://en.wikipedia.org/wiki/Tom%C3%A1%C5%A1ovce%2C%20Lu%C4%8Denec%20District
|
Tomášovce () is a village and municipality in the Lučenec District in the Banská Bystrica Region of Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Lučenec District
|
https://en.wikipedia.org/wiki/Trebe%C4%BEovce
|
Trebeľovce () is a village and municipality in the Lučenec
District in the Banská Bystrica Region of Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Lučenec District
|
https://en.wikipedia.org/wiki/Vidin%C3%A1
|
Vidiná () is a village and municipality in the Lučenec District in the Banská Bystrica Region of Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Lučenec District
|
https://en.wikipedia.org/wiki/Ve%C4%BEk%C3%A1%20nad%20Ip%C4%BEom
|
Veľká nad Ipľom () is a village and municipality in the Lučenec District in the Banská Bystrica Region of Slovakia.
External links
https://web.archive.org/web/20080111223415/http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Lučenec District
|
https://en.wikipedia.org/wiki/To%C4%8Dnica
|
Točnica () is a village and municipality in the Lučenec District in the Banská Bystrica Region of Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Lučenec District
|
https://en.wikipedia.org/wiki/Tren%C4%8D
|
Trenč () is a village and municipality in the Lučenec District in the Banská Bystrica Region of Slovakia.
References
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Lučenec District
|
https://en.wikipedia.org/wiki/GridWars
|
GridWars or GridWars 2 (stylized as |g|ridwars) is a video game inspired by Bizarre Creations' Geometry Wars by Canadian developer Marco Incitti. The first version of the game was released as freeware on December 21, 2005 on Marco Incitti's Blitz Creations website.
Although other freeware games similar to Geometry Wars exist, GridWars eventually became so popular that Bizarre emailed Incitti and asked him to take the game down, citing that they were "beginning to feel the Geometry Wars clone's effects on our sales via Microsoft now and are beginning a process to begin to more robustly protect our copyright and intellectual property". The link to the download was pulled from the site on August 8, 2006. The game was subsequently released on a German website dedicated to Flash and freeware games and is available for Windows, OS X and Linux.
Unlike Geometry Wars which eventually plateaus in difficulty facilitating high scores in the range of hundreds of millions of points, GridWars 2 has an exponentially increasing difficulty curve. Eventually, the enemy spawn sites become so large that they nearly envelop the entire grid.
Another important difference from Geometry Wars is the ability to "farm" black holes. Besides distracting and swallowing enemies, the black holes can net points (and thus bombs/lives) based on the square of the number of enemies they have swallowed.
References
Sinclair, Brandan. "Bizarre begins Geometry Wars clone wars". GameSpot. August 15, 2006.
Bramwell, Tom. "Bizarre finally tackles Grid Wars ". Eurogamer. August 16, 2006.
Bramwell, Tom. "Grid Wars author's reflections ". Eurogamer. August 18, 2006.
External links
My Blitz Creations - Marco Incitti's page with his other games, and where GridWars was formerly hosted.
2005 video games
Multidirectional shooters
Windows games
MacOS games
Linux games
Video game clones
Unauthorized video games
Video games developed in Canada
|
https://en.wikipedia.org/wiki/Ratkovsk%C3%A9%20Bystr%C3%A9
|
Ratkovské Bystré () is a village and municipality in Revúca District in the Banská Bystrica Region of Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Revúca District
|
https://en.wikipedia.org/wiki/Mur%C3%A1nska%20Huta
|
Muránska Huta () is a village and municipality in Revúca District in the Banská Bystrica Region of Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Revúca District
|
https://en.wikipedia.org/wiki/Rev%C3%BAcka%20Lehota
|
Revúcka Lehota () is a village and municipality in Revúca District in the Banská Bystrica Region of Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Revúca District
|
https://en.wikipedia.org/wiki/NCCS
|
NCCS may refer to:
National Cancer Centre Singapore, a Cancer specialist medical centre in Singapore
National Catholic Community Service
National Center for Charitable Statistics
National Center for Computational Sciences, at Oak Ridge National Laboratory
National Center for Constitutional Studies, American conservative organization
National Centre For Cell Science, University of Pune, India
National Coalition for Cancer Survivorship, American cancer survivor advocacy organization
New Canaan Country School, a K-9 school in New Canaan, Connecticut
North County Christian School, K-12 Christian school in Florissant, Missouri
North Cow Creek School, K-8 public school in Palo Cedro, California
National Center for Cyber Security (Pakistan)
See also
NCC (disambiguation)
|
https://en.wikipedia.org/wiki/Parker%E2%80%93Sochacki%20method
|
In mathematics, the Parker–Sochacki method is an algorithm for solving systems of ordinary differential equations (ODEs), developed by G. Edgar Parker and James Sochacki, of the James Madison University Mathematics Department. The method produces Maclaurin series solutions to systems of differential equations, with the coefficients in either algebraic or numerical form.
Summary
The Parker–Sochacki method rests on two simple observations:
If a set of ODEs has a particular form, then the Picard method can be used to find their solution in the form of a power series.
If the ODEs do not have the required form, it is nearly always possible to find an expanded set of equations that do have the required form, such that a subset of the solution is a solution of the original ODEs.
Several coefficients of the power series are calculated in turn, a time step is chosen, the series is evaluated at that time, and the process repeats.
The end result is a high order piecewise solution to the original ODE problem. The order of the solution desired is an adjustable variable in the program that can change between steps. The order of the solution is only limited by the floating point representation on the machine running the program. And in some cases can be either extended by using arbitrary precision floating point numbers, or for special cases by finding solution with only integer or rational coefficients.
Advantages
The method requires only addition, subtraction, and multiplication, making it very convenient for high-speed computation. (The only divisions are inverses of small integers, which can be precomputed.)
Use of a high order—calculating many coefficients of the power series—is convenient. (Typically a higher order permits a longer time step without loss of accuracy, which improves efficiency.)
The order and step size can be easily changed from one step to the next.
It is possible to calculate a guaranteed error bound on the solution.
Arbitrary precision floating point libraries allow this method to compute arbitrarily accurate solutions.
With the Parker–Sochacki method, information between integration steps is developed at high order. As the Parker–Sochacki method integrates, the program can be designed to save the power series coefficients that provide a smooth solution between points in time. The coefficients can be saved and used so that polynomial evaluation provides the high order solution between steps. With most other classical integration methods, one would have to resort to interpolation to get information between integration steps, leading to an increase of error.
There is an A-priori error bound for a single step with the Parker–Sochacki method. This allows a Parker–Sochacki program to calculate the step size that guarantees that the error is below any non-zero given tolerance. Using this calculated step size with an error tolerance of less than half of the machine epsilon yields a symplectic integration.
Disadvantages
Most met
|
https://en.wikipedia.org/wiki/Tightness%20of%20measures
|
In mathematics, tightness is a concept in measure theory. The intuitive idea is that a given collection of measures does not "escape to infinity".
Definitions
Let be a Hausdorff space, and let be a σ-algebra on that contains the topology . (Thus, every open subset of is a measurable set and is at least as fine as the Borel σ-algebra on .) Let be a collection of (possibly signed or complex) measures defined on . The collection is called tight (or sometimes uniformly tight) if, for any , there is a compact subset of such that, for all measures ,
where is the total variation measure of . Very often, the measures in question are probability measures, so the last part can be written as
If a tight collection consists of a single measure , then (depending upon the author) may either be said to be a tight measure or to be an inner regular measure.
If is an -valued random variable whose probability distribution on is a tight measure then is said to be a separable random variable or a Radon random variable.
Examples
Compact spaces
If is a metrisable compact space, then every collection of (possibly complex) measures on is tight. This is not necessarily so for non-metrisable compact spaces. If we take with its order topology, then there exists a measure on it that is not inner regular. Therefore, the singleton is not tight.
Polish spaces
If is a Polish space, then every probability measure on is tight. Furthermore, by Prokhorov's theorem, a collection of probability measures on is tight if and only if
it is precompact in the topology of weak convergence.
