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https://en.wikipedia.org/wiki/Jacobi%20elliptic%20functions
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In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions. They are found in the description of the motion of a pendulum (see also pendulum (mathematics)), as well as in the design of electronic elliptic filters. While trigonometric functions are defined with reference to a circle, the Jacobi elliptic functions are a generalization which refer to other conic sections, the ellipse in particular. The relation to trigonometric functions is contained in the notation, for example, by the matching notation for . The Jacobi elliptic functions are used more often in practical problems than the Weierstrass elliptic functions as they do not require notions of complex analysis to be defined and/or understood. They were introduced by . Carl Friedrich Gauss had already studied special Jacobi elliptic functions in 1797, the lemniscate elliptic functions in particular, but his work was published much later.
Overview
There are twelve Jacobi elliptic functions denoted by , where and are any of the letters , , , and . (Functions of the form are trivially set to unity for notational completeness.) is the argument, and is the parameter, both of which may be complex. In fact, the Jacobi elliptic functions are meromorphic in both and . The distribution of the zeros and poles in the -plane is well-known. However, questions of the distribution of the zeros and poles in the -plane remain to be investigated.
In the complex plane of the argument , the twelve functions form a repeating lattice of simple poles and zeroes. Depending on the function, one repeating parallelogram, or unit cell, will have sides of length or on the real axis, and or on the imaginary axis, where and are known as the quarter periods with being the elliptic integral of the first kind. The nature of the unit cell can be determined by inspecting the "auxiliary rectangle" (generally a parallelogram), which is a rectangle formed by the origin at one corner, and as the diagonally opposite corner. As in the diagram, the four corners of the auxiliary rectangle are named , , , and , going counter-clockwise from the origin. The function will have a zero at the corner and a pole at the corner. The twelve functions correspond to the twelve ways of arranging these poles and zeroes in the corners of the rectangle.
When the argument and parameter are real, with , and will be real and the auxiliary parallelogram will in fact be a rectangle, and the Jacobi elliptic functions will all be real valued on the real line.
Since the Jacobian elliptic functions are doubly periodic in , they factor through a torus – in effect, their domain can be taken to be a torus, just as cosine and sine are in effect defined on a circle. Instead of having only one circle, we now have the product of two circles, one real and the other imaginary. The complex plane can be replaced by a complex torus. The circumference of the first circle is and the second , where and are the quarter
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https://en.wikipedia.org/wiki/CBA
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CBA may refer to:
Maths and science
Casei Bifidus Acidophilus, a bacterium
Colicin, activity protein
Complete Boolean algebra, a concept from mathematics
Cytometric Bead Array, a bead-based immunoassay
Cell Based Assay, also a kind of immunoassay
4-Carboxybenzaldehyde, a byproduct in the industrial production of terephthalic acid
Congenital bronchial atresia, a rare congenital abnormality
Organizations
Academic
Catholic Biblical Association
Center for Bits and Atoms, a research institution at the Massachusetts Institute of Technology, United States
Christian Brothers Academy, schools run by the Institute of the Brothers of the Christian Schools, including:
Christian Brothers Academy (New Jersey), in Lincroft, New Jersey
Christian Brothers Academy (Albany, New York)
Christian Brothers Academy (Syracuse, New York)
College of Business Administration (Saudi Arabia), private college in Saudi Arabia
Corby Business Academy, in Corby, England
Banks
Central Bank of Armenia
Centrale Bank van Aruba, the central bank of Aruba
Commercial Bank of Africa, headquartered in Nairobi, Kenya
Commercial Bank of Australia (1866–1982), merged into the Wales bank to form Westpac
Commonwealth Bank of Australia
Professional and interest
California Bluegrass Association
Cambridge Buddhist Association
Canadian Bankers Association
Canadian Bar Association
CBA (Christian trade association), established in 1950 by bookstores
Chicago Bar Association
Chinese Benevolent Association
Chinese Benevolent Association of Vancouver
Christian Bowhunters of America
Commonwealth Broadcasting Association
Consumer Bankers Association, lobbying voice on retail banking issues in the United States
Council for British Archaeology
Sports
California Basketball Association, the original name the West Coast Conference
Chinese Badminton Association
Chinese Baseball Association
Chinese Basketball Alliance, a professional men's league (1994 to 1999) in Taiwan
Chinese Basketball Association, the pre-eminent professional men's basketball league in China
Chinese Basketball Association (organisation), the national basketball association of China
Christian Bowhunters of America
Continental Basketball Association, a defunct professional men's basketball minor league in the United States
Continental Basketball Association (1969–1974), defunct semi-pro basketball league in the United States
Continental Basketball Association, a defunct semi-professional men's league, renamed Australian Basketball Association in 1999
Other organizations
CBA (food retail), a Hungarian food-retail network
Companhia Brasileira de Aluminio, the largest aluminium producer in Brazil
CBA (AM), former CBC Radio One AM station in Moncton, New Brunswick, now known as CBAM-FM
CBA-FM, the CBC Radio Two station in Moncton
Central Anticorruption Bureau of Poland
Commonwealth Broadcasting Association
Luis A. Ferré Performing Arts Center (), a multi-use performance centre in San Ju
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https://en.wikipedia.org/wiki/Azuma%27s%20inequality
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In probability theory, the Azuma–Hoeffding inequality (named after Kazuoki Azuma and Wassily Hoeffding) gives a concentration result for the values of martingales that have bounded differences.
Suppose is a martingale (or super-martingale) and
almost surely. Then for all positive integers N and all positive reals ,
And symmetrically (when Xk is a sub-martingale):
If X is a martingale, using both inequalities above and applying the union bound allows one to obtain a two-sided bound:
Proof
The proof shares similar idea of the proof for the general form of Azuma's inequality listed below. Actually, this can be viewed as a direct corollary of the general form of Azuma's inequality.
A general form of Azuma's inequality
Limitation of the vanilla Azuma's inequality
Note that the vanilla Azuma's inequality requires symmetric bounds on martingale increments, i.e. . So, if known bound is asymmetric, e.g. , to use Azuma's inequality, one need to choose which might be a waste of information on the boundedness of . However, this issue can be resolved and one can obtain a tighter probability bound with the following general form of Azuma's inequality.
Statement
Let be a martingale (or supermartingale) with respect to filtration . Assume there are predictable processes and with respect to , i.e. for all , are -measurable, and constants such that
almost surely. Then for all ,
Since a submartingale is a supermartingale with signs reversed, we have if instead is a martingale (or submartingale),
If is a martingale, since it is both a supermartingale and submartingale, by applying union bound to the two inequalities above, we could obtain the two-sided bound:
Proof
We will prove the supermartingale case only as the rest are self-evident. By Doob decomposition, we could decompose supermartingale as where is a martingale and is a nonincreasing predictable sequence (Note that if itself is a martingale, then ). From , we have
Applying Chernoff bound to , we have for ,
For the inner expectation term, since
(i) as is a martingale;
(ii) ;
(iii) and are both -measurable as is a predictable process;
(iv) ;
By applying Hoeffding's lemma, we have
Repeating this step, one could get
Note that the minimum is achieved at , so we have
Finally, since and as is nonincreasing, so event implies , and therefore
Remark
Note that by setting , we could obtain the vanilla Azuma's inequality.
Note that for either submartingale or supermartingale, only one side of Azuma's inequality holds. We can't say much about how fast a submartingale with bounded increments rises (or a supermartingale falls).
This general form of Azuma's inequality applied to the Doob martingale gives McDiarmid's inequality which is common in the analysis of randomized algorithms.
Simple example of Azuma's inequality for coin flips
Let Fi be a sequence of independent and identically distributed random coin flips (i.e., let Fi be equally likely to be −1 or 1 independent
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https://en.wikipedia.org/wiki/Field%20norm
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In mathematics, the (field) norm is a particular mapping defined in field theory, which maps elements of a larger field into a subfield.
Formal definition
Let K be a field and L a finite extension (and hence an algebraic extension) of K.
The field L is then a finite dimensional vector space over K.
Multiplication by α, an element of L,
,
is a K-linear transformation of this vector space into itself.
The norm, NL/K(α), is defined as the determinant of this linear transformation.
If L/K is a Galois extension, one may compute the norm of α ∈ L as the product of all the Galois conjugates of α:
where Gal(L/K) denotes the Galois group of L/K. (Note that there may be a repetition in the terms of the product.)
For a general field extension L/K, and nonzero α in L, let σ(α), ..., σ(α) be the roots of the minimal polynomial of α over K (roots listed with multiplicity and lying in some extension field of L); then
.
If L/K is separable, then each root appears only once in the product (though the exponent, the degree [L:K(α)], may still be greater than 1).
Examples
Quadratic field extensions
One of the basic examples of norms comes from quadratic field extensions where is a square-free integer.
Then, the multiplication map by on an element is
The element can be represented by the vector
since there is a direct sum decomposition as a -vector space.
The matrix of is then
and the norm is , since it is the determinant of this matrix.
Norm of Q(√2)
Consider the number field .
The Galois group of over has order and is generated by the element which sends to . So the norm of is:
The field norm can also be obtained without the Galois group.
Fix a -basis of , say:
.
Then multiplication by the number sends
1 to and
to .
So the determinant of "multiplying by " is the determinant of the matrix which sends the vector
(corresponding to the first basis element, i.e., 1) to ,
(corresponding to the second basis element, i.e., ) to ,
viz.:
The determinant of this matrix is −1.
p-th root field extensions
Another easy class of examples comes from field extensions of the form where the prime factorization of contains no -th powers, for a fixed odd prime.
The multiplication map by of an element isgiving the matrixThe determinant gives the norm
Complex numbers over the reals
The field norm from the complex numbers to the real numbers sends
to
,
because the Galois group of over has two elements,
the identity element and
complex conjugation,
and taking the product yields .
Finite fields
Let L = GF(qn) be a finite extension of a finite field K = GF(q).
Since L/K is a Galois extension, if α is in L, then the norm of α is the product of all the Galois conjugates of α, i.e.
In this setting we have the additional properties,
Properties of the norm
Several properties of the norm function hold for any finite extension.
Group homomorphism
The norm N : L* → K* is a group homomorphism from the multiplicative group of L
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https://en.wikipedia.org/wiki/Jacobi
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Jacobi may refer to:
People with the surname Jacobi
Mathematics:
Jacobi sum, a type of character sum
Jacobi method, a method for determining the solutions of a diagonally dominant system of linear equations
Jacobi eigenvalue algorithm, a method for calculating the eigenvalues and eigenvectors of a real symmetric matrix
Jacobi elliptic functions, a set of doubly-periodic functions
Jacobi polynomials, a class of orthogonal polynomials
Jacobi symbol, a generalization of the Legendre symbol
Jacobi coordinates, a simplification of coordinates for an n-body system
Jacobi identity for non-associative binary operations
Jacobi's formula for the derivative of the determinant of a matrix
Jacobi triple product an identity in the theory of theta functions
Jacobi's theorem (disambiguation) (various)
Other:
Jacobi Medical Center, New York
Jacobi (grape), another name for the French/German wine grape Pinot Noir Précoce
Jacobi (crater), a lunar impact crater in the southern highlands on the near side of the Moon
Software_for_handling_chess_problems#Jacobi, chess software
See also
Jacoby (disambiguation)
Jacob
Jakob (disambiguation)
Jacobs (disambiguation)
Jacobite (disambiguation)
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https://en.wikipedia.org/wiki/Cube%20%28algebra%29
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In arithmetic and algebra, the cube of a number is its third power, that is, the result of multiplying three instances of together.
The cube of a number or any other mathematical expression is denoted by a superscript 3, for example or .
The cube is also the number multiplied by its square:
.
The cube function is the function (often denoted ) that maps a number to its cube. It is an odd function, as
.
The volume of a geometric cube is the cube of its side length, giving rise to the name. The inverse operation that consists of finding a number whose cube is is called extracting the cube root of . It determines the side of the cube of a given volume. It is also raised to the one-third power.
The graph of the cube function is known as the cubic parabola. Because the cube function is an odd function, this curve has a center of symmetry at the origin, but no axis of symmetry.
In integers
A cube number, or a perfect cube, or sometimes just a cube, is a number which is the cube of an integer.
The non-negative perfect cubes up to 603 are :
Geometrically speaking, a positive integer is a perfect cube if and only if one can arrange solid unit cubes into a larger, solid cube. For example, 27 small cubes can be arranged into one larger one with the appearance of a Rubik's Cube, since .