A collection of point masses
Consider the real line with its usual Borel topology. Let denote the Dirac measure, a unit mass at the point in . The collection
is not tight, since the compact subsets of are precisely the closed and bounded subsets, and any such set, since it is bounded, has -measure zero for large enough . On the other hand, the collection
is tight: the compact interval will work as for any . In general, a collection of Dirac delta measures on is tight if, and only if, the collection of their supports is bounded.
A collection of Gaussian measures
Consider -dimensional Euclidean space with its usual Borel topology and σ-algebra. Consider a collection of Gaussian measures
where the measure has expected value (mean) and covariance matrix . Then the collection is tight if, and only if, the collections and are both bounded.
Tightness and convergence
Tightness is often a necessary criterion for proving the weak convergence of a sequence of probability measures, especially when the measure space has infinite dimension. See
Finite-dimensional distribution
Prokhorov's theorem
Lévy–Prokhorov metric
Weak convergence of measures
Tightness in classical Wiener space
Tightness in Skorokhod space
Exponential tightness
A strengthening of tightness is the concept of exponential tightness, which has applications in large deviations theory. A famil
|
https://en.wikipedia.org/wiki/Prokhorov%27s%20theorem
|
In measure theory Prokhorov's theorem relates tightness of measures to relative compactness (and hence weak convergence) in the space of probability measures. It is credited to the Soviet mathematician Yuri Vasilyevich Prokhorov, who considered probability measures on complete separable metric spaces. The term "Prokhorov’s theorem" is also applied to later generalizations to either the direct or the inverse statements.
Statement
Let be a separable metric space.
Let denote the collection of all probability measures defined on (with its Borel σ-algebra).
Theorem.
A collection of probability measures is tight if and only if the closure of is sequentially compact in the space equipped with the topology of weak convergence.
The space with the topology of weak convergence is metrizable.
Suppose that in addition, is a complete metric space (so that is a Polish space). There is a complete metric on equivalent to the topology of weak convergence; moreover, is tight if and only if the closure of in is compact.
Corollaries
For Euclidean spaces we have that:
If is a tight sequence in (the collection of probability measures on -dimensional Euclidean space), then there exist a subsequence and a probability measure such that converges weakly to .
If is a tight sequence in such that every weakly convergent subsequence has the same limit , then the sequence converges weakly to .
Extension
Prokhorov's theorem can be extended to consider complex measures or finite signed measures.
Theorem:
Suppose that is a complete separable metric space and is a family of Borel complex measures on . The following statements are equivalent:
is sequentially precompact; that is, every sequence has a weakly convergent subsequence.
is tight and uniformly bounded in total variation norm.
Comments
Since Prokhorov's theorem expresses tightness in terms of compactness, the Arzelà–Ascoli theorem is often used to substitute for compactness: in function spaces, this leads to a characterization of tightness in terms of the modulus of continuity or an appropriate analogue—see tightness in classical Wiener space and tightness in Skorokhod space.
There are several deep and non-trivial extensions to Prokhorov's theorem. However, those results do not overshadow the importance and the relevance to applications of the original result.
See also
References
Compactness theorems
Theorems in measure theory
|
https://en.wikipedia.org/wiki/William%20Eckhardt%20%28trader%29
|
William Eckhardt is a trader.
Education
Eckhardt never finished his PhD in mathematics, claiming that he left graduate school for the trading pits after an unexpected change of thesis advisors. Despite leaving academia prematurely, Eckhardt has published several papers in academic journals. In 1993, Eckhardt's article "Probability Theory and the Doomsday Argument" was published in the philosophical journal Mind. His follow-up article, "A Shooting-Room view of Doomsday" was published in The Journal of Philosophy in 1997. Both articles make arguments skeptical of the Doomsday Argument as formulated by John Leslie. In 2006, he published "Causal time asymmetry" in the journal Studies In History and Philosophy of Modern Physics.
In 2013, he published a book "Paradoxes in Probability Theory"
References
Notes
Further reading
1955 births
American money managers
American derivatives traders
American financial analysts
American hedge fund managers
Living people
Stock and commodity market managers
20th-century American businesspeople
|
https://en.wikipedia.org/wiki/Outline%20of%20calculus
|
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of contemporary mathematics education. Calculus has widespread applications in science, economics, and engineering and can solve many problems for which algebra alone is insufficient.
Branches of calculus
Differential calculus
Integral calculus
Multivariable calculus
Fractional calculus
Differential Geometry
History of calculus
History of calculus
Important publications in calculus
General calculus concepts
Continuous function
Derivative
Fundamental theorem of calculus
Integral
Limit
Non-standard analysis
Partial derivative
Infinite Series
Calculus scholars
Sir Isaac Newton
Gottfried Leibniz
Calculus lists
List of calculus topics
See also
Glossary of calculus
Table of mathematical symbols
References
External links
Calculus Made Easy (1914) by Silvanus P. Thompson Full text in PDF
Calculus.org: The Calculus page at University of California, Davis – contains resources and links to other sites
COW: Calculus on the Web at Temple University - contains resources ranging from pre-calculus and associated algebra
Online Integrator (WebMathematica) from Wolfram Research
The Role of Calculus in College Mathematics from ERICDigests.org
OpenCourseWare Calculus from the Massachusetts Institute of Technology
Infinitesimal Calculus – an article on its historical development, in Encyclopaedia of Mathematics, Michiel Hazewinkel ed.
Calculus
Calculus
Calculus
|
https://en.wikipedia.org/wiki/AMT%20Skipper
|
The AMT Skipper was a stainless steel copy of the .45 ACP Colt Combat Commander made by Arcadia Machine and Tool.
Statistics
SKIPPER
Chambering: .45 ACP., .40 S&W
Barrel Length: 4 inches.
Overall Length: 7.5 inches.
Weight: ?
Magazine: 6-round single-column box magazine.
Sights: Adjustable for windage and elevation
Finish:
Furniture:Plastic
Features: Adjustable Trigger & Sights
Production: 1978-79 (Very limited ) Manufactured in El Monte, CA.
COMBAT SKIPPER
Chambering: .45 ACP, .40 S&W
Barrel Length: 4.5 inches
Overall Length: 8 inches
Weight: 32 ounces
Magazine: 6-round single-stack magazine
Sights: Adjustable
Finish: Stainless Steel
Stocks: Plastic or Redwood
Production: 1978-79 ( Very Limited ) Manufactured in El Monte, CA.
See AMT Hardballer for additional info
External links
AMT Skipper Owners Manual
AMT semi-automatic pistols
.45 ACP semi-automatic pistols
Semi-automatic pistols of the United States
|
https://en.wikipedia.org/wiki/Andr%C3%A1s%20Kornai
|
András Kornai (born 1957 in Budapest), son of economist János Kornai, is a mathematical linguist. He has earned two PhDs. He earned his first in Mathematics in 1983 from Eötvös Loránd University in Budapest, where his advisor was Miklós Ajtai, and his second in Linguistics in 1991 from Stanford University, where his advisor was Paul Kiparsky.
He is a professor in the Department of Algebra at the Budapest Institute of Technology, where he works on an open source Hungarian morphological analyzer. He was Chief Scientist at MetaCarta, where he worked on information extraction before the company was acquired by Nokia. Prior to MetaCarta, he was Chief Scientist at Northern Light.
He is on the board of the journal Grammars and YourAmigo PLC. His research interests include all mathematical aspects of natural language processing, speech recognition, and OCR.
As Area Editor he was responsible for the Mathematical Linguistics area of the Oxford International Encyclopedia of Linguistics, and his joint work with Geoffrey Pullum, "The X-bar Theory of Phrase Structure", formally reconstructed that then-popular linguistic theory.
Monographs
Semantics. Springer Nature, 2020.
Mathematical Linguistics. Springer Verlag, in the series Advanced Information and Knowledge Processing, November 2007. Hardbound, approximately 300 pages. See description.
Formal Phonology. In the series Outstanding Dissertations in Linguistics, Garland Publishing, 1994, , hardbound, 240 pages Contents, Preface, Introduction (20 pages)
On Hungarian Morphology. In the series Linguistica, Hungarian Academy of Sciences, 1994, , paperbound, 174 pages Contents, Preface, Introduction (10 pages)
Books edited
Oxford International Encyclopedia of Linguistics (Mathematical Linguistics Area Editor under Editor in Chief William Frawley). 4 volumes, Oxford University Press, 2003, .
Proceedings of the HLT-NAACL Workshop on the Analysis of Geographic References. Jointly with Beth Sundheim. Association for Computational Linguistics, 2003, (WS9), paperbound, vi+81 pages. See related material.
Extended Finite State Models of Language (editor). In the series Studies in Natural Language Processing, Cambridge University Press, 1999, , hardbound, x+278 pages Contents, Introduction (7 pages).
Selected papers
Digital Language Death. PLoS ONE 8(10): e77056, 2012.
Hunmorph: open source word analysis (Jointly with V. Tron, Gy. Gyepesi, P. Halacsy, L. Nemeth, and D. Varga). In Proc. ACL 2005 Software Workshop 77-85
Leveraging the open source ispell codebase for minority language analysis (Jointly with P. Halacsy, L. Nemeth, A. Rung, I. Szakadat, and V. Tron). In J. Carson-Berndsen (ed): Proc. SALTMIL 2004 56-59
Explicit Finitism, International Journal of Theoretical Physics 2003/2 301-307
Mathematical Linguistics (Jointly with G.K. Pullum) In W. Frawley (ed): Oxford International Encyclopedia of Linguistics, Oxford University Press 2003, v3 17-20
Optical Character Recognition, In W. Frawley (ed):
|
https://en.wikipedia.org/wiki/Ham-Nord
|
Ham-Nord, Quebec is a township municipality in the Centre-du-Québec region of Quebec, Canada.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Ham-Nord 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
(Google Maps)
External links
Township municipalities in Quebec
Incorporated places in Centre-du-Québec
|
https://en.wikipedia.org/wiki/GEUP
|
GEUP is a commercial interactive geometry software program, similar to Cabri Geometry. Originally using the Spanish language, it was programmed by Ramón
Alvarez Galván. Recent versions include support for three-dimensional geometry.
References
Further reading
.
External links
GEUP.net
Mathematical software
Interactive geometry software
Science software for Windows
|
https://en.wikipedia.org/wiki/Lev%C3%A1re
|
Levare () is a village and municipality in Revúca District in the Banská Bystrica Region of Slovakia.
External links
https://web.archive.org/web/20061230185723/http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Revúca District
|
https://en.wikipedia.org/wiki/Polina%2C%20Rev%C3%BAca%20District
|
Polina () is a hamlet in the Revúca District, Banská Bystrica Region, Slovakia.
External links
https://web.archive.org/web/20071027094149/http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Revúca District
|
https://en.wikipedia.org/wiki/Avi%20Wigderson
|
Avi Wigderson (; born 9 September 1956) is an Israeli mathematician and computer scientist. He is the Herbert H. Maass Professor in the school of mathematics at the Institute for Advanced Study in Princeton, New Jersey, United States of America. His research interests include complexity theory, parallel algorithms, graph theory, cryptography, distributed computing, and neural networks. Wigderson received the Abel Prize in 2021 for his work in theoretical computer science.
Biography
Avi Wigderson was born in Haifa, Israel, to Holocaust survivors. Wigderson is a graduate of the Hebrew Reali School in Haifa, and did his undergraduate studies at the Technion in Haifa, Israel, graduating in 1980, and went on to graduate study at Princeton University. He received his PhD in computer science in 1983 after completing a doctoral dissertation, titled "Studies in computational complexity", under the supervision of Richard Lipton. After short-term positions at the University of California, Berkeley, the IBM Almaden Research Center in San Jose, California, and the Mathematical Sciences Research Institute in Berkeley, he joined the faculty of Hebrew University in 1986. In 1999 he also took a position at the Institute for Advanced Study, and in 2003 he gave up his Hebrew University position to take up full-time residence at the IAS.
Awards and honors
Wigderson received the Nevanlinna Prize in 1994 for his work on computational complexity. Along with Omer Reingold and Salil Vadhan he won the 2009 Gödel Prize for work on the zig-zag product of graphs, a method of combining smaller graphs to produce larger ones used in the construction of expander graphs. Wigderson was elected as a member of the American Academy of Arts and Sciences in 2011. He was elected to the National Academy of Sciences in 2013.
He was elected as an ACM Fellow in 2018 for "contributions to theoretical computer science and mathematics".
In 2019, Wigderson was awarded the Knuth Prize for his contributions to "the foundations of computer science in areas including randomized computation, cryptography, circuit complexity, proof complexity, parallel computation, and our understanding of fundamental graph properties".
In 2021 Wigderson shared the Abel Prize with László Lovász "for their foundational contributions to theoretical computer science and discrete mathematics, and their leading role in shaping them into central fields of modern mathematics."
References
External links
Avi Wigderson's home page
1956 births
Living people
20th-century American scientists
20th-century Israeli engineers
21st-century American scientists
21st-century engineers
Abel Prize laureates
American people of Israeli descent
Fellows of the Association for Computing Machinery
Gödel Prize laureates
Knuth Prize laureates
Academic staff of the Hebrew University of Jerusalem
Institute for Advanced Study faculty
Israeli computer scientists
20th-century Israeli mathematicians
Members of the United States National Academy of
|
https://en.wikipedia.org/wiki/Dennis%20Gaitsgory
|
Dennis Gaitsgory is a professor of mathematics at Harvard University known for his research on the geometric Langlands program.
Born in Chișinău, now in Moldova, he grew up in Tajikistan, before studying at Tel Aviv University under Joseph Bernstein (1990–1996). He received his doctorate in 1997 for a thesis entitled "Automorphic Sheaves and Eisenstein Series". He has been awarded a Harvard Junior Fellowship, a Clay Research Fellowship, and the prize of the European Mathematical Society for his work.
His work in geometric Langlands culminated in a joint 2002 paper with Edward Frenkel and Kari Vilonen, establishing the conjecture for finite fields, and a separate 2004 paper, generalizing the proof to include the field of complex numbers as well.
Prior to his 2005 appointment at Harvard, he was an associate professor at the University of Chicago from 2001–2005.
Selected publications
References
External links
Gaitsgory's thesis
Living people
Year of birth missing (living people)
Scientists from Chișinău
Tel Aviv University alumni
University of Chicago faculty
Harvard University Department of Mathematics faculty
Harvard University faculty
20th-century American mathematicians
21st-century American mathematicians
Israeli mathematicians
Israeli people of Russian-Jewish descent
Max Planck Institute directors
|
https://en.wikipedia.org/wiki/Cylinder%20set%20measure
|
In mathematics, cylinder set measure (or promeasure, or premeasure, or quasi-measure, or CSM) is a kind of prototype for a measure on an infinite-dimensional vector space. An example is the Gaussian cylinder set measure on Hilbert space.
Cylinder set measures are in general not measures (and in particular need not be countably additive but only finitely additive), but can be used to define measures, such as classical Wiener measure on the set of continuous paths starting at the origin in Euclidean space.
Definition
Let be a separable real topological vector space. Let denote the collection of all surjective continuous linear maps defined on whose image is some finite-dimensional real vector space :
A cylinder set measure on is a collection of probability measures
where is a probability measure on These measures are required to satisfy the following consistency condition: if is a surjective projection, then the push forward of the measure is as follows:
Remarks
The consistency condition
is modelled on the way that true measures push forward (see the section cylinder set measures versus true measures). However, it is important to understand that in the case of cylinder set measures, this is a requirement that is part of the definition, not a result.