The difference between the cubes of consecutive integers can be expressed as follows:
.
or
.
There is no minimum perfect cube, since the cube of a negative integer is negative. For example, .
Base ten
Unlike perfect squares, perfect cubes do not have a small number of possibilities for the last two digits. Except for cubes divisible by 5, where only 25, 75 and 00 can be the last two digits, any pair of digits with the last digit odd can occur as the last digits of a perfect cube. With even cubes, there is considerable restriction, for only 00, 2, 4, 6 and 8 can be the last two digits of a perfect cube (where stands for any odd digit and for any even digit). Some cube numbers are also square numbers; for example, 64 is a square number and a cube number . This happens if and only if the number is a perfect sixth power (in this case 2).
The last digits of each 3rd power are:
It is, however, easy to show that most numbers are not perfect cubes because all perfect cubes must have digital root 1, 8 or 9. That is their values modulo 9 may be only 0, 1, and 8. Moreover, the digital root of any number's cube can be determined by the remainder the number gives when divided by 3:
If the number x is divisible by 3, its cube has digital root 9; that is,
If it has a remainder of 1 when divided by 3, its cube has digital root 1; that is,
If it has a remainder of 2 when divided by 3, its cube has digital root 8; that is,
Waring's problem for cubes
Every positive integer can be written as the sum of nine (or fewer) positive cubes. This upper limit of nine cubes cannot be reduced because, for example, 23 cannot be written as the sum of fewer than nine posit
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https://en.wikipedia.org/wiki/Historiometry
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Historiometry is the historical study of human progress or individual personal characteristics, using statistics to analyze references to geniuses, their statements, behavior and discoveries in relatively neutral texts. Historiometry combines techniques from cliometrics, which studies economic history and from psychometrics, the psychological study of an individual's personality and abilities.
Origins
Historiometry started in the early 19th century with studies on the relationship between age and achievement by Belgian mathematician Adolphe Quetelet in the careers of prominent French and English playwrights but it was Sir Francis Galton, an English polymath who popularized historiometry in his 1869 work, Hereditary Genius. It was further developed by Frederick Adams Woods (who coined the term historiometry) in the beginning of the 20th century. Also psychologist Paul E. Meehl published several papers on historiometry later in his career, mainly in the area of medical history, although it is usually referred to as cliometric metatheory by him.
Historiometry was the first field studying genius by using scientific methods.
Current research
Prominent current historiometry researchers include Dean Keith Simonton and Charles Murray.
Historiometry is defined by Dean Keith Simonton as: a quantitative method of statistical analysis for retrospective data. In Simonton's work the raw data comes from psychometric assessment of famous personalities, often already deceased, in an attempt to assess creativity, genius and talent development.
Charles Murray's Human Accomplishment is one example of this approach to quantify the impact of individuals on technology, science and the arts. This work tracks many famous innovators in these areas, and quantifies how much attention to them has been paid by past historians, in terms of the number of references and the number of pages of reference material devoted to each subject. However, this work has been criticized for manipulating its data to derive conclusions that would not follow from unmanipulated data.
Examples of research
Since historiometry deals with subjective personal traits as creativity, charisma or openness most studies deal with the comparison of scientists, artists or politicians. The study (Human Accomplishment) by Charles Murray classifies, for example, Einstein and Newton as the most important physicists and Michelangelo as the top ranking western artist. As another example, several studies have compared charisma and even the IQ of presidents and presidential candidates of the United States. The latter study classifies John Quincy Adams as the most clever US president, with an estimated IQ between 165 and 175.
A historiometric analysis has also been applied successfully in the field of musicology. In one groundbreaking study, researchers analyzed statistically a collection of over 1,300 printed program leaflets (playbills) of concerts given by Clara Schumann (1819–1896) throughout her lifetime
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https://en.wikipedia.org/wiki/1742%20in%20science
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The year 1742 in science and technology involved some significant events.
Astronomy
January 14 – Death of Edmond Halley; James Bradley succeeds him as Astronomer Royal in Great Britain.
Mathematics
June – Christian Goldbach produces Goldbach's conjecture.
Colin Maclaurin publishes his Treatise on Fluxions in Great Britain, the first systematic exposition of Newton's methods.
Metrology
Anders Celsius publishes his proposal for a centigrade temperature scale originated in 1741.
Physiology and medicine
Surgeon Joseph Hurlock publishes his A Practical Treatise upon Dentition, or The breeding of teeth in children in London, the first treatise in English on dentition.
Technology
Benjamin Robins publishes his New Principles of Gunnery, containing the determination of the force of gun-powder and an investigation of the difference in the resisting power of the air to swift and slow motions in London, containing a description of his ballistic pendulum and the results of his scientific experiments into improvements in ballistics.
The first large (12 ft focal length) reflecting telescope is made, in Gregorian form, by James Short, for use by Charles Spencer, 3rd Duke of Marlborough, in London.
Awards
Copley Medal: Christopher Middleton.
Births
March 15 (bapt.) – John Stackhouse, English botanist (died 1819).
May 18 – Lionel Lukin, English inventor (died 1834).
December 3 – James Rennell, English geographer, historian and oceanographer (died 1830).
December 9 – Carl Wilhelm Scheele, Swedish chemist (died 1786).
December 26 – Ignaz von Born, Hungarian metallurgist (died 1791).
Deaths
January 14 – Edmond Halley, English astronomer, geophysicist, mathematician, meteorologist, and physicist (born 1656).
February 28 – Willem 's Gravesande, Dutch polymath (born 1688).
May 13 – Nicolas Andry, French physician (born 1658).
September 22 – Frederic Louis Norden, Danish explorer (born 1708).
References
18th century in science
1740s in science
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https://en.wikipedia.org/wiki/J-invariant
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In mathematics, Felix Klein's -invariant or function, regarded as a function of a complex variable , is a modular function of weight zero for defined on the upper half-plane of complex numbers. It is the unique such function which is holomorphic away from a simple pole at the cusp such that
Rational functions of are modular, and in fact give all modular functions. Classically, the -invariant was studied as a parameterization of elliptic curves over , but it also has surprising connections to the symmetries of the Monster group (this connection is referred to as monstrous moonshine).
Definition
The -invariant can be defined as a function on the upper half-plane
with the third definition implying can be expressed as a cube, also since 1728.
The given functions are the modular discriminant , Dedekind eta function , and modular invariants,
where , are Fourier series,
and , are Eisenstein series,
and (the square of the nome). The -invariant can then be directly expressed in terms of the Eisenstein series as,
with no numerical factor other than 1728. This implies a third way to define the modular discriminant,
For example, using the definitions above and , then the Dedekind eta function has the exact value,
implying the transcendental numbers,
but yielding the algebraic number (in fact, an integer),
In general, this can be motivated by viewing each as representing an isomorphism class of elliptic curves. Every elliptic curve over is a complex torus, and thus can be identified with a rank 2 lattice; that is, a two-dimensional lattice of . This lattice can be rotated and scaled (operations that preserve the isomorphism class), so that it is generated by and . This lattice corresponds to the elliptic curve (see Weierstrass elliptic functions).
Note that is defined everywhere in as the modular discriminant is non-zero. This is due to the corresponding cubic polynomial having distinct roots.
The fundamental region
It can be shown that is a modular form of weight twelve, and one of weight four, so that its third power is also of weight twelve. Thus their quotient, and therefore , is a modular function of weight zero, in particular a holomorphic function invariant under the action of . Quotienting out by its centre yields the modular group, which we may identify with the projective special linear group .
By a suitable choice of transformation belonging to this group,
we may reduce to a value giving the same value for , and lying in the fundamental region for , which consists of values for satisfying the conditions
The function when restricted to this region still takes on every value in the complex numbers exactly once. In other words, for every in , there is a unique τ in the fundamental region such that . Thus, has the property of mapping the fundamental region to the entire complex plane.
Additionally two values produce the same elliptic curve iff for some . This means provides a bijection from the set of
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https://en.wikipedia.org/wiki/Linking%20number
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In mathematics, the linking number is a numerical invariant that describes the linking of two closed curves in three-dimensional space. Intuitively, the linking number represents the number of times that each curve winds around the other. In Euclidean space, the linking number is always an integer, but may be positive or negative depending on the orientation of the two curves (this is not true for curves in most 3-manifolds, where linking numbers can also be fractions or just not exist at all).
The linking number was introduced by Gauss in the form of the linking integral. It is an important object of study in knot theory, algebraic topology, and differential geometry, and has numerous applications in mathematics and science, including quantum mechanics, electromagnetism, and the study of DNA supercoiling.
Definition
Any two closed curves in space, if allowed to pass through themselves but not each other, can be moved into exactly one of the following standard positions. This determines the linking number:
Each curve may pass through itself during this motion, but the two curves must remain separated throughout. This is formalized as regular homotopy, which further requires that each curve be an immersion, not just any map. However, this added condition does not change the definition of linking number (it does not matter if the curves are required to always be immersions or not), which is an example of an h-principle (homotopy-principle), meaning that geometry reduces to topology.
Proof
This fact (that the linking number is the only invariant) is most easily proven by placing one circle in standard position, and then showing that linking number is the only invariant of the other circle. In detail:
A single curve is regular homotopic to a standard circle (any knot can be unknotted if the curve is allowed to pass through itself). The fact that it is homotopic is clear, since 3-space is contractible and thus all maps into it are homotopic, though the fact that this can be done through immersions requires some geometric argument.
The complement of a standard circle is homeomorphic to a solid torus with a point removed (this can be seen by interpreting 3-space as the 3-sphere with the point at infinity removed, and the 3-sphere as two solid tori glued along the boundary), or the complement can be analyzed directly.
The fundamental group of 3-space minus a circle is the integers, corresponding to linking number. This can be seen via the Seifert–Van Kampen theorem (either adding the point at infinity to get a solid torus, or adding the circle to get 3-space, allows one to compute the fundamental group of the desired space).
Thus homotopy classes of a curve in 3-space minus a circle are determined by linking number.
It is also true that regular homotopy classes are determined by linking number, which requires additional geometric argument.
Computing the linking number
There is an algorithm to compute the linking number of two curves from
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https://en.wikipedia.org/wiki/Eric%20de%20Sturler
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Eric de Sturler (born 15 January 1966, Groningen) is a Professor of Mathematics at Virginia Tech in Blacksburg, Virginia. He is on the editorial board of Applied Numerical Mathematics and the Open Applied Mathematics Journal.
Prof. de Sturler completed his Ph.D. under the direction of Henk van der Vorst at Technische Universiteit Delft in 1994. His thesis is entitled Iterative Methods on Distributive Memory Computers. He was a second-place winner of the Leslie Fox Prize for Numerical Analysis in 1997.
His research focuses on preconditioned iterative methods for solving linear and nonlinear systems, with applications in computational physics, material science, and mathematical biology.
References
External links
Eric de Sturler's personal webpage
1966 births
Living people
Dutch mathematicians
De Sturler, Eric
De Sturler, Eric
Delft University of Technology alumni
Scientists from Groningen (city)
Dutch expatriates in the United States
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https://en.wikipedia.org/wiki/Limit%20cycle
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In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity or as time approaches negative infinity. Such behavior is exhibited in some nonlinear systems. Limit cycles have been used to model the behavior of many real-world oscillatory systems. The study of limit cycles was initiated by Henri Poincaré (1854–1912).
Definition
We consider a two-dimensional dynamical system of the form
where
is a smooth function. A trajectory of this system is some smooth function with values in which satisfies this differential equation. Such a trajectory is called closed (or periodic) if it is not constant but returns to its starting point, i.e. if there exists some such that for all . An orbit is the image of a trajectory, a subset of . A closed orbit, or cycle, is the image of a closed trajectory. A limit cycle is a cycle which is the limit set of some other trajectory.
Properties
By the Jordan curve theorem, every closed trajectory divides the plane into two regions, the interior and the exterior of the curve.
Given a limit cycle and a trajectory in its interior that approaches the limit cycle for time approaching , then there is a neighborhood around the limit cycle such that all trajectories in the interior that start in the neighborhood approach the limit cycle for time approaching . The corresponding statement holds for a trajectory in the interior that approaches the limit cycle for time approaching , and also for trajectories in the exterior approaching the limit cycle.
Stable, unstable and semi-stable limit cycles
In the case where all the neighboring trajectories approach the limit cycle as time approaches infinity, it is called a stable or attractive limit cycle (ω-limit cycle). If instead, all neighboring trajectories approach it as time approaches negative infinity, then it is an unstable limit cycle (α-limit cycle). If there is a neighboring trajectory which spirals into the limit cycle as time approaches infinity, and another one which spirals into it as time approaches negative infinity, then it is a semi-stable limit cycle. There are also limit cycles that are neither stable, unstable nor semi-stable: for instance, a neighboring trajectory may approach the limit cycle from the outside, but the inside of the limit cycle is approached by a family of other cycles (which wouldn't be limit cycles).
Stable limit cycles are examples of attractors. They imply self-sustained oscillations: the closed trajectory describes the perfect periodic behavior of the system, and any small perturbation from this closed trajectory causes the system to return to it, making the system stick to the limit cycle.
Finding limit cycles
Every closed trajectory contains within its interior a stationary point of the system, i.e. a point where . The Bendixson–Dulac theorem and the Poin
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https://en.wikipedia.org/wiki/Orthogonal%20functions
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In mathematics, orthogonal functions belong to a function space that is a vector space equipped with a bilinear form. When the function space has an interval as the domain, the bilinear form may be the integral of the product of functions over the interval:
The functions and are orthogonal when this integral is zero, i.e. whenever . As with a basis of vectors in a finite-dimensional space, orthogonal functions can form an infinite basis for a function space. Conceptually, the above integral is the equivalent of a vector dot product; two vectors are mutually independent (orthogonal) if their dot-product is zero.
Suppose is a sequence of orthogonal functions of nonzero L2-norms . It follows that the sequence is of functions of L2-norm one, forming an orthonormal sequence. To have a defined L2-norm, the integral must be bounded, which restricts the functions to being square-integrable.
Trigonometric functions
Several sets of orthogonal functions have become standard bases for approximating functions. For example, the sine functions and are orthogonal on the interval when and n and m are positive integers. For then
and the integral of the product of the two sine functions vanishes. Together with cosine functions, these orthogonal functions may be assembled into a trigonometric polynomial to approximate a given function on the interval with its Fourier series.
Polynomials
If one begins with the monomial sequence on the interval and applies the Gram–Schmidt process, then one obtains the Legendre polynomials. Another collection of orthogonal polynomials are the associated Legendre polynomials.
The study of orthogonal polynomials involves weight functions that are inserted in the bilinear form:
For Laguerre polynomials on the weight function is .
Both physicists and probability theorists use Hermite polynomials on , where the weight function is or .
Chebyshev polynomials are defined on and use weights or .
Zernike polynomials are defined on the unit disk and have orthogonality of both radial and angular parts.
Binary-valued functions
Walsh functions and Haar wavelets are examples of orthogonal functions with discrete ranges.
Rational functions
Legendre and Chebyshev polynomials provide orthogonal families for the interval while occasionally orthogonal families are required on . In this case it is convenient to apply the Cayley transform first, to bring the argument into . This procedure results in families of rational orthogonal functions called Legendre rational functions and Chebyshev rational functions.
In differential equations
Solutions of linear differential equations with boundary conditions can often be written as a weighted sum of orthogonal solution functions (a.k.a. eigenfunctions), leading to generalized Fourier series.
See also
Eigenvalues and eigenvectors
Hilbert space
Karhunen–Loève theorem
Lauricella's theorem
Wannier function
References
George B. Arfken & Hans J. Weber (2005) Mathematical Met
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https://en.wikipedia.org/wiki/Gambler%27s%20ruin
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In statistics, gambler's ruin is the fact that a gambler playing a game with negative expected value will eventually go broke, regardless of their betting system.
The concept was initially stated: A persistent gambler who raises their bet to a fixed fraction of the gambler's bankroll after a win, but does not reduce it after a loss, will eventually and inevitably go broke, even if each bet has a positive expected value.
Another statement of the concept is that a persistent gambler with finite wealth, playing a fair game (that is, each bet has expected value of zero to both sides) will eventually and inevitably go broke against an opponent with infinite wealth. Such a situation can be modeled by a random walk on the real number line. In that context, it is probable that the gambler will, with virtual certainty, return to their point of origin, which means going broke, and is ruined an infinite number of times if the random walk continues forever. This is a corollary of a general theorem by Christiaan Huygens, which is also known as gambler's ruin. That theorem shows how to compute the probability of each player winning a series of bets that continues until one's entire initial stake is lost, given the initial stakes of the two players and the constant probability of winning. This is the oldest mathematical idea that goes by the name gambler's ruin, but not the first idea to which the name was applied. The term's common usage today is another corollary to Huygens's result.
The concept has specific relevance for gamblers. However it also leads to mathematical theorems with wide application and many related results in probability and statistics. Huygens's result in particular led to important advances in the mathematical theory of probability.
History
The earliest known mention of the gambler's ruin problem is a letter from Blaise Pascal to Pierre Fermat in 1656 (two years after the more famous correspondence on the problem of points). Pascal's version was summarized in a 1656 letter from Pierre de Carcavi to Huygens:
Let two men play with three dice, the first player scoring a point whenever 11 is thrown, and the second whenever 14 is thrown. But instead of the points accumulating in the ordinary way, let a point be added to a player's score only if his opponent's score is nil, but otherwise let it be subtracted from his opponent's score. It is as if opposing points form pairs, and annihilate each other, so that the trailing player always has zero points. The winner is the first to reach twelve points; what are the relative chances of each player winning?
Huygens reformulated the problem and published it in De ratiociniis in ludo aleae ("On Reasoning in Games of Chance", 1657):
Problem (2-1) Each player starts with 12 points, and a successful roll of the three dice for a player (getting an 11 for the first player or a 14 for the second) adds one to that player's score and subtracts one from the other player's score; the loser of the game is the
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https://en.wikipedia.org/wiki/Parametrization%20%28geometry%29
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In mathematics, and more specifically in geometry, parametrization (or parameterization; also parameterisation, parametrisation) is the process of finding parametric equations of a curve, a surface, or, more generally, a manifold or a variety, defined by an implicit equation. The inverse process is called implicitization. "To parameterize" by itself means "to express in terms of parameters".
Parametrization is a mathematical process consisting of expressing the state of a system, process or model as a function of some independent quantities called parameters. The state of the system is generally determined by a finite set of coordinates, and the parametrization thus consists of one function of several real variables for each coordinate. The number of parameters is the number of degrees of freedom of the system.
For example, the position of a point that moves on a curve in three-dimensional space is determined by the time needed to reach the point when starting from a fixed origin. If are the coordinates of the point, the movement is thus described by a parametric equation
where is the parameter and denotes the time. Such a parametric equation completely determines the curve, without the need of any interpretation of as time, and is thus called a parametric equation of the curve (this is sometimes abbreviated by saying that one has a parametric curve). One similarly gets the parametric equation of a surface by considering functions of two parameters and .
Non-uniqueness
Parametrizations are not generally unique. The ordinary three-dimensional object can be parametrized (or "coordinatized") equally efficiently with Cartesian coordinates (x, y, z), cylindrical polar coordinates (ρ, φ, z), spherical coordinates (r, φ, θ) or other coordinate systems.
Similarly, the color space of human trichromatic color vision can be parametrized in terms of the three colors red, green and blue, RGB, or with cyan, magenta, yellow and black, CMYK.
Dimensionality
Generally, the minimum number of parameters required to describe a model or geometric object is equal to its dimension, and the scope of the parameters—within their allowed ranges—is the parameter space. Though a good set of parameters permits identification of every point in the object space, it may be that, for a given parametrization, different parameter values can refer to the same point. Such mappings are surjective but not injective. An example is the pair of cylindrical polar coordinates (ρ, φ, z) and (ρ, φ + 2π, z).
Invariance
As indicated above, there is arbitrariness in the choice of parameters of a given model, geometric object, etc. Often, one wishes to determine intrinsic properties of an object that do not depend on this arbitrariness, which are therefore independent of any particular choice of parameters. This is particularly the case in physics, wherein parametrization invariance (or 'reparametrization invariance') is a guiding principle in the search for physically acceptable t
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https://en.wikipedia.org/wiki/Glossary%20of%20order%20theory
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This is a glossary of some terms used in various branches of mathematics that are related to the fields of order, lattice, and domain theory. Note that there is a structured list of order topics available as well. Other helpful resources might be the following overview articles:
completeness properties of partial orders
distributivity laws of order theory
preservation properties of functions between posets.
In the following, partial orders will usually just be denoted by their carrier sets. As long as the intended meaning is clear from the context, will suffice to denote the corresponding relational symbol, even without prior introduction. Furthermore, < will denote the strict order induced by
A
Acyclic. A binary relation is acyclic if it contains no "cycles": equivalently, its transitive closure is antisymmetric.
Adjoint. See Galois connection.
Alexandrov topology. For a preordered set P, any upper set O is Alexandrov-open. Inversely, a topology is Alexandrov if any intersection of open sets is open.
Algebraic poset. A poset is algebraic if it has a base of compact elements.
Antichain. An antichain is a poset in which no two elements are comparable, i.e., there are no two distinct elements x and y such that x ≤ y. In other words, the order relation of an antichain is just the identity relation.
Approximates relation. See way-below relation.
Antisymmetric relation. A homogeneous relation R on a set X is antisymmetric, if x R y and y R x implies x = y, for all elements x, y in X.
Antitone. An antitone function f between posets P and Q is a function for which, for all elements x, y of P, x ≤ y (in P) implies f(y) ≤ f(x) (in Q). Another name for this property is order-reversing. In analysis, in the presence of total orders, such functions are often called monotonically decreasing, but this is not a very convenient description when dealing with non-total orders. The dual notion is called monotone or order-preserving.
Asymmetric relation. A homogeneous relation R on a set X is asymmetric, if x R y implies not y R x, for all elements x, y in X.
Atom. An atom in a poset P with least element 0, is an element that is minimal among all elements that are unequal to 0.
Atomic. An atomic poset P with least element 0 is one in which, for every non-zero element x of P, there is an atom a of P with a ≤ x.
B
Base. See continuous poset.
Binary relation. A binary relation over two sets is a subset of their Cartesian product
Boolean algebra. A Boolean algebra is a distributive lattice with least element 0 and greatest element 1, in which every element x has a complement ¬x, such that x ∧ ¬x = 0 and x ∨ ¬x = 1.
Bounded poset. A bounded poset is one that has a least element and a greatest element.
Bounded complete. A poset is bounded complete if every of its subsets with some upper bound also has a least such upper bound. The dual notion is not common.
C
Chain. A chain is a totally ordered set or a totally ordered subset of a poset
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https://en.wikipedia.org/wiki/127%20%28number%29
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127 (one hundred [and] twenty-seven') is the natural number following 126 and preceding 128. It is also a prime number.
In mathematics
As a Mersenne prime, 127 is related to the perfect number 8128. 127 is also the largest known Mersenne prime exponent for a Mersenne number, , which is also a Mersenne prime. It was discovered by Édouard Lucas in 1876 and held the record for the largest known prime for 75 years.
is the largest prime ever discovered by hand calculations as well as the largest known double Mersenne prime.
Furthermore, 127 is equal to , and 7 is equal to , and 3 is the smallest Mersenne prime, making 7 the smallest double Mersenne prime and 127 the smallest triple Mersenne prime.
There are a total of 127 prime numbers between 2,000 and 3,000.
127 is also a cuban prime of the form , . The next prime is 131, with which it comprises a cousin prime. Because the next odd number, 129, is a semiprime, 127 is a Chen prime. 127 is greater than the arithmetic mean of its two neighboring primes; thus, it is a strong prime.
127 is a centered hexagonal number.
It is the seventh Motzkin number.
127 is a palindromic prime in nonary and binary.
127 is the first Friedman prime in decimal. It is also the first nice Friedman number in decimal, since , as well as binary since .
127 is the sum of the sums of the divisors of the first twelve positive integers.
127 is the smallest prime that can be written as the sum of the first two or more odd primes: .
127 is the smallest odd number that cannot be written in the form , for is a prime number, and is an integer, since and are all composite numbers.
127 is an isolated prime where neither nor is prime.
127 is the smallest digitally delicate prime in binary.