A cylinder set measure can be intuitively understood as defining a finitely additive function on the cylinder sets of the topological vector space The cylinder sets are the pre-images in of measurable sets in : if denotes the -algebra on on which is defined, then
In practice, one often takes to be the Borel -algebra on In this case, one can show that when is a separable Banach space, the σ-algebra generated by the cylinder sets is precisely the Borel -algebra of :
Cylinder set measures versus measures
A cylinder set measure on is not actually a measure on : it is a collection of measures defined on all finite-dimensional images of If has a probability measure already defined on it, then gives rise to a cylinder set measure on using the push forward: set on
When there is a measure on such that in this way, it is customary to abuse notation slightly and say that the cylinder set measure "is" the measure
Cylinder set measures on Hilbert spaces
When the Banach space is actually a Hilbert space there is a arising from the inner product structure on Specifically, if denotes the inner product on let denote the quotient inner product on The measure on is then defined to be the canonical Gaussian measure on :
where is an isometry of Hilbert spaces taking the Euclidean inner product on to the inner product on and is the standard Gaussian measure on
The canonical Gaussian cylinder set measure on an infinite-dimensional separable Hilbert space does not correspond to a true measure on The proof is quite simple: the ball of radius (and center 0) has measure at most equal to that of the ball of radius in an -dimensional Hilbert space, and this tends to
|
https://en.wikipedia.org/wiki/The%20Sixth%20Dimension
|
The Sixth Dimension or Sixth Dimension may refer to:
Six-dimensional space, a concept in mathematics and physics
Sixth Dimension, a 2017 album by Power Quest
The Sixth Dimension, a fictional place in the 1982 film Forbidden Zone
The Sixth Dimension, a fictional place in the British-Canadian TV series Ace Lightning
|
https://en.wikipedia.org/wiki/John%20Panaretos
|
John Panaretos (; born 1948 in Kythera) is a Greek educator and statistician. He is Professor of Probability and Statistics at the Athens University of Economics and Business. He was Deputy Minister of Education, Lifelong Learning and Religious Affairs (6 October 2009 – 17 June 2011). He has also been appointed by the Prime Minister to be in charge of the Open Government project.
Education and career
Before joining the Athens University of Economics and Business, he taught at the universities of Patras and Crete in Greece, University of Iowa and University of Missouri in the United States, and at Trinity College, Dublin in Ireland.
He has been Director of the Institute of Statistical Documentation Research and Analysis since 1996, a life member of the Scientific Council of the Greek Parliament since 1987, and a member of the governing board of the Institute of Strategic and Development Studies (ISTAME) - Andreas Papandreou (2005–2008). Since 2004, he is the education adviser of George Papandreou, leader of Panhellenic Socialist Movement (PASOK, the main opposition party of Greece) and the Socialist International. He was a member of the National Council of Education, the Council of University Education, the Council of Technical Education and the Council of Primary and Secondary Education.
He has acted as Chairman of the Department of Statistics (1993–1996 & 2000-2002) and as member of the research committee of the University (1993–1996). At the University of Patras, he served as Vice Rector for Academic Affairs, as Chairman of the Research Committee, as Associate Dean of the Engineering School and as Chairman of the Division of Mathematics of the School.
Panaretos has served as Vice President of the European Network of the National Councils of Education (1997–2000), Chairman of the National Council of Education of Greece (1996–2000), the Council of Higher Education (1988–1989), and the Secretary General of the Ministry of Education and Religious Affairs in Greece (1988–1989 and 1995–1996), and member of the governing board of the Institute of Strategic and Development Studies (ISTAME) - Andreas Papandreou (2004–2008).
He has published more than fifty papers in international scientific journals, was an invited speaker in many scientific meetings, and served as Associate Editor of Communications in Statistics (A), Theory and Methods.
Initiatives as Deputy Minister of Education
The preparation of a new Law of Higher Education in Greece that introduced, for the first time in Greece, a new model of Governance (Board of Trustees) and a selection of the Rector and the deans by means of an open call of interest, at an international level.
The formation of an international advisory committee for the reforms on higher education consisting of nine internationally known figures in HE from all over the world (Linda P.B. Katehi (Chair), Chancellor, University of California, Davis, USA – Patrick Aebischer, President, École Polytechnique Fédérale de La
|
https://en.wikipedia.org/wiki/Gaussian%20measure
|
In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space Rn, closely related to the normal distribution in statistics. There is also a generalization to infinite-dimensional spaces. Gaussian measures are named after the German mathematician Carl Friedrich Gauss. One reason why Gaussian measures are so ubiquitous in probability theory is the central limit theorem. Loosely speaking, it states that if a random variable X is obtained by summing a large number N of independent random variables of order 1, then X is of order and its law is approximately Gaussian.
Definitions
Let n ∈ N and let B0(Rn) denote the completion of the Borel σ-algebra on Rn. Let λn : B0(Rn) → [0, +∞] denote the usual n-dimensional Lebesgue measure. Then the standard Gaussian measure γn : B0(Rn) → [0, 1] is defined by
for any measurable set A ∈ B0(Rn). In terms of the Radon–Nikodym derivative,
More generally, the Gaussian measure with mean μ ∈ Rn and variance σ2 > 0 is given by
Gaussian measures with mean μ = 0 are known as centred Gaussian measures.
The Dirac measure δμ is the weak limit of as σ → 0, and is considered to be a degenerate Gaussian measure; in contrast, Gaussian measures with finite, non-zero variance are called non-degenerate Gaussian measures.
Properties
The standard Gaussian measure γn on Rn
is a Borel measure (in fact, as remarked above, it is defined on the completion of the Borel sigma algebra, which is a finer structure);
is equivalent to Lebesgue measure: , where stands for absolute continuity of measures;
is supported on all of Euclidean space: supp(γn) = Rn;
is a probability measure (γn(Rn) = 1), and so it is locally finite;
is strictly positive: every non-empty open set has positive measure;
is inner regular: for all Borel sets A, so Gaussian measure is a Radon measure;
is not translation-invariant, but does satisfy the relation where the derivative on the left-hand side is the Radon–Nikodym derivative, and (Th)∗(γn) is the push forward of standard Gaussian measure by the translation map Th : Rn → Rn, Th(x) = x + h;
is the probability measure associated to a normal probability distribution:
Infinite-dimensional spaces
It can be shown that there is no analogue of Lebesgue measure on an infinite-dimensional vector space. Even so, it is possible to define Gaussian measures on infinite-dimensional spaces, the main example being the abstract Wiener space construction. A Borel measure γ on a separable Banach space E is said to be a non-degenerate (centered) Gaussian measure if, for every linear functional L ∈ E∗ except L = 0, the push-forward measure L∗(γ) is a non-degenerate (centered) Gaussian measure on R in the sense defined above.
For example, classical Wiener measure on the space of continuous paths is a Gaussian measure.
References
See also
- a generalisation of Gaussian measure
Measures (measure theory)
Stochastic processes
|
https://en.wikipedia.org/wiki/Cameron%E2%80%93Martin%20theorem
|
In mathematics, the Cameron–Martin theorem or Cameron–Martin formula (named after Robert Horton Cameron and W. T. Martin) is a theorem of measure theory that describes how abstract Wiener measure changes under translation by certain elements of the Cameron–Martin Hilbert space.
Motivation
The standard Gaussian measure on -dimensional Euclidean space is not translation-invariant. (In fact, there is a unique translation invariant Radon measure up to scale by Haar's theorem: the -dimensional Lebesgue measure, denoted here .) Instead, a measurable subset has Gaussian measure
Here refers to the standard Euclidean dot product in . The Gaussian measure of the translation of by a vector is
So under translation through , the Gaussian measure scales by the distribution function appearing in the last display:
The measure that associates to the set the number is the pushforward measure, denoted . Here refers to the translation map: . The above calculation shows that the Radon–Nikodym derivative of the pushforward measure with respect to the original Gaussian measure is given by
The abstract Wiener measure on a separable Banach space , where is an abstract Wiener space, is also a "Gaussian measure" in a suitable sense. How does it change under translation? It turns out that a similar formula to the one above holds if we consider only translations by elements of the dense subspace .
Statement of the theorem
Let be an abstract Wiener space with abstract Wiener measure . For , define by . Then is equivalent to with Radon–Nikodym derivative
where
denotes the Paley–Wiener integral.
The Cameron–Martin formula is valid only for translations by elements of the dense subspace , called Cameron–Martin space, and not by arbitrary elements of . If the Cameron–Martin formula did hold for arbitrary translations, it would contradict the following result:
If is a separable Banach space and is a locally finite Borel measure on that is equivalent to its own push forward under any translation, then either has finite dimension or is the trivial (zero) measure. (See quasi-invariant measure.)
In fact, is quasi-invariant under translation by an element if and only if . Vectors in are sometimes known as Cameron–Martin directions.
Integration by parts
The Cameron–Martin formula gives rise to an integration by parts formula on : if has bounded Fréchet derivative , integrating the Cameron–Martin formula with respect to Wiener measure on both sides gives
for any . Formally differentiating with respect to and evaluating at gives the integration by parts formula
Comparison with the divergence theorem of vector calculus suggests
where is the constant "vector field" for all . The wish to consider more general vector fields and to think of stochastic integrals as "divergences" leads to the study of stochastic processes and the Malliavin calculus, and, in particular, the Clark–Ocone theorem and its associated integration by parts formula.
An
|
https://en.wikipedia.org/wiki/Exceptional%20Lie%20algebra
|
In mathematics, an exceptional Lie algebra is a complex simple Lie algebra whose Dynkin diagram is of exceptional (nonclassical) type. There are exactly five of them: ; their respective dimensions are 14, 52, 78, 133, 248. The corresponding diagrams are:
G2 :
F4 :
E6 :
E7 :
E8 :
In contrast, simple Lie algebras that are not exceptional are called classical Lie algebras (there are infinitely many of them).
Construction
There is no simple universally accepted way to construct exceptional Lie algebras; in fact, they were discovered only in the process of the classification program. Here are some constructions:
§ 22.1-2 of give a detailed construction of .
Exceptional Lie algebras may be realized as the derivation algebras of appropriate nonassociative algebras.
Construct first and then find as subalgebras.
Tits has given a uniformed construction of the five exceptional Lie algebras.
References
Further reading
https://www.encyclopediaofmath.org/index.php/Lie_algebra,_exceptional
http://math.ucr.edu/home/baez/octonions/node13.html
Lie algebras
|
https://en.wikipedia.org/wiki/Masoud%20Zarei
|
Masoud Zarei (, born August 25, 1981, in Tehran, Iran) is an Iranian footballer, currently a member of the IPL club Mes Kerman.
Club career
Club career Statistics
Last Update 16 December 2009
Assist Goals
Honours
Azadegan League
Winner: 1
2003/04 with Saba Battery
Hazfi Cup
Winner: 1
2005 with Saba Battery
Iran's Premier Football League
Winner: 1
2007/08 with Persepolis
External links
1981 births
Living people
Saba Qom F.C. players
Iranian men's footballers
Persepolis F.C. players
Sanat Mes Kerman F.C. players
Men's association football defenders
Footballers from Tehran
Persepolis F.C. non-playing staff
|
https://en.wikipedia.org/wiki/Ebrahim%20Asadi
|
Ebrahim Asadi (, born June 8, 1979) is a retired Iranian footballer who played for Persepolis.
Club career
Club Career Statistics
Last Update 18 September 2010
Assist Goals
Honours
Club
Persepolis
Iranian Football League (2) : 1999–2000, 2001–02
References
External links
Persian League Profile
1979 births
Living people
Zob Ahan Esfahan F.C. players
Steel Azin F.C. players
F.C. Nassaji Mazandaran players
Iranian men's footballers
Persepolis F.C. players
Men's association football midfielders
Footballers from Tehran
|
https://en.wikipedia.org/wiki/Regge
|
Regge may refer to
Tullio Regge (1931-2014), Italian physicist, developer of Regge calculus and Regge theory
Regge calculus, formalism for producing simplicial approximations of spacetimes
Regge theory, study of the analytic properties of scattering
3778 Regge, main-belt asteroid
Regge (river), river in Overijssel, the Netherlands
Surnames of Italian origin
|
https://en.wikipedia.org/wiki/Lajos%20Steiner
|
Lajos Steiner (14 June 1903, in Nagyvárad (Oradea) – 22 April 1975, in Sydney) was a Hungarian–born Australian chess master.
Steiner was one of four children of Bernat Steiner, a mathematics teacher, and his wife Cecilia,(née Schwarz). His elder brother was Endre Steiner. He was educated at the Technical High School in Budapest, and graduated in 1926 with a diploma in mechanical engineering from the Technikum Mittweida in Germany.
In 1923, he tied for 4-5th in Vienna. In 1925 he took 2nd, behind Sándor Takács, in Budapest. In 1927, he won in Schandau and tied for 2nd-3rd in Kecskemét. In 1927/28, he took 2nd. In 1929, he took 2nd in Bradley Beach. In 1931, he won in Budapest (HUN-ch), took 5th in Vienna, and tied for 5-6th in Berlin. The event was won by Herman Steiner. In 1932/33, he tied for 3rd-4th in Hastings (Salo Flohr won). In 1933, he tied for 2nd-3rd in Maehrisch-Ostrau (Ostrava). The event was won by Ernst Grünfeld. In 1933, he took 4th in Budapest.
In 1934, he tied for 1st-2nd with Vasja Pirc in Maribor (Marburg). In 1935, he tied for 1st-2nd with Erich Eliskases in Vienna (the 18th Trebitsch Memorial). In 1935, he tied for 5-6th in Łódź (Savielly Tartakower won) and took 4th in Tatatovaros (László Szabó won). In 1936, he won, with Mieczysław Najdorf, in Budapest (HUN-ch). In 1937, he took 2nd in Brno (Brunn), and took 3rd in Zoppot (Sopot). In 1937/38, he won in Vienna (the 20th Trebitsch Memorial). In 1938, he tied for 3rd-4th in Ljubljana (Laibach). The event was won by Borislav Kostić. In 1938, he tied for 8-9th in Łódź where Pirc won.
Lajos Steiner played a few matches. In 1930, he lost (+3 –5 =2) to Isaac Kashdan. In 1934, he won (+7 –3) against Pál Réthy. In 1935, he won (+3 –1) vs Henri Grob.
He played for Hungary in four Chess Olympiads:
In 1931, he played at second board at 4th Chess Olympiad in Prague (+10 –3 =4).
In 1933, he played at second board at 5th Chess Olympiad in Folkestone (+5 –4 =5).
In 1935, he played at first board at 6th Chess Olympiad in Warsaw (+7 –4 =7).
In 1936, he played at second board at 3rd unofficial Chess Olympiad in Munich (+13 –2 =5).
He won individual bronze medal in Prague, and team gold medal and individual silver medal in Munich.
Steiner emigrated to Australia in 1939. He won the Australian Chess Championship four times in 1945, 1946/47, 1952/53, and 1958/59. He also won nine of his ten attempts at the New South Wales title (1940–41, 1943, 1944, 1945–46, 1953, 1955, 1958). He took 3rd in Karlovy Vary – Mariánské Lázně in 1948. The event was won by Jan Foltys. He took 19th at the 1st Interzonal Tournament in Saltsjöbaden in 1948. The event was won by David Bronstein.
He was awarded the International Master (IM) title in 1950.
See also
List of Jewish chess players
References
External links
ADB bio
1903 births
Hungarian Jews
Hungarian emigrants to Australia
Hungarian chess players
Australian chess players
Jewish chess players
Chess International Masters
Chess Olympiad compe
|
https://en.wikipedia.org/wiki/Paley%E2%80%93Wiener%20integral
|
In mathematics, the Paley–Wiener integral is a simple stochastic integral. When applied to classical Wiener space, it is less general than the Itō integral, but the two agree when they are both defined.