127 is the 31st prime number and therefore it is the smallest Mersenne prime with a Mersenne prime index.
127 is the largest number with the property where is the th prime number. There are only two numbers with that property; the other one is 43.
127 is equal to where is the th prime number.
127 is the number of non-equivalent ways of expressing 10,000 as the sum of two prime numbers
In the military
was a Mission Buenaventura-class fleet oilers during World War II
was a United States Navy transport ship
was a United States Navy
was a United States Navy in World War II
was a United States Navy
was a United States Navy for removing naval mines
In religion
The biblical figure Sarah died at the age of 127.
According to the Book of Esther, the Persian Empire under Ahasuerus consisted of 127 provinces "from India to Ethiopia".
Havamal Stanza 127 is used as a declaration against folkish traditions of Heathenry and specifically the Asatru Folk Assembly.
In transportation
The small Fiat 127 automobile
London Buses route 127 is a Transport for London contracted bus route in London
127 is the number of many roads, including U.S. Route 127
STS-127 was a Space Shuttle Endeavour mission to the International Space Stati
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https://en.wikipedia.org/wiki/Thoralf%20Skolem
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Thoralf Albert Skolem (; 23 May 1887 – 23 March 1963) was a Norwegian mathematician who worked in mathematical logic and set theory.
Life
Although Skolem's father was a primary school teacher, most of his extended family were farmers. Skolem attended secondary school in Kristiania (later renamed Oslo), passing the university entrance examinations in 1905. He then entered Det Kongelige Frederiks Universitet to study mathematics, also taking courses in physics, chemistry, zoology and botany.
In 1909, he began working as an assistant to the physicist Kristian Birkeland, known for bombarding magnetized spheres with electrons and obtaining aurora-like effects; thus Skolem's first publications were physics papers written jointly with Birkeland. In 1913, Skolem passed the state examinations with distinction, and completed a dissertation titled Investigations on the Algebra of Logic. He also traveled with Birkeland to the Sudan to observe the zodiacal light. He spent the winter semester of 1915 at the University of Göttingen, at the time the leading research center in mathematical logic, metamathematics, and abstract algebra, fields in which Skolem eventually excelled. In 1916 he was appointed a research fellow at Det Kongelige Frederiks Universitet. In 1918, he became a Docent in Mathematics and was elected to the Norwegian Academy of Science and Letters.
Skolem did not at first formally enroll as a Ph.D. candidate, believing that the Ph.D. was unnecessary in Norway. He later changed his mind and submitted a thesis in 1926, titled Some theorems about integral solutions to certain algebraic equations and inequalities. His notional thesis advisor was Axel Thue, even though Thue had died in 1922.
In 1927, he married Edith Wilhelmine Hasvold.
Skolem continued to teach at Det kongelige Frederiks Universitet (renamed the University of Oslo in 1939) until 1930 when he became a Research Associate in Chr. Michelsen Institute in Bergen. This senior post allowed Skolem to conduct research free of administrative and teaching duties. However, the position also required that he reside in Bergen, a city which then lacked a university and hence had no research library, so that he was unable to keep abreast of the mathematical literature. In 1938, he returned to Oslo to assume the Professorship of Mathematics at the university. There he taught the graduate courses in algebra and number theory, and only occasionally on mathematical logic. Skolem's Ph.D. student Øystein Ore went on to a career in the USA.
Skolem served as president of the Norwegian Mathematical Society, and edited the Norsk Matematisk Tidsskrift ("The Norwegian Mathematical Journal") for many years. He was also the founding editor of Mathematica Scandinavica.
After his 1957 retirement, he made several trips to the United States, speaking and teaching at universities there. He remained intellectually active until his sudden and unexpected death.
For more on Skolem's academic life, see Fenstad (197
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https://en.wikipedia.org/wiki/Exact%20functor
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In mathematics, particularly homological algebra, an exact functor is a functor that preserves short exact sequences. Exact functors are convenient for algebraic calculations because they can be directly applied to presentations of objects. Much of the work in homological algebra is designed to cope with functors that fail to be exact, but in ways that can still be controlled.
Definitions
Let P and Q be abelian categories, and let be a covariant additive functor (so that, in particular, F(0) = 0). We say that F is an exact functor if whenever
is a short exact sequence in P then
is a short exact sequence in Q. (The maps are often omitted and implied, and one says: "if 0→A→B→C→0 is exact, then 0→F(A)→F(B)→F(C)→0 is also exact".)
Further, we say that F is
left-exact if whenever 0→A→B→C→0 is exact then 0→F(A)→F(B)→F(C) is exact;
right-exact if whenever 0→A→B→C→0 is exact then F(A)→F(B)→F(C)→0 is exact;
half-exact if whenever 0→A→B→C→0 is exact then F(A)→F(B)→F(C) is exact. This is distinct from the notion of a topological half-exact functor.
If G is a contravariant additive functor from P to Q, we similarly define G to be
exact if whenever 0→A→B→C→0 is exact then 0→G(C)→G(B)→G(A)→0 is exact;
left-exact if whenever 0→A→B→C→0 is exact then 0→G(C)→G(B)→G(A) is exact;
right-exact if whenever 0→A→B→C→0 is exact then G(C)→G(B)→G(A)→0 is exact;
half-exact if whenever 0→A→B→C→0 is exact then G(C)→G(B)→G(A) is exact.
It is not always necessary to start with an entire short exact sequence 0→A→B→C→0 to have some exactness preserved. The following definitions are equivalent to the ones given above:
F is exact if and only if A→B→C exact implies F(A)→F(B)→F(C) exact;
F is left-exact if and only if 0→A→B→C exact implies 0→F(A)→F(B)→F(C) exact (i.e. if "F turns kernels into kernels");
F is right-exact if and only if A→B→C→0 exact implies F(A)→F(B)→F(C)→0 exact (i.e. if "F turns cokernels into cokernels");
G is left-exact if and only if A→B→C→0 exact implies 0→G(C)→G(B)→G(A) exact (i.e. if "G turns cokernels into kernels");
G is right-exact if and only if 0→A→B→C exact implies G(C)→G(B)→G(A)→0 exact (i.e. if "G turns kernels into cokernels").
Examples
Every equivalence or duality of abelian categories is exact.
The most basic examples of left exact functors are the Hom functors: if A is an abelian category and A is an object of A, then FA(X) = HomA(A,X) defines a covariant left-exact functor from A to the category Ab of abelian groups. The functor FA is exact if and only if A is projective. The functor GA(X) = HomA(X,A) is a contravariant left-exact functor; it is exact if and only if A is injective.
If k is a field and V is a vector space over k, we write V * = Homk(V,k) (this is commonly known as the dual space). This yields a contravariant exact functor from the category of k-vector spaces to itself. (Exactness follows from the above: k is an injective k-module. Alternatively, one can argue that every short exact sequence of k-vector
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https://en.wikipedia.org/wiki/Integral%20test%20for%20convergence
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In mathematics, the integral test for convergence is a method used to test infinite series of monotonous terms for convergence. It was developed by Colin Maclaurin and Augustin-Louis Cauchy and is sometimes known as the Maclaurin–Cauchy test.
Statement of the test
Consider an integer and a function defined on the unbounded interval , on which it is monotone decreasing. Then the infinite series
converges to a real number if and only if the improper integral
is finite. In particular, if the integral diverges, then the series diverges as well.
Remark
If the improper integral is finite, then the proof also gives the lower and upper bounds
for the infinite series.
Note that if the function is increasing, then the function is decreasing and the above theorem applies.
Proof
The proof basically uses the comparison test, comparing the term with the integral of over the intervals
and , respectively.
The monotonous function is continuous almost everywhere. To show this, let . For every , there exists by the density of a so that . Note that this set contains an open non-empty interval precisely if is discontinuous at . We can uniquely identify as the rational number that has the least index in an enumeration and satisfies the above property. Since is monotone, this defines an injective mapping and thus is countable. It follows that is continuous almost everywhere. This is sufficient for Riemann integrability.
Since is a monotone decreasing function, we know that
and
Hence, for every integer ,
and, for every integer ,
By summation over all from to some larger integer , we get from ()
and from ()
Combining these two estimates yields
Letting tend to infinity, the bounds in () and the result follow.
Applications
The harmonic series
diverges because, using the natural logarithm, its antiderivative, and the fundamental theorem of calculus, we get
On the other hand, the series
(cf. Riemann zeta function)
converges for every , because by the power rule
From () we get the upper estimate
which can be compared with some of the particular values of Riemann zeta function.
Borderline between divergence and convergence
The above examples involving the harmonic series raise the question of whether there are monotone sequences such that decreases to 0 faster than but slower than in the sense that
for every , and whether the corresponding series of the still diverges. Once such a sequence is found, a similar question can be asked with taking the role of , and so on. In this way it is possible to investigate the borderline between divergence and convergence of infinite series.
Using the integral test for convergence, one can show (see below) that, for every natural number , the series
still diverges (cf. proof that the sum of the reciprocals of the primes diverges for ) but
converges for every . Here denotes the -fold composition of the natural logarithm defined recursively by
Furthermore, denotes the smallest natural num
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https://en.wikipedia.org/wiki/Michel%20Plancherel
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Michel Plancherel (16 January 1885 – 4 March 1967) was a Swiss mathematician. He was born in Bussy (Fribourg, Switzerland) and obtained his Diplom in mathematics from the University of Fribourg and then his doctoral degree in 1907 with a thesis written under the supervision of Mathias Lerch. Plancherel was a professor in Fribourg (1911), and from 1920 at ETH Zurich.
He worked in the areas of mathematical analysis, mathematical physics and algebra, and is known for the Plancherel theorem in harmonic analysis. He was an Invited Speaker of the ICM in 1924 at Toronto and in 1928 at Bologna.
He was married to Cécile Tercier, had nine children, and presided at the Mission Catholique Française in Zürich.
References
External links
Short biography, Department of mathematics, University of Fribourg
1885 births
1967 deaths
20th-century Swiss mathematicians
Swiss Roman Catholics
Academic staff of ETH Zurich
University of Fribourg alumni
Academic staff of the University of Fribourg
People from the canton of Fribourg
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https://en.wikipedia.org/wiki/Plancherel%20theorem
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In mathematics, the Plancherel theorem (sometimes called the Parseval–Plancherel identity) is a result in harmonic analysis, proven by Michel Plancherel in 1910. It states that the integral of a function's squared modulus is equal to the integral of the squared modulus of its frequency spectrum. That is, if is a function on the real line, and is its frequency spectrum, then
A more precise formulation is that if a function is in both Lp spaces and , then its Fourier transform is in , and the Fourier transform map is an isometry with respect to the L2 norm. This implies that the Fourier transform map restricted to has a unique extension to a linear isometric map , sometimes called the Plancherel transform. This isometry is actually a unitary map. In effect, this makes it possible to speak of Fourier transforms of quadratically integrable functions.
Plancherel's theorem remains valid as stated on n-dimensional Euclidean space . The theorem also holds more generally in locally compact abelian groups. There is also a version of the Plancherel theorem which makes sense for non-commutative locally compact groups satisfying certain technical assumptions. This is the subject of non-commutative harmonic analysis.
The unitarity of the Fourier transform is often called Parseval's theorem in science and engineering fields, based on an earlier (but less general) result that was used to prove the unitarity of the Fourier series.
Due to the polarization identity, one can also apply Plancherel's theorem to the inner product of two functions. That is, if and are two functions, and denotes the Plancherel transform, then
and if and are furthermore functions, then
and
so
See also
Plancherel theorem for spherical functions
References
.
.
.
External links
Plancherel's Theorem on Mathworld
Theorems in functional analysis
Theorems in harmonic analysis
Theorems in Fourier analysis
Lp spaces
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https://en.wikipedia.org/wiki/Locally%20compact%20abelian%20group
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In several mathematical areas, including harmonic analysis, topology, and number theory, locally compact abelian groups are abelian groups which have a particularly convenient topology on them. For example, the group of integers (equipped with the discrete topology), or the real numbers or the circle (both with their usual topology) are locally compact abelian groups.
Definition and examples
A topological group is called locally compact if the underlying topological space is locally compact and Hausdorff; the topological group is called abelian if the underlying group is abelian.
Examples of locally compact abelian groups include:
for n a positive integer, with vector addition as group operation.
The positive real numbers with multiplication as operation. This group is isomorphic to by the exponential map.
Any finite abelian group, with the discrete topology. By the structure theorem for finite abelian groups, all such groups are products of cyclic groups.
The integers under addition, again with the discrete topology.
The circle group, denoted for torus. This is the group of complex numbers of modulus 1. is isomorphic as a topological group to the quotient group .
The field of p-adic numbers under addition, with the usual p-adic topology.
The dual group
If is a locally compact abelian group, a character of is a continuous group homomorphism from with values in the circle group . The set of all characters on can be made into a locally compact abelian group, called the dual group of and denoted . The group operation on the dual group is given by pointwise multiplication of characters, the inverse of a character is its complex conjugate and the topology on the space of characters is that of uniform convergence on compact sets (i.e., the compact-open topology, viewing as a subset of the space of all continuous functions from to .). This topology is in general not metrizable. However, if the group is a separable locally compact abelian group, then the dual group is metrizable.
This is analogous to the dual space in linear algebra: just as for a vector space over a field , the dual space is , so too is the dual group . More abstractly, these are both examples of representable functors, being represented respectively by and .
A group that is isomorphic (as topological groups) to its dual group is called self-dual. While the reals and finite cyclic groups are self-dual, the group and the dual group are not naturally isomorphic, and should be thought of as two different groups.
Examples of dual groups
The dual of is isomorphic to the circle group . A character on the infinite cyclic group of integers under addition is determined by its value at the generator 1. Thus for any character on , . Moreover, this formula defines a character for any choice of in . The topology of uniform convergence on compact sets is in this case the topology of pointwise convergence. This is the topology of the circle group inherit
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https://en.wikipedia.org/wiki/Abstraction%20%28mathematics%29
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Abstraction in mathematics is the process of extracting the underlying structures, patterns or properties of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected, and generalizing it so that it has wider applications or matching among other abstract descriptions of equivalent phenomena. Two of the most highly abstract areas of modern mathematics are category theory and model theory.
Description
Many areas of mathematics began with the study of real world problems, before the underlying rules and concepts were identified and defined as abstract structures. For example, geometry has its origins in the calculation of distances and areas in the real world, and algebra started with methods of solving problems in arithmetic.
Abstraction is an ongoing process in mathematics and the historical development of many mathematical topics exhibits a progression from the concrete to the abstract. For example, the first steps in the abstraction of geometry were historically made by the ancient Greeks, with Euclid's Elements being the earliest extant documentation of the axioms of plane geometry—though Proclus tells of an earlier axiomatisation by Hippocrates of Chios. In the 17th century, Descartes introduced Cartesian co-ordinates which allowed the development of analytic geometry. Further steps in abstraction were taken by Lobachevsky, Bolyai, Riemann and Gauss, who generalised the concepts of geometry to develop non-Euclidean geometries. Later in the 19th century, mathematicians generalised geometry even further, developing such areas as geometry in n dimensions, projective geometry, affine geometry and finite geometry. Finally Felix Klein's "Erlangen program" identified the underlying theme of all of these geometries, defining each of them as the study of properties invariant under a given group of symmetries. This level of abstraction revealed connections between geometry and abstract algebra.
In mathematics, abstraction can be advantageous in the following ways:
It reveals deep connections between different areas of mathematics.
Known results in one area can suggest conjectures in another related area.
Techniques and methods from one area can be applied to prove results in other related areas.
Patterns from one mathematical object can be generalized to other similar objects in the same class.
On the other hand, abstraction can also be disadvantageous in that highly abstract concepts can be difficult to learn. A degree of mathematical maturity and experience may be needed for conceptual assimilation of abstractions.
Bertrand Russell, in The Scientific Outlook (1931), writes that "Ordinary language is totally unsuited for expressing what physics really asserts, since the words of everyday life are not sufficiently abstract. Only mathematics and mathematical logic can say as little as the physicist means to say."
See also
Abstract detail
Generalization
Abstract thinking
Abstra
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https://en.wikipedia.org/wiki/Nine%20lemma
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In mathematics, the nine lemma (or 3×3 lemma) is a statement about commutative diagrams and exact sequences valid in the category of groups and any abelian category. It states: if the diagram to the right is a commutative diagram and all columns as well as the two bottom rows are exact, then the top row is exact as well. Likewise, if all columns as well as the two top rows are exact, then the bottom row is exact as well. Similarly, because the diagram is symmetric about its diagonal, rows and columns may be interchanged in the above as well.
The nine lemma can be proved by direct diagram chasing, or by applying the snake lemma (to the two bottom rows in the first case, and to the two top rows in the second case).
Linderholm (p. 201) offers a satirical view of the nine lemma:
"Draw a noughts-and-crosses board... Do not fill it in with noughts and crosses... Instead, use curved arrows... Wave your hands about in complicated patterns over this board. Make some noughts, but not in the squares; put them at both ends of the horizontal and vertical lines. Make faces. You have now proved:
(a) the Nine Lemma
(b) the Sixteen Lemma
(c) the Twenty-five Lemma..."
There are two variants of nine lemma: sharp nine lemma and symmetric nine lemma (see Lemmas 3.3, 3.4 in Chapter XII of ).
References
Homological algebra
Lemmas in category theory
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https://en.wikipedia.org/wiki/1740%20in%20science
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The year 1740 in science and technology involved some significant events.
Mathematics
Jean Paul de Gua de Malves publishes his work of analytic geometry, .
Metallurgy
Benjamin Huntsman develops the technique of crucible steel production at Handsworth, South Yorkshire, England.
Physics
Jacques-Barthélemy Micheli du Crest creates a spirit thermometer, making use of two fixed points, 0 for "Temperature of earth" based on a cave at Paris Observatory and 100 for the heat of boiling water.
Émilie du Châtelet publishes Institutions de Physique, including a demonstration that the energy of a moving object is proportional to the square of its velocity (Ek = mv²).
Louis Bertrand Castel publishes L'Optique des couleurs in Paris, including the observation that the colours of white light split by a prism depend on distance from the prism.
Technology
Henry Hindley of Yorkshire invents a device to cut the teeth of clock wheels.
Awards
Copley Medal: Alexander Stuart
Births
February 17 – Horace Bénédict de Saussure, Genevan pioneer of Alpine studies (died 1799)
March 28 (bapt.) – James Small, Scottish inventor (died 1793)
June 27 – John Latham, English physician and naturalist, "grandfather of Australian ornithology" (died 1837)
July 1 – Franz-Joseph Müller von Reichenstein, Austrian mineralogist and discoverer of tellurium (died 1825)
August 26 – Joseph Michel Montgolfier, French pioneer balloonist (died 1810)
September 29 – Thomas Percival, English reforming physician and medical ethicist (died 1804)
December 24 – Anders Johan Lexell, Finnish-Swedish astronomer and mathematician (died 1784)
unknown – William Smellie, Scottish naturalist and encyclopedist (died 1795)
Deaths
March 23 – Olof Rudbeck the Younger, Swedish naturalist (born 1660).
References
18th century in science
1740s in science
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https://en.wikipedia.org/wiki/Category%20of%20abelian%20groups
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In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every small abelian category can be embedded in Ab.
Properties
The zero object of Ab is the trivial group {0} which consists only of its neutral element.
The monomorphisms in Ab are the injective group homomorphisms, the epimorphisms are the surjective group homomorphisms, and the isomorphisms are the bijective group homomorphisms.
Ab is a full subcategory of Grp, the category of all groups. The main difference between Ab and Grp is that the sum of two homomorphisms f and g between abelian groups is again a group homomorphism:
(f+g)(x+y) = f(x+y) + g(x+y) = f(x) + f(y) + g(x) + g(y)
= f(x) + g(x) + f(y) + g(y) = (f+g)(x) + (f+g)(y)
The third equality requires the group to be abelian. This addition of morphism turns Ab into a preadditive category, and because the direct sum of finitely many abelian groups yields a biproduct, we indeed have an additive category.
In Ab, the notion of kernel in the category theory sense coincides with kernel in the algebraic sense, i.e. the categorical kernel of the morphism f : A → B is the subgroup K of A defined by K = {x ∈ A : f(x) = 0}, together with the inclusion homomorphism i : K → A. The same is true for cokernels; the cokernel of f is the quotient group C = B / f(A) together with the natural projection p : B → C. (Note a further crucial difference between Ab and Grp: in Grp it can happen that f(A) is not a normal subgroup of B, and that therefore the quotient group B / f(A) cannot be formed.) With these concrete descriptions of kernels and cokernels, it is quite easy to check that Ab is indeed an abelian category.
The product in Ab is given by the product of groups, formed by taking the cartesian product of the underlying sets and performing the group operation componentwise. Because Ab has kernels, one can then show that Ab is a complete category. The coproduct in Ab is given by the direct sum; since Ab has cokernels, it follows that Ab is also cocomplete.
We have a forgetful functor Ab → Set which assigns to each abelian group the underlying set, and to each group homomorphism the underlying function. This functor is faithful, and therefore Ab is a concrete category. The forgetful functor has a left adjoint (which associates to a given set the free abelian group with that set as basis) but does not have a right adjoint.
Taking direct limits in Ab is an exact functor. Since the group of integers Z serves as a generator, the category Ab is therefore a Grothendieck category; indeed it is the prototypical example of a Grothendieck category.
An object in Ab is injective if and only if it is a divisible group; it is projective if and only if it is a free abelian group. The category has a projective generator (Z) and an injective cogenerator (Q/Z).
Given two abelian groups A and B, their tensor product A⊗B is defined; it is again an ab
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https://en.wikipedia.org/wiki/Quadratic%20field
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In algebraic number theory, a quadratic field is an algebraic number field of degree two over , the rational numbers.
Every such quadratic field is some where is a (uniquely defined) square-free integer different from and . If , the corresponding quadratic field is called a real quadratic field, and, if , it is called an imaginary quadratic field or a complex quadratic field, corresponding to whether or not it is a subfield of the field of the real numbers.
Quadratic fields have been studied in great depth, initially as part of the theory of binary quadratic forms. There remain some unsolved problems. The class number problem is particularly important.
Ring of integers
Discriminant
For a nonzero square free integer , the discriminant of the quadratic field is if is congruent to modulo , and otherwise . For example, if is , then is the field of Gaussian rationals and the discriminant is . The reason for such a distinction is that the ring of integers of is generated by in the first case and by in the second case.
The set of discriminants of quadratic fields is exactly the set of fundamental discriminants.
Prime factorization into ideals
Any prime number gives rise to an ideal in the ring of integers of a quadratic field . In line with general theory of splitting of prime ideals in Galois extensions, this may be
is inert is a prime ideal.
The quotient ring is the finite field with elements: .
splits is a product of two distinct prime ideals of .
The quotient ring is the product .
is ramified is the square of a prime ideal of .
The quotient ring contains non-zero nilpotent elements.
The third case happens if and only if divides the discriminant . The first and second cases occur when the Kronecker symbol equals and , respectively. For example, if is an odd prime not dividing , then splits if and only if is congruent to a square modulo . The first two cases are, in a certain sense, equally likely to occur as runs through the primes—see Chebotarev density theorem.
The law of quadratic reciprocity implies that the splitting behaviour of a prime in a quadratic field depends only on modulo , where is the field discriminant.
Class group
Determining the class group of a quadratic field extension can be accomplished using Minkowski's bound and the Kronecker symbol because of the finiteness of the class group. A quadratic field has discriminant
so the Minkowski bound is
Then, the ideal class group is generated by the prime ideals whose norm is less than . This can be done by looking at the decomposition of the ideals for prime where page 72 These decompositions can be found using the Dedekind–Kummer theorem.