The integral is named after its discoverers, Raymond Paley and Norbert Wiener.
Definition
Let be an abstract Wiener space with abstract Wiener measure on . Let be the adjoint of . (We have abused notation slightly: strictly speaking, , but since is a Hilbert space, it is isometrically isomorphic to its dual space , by the Riesz representation theorem.)
It can be shown that is an injective function and has dense image in . Furthermore, it can be shown that every linear functional is also square-integrable: in fact,
This defines a natural linear map from to , under which goes to the equivalence class of in . This is well-defined since is injective. This map is an isometry, so it is continuous.
However, since a continuous linear map between Banach spaces such as and is uniquely determined by its values on any dense subspace of its domain, there is a unique continuous linear extension of the above natural map to the whole of .
This isometry is known as the Paley–Wiener map. , also denoted , is a function on and is known as the Paley–Wiener integral (with respect to ).
It is important to note that the Paley–Wiener integral for a particular element is a function on . The notation does not really denote an inner product (since and belong to two different spaces), but is a convenient abuse of notation in view of the Cameron–Martin theorem. For this reason, many authors prefer to write or rather than using the more compact but potentially confusing notation.
See also
Other stochastic integrals:
Itō integral
Skorokhod integral
Stratonovich integral
References
(Section 6)
Definitions of mathematical integration
Stochastic calculus
|
https://en.wikipedia.org/wiki/R%C3%ADo%20Claro
|
Río Claro is a commune of the Talca Province in Chile's Maule Region. The municipal seat is the town of Cumpeo.
Demographics
According to the 2002 census of the National Statistics Institute, Río Claro spans an area of and has 12,698 inhabitants (6,716 men and 5,982 women). Of these, 2,651 (20.9%) lived in urban areas and 10,047 (79.1%) in rural areas. The population grew by 0.8% (107 persons) between the 1992 and 2002 censuses.
Administration
As a commune, Río Claro is a third-level administrative division of Chile administered by a municipal council, headed by an alcalde who is directly elected every four years. The 2008-2012 alcalde is Claudio Guajardo Oyarce.
Within the electoral divisions of Chile, Río Claro is represented in the Chamber of Deputies by Pablo Lorenzini (PDC) and Pedro Pablo Alvarez-Salamanca (UDI) as part of the 38th electoral district, together with Curepto, Constitución, Empedrado, Pencahue, Maule, San Clemente, Pelarco and San Rafael. The commune is represented in the Senate by Juan Antonio Coloma Correa (UDI) and Andrés Zaldívar Larraín (PDC) as part of the 10th senatorial constituency (Maule-North).
References
External links
Municipality of Río Claro
Communes of Chile
Populated places in Talca Province
|
https://en.wikipedia.org/wiki/Sagrada%20Familia%2C%20Chile
|
Sagrada Familia (Spanish meaning "Holy Family") is a Chilean town and commune in Curicó Province, Maule Region.
Demographics
According to the 2002 census of the National Statistics Institute, Sagrada Familia spans an area of and has 17,519 inhabitants (9,108 men and 8,411 women). Of these, 5,080 (29%) lived in urban areas and 12,439 (71%) in rural areas. The population grew by 3.7% (625 persons) between the 1992 and 2002 censuses.
Administration
As a commune, Sagrada Familia is a third-level administrative division of Chile administered by a municipal council, headed by an alcalde who is directly elected every four years. The 2008-2012 alcalde is Francisco Meléndez Rojas (PS).
Within the electoral divisions of Chile, Sagrada Familia is represented in the Chamber of Deputies by Roberto León (PDC) and Celso Morales (UDI) as part of the 36th electoral district, together with Curicó, Teno, Romeral, Molina, Hualañé, Licantén, Vichuquén and Rauco. The commune is represented in the Senate by Juan Antonio Coloma Correa (UDI) and Andrés Zaldívar Larraín (PDC) as part of the 10th senatorial constituency (Maule-North).
References
External links
Municipality of Sagrada Familia
Populated places in Curicó Province
Communes of Chile
|
https://en.wikipedia.org/wiki/Romeral
|
Romeral is a Chilean town and commune in Curicó Province, Maule Region. The commune spans and area of .
Demographics
According to the 2002 census of Population and Housing by the National Statistics Institute, the Romeral commune had 12,707 inhabitants; of these, 3,675 (28.9%) lived in urban areas and 9,032 (71.1%) in rural areas. At that time, there were 6,596 men and 6,111 women. The population grew by 10.6% (1,217 persons) between the 1992 and 2002 censuses.
Administration
As a commune, Romeral is a third-level administrative division of Chile administered by a municipal council, headed by an alcalde who is directly elected every four years. The 2008-2012 alcalde is Carlos Cisterna Negrete (PDC).
Within the electoral divisions of Chile, Romeral is represented in the Chamber of Deputies by Roberto León (PDC) and Celso Morales (UDI) as part of the 36th electoral district, together with Curicó, Teno, Molina, Sagrada Familia, Hualañé, Licantén, Vichuquén and Rauco. The commune is represented in the Senate by Juan Antonio Coloma Correa (UDI) and Andrés Zaldívar Larraín (PDC) as part of the 10th senatorial constituency (Maule-North).
References
Populated places in Curicó Province
Communes of Chile
|
https://en.wikipedia.org/wiki/Banach%E2%80%93Mazur%20theorem
|
In functional analysis, a field of mathematics, the Banach–Mazur theorem is a theorem roughly stating that most well-behaved normed spaces are subspaces of the space of continuous paths. It is named after Stefan Banach and Stanisław Mazur.
Statement
Every real, separable Banach space is isometrically isomorphic to a closed subspace of , the space of all continuous functions from the unit interval into the real line.
Comments
On the one hand, the Banach–Mazur theorem seems to tell us that the seemingly vast collection of all separable Banach spaces is not that vast or difficult to work with, since a separable Banach space is "only" a collection of continuous paths. On the other hand, the theorem tells us that is a "really big" space, big enough to contain every possible separable Banach space.
Non-separable Banach spaces cannot embed isometrically in the separable space , but for every Banach space , one can find a compact Hausdorff space and an isometric linear embedding of into the space of scalar continuous functions on . The simplest choice is to let be the unit ball of the continuous dual , equipped with the w*-topology. This unit ball is then compact by the Banach–Alaoglu theorem. The embedding is introduced by saying that for every , the continuous function on is defined by
The mapping is linear, and it is isometric by the Hahn–Banach theorem.
Another generalization was given by Kleiber and Pervin (1969): a metric space of density equal to an infinite cardinal is isometric to a subspace of , the space of real continuous functions on the product of copies of the unit interval.
Stronger versions of the theorem
Let us write for . In 1995, Luis Rodríguez-Piazza proved that the isometry can be chosen so that every non-zero function in the image is nowhere differentiable. Put another way, if consists of functions that are differentiable at at least one point of , then can be chosen so that This conclusion applies to the space itself, hence there exists a linear map that is an isometry onto its image, such that image under of (the subspace consisting of functions that are everywhere differentiable with continuous derivative) intersects only at : thus the space of smooth functions (with respect to the uniform distance) is isometrically isomorphic to a space of nowhere-differentiable functions. Note that the (metrically incomplete) space of smooth functions is dense in .
References
Theory of continuous functions
Functional analysis
Theorems in functional analysis
|
https://en.wikipedia.org/wiki/Progressively%20measurable%20process
|
In mathematics, progressive measurability is a property in the theory of stochastic processes. A progressively measurable process, while defined quite technically, is important because it implies the stopped process is measurable. Being progressively measurable is a strictly stronger property than the notion of being an adapted process. Progressively measurable processes are important in the theory of Itô integrals.
Definition
Let
be a probability space;
be a measurable space, the state space;
be a filtration of the sigma algebra ;
be a stochastic process (the index set could be or instead of );
be the Borel sigma algebra on .