Quadratic subfields of cyclotomic fields
The quadratic subfield of the prime cyclotomic field
A classical example of the construction of a quadratic field is to take the unique quadratic field inside the cyclotomic field generated by a primitive th root of unity, with an odd prime number. The unique
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https://en.wikipedia.org/wiki/Set
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Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics
Mathematics
Set (mathematics), a collection of elements
Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electronics and computing
Set (abstract data type), a data type in computer science that is a collection of unique values
Set (C++), a set implementation in the C++ Standard Library
Set (command), a command for setting values of environment variables in Unix and Microsoft operating-systems
Secure Electronic Transaction, a standard protocol for securing credit card transactions over insecure networks
Single-electron transistor, a device to amplify currents in nanoelectronics
Single-ended triode, a type of electronic amplifier
Set!, a programming syntax in the scheme programming language
Biology and psychology
Set (psychology), a set of expectations which shapes perception or thought
Set or sett, a badger's den
Set, a small tuber or bulb used instead of seed, especially:
Potato set
Onion set
SET (gene), gene for a human protein involved in apoptosis, transcription and nucleosome assembly
Single Embryo Transfer, used in in vitro fertilization
Physics and chemistry
A chemical change in an adhesive from unbonded to bonded
Set, to make/become solid; see Solidification
Stress–energy tensor, a physical quantity in the theory of fields
Single electron transfer
Other uses in science and technology
Saw set, the process of setting the teeth of a saw so each tooth protrudes to the side of the blade
Scalar expectancy theory, a model of the processes that govern behavior controlled by time
Science, Engineering & Technology, e.g. The Science, Engineering & Technology Student of the Year Awards
Setting (typesetting), the act of typesetting a publication for print or display
Simulated Emergency Test, an amateur radio training exercise
Software Engineer in Test, a Quality Assurance job title in some software companies
Strategic Energy Technologies Plan of the European Union
Suzuki SET, Suzuki Exhaust Tuning of motorcycles
Arts and entertainment
Dance
Set, the basic square formation in square dancing
Set, the basic longwise, square or triangular formation in Scottish Country dancing
Set, the basic formation of more than one couple in Scottish, English and Irish Céilidh
Film, television and theatre
Set (film and TV scenery)
Theatrical scenery
Set construction, construction of scenery for theatrical, movie, television production, and video game production
The Set (film), a 1970 Australian movie
The Set (TV series), an Australian music television show
Sanlih Entertainment Television, a television channel in Taiwan
Sony Entertainment Television, a Hindi-language television channel
Music
DJ set or DJ mix, a musical performance by a DJ
Set theory (music), dealing with concepts for categorizing musical objects and describing their relationships
Set (music), a collection of discrete entities, for example pitch
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https://en.wikipedia.org/wiki/Positive%20linear%20functional
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In mathematics, more specifically in functional analysis, a positive linear functional on an ordered vector space is a linear functional on so that for all positive elements that is it holds that
In other words, a positive linear functional is guaranteed to take nonnegative values for positive elements. The significance of positive linear functionals lies in results such as Riesz–Markov–Kakutani representation theorem.
When is a complex vector space, it is assumed that for all is real. As in the case when is a C*-algebra with its partially ordered subspace of self-adjoint elements, sometimes a partial order is placed on only a subspace and the partial order does not extend to all of in which case the positive elements of are the positive elements of by abuse of notation. This implies that for a C*-algebra, a positive linear functional sends any equal to for some to a real number, which is equal to its complex conjugate, and therefore all positive linear functionals preserve the self-adjointness of such This property is exploited in the GNS construction to relate positive linear functionals on a C*-algebra to inner products.
Sufficient conditions for continuity of all positive linear functionals
There is a comparatively large class of ordered topological vector spaces on which every positive linear form is necessarily continuous.
This includes all topological vector lattices that are sequentially complete.
Theorem Let be an Ordered topological vector space with positive cone and let denote the family of all bounded subsets of
Then each of the following conditions is sufficient to guarantee that every positive linear functional on is continuous:
has non-empty topological interior (in ).
is complete and metrizable and
is bornological and is a semi-complete strict -cone in
is the inductive limit of a family of ordered Fréchet spaces with respect to a family of positive linear maps where for all where is the positive cone of
Continuous positive extensions
The following theorem is due to H. Bauer and independently, to Namioka.
Theorem: Let be an ordered topological vector space (TVS) with positive cone let be a vector subspace of and let be a linear form on Then has an extension to a continuous positive linear form on if and only if there exists some convex neighborhood of in such that is bounded above on
Corollary: Let be an ordered topological vector space with positive cone let be a vector subspace of If contains an interior point of then every continuous positive linear form on has an extension to a continuous positive linear form on
Corollary: Let be an ordered vector space with positive cone let be a vector subspace of and let be a linear form on Then has an extension to a positive linear form on if and only if there exists some convex absorbing subset in containing the origin of such that is bounded above on
Proof: It suffices to endow with the finest loc
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https://en.wikipedia.org/wiki/Positive%20operator%20%28Hilbert%20space%29
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In mathematics (specifically linear algebra, operator theory, and functional analysis) as well as physics, a linear operator acting on an inner product space is called positive-semidefinite (or non-negative) if, for every , and , where is the domain of . Positive-semidefinite operators are denoted as . The operator is said to be positive-definite, and written , if for all .
In physics (specifically quantum mechanics), such operators represent quantum states, via the density matrix formalism.
Cauchy–Schwarz inequality
If then
Indeed, let Applying Cauchy–Schwarz inequality to the inner product
as proves the claim.
It follows that If is defined everywhere, and then
On a complex Hilbert space, if A ≥ 0 then A is symmetric
Without loss of generality, let the inner product be anti-linear on the first argument and linear on the second. (If the reverse is true, then we work with instead). For the polarization identity
and the fact that for positive operators, show that so is symmetric.
In contrast with the complex case, a positive-semidefinite operator on a real Hilbert space may not be symmetric. As a counterexample, define to be an operator of rotation by an acute angle Then but so is not symmetric.
If A ≥ 0 and Dom A = , then A is self-adjoint and bounded
The symmetry of implies that and For to be self-adjoint, it is necessary that In our case, the equality of domains holds because so is indeed self-adjoint. The fact that is bounded now follows from the Hellinger–Toeplitz theorem.
This property does not hold on
Order in self-adjoint operators on
A natural ordering of self-adjoint operators arises from the definition of positive operators. Define if the following hold:
and are self-adjoint
It can be seen that a similar result as the Monotone convergence theorem holds for monotone increasing, bounded, self-adjoint operators on Hilbert spaces.
Application to physics: quantum states
The definition of a quantum system includes a complex separable Hilbert space and a set of positive trace-class operators on for which The set is the set of states. Every is called a state or a density operator. For where the operator of projection onto the span of is called a pure state. (Since each pure state is identifiable with a unit vector some sources define pure states to be unit elements from States that are not pure are called mixed.
References
Operator theory
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https://en.wikipedia.org/wiki/Extreme%20point
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In mathematics, an extreme point of a convex set in a real or complex vector space is a point in that does not lie in any open line segment joining two points of In linear programming problems, an extreme point is also called vertex or corner point of
Definition
Throughout, it is assumed that is a real or complex vector space.
For any say that and if and there exists a such that
If is a subset of and then is called an of if it does not lie between any two points of That is, if there does exist and such that and The set of all extreme points of is denoted by
Generalizations
If is a subset of a vector space then a linear sub-variety (that is, an affine subspace) of the vector space is called a if meets (that is, is not empty) and every open segment whose interior meets is necessarily a subset of A 0-dimensional support variety is called an extreme point of
Characterizations
The of two elements and in a vector space is the vector
For any elements and in a vector space, the set is called the or between and The or between and is when while it is when The points and are called the of these interval. An interval is said to be a or a if its endpoints are distinct. The is the midpoint of its endpoints.
The closed interval is equal to the convex hull of if (and only if) So if is convex and then
If is a nonempty subset of and is a nonempty subset of then is called a of if whenever a point lies between two points of then those two points necessarily belong to
Examples
If are two real numbers then and are extreme points of the interval However, the open interval has no extreme points.
Any open interval in has no extreme points while any non-degenerate closed interval not equal to does have extreme points (that is, the closed interval's endpoint(s)). More generally, any open subset of finite-dimensional Euclidean space has no extreme points.
The extreme points of the closed unit disk in is the unit circle.
The perimeter of any convex polygon in the plane is a face of that polygon.
The vertices of any convex polygon in the plane are the extreme points of that polygon.
An injective linear map sends the extreme points of a convex set to the extreme points of the convex set This is also true for injective affine maps.
Properties
The extreme points of a compact convex set form a Baire space (with the subspace topology) but this set may to be closed in
Theorems
Krein–Milman theorem
The Krein–Milman theorem is arguably one of the most well-known theorems about extreme points.
For Banach spaces
These theorems are for Banach spaces with the Radon–Nikodym property.
A theorem of Joram Lindenstrauss states that, in a Banach space with the Radon–Nikodym property, a nonempty closed and bounded set has an extreme point. (In infinite-dimensional spaces, the property of compactness is stronger than the joint properties of being closed and being bounded.)
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https://en.wikipedia.org/wiki/160%20%28number%29
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160 (one hundred [and] sixty) is the natural number following 159 and preceding 161.
In mathematics
160 is the sum of the first 11 primes, as well as the sum of the cubes of the first three primes.
Given 160, the Mertens function returns 0. 160 is the smallest number n with exactly 12 solutions to the equation φ(x) = n.
In telecommunications
The number of characters permitted in a standard short message service
The number for Dial-a-Disc (1966–1991), a telephone number operated by the General Post Office in the United Kingdom, which enabled callers to hear the latest chart hits
See also
160s
List of highways numbered 160
United Nations Security Council Resolution 160
United States Supreme Court cases, Volume 160
Article 160 of the Constitution of Malaysia
Norris School District 160, Lancaster County, Nebraska
References
External links
Number Facts and Trivia: 160
Integers
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https://en.wikipedia.org/wiki/170%20%28number%29
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170 (one hundred [and] seventy) is the natural number following 169 and preceding 171.
In mathematics
170 is the smallest n for which φ(n) and σ(n) are both square (64 and 324 respectively). But 170 is never a solution for φ(x), making it a nontotient. Nor is it ever a solution to x - φ(x), making it a noncototient.
170 is a repdigit in base 4 (2222) and base 16 (AA), as well as in bases 33, 84, and 169. It is also a sphenic number.
170 is the largest integer for which its factorial can be stored in IEEE 754 double-precision floating-point format. This is probably why it is also the largest factorial that Google's built-in calculator will calculate, returning the answer as 170! = 7.25741562 × 10306.
There are 170 different cyclic Gilbreath permutations on 12 elements, and therefore there are 170 different real periodic points of order 12 on the Mandelbrot set.
See also
170s
E170 (disambiguation)
F170 (disambiguation)
List of highways numbered 170
United States Supreme Court cases, Volume 170
United Nations Security Council Resolution 170
Pennsylvania House of Representatives, District 170
References
External links
The Number 170
Number Facts and Trivia: 170
The Positive Integer 170
Prime curiosities: 170
Integers
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https://en.wikipedia.org/wiki/180%20%28number%29
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180 (one hundred [and] eighty) is the natural number following 179 and preceding 181.
In mathematics
180 is an abundant number, with its proper divisors summing up to 366. 180 is also a highly composite number, a positive integer with more divisors than any smaller positive integer. One of the consequences of 180 having so many divisors is that it is a practical number, meaning that any positive number smaller than 180 that is not a divisor of 180 can be expressed as the sum of some of 180's divisors. 180 is a Harshad number and a refactorable number.
180 is the sum of two square numbers: 122 + 62. It can be expressed as either the sum of six consecutive prime numbers: 19 + 23 + 29 + 31 + 37 + 41, or the sum of eight consecutive prime numbers: 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37. 180 is an Ulam number, which can be expressed as a sum of earlier terms in the Ulam sequence only as 177 + 3.
180 is a 61-gonal number, while 61 is the 18th prime number.
Half a circle has 180 degrees, and thus a U-turn is also referred to as a 180.
Summing Euler's totient function φ(x) over the first + 24 integers gives 180.
In binary it is a digitally balanced number, since its binary representation has the same number of zeros as ones (10110100).
A triangle has three interior angles that collectively total 180 degrees. In general, the interior angles of an -sided polygon add to degrees.
In religion
The Book of Genesis says that Isaac died at the age of 180.
Other
180 is the highest score possible with three darts.
See also
List of highways numbered 180
United Nations Security Council Resolution 180
United States Supreme Court cases, Volume 180
Pennsylvania House of Representatives, District 180
References
External links
Integers
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https://en.wikipedia.org/wiki/190%20%28number%29
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190 (one hundred [and] ninety) is the natural number following 189 and preceding 191.
In mathematics
190 is a triangular number, a hexagonal number, and a centered nonagonal number, the fourth figurate number (after 1, 28, and 91) with that combination of properties. It is also a truncated square pyramid number.
Integers from 191 to 199
191
191 is a prime number.