The process is said to be progressively measurable (or simply progressive) if, for every time , the map defined by is -measurable. This implies that is -adapted.
A subset is said to be progressively measurable if the process is progressively measurable in the sense defined above, where is the indicator function of . The set of all such subsets form a sigma algebra on , denoted by , and a process is progressively measurable in the sense of the previous paragraph if, and only if, it is -measurable.
Properties
It can be shown that , the space of stochastic processes for which the Itô integral
with respect to Brownian motion is defined, is the set of equivalence classes of -measurable processes in .
Every adapted process with left- or right-continuous paths is progressively measurable. Consequently, every adapted process with càdlàg paths is progressively measurable.
Every measurable and adapted process has a progressively measurable modification.
References
Stochastic processes
Measure theory
|
https://en.wikipedia.org/wiki/San%20Rafael%2C%20Chile
|
San Rafael is a town and commune of the Talca Province in the Maule Region of Chile. The town serves as the communal capital.
Demographics
According to the 2002 census of the National Statistics Institute, San Rafael spans an area of and has 7,674 inhabitants (3,903 men and 3,771 women). Of these, 3,482 (45.4%) lived in urban areas and 4,192 (54.6%) in rural areas. The population grew by 6.5% (465 persons) between the 1992 and 2002 censuses.
Administration
As a commune, San Rafael is a third-level administrative division of Chile administered by a municipal council, headed by an alcalde who is directly elected every four years. The 2008-2012 alcaldesa is CLAUDIA ALEJANDRA DIAZ BRAVO. (UDI).
Within the electoral divisions of Chile, San Rafael is represented in the Chamber of Deputies by Pablo Lorenzini (PDC) and Pedro Pablo Alvarez-Salamanca (UDI) as part of the 38th electoral district, together with Curepto, Constitución, Empedrado, Pencahue, Maule, San Clemente, Pelarco and Río Claro. The commune is represented in the Senate by Juan Antonio Coloma Correa (UDI) and Andrés Zaldívar Larraín (PDC) as part of the 10th senatorial constituency (Maule-North).
References
External links
Municipality of San Rafael
Populated places in Talca Province
Communes of Chile
|
https://en.wikipedia.org/wiki/Pompeiu%20problem
|
In mathematics, the Pompeiu problem is a conjecture in integral geometry, named for Dimitrie Pompeiu, who posed the problem in 1929,
as follows. Suppose f is a nonzero continuous function defined on a Euclidean space, and K is a simply connected Lipschitz domain, so that the integral of f vanishes on every congruent copy of K. Then the domain is a ball.
A special case is Schiffer's conjecture.
References
External links
Pompeiu problem at Department of Geometry, Bolyai Institute, University of Szeged, Hungary
Pompeiu problem at SpringerLink encyclopaedia of mathematics
The Pompeiu problem,
Schiffer's conjecture,
Mathematical analysis
Integral geometry
Conjectures
Unsolved problems in geometry
|
https://en.wikipedia.org/wiki/Support%20%28measure%20theory%29
|
In mathematics, the support (sometimes topological support or spectrum) of a measure on a measurable topological space is a precise notion of where in the space the measure "lives". It is defined to be the largest (closed) subset of for which every open neighbourhood of every point of the set has positive measure.
Motivation
A (non-negative) measure on a measurable space is really a function Therefore, in terms of the usual definition of support, the support of is a subset of the σ-algebra
where the overbar denotes set closure. However, this definition is somewhat unsatisfactory: we use the notion of closure, but we do not even have a topology on What we really want to know is where in the space the measure is non-zero. Consider two examples:
Lebesgue measure on the real line It seems clear that "lives on" the whole of the real line.
A Dirac measure at some point Again, intuition suggests that the measure "lives at" the point and nowhere else.
In light of these two examples, we can reject the following candidate definitions in favour of the one in the next section:
We could remove the points where is zero, and take the support to be the remainder This might work for the Dirac measure but it would definitely not work for since the Lebesgue measure of any singleton is zero, this definition would give empty support.
By comparison with the notion of strict positivity of measures, we could take the support to be the set of all points with a neighbourhood of positive measure: (or the closure of this). It is also too simplistic: by taking for all points this would make the support of every measure except the zero measure the whole of
However, the idea of "local strict positivity" is not too far from a workable definition.
Definition
Let be a topological space; let denote the Borel σ-algebra on i.e. the smallest sigma algebra on that contains all open sets Let be a measure on Then the support (or spectrum) of is defined as the set of all points in for which every open neighbourhood of has positive measure:
Some authors prefer to take the closure of the above set. However, this is not necessary: see "Properties" below.
An equivalent definition of support is as the largest (with respect to inclusion) such that every open set which has non-empty intersection with has positive measure, i.e. the largest such that:
Signed and complex measures
This definition can be extended to signed and complex measures.
Suppose that is a signed measure. Use the Hahn decomposition theorem to write
where are both non-negative measures. Then the support of is defined to be
Similarly, if is a complex measure, the support of is defined to be the union of the supports of its real and imaginary parts.
Properties
holds.
A measure on is strictly positive if and only if it has support If is strictly positive and is arbitrary, then any open neighbourhood of since it is an open set, has positive measure; hence,
|
https://en.wikipedia.org/wiki/Helmut%20Maier
|
Helmut Maier (born 17 October 1953) is a German mathematician and professor at the University of Ulm, Germany. He is known for his contributions in analytic number theory and mathematical analysis and particularly for the so-called Maier's matrix method as well as Maier's theorem for primes in short intervals. He has also done important work in exponential sums and trigonometric sums over special sets of integers and the Riemann zeta function.
Education
Helmut Maier graduated with a Diploma in Mathematics from the University of Ulm in 1976, under the supervision of Hans-Egon Richert. He received his PhD from the University of Minnesota in 1981, under the supervision of J. Ian Richards.
Research and academic positions
Maier's PhD thesis was an extension of his paper Chains of large gaps between consecutive primes. In this paper Maier applied for the first time what is now known as Maier's matrix method. This method later on led him and other mathematicians to the discovery of unexpected irregularities in the distribution of prime numbers. There have been various other applications of Maier's Matrix Method, such as on irreducible polynomials and on strings of consecutive primes in the same residue class.
After postdoctoral positions at the University of Michigan and the Institute for Advanced Study, Princeton, Maier obtained a permanent position at the University of Georgia. While in Georgia he proved that the usual formulation of the Cramér model for the distribution of prime numbers is wrong. This was a completely unexpected result. Jointly with Carl Pomerance he studied the values of Euler's -function
and large gaps between primes. During the same period Maier investigated as well
the size of the coefficients of cyclotomic polynomials and later collaborated with Sergei Konyagin
and Eduard Wirsing on this topic. He also collaborated with Hugh Lowell Montgomery on the size of the sum of the
Möbius function under the assumption of the Riemann Hypothesis. Maier and Gérald Tenenbaum
in joint work investigated the sequence of divisors of integers, solving the famous propinquity problem of
Paul Erdős. Since 1993 Maier is a professor at the University of Ulm, Germany.
Collaborators of Helmut Maier
include Paul Erdős, C. Feiler, John Friedlander, Andrew Granville, D. Haase, A. J. Hildebrand,
, J. W. Neuberger, A. Sankaranarayanan, A. Sárközy, Wolfgang P. Schleich, Cameron Leigh Stewart.
See also
Maier's matrix method
Maier's theorem
References
External links
Maier's webpage
20th-century German mathematicians
21st-century German mathematicians
Number theorists
University of Minnesota alumni
Academic staff of the University of Ulm
Living people
1953 births
|
https://en.wikipedia.org/wiki/Structure%20theorem%20for%20Gaussian%20measures
|
In mathematics, the structure theorem for Gaussian measures shows that the abstract Wiener space construction is essentially the only way to obtain a strictly positive Gaussian measure on a separable Banach space. It was proved in the 1970s by Kallianpur–Sato–Stefan and Dudley–Feldman–le Cam.