192
192 = 26 × 3 is a 3-smooth number, the smallest number with 14 divisors.
193
193 is a prime number.
194
194 = 2 × 97 is a Markov number, the smallest number written as the sum of three squares in five ways, and the number of irreducible representations of the Monster group.
195
195 = 3 × 5 × 13 is the smallest number expressed as a sum of distinct squares in 16 different ways.
196
196 = 22 × 72 is a square number.
197
197 is a prime number and a Schröder–Hipparchus number.
198
198 = 2 × 32 × 11 is the smallest number written as the sum of four squares in ten ways.
No integer factorial ever ends in exactly 198 zeroes in base 10 or in base 12.
There are 198 ridges on a U.S. dollar coin.
199
199 is a prime number and a centered triangular number.
In other fields
190 is the telephonic number of the 27 Brazilian Military Polices.
See also
190 (disambiguation)
References
Integers
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https://en.wikipedia.org/wiki/Color%20theory
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In the visual arts, color theory is the body of practical guidance for color mixing and the visual effects of a specific color combination. Color terminology based on the color wheel and its geometry separates colors into primary color, secondary color, and tertiary color. The understanding of color theory dates to antiquity. Aristotle (d. 322 BCE) and Claudius Ptolemy (d. 168 CE) already discussed which and how colors can be produced by mixing other colors. The influence of light on color was investigated and revealed further by al-Kindi (d. 873) and Ibn al-Haytham (d.1039). Ibn Sina (d. 1037), Nasir al-Din al-Tusi (d. 1274), and Robert Grosseteste (d. 1253) discovered that contrary to the teachings of Aristotle, there are multiple color paths to get from black to white. More modern approaches to color theory principles can be found in the writings of Leone Battista Alberti (c. 1435) and the notebooks of Leonardo da Vinci (c. 1490). A formalization of "color theory" began in the 18th century, initially within a partisan controversy over Isaac Newton's theory of color (Opticks, 1704) and the nature of primary colors. From there it developed as an independent artistic tradition with only superficial reference to colorimetry and vision science.
Classifications
Colors can be classified as:
Warm and cold
Receding and advancing
Positive and negative
Subtractive and additive
Color abstractions
The foundations of pre-20th-century color theory were built around "pure" or ideal colors, characterized by different sensory experiences rather than attributes of the physical world. This has led to several inaccuracies in traditional color theory principles that are not always remedied in modern formulations.
Another issue has been the tendency to describe color effects holistically or categorically, for example as a contrast between "yellow" and "blue" conceived as generic colors, when most color effects are due to contrasts on three relative attributes which define all colors:
Value (light vs. dark, or white vs. black),
Chroma [saturation, purity, strength, intensity] (intense vs. dull), and
Hue (e.g. the name of the color family: red, yellow, green, cyan, blue, magenta).
The visual impact of "yellow" vs. "blue" hues in visual design depends on the relative lightness and saturation of the hues.
These confusions are partly historical and arose in scientific uncertainty about color perception that was not resolved until the late 19th century when artistic notions were already entrenched. They also arise from the attempt to describe the highly contextual and flexible behavior of color perception in terms of abstract color sensations that can be generated equivalently by any visual media.
Many historical "color theorists" have assumed that three "pure" primary colors can mix into all possible colors, and any failure of specific paints or inks to match this ideal performance is due to the impurity or imperfection of the colorants. In reality, onl
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https://en.wikipedia.org/wiki/Lusser%27s%20law
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Lusser's law in systems engineering is a prediction of reliability. Named after engineer Robert Lusser, and also known as Lusser's product law or the probability product law of series components, it states that the reliability of a series of components is equal to the product of the individual reliabilities of the components, if their failure modes are known to be statistically independent. For a series of N components, this is expressed as:
where Rs is the overall reliability of the system, and rn is the reliability of the nth component.
If the failure probabilities of all components are equal, then as Lusser's colleague Erich Pieruschka observed, this can be expressed simply as:
Lusser's law has been described as the idea that a series system is "weaker than its weakest link", as the product reliability of a series of components can be less than the lowest-value component.
For example, given a series system of two components with different reliabilities — one of 0.95 and the other of 0.8 — Lusser's law will predict a reliability of
which is lower than either of the individual components.
References
Engineering failures
Reliability analysis
Reliability engineering
Statistics articles needing expert attention
Survival analysis
Systems analysis
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https://en.wikipedia.org/wiki/Von%20Neumann%20bicommutant%20theorem
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In mathematics, specifically functional analysis, the von Neumann bicommutant theorem relates the closure of a set of bounded operators on a Hilbert space in certain topologies to the bicommutant of that set. In essence, it is a connection between the algebraic and topological sides of operator theory.
The formal statement of the theorem is as follows:
Von Neumann bicommutant theorem. Let be an algebra consisting of bounded operators on a Hilbert space , containing the identity operator, and closed under taking adjoints. Then the closures of in the weak operator topology and the strong operator topology are equal, and are in turn equal to the bicommutant of .
This algebra is called the von Neumann algebra generated by .
There are several other topologies on the space of bounded operators, and one can ask what are the *-algebras closed in these topologies. If is closed in the norm topology then it is a C*-algebra, but not necessarily a von Neumann algebra. One such example is the C*-algebra of compact operators (on an infinite dimensional Hilbert space). For most other common topologies the closed *-algebras containing 1 are von Neumann algebras; this applies in particular to the weak operator, strong operator, *-strong operator, ultraweak, ultrastrong, and *-ultrastrong topologies.
It is related to the Jacobson density theorem.
Proof
Let be a Hilbert space and the bounded operators on . Consider a self-adjoint unital subalgebra of (this means that contains the adjoints of its members, and the identity operator on ).
The theorem is equivalent to the combination of the following three statements:
(i)
(ii)
(iii)
where the and subscripts stand for closures in the weak and strong operator topologies, respectively.
Proof of (i)
By definition of the weak operator topology, for any and in , the map T → <Tx, y> is continuous in this topology. Therefore, for any operator (and by substituting once and once ), so is the map
Let S be any subset of , and S′ its commutant. For any operator not in S′, <OTx, y> - <TOx, y> is nonzero for some O in S and some x and y in . By the continuity of the abovementioned mapping, there is an open neighborhood of in the weak operator topology for which this is nonzero, therefore this open neighborhood is also not in S′. Thus S′ is closed in the weak operator, i.e. S′ is weakly closed. Thus every commutant is weakly closed, and so is ; since it contains , it also contains its weak closure.
Proof of (ii)
This follows directly from the weak operator topology being coarser than the strong operator topology: for every point in , every open neighborhood of in the weak operator topology is also open in the strong operator topology and therefore contains a member of ; therefore is also a member of .
Proof of (iii)
Fix . We will show .
Fix an open neighborhood of in the strong operator topology. By definition of the strong operator topology, U contains a finite intersection U(h1,ε1) ∩...∩U(hn,
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https://en.wikipedia.org/wiki/Bicommutant
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In algebra, the bicommutant of a subset S of a semigroup (such as an algebra or a group) is the commutant of the commutant of that subset. It is also known as the double commutant or second commutant and is written .
The bicommutant is particularly useful in operator theory, due to the von Neumann double commutant theorem, which relates the algebraic and analytic structures of operator algebras. Specifically, it shows that if M is a unital, self-adjoint operator algebra in the C*-algebra B(H), for some Hilbert space H, then the weak closure, strong closure and bicommutant of M are equal. This tells us that a unital C*-subalgebra M of B(H) is a von Neumann algebra if, and only if, , and that if not, the von Neumann algebra it generates is .
The bicommutant of S always contains S. So . On the other hand, . So , i.e. the commutant of the bicommutant of S is equal to the commutant of S. By induction, we have:
and
for n > 1.
It is clear that, if S1 and S2 are subsets of a semigroup,
If it is assumed that and (this is the case, for instance, for von Neumann algebras), then the above equality gives
See also
von Neumann double commutant theorem
References
J. Dixmier, Von Neumann Algebras, North-Holland, Amsterdam, 1981.
Group theory
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https://en.wikipedia.org/wiki/Operator%20algebra
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In functional analysis, a branch of mathematics, an operator algebra is an algebra of continuous linear operators on a topological vector space, with the multiplication given by the composition of mappings.
The results obtained in the study of operator algebras are often phrased in algebraic terms, while the techniques used are often highly analytic. Although the study of operator algebras is usually classified as a branch of functional analysis, it has direct applications to representation theory, differential geometry, quantum statistical mechanics, quantum information, and quantum field theory.
Overview
Operator algebras can be used to study arbitrary sets of operators with little algebraic relation simultaneously. From this point of view, operator algebras can be regarded as a generalization of spectral theory of a single operator. In general operator algebras are non-commutative rings.
An operator algebra is typically required to be closed in a specified operator topology inside the whole algebra of continuous linear operators. In particular, it is a set of operators with both algebraic and topological closure properties. In some disciplines such properties are axiomized and algebras with certain topological structure become the subject of the research.
Though algebras of operators are studied in various contexts (for example, algebras of pseudo-differential operators acting on spaces of distributions), the term operator algebra is usually used in reference to algebras of bounded operators on a Banach space or, even more specially in reference to algebras of operators on a separable Hilbert space, endowed with the operator norm topology.
In the case of operators on a Hilbert space, the Hermitian adjoint map on operators gives a natural involution, which provides an additional algebraic structure that can be imposed on the algebra. In this context, the best studied examples are self-adjoint operator algebras, meaning that they are closed under taking adjoints. These include C*-algebras, von Neumann algebras, and AW*-algebra. C*-algebras can be easily characterized abstractly by a condition relating the norm, involution and multiplication. Such abstractly defined C*-algebras can be identified to a certain closed subalgebra of the algebra of the continuous linear operators on a suitable Hilbert space. A similar result holds for von Neumann algebras.
Commutative self-adjoint operator algebras can be regarded as the algebra of complex-valued continuous functions on a locally compact space, or that of measurable functions on a standard measurable space. Thus, general operator algebras are often regarded as a noncommutative generalizations of these algebras, or the structure of the base space on which the functions are defined. This point of view is elaborated as the philosophy of noncommutative geometry, which tries to study various non-classical and/or pathological objects by noncommutative operator algebras.
Examples of operator algebra
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https://en.wikipedia.org/wiki/Incomplete%20gamma%20function
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In mathematics, the upper and lower incomplete gamma functions are types of special functions which arise as solutions to various mathematical problems such as certain integrals.
Their respective names stem from their integral definitions, which are defined similarly to the gamma function but with different or "incomplete" integral limits. The gamma function is defined as an integral from zero to infinity. This contrasts with the lower incomplete gamma function, which is defined as an integral from zero to a variable upper limit. Similarly, the upper incomplete gamma function is defined as an integral from a variable lower limit to infinity.
Definition
The upper incomplete gamma function is defined as:
whereas the lower incomplete gamma function is defined as:
In both cases is a complex parameter, such that the real part of is positive.
Properties
By integration by parts we find the recurrence relations
and
Since the ordinary gamma function is defined as
we have
and
Continuation to complex values
The lower incomplete gamma and the upper incomplete gamma function, as defined above for real positive and , can be developed into holomorphic functions, with respect both to and , defined for almost all combinations of complex and . Complex analysis shows how properties of the real incomplete gamma functions extend to their holomorphic counterparts.
Lower incomplete gamma function
Holomorphic extension
Repeated application of the recurrence relation for the lower incomplete gamma function leads to the power series expansion:
Given the rapid growth in absolute value of when , and the fact that the reciprocal of is an entire function, the coefficients in the rightmost sum are well-defined, and locally the sum converges uniformly for all complex and . By a theorem of Weierstraß, the limiting function, sometimes denoted as ,
is entire with respect to both (for fixed ) and (for fixed ), and, thus, holomorphic on by Hartog's theorem. Hence, the following decomposition
extends the real lower incomplete gamma function as a holomorphic function, both jointly and separately in and . It follows from the properties of and the Γ-function, that the first two factors capture the singularities of (at or a non-positive integer), whereas the last factor contributes to its zeros.
Multi-valuedness
The complex logarithm is determined up to a multiple of only, which renders it multi-valued. Functions involving the complex logarithm typically inherit this property. Among these are the complex power, and, since appears in its decomposition, the -function, too.