There is the earlier result due to H. Satô (1969) which proves that "any Gaussian measure on a separable Banach space is an abstract Wiener measure in the sense of L. Gross". The result by Dudley et al. generalizes this result to the setting of Gaussian measures on a general topological vector space.
Statement of the theorem
Let γ be a strictly positive Gaussian measure on a separable Banach space (E, || ||). Then there exists a separable Hilbert space (H, 〈 , 〉) and a map i : H → E such that i : H → E is an abstract Wiener space with γ = i∗(γH), where γH is the canonical Gaussian cylinder set measure on H.
References
Banach spaces
Probability theorems
Theorems in measure theory
|
https://en.wikipedia.org/wiki/Finite-dimensional%20distribution
|
In mathematics, finite-dimensional distributions are a tool in the study of measures and stochastic processes. A lot of information can be gained by studying the "projection" of a measure (or process) onto a finite-dimensional vector space (or finite collection of times).
Finite-dimensional distributions of a measure
Let be a measure space. The finite-dimensional distributions of are the pushforward measures , where , , is any measurable function.
Finite-dimensional distributions of a stochastic process
Let be a probability space and let be a stochastic process. The finite-dimensional distributions of are the push forward measures on the product space for defined by
Very often, this condition is stated in terms of measurable rectangles:
The definition of the finite-dimensional distributions of a process is related to the definition for a measure in the following way: recall that the law of is a measure on the collection of all functions from into . In general, this is an infinite-dimensional space. The finite dimensional distributions of are the push forward measures on the finite-dimensional product space , where
is the natural "evaluate at times " function.
Relation to tightness
It can be shown that if a sequence of probability measures is tight and all the finite-dimensional distributions of the converge weakly to the corresponding finite-dimensional distributions of some probability measure , then converges weakly to .
See also
Law (stochastic processes)
Measure theory
Stochastic processes
|
https://en.wikipedia.org/wiki/Hewitt%E2%80%93Savage%20zero%E2%80%93one%20law
|
The Hewitt–Savage zero–one law is a theorem in probability theory, similar to Kolmogorov's zero–one law and the Borel–Cantelli lemma, that specifies that a certain type of event will either almost surely happen or almost surely not happen. It is sometimes known as the Savage-Hewitt law for symmetric events. It is named after Edwin Hewitt and Leonard Jimmie Savage.
Statement of the Hewitt-Savage zero-one law
Let be a sequence of independent and identically-distributed random variables taking values in a set . The Hewitt-Savage zero–one law says that any event whose occurrence or non-occurrence is determined by the values of these random variables and whose occurrence or non-occurrence is unchanged by finite permutations of the indices, has probability either 0 or 1 (a "finite" permutation is one that leaves all but finitely many of the indices fixed).
Somewhat more abstractly, define the exchangeable sigma algebra or sigma algebra of symmetric events to be the set of events (depending on the sequence of variables ) which are invariant under finite permutations of the indices in the sequence . Then .
Since any finite permutation can be written as a product of transpositions, if we wish to check whether or not an event is symmetric (lies in ), it is enough to check if its occurrence is unchanged by an arbitrary transposition , .
Examples
Example 1
Let the sequence of independent and identically distributed random variables take values in . Then the event that the series converges (to a finite value) is a symmetric event in , since its occurrence is unchanged under transpositions (for a finite re-ordering, the convergence or divergence of the series—and, indeed, the numerical value of the sum itself—is independent of the order in which we add up the terms). Thus, the series either converges almost surely or diverges almost surely. If we assume in addition that the common expected value (which essentially means that because of the random variables' non-negativity), we may conclude that
i.e. the series diverges almost surely. This is a particularly simple application of the Hewitt–Savage zero–one law. In many situations, it can be easy to apply the Hewitt–Savage zero–one law to show that some event has probability 0 or 1, but surprisingly hard to determine which of these two extreme values is the correct one.
Example 2
Continuing with the previous example, define
which is the position at step N of a random walk with the iid increments Xn. The event { SN = 0 infinitely often } is invariant under finite permutations. Therefore, the zero–one law is applicable and one infers that the probability of a random walk with real iid increments visiting the origin infinitely often is either one or zero. Visiting the origin infinitely often is a tail event with respect to the sequence (SN), but SN are not independent and therefore the Kolmogorov's zero–one law is not directly applicable here.
References
Probability theorems
Covering lemmas
|
https://en.wikipedia.org/wiki/Identity%20theorem%20for%20Riemann%20surfaces
|
In mathematics, the identity theorem for Riemann surfaces is a theorem that states that a holomorphic function is completely determined by its values on any subset of its domain that has a limit point.
Statement of the theorem
Let and be Riemann surfaces, let be connected, and let be holomorphic. Suppose that for some subset that has a limit point, where denotes the restriction of to . Then (on the whole of ).
References
Theorems in complex analysis
Riemann surfaces
|
https://en.wikipedia.org/wiki/Branching%20theorem
|
In mathematics, the branching theorem is a theorem about Riemann surfaces. Intuitively, it states that every non-constant holomorphic function is locally a polynomial.
Statement of the theorem
Let and be Riemann surfaces, and let be a non-constant holomorphic map. Fix a point and set . Then there exist and charts on and on such that
; and
is
This theorem gives rise to several definitions:
We call the multiplicity of at . Some authors denote this .
If , the point is called a branch point of .
If has no branch points, it is called unbranched. See also unramified morphism.
References
.
Theorems in complex analysis
Riemann surfaces
|
https://en.wikipedia.org/wiki/Clark%E2%80%93Ocone%20theorem
|
In mathematics, the Clark–Ocone theorem (also known as the Clark–Ocone–Haussmann theorem or formula) is a theorem of stochastic analysis. It expresses the value of some function F defined on the classical Wiener space of continuous paths starting at the origin as the sum of its mean value and an Itô integral with respect to that path. It is named after the contributions of mathematicians J.M.C. Clark (1970), Daniel Ocone (1984) and U.G. Haussmann (1978).
Statement of the theorem
Let C0([0, T]; R) (or simply C0 for short) be classical Wiener space with Wiener measure γ. Let F : C0 → R be a BC1 function, i.e. F is bounded and Fréchet differentiable with bounded derivative DF : C0 → Lin(C0; R). Then
In the above
F(σ) is the value of the function F on some specific path of interest, σ;
the first integral,
is the expected value of F over the whole of Wiener space C0;
the second integral,
is an Itô integral;
Σ∗ is the natural filtration of Brownian motion B : [0, T] × Ω → R: Σt is the smallest σ-algebra containing all Bs−1(A) for times 0 ≤ s ≤ t and Borel sets A ⊆ R;
E[·|Σt] denotes conditional expectation with respect to the sigma algebra Σt;
∂/∂t denotes differentiation with respect to time t; ∇H denotes the H-gradient; hence, ∂/∂t∇H is the Malliavin derivative.
More generally, the conclusion holds for any F in L2(C0; R) that is differentiable in the sense of Malliavin.
Integration by parts on Wiener space
The Clark–Ocone theorem gives rise to an integration by parts formula on classical Wiener space, and to write Itô integrals as divergences:
Let B be a standard Brownian motion, and let L02,1 be the Cameron–Martin space for C0 (see abstract Wiener space. Let V : C0 → L02,1 be a vector field such that
is in L2(B) (i.e. is Itô integrable, and hence is an adapted process). Let F : C0 → R be BC1 as above. Then
i.e.
or, writing the integrals over C0 as expectations:
where the "divergence" div(V) : C0 → R is defined by
The interpretation of stochastic integrals as divergences leads to concepts such as the Skorokhod integral and the tools of the Malliavin calculus.
See also
Integral representation theorem for classical Wiener space, which uses the Clark–Ocone theorem in its proof
Integration by parts operator
Malliavin calculus
References
External links
Theorems regarding stochastic processes
Theorems in measure theory
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.