The indeterminacy of multi-valued functions introduces complications, since it must be stated how to select a value. Strategies to handle this are:
(the most general way) replace the domain of multi-valued functions by a suitable manifold in called Riemann surface. While this removes multi-valuedness, one has to know the theory behind it;
restrict the domain such that a multi-v
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https://en.wikipedia.org/wiki/Gaussian%20period
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In mathematics, in the area of number theory, a Gaussian period is a certain kind of sum of roots of unity. The periods permit explicit calculations in cyclotomic fields connected with Galois theory and with harmonic analysis (discrete Fourier transform). They are basic in the classical theory called cyclotomy. Closely related is the Gauss sum, a type of exponential sum which is a linear combination of periods.
History
As the name suggests, the periods were introduced by Gauss and were the basis for his theory of compass and straightedge construction. For example, the construction of the heptadecagon (a formula that furthered his reputation) depended on the algebra of such periods, of which
is an example involving the seventeenth root of unity
General definition
Given an integer n > 1, let H be any subgroup of the multiplicative group
of invertible residues modulo n, and let
A Gaussian period P is a sum of the primitive n-th roots of unity , where runs through all of the elements in a fixed coset of H in G.
The definition of P can also be stated in terms of the field trace. We have
for some subfield L of Q(ζ) and some j coprime to n. This corresponds to the previous definition by identifying G and H with the Galois groups of Q(ζ)/Q and Q(ζ)/L, respectively. The choice of j determines the choice of coset of H in G in the previous definition.
Example
The situation is simplest when n is a prime number p > 2. In that case G is cyclic of order p − 1, and has one subgroup H of order d for every factor d of p − 1. For example, we can take H of index two. In that case H consists of the quadratic residues modulo p. Corresponding to this H we have the Gaussian period
summed over (p − 1)/2 quadratic residues, and the other period P* summed over the (p − 1)/2 quadratic non-residues. It is easy to see that
since the left-hand side adds all the primitive p-th roots of 1. We also know, from the trace definition, that P lies in a quadratic extension of Q. Therefore, as Gauss knew, P satisfies a quadratic equation with integer coefficients. Evaluating the square of the sum P is connected with the problem of counting how many quadratic residues between 1 and p − 1 are succeeded by quadratic residues. The solution is elementary (as we would now say, it computes a local zeta-function, for a curve that is a conic). One has
(P − P*)2 = p or −p, for p = 4m + 1 or 4m + 3 respectively.
This therefore gives us the precise information about which quadratic field lies in Q(ζ). (That could be derived also by ramification arguments in algebraic number theory; see quadratic field.)
As Gauss eventually showed, to evaluate P − P*, the correct square root to take is the positive (resp. i times positive real) one, in the two cases. Thus the explicit value of the period P is given by
Gauss sums
As is discussed in more detail below, the Gaussian periods are closely related to another class of sums of roots of unity, now generally called Gauss su
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https://en.wikipedia.org/wiki/496%20%28number%29
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496 (four hundred [and] ninety-six) is the natural number following 495 and preceding 497.
In mathematics
496 is most notable for being a perfect number, and one of the earliest numbers to be recognized as such. As a perfect number, it is tied to the Mersenne prime 31, 25 − 1, with 24 (25 − 1) yielding 496. Also related to its being a perfect number, 496 is a harmonic divisor number, since the number of proper divisors of 496 divided by the sum of the reciprocals of its divisors, 1, 2, 4, 8, 16, 31, 62, 124, 248 and 496, (the harmonic mean), yields an integer, 5 in this case.
A triangular number and a hexagonal number, 496 is also a centered nonagonal number. Being the 31st triangular number, 496 is the smallest counterexample to the hypothesis that one more than an even triangular prime-indexed number is a prime number. It is the largest happy number less than 500.
There is no solution to the equation φ(x) = 496, making 496 a nontotient.
E8 has real dimension 496.
In physics
The number 496 is a very important number in superstring theory. In 1984, Michael Green and John H. Schwarz realized that one of the necessary conditions for a superstring theory to make sense is that the dimension of the gauge group of type I string theory must be 496. The group is therefore SO(32). Their discovery started the first superstring revolution. It was realized in 1985 that the heterotic string can admit another possible gauge group, namely E8 x E8.
Telephone numbers
The UK's Ofcom reserves telephone numbers in many dialing areas in the 496 local block for fictional purposes, such as 0114 496-1234.
See also
AD 496
References
Integers
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https://en.wikipedia.org/wiki/Kerr%20metric
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The Kerr metric or Kerr geometry describes the geometry of empty spacetime around a rotating uncharged axially symmetric black hole with a quasispherical event horizon. The Kerr metric is an exact solution of the Einstein field equations of general relativity; these equations are highly non-linear, which makes exact solutions very difficult to find.
Overview
The Kerr metric is a generalization to a rotating body of the Schwarzschild metric, discovered by Karl Schwarzschild in 1915, which described the geometry of spacetime around an uncharged, spherically symmetric, and non-rotating body. The corresponding solution for a charged, spherical, non-rotating body, the Reissner–Nordström metric, was discovered soon afterwards (1916–1918). However, the exact solution for an uncharged, rotating black hole, the Kerr metric, remained unsolved until 1963, when it was discovered by Roy Kerr. The natural extension to a charged, rotating black hole, the Kerr–Newman metric, was discovered shortly thereafter in 1965. These four related solutions may be summarized by the following table, where Q represents the body's electric charge and J represents its spin angular momentum:
{| class="wikitable"
!
! Non-rotating (J = 0)
! Rotating (J ≠ 0)
|-
! Uncharged (Q = 0)
| Schwarzschild
| Kerr
|-
! Charged (Q ≠ 0)
| Reissner–Nordström
| Kerr–Newman
|}
According to the Kerr metric, a rotating body should exhibit frame-dragging (also known as Lense–Thirring precession), a distinctive prediction of general relativity. The first measurement of this frame dragging effect was done in 2011 by the Gravity Probe B experiment. Roughly speaking, this effect predicts that objects coming close to a rotating mass will be entrained to participate in its rotation, not because of any applied force or torque that can be felt, but rather because of the swirling curvature of spacetime itself associated with rotating bodies. In the case of a rotating black hole, at close enough distances, all objects – even light – must rotate with the black hole; the region where this holds is called the ergosphere.
The light from distant sources can travel around the event horizon several times (if close enough); creating multiple images of the same object. To a distant viewer, the apparent perpendicular distance between images decreases at a factor of 2 (about 500). However, fast spinning black holes have less distance between multiplicity images.
Rotating black holes have surfaces where the metric seems to have apparent singularities; the size and shape of these surfaces depends on the black hole's mass and angular momentum. The outer surface encloses the ergosphere and has a shape similar to a flattened sphere. The inner surface marks the event horizon; objects passing into the interior of this horizon can never again communicate with the world outside that horizon. However, neither surface is a true singularity, since their apparent singularity can be eliminated in a different coordinate system
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https://en.wikipedia.org/wiki/Second%20%28disambiguation%29
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A second is the base unit of time in the International System of Units (SI).
Second, Seconds or 2nd may also refer to:
Mathematics
2 (number), as an ordinal (also written as 2nd or 2d)
Second of arc, an angular measurement unit, of a degree
Seconds (angle), units of angular measurement
Music
Notes and intervals
Augmented second, an interval in classical music
Diminished second, unison
Major second, a whole tone
Minor second, semitone
Neutral second one-and-a-half semitones
Albums and EPs
2nd (The Rasmus EP), 1996
Second (Baroness EP), 2005
Second (Raye EP), 2014
The Second, second studio album by rock band Steppenwolf
Seconds (The Dogs D'Amour album), released in 2000
Seconds (Kate Rogers album), released in 2005
Seconds (Tim Berne album), released in 2007
The 2nd (album), a 2006 album by Hater
Songs
"Second" (song), a 2021 song by Hyoyeon
"Second", a 2020 song by Hope D
"Second", a 2019 song by Erika Costell
"Second", a song from Sleaford Mods' 2020 compilation album All That Glue
"Seconds", from The Human League's 1981 album Dare
"Seconds" (song), from U2's 1983 album War
"Seconds", from Le Tigre's 2004 album This Island
Film
The 2nd (film), an American 2020 film starring Ryan Phillippe
Seconds (1966 film), a US thriller directed by John Frankenheimer
Seconds (2014 film), an Indian Malayalam-language thriller film by Aneesh Upasana
"Seconds" (The Batman), an episode in the American animated TV series
"Seconds", an episode of the American TV series Lois & Clark: The New Adventures of Superman
People
Albéric Second, a French journalist and writer
Second-in-command, a deputy commander in British and Commonwealth armies
Science
Second of right ascension, in astronomy
Specific Impulse (rocket engine)
Sports, games, and dueling
Second (climbing), the climber who belays the lead climber in lead climbing
Second dealing, a way of cheating in card games
Second (chess), assistant to a chess player
Second (curling), delivers the second set of stones in curling
Second (duel), the agent of the participant
Second, the cornerman in combative sports such as boxing
Other
Second (parliamentary procedure), to formally support a motion or resolution
Factory second, a new product sold for a discount because of minor imperfections
Second hand or used goods, items that have been used before being resold
Academic degree, second-class degree, divided into upper-second and lower-second, or 2.1 and 2.2, in the British undergraduate degree classification
Educational stage in North American elementary schools
Seconds, an interview magazine published from 1987–2000.
Seconds (comics), a 2014 graphic novel by Bryan Lee O'Malley.
See also
Secondment, a transfer of an employee, usually within an organization
Segundo (disambiguation)
SND (disambiguation)
Secondary (disambiguation)
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https://en.wikipedia.org/wiki/Sulfur%20hexafluoride
|
Sulfur hexafluoride or sulphur hexafluoride (British spelling) is an inorganic compound with the formula SF6. It is a colorless, odorless, non-flammable, and non-toxic gas. has an octahedral geometry, consisting of six fluorine atoms attached to a central sulfur atom. It is a hypervalent molecule.
Typical for a nonpolar gas, is poorly soluble in water but quite soluble in nonpolar organic solvents. It has a density of 6.12 g/L at sea level conditions, considerably higher than the density of air (1.225 g/L). It is generally transported as a liquefied compressed gas.
is 23,500 times more potent than as a greenhouse gas but exists in relatively minor concentrations in the atmosphere. Its concentration in Earth's troposphere reached 10.63 parts per trillion (ppt) in 2021, rising at 0.39 ppt/year. The increase over the prior 40 years was driven in large part by the expanding electric power sector, including fugitive emissions from banks of gas contained in its medium- and high-voltage switchgear. Uses in magnesium, aluminium, and electronics manufacturing also hastened atmospheric growth.
Synthesis and reactions
Sulfur hexafluoride on Earth exists primarily as a man-made industrial gas, but has also been found to occur naturally.
can be prepared from the elements through exposure of to . This was also the method used by the discoverers Henri Moissan and Paul Lebeau in 1901. Some other sulfur fluorides are cogenerated, but these are removed by heating the mixture to disproportionate any (which is highly toxic) and then scrubbing the product with NaOH to destroy remaining .
Alternatively, using bromine, sulfur hexafluoride can be synthesized from SF4 and CoF3 at lower temperatures (e.g. 100 °C), as follows:
There is virtually no reaction chemistry for . A main contribution to the inertness of SF6 is the steric hindrance of the sulfur atom, whereas its heavier group 16 counterparts, such as SeF6 are more reactive than SF6 as a result of less steric hindrance (See hydrolysis example). It does not react with molten sodium below its boiling point, but reacts exothermically with lithium.
Applications
The electrical power industry used about 80% of the sulfur hexafluoride produced in 2000, mostly as a gaseous dielectric medium. Other main uses as of 2015 included a silicon etchant for semiconductor manufacturing, and an inert gas for the casting of magnesium.
Dielectric medium
is used in the electrical industry as a gaseous dielectric medium for high-voltage sulfur hexafluoride circuit breakers, switchgear, and other electrical equipment, often replacing oil-filled circuit breakers (OCBs) that can contain harmful polychlorinated biphenyls (PCBs). gas under pressure is used as an insulator in gas insulated switchgear (GIS) because it has a much higher dielectric strength than air or dry nitrogen. The high dielectric strength is a result of the gas's high electronegativity and density. This property makes it possible to significantly redu
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https://en.wikipedia.org/wiki/Galois%20extension
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In mathematics, a Galois extension is an algebraic field extension E/F that is normal and separable; or equivalently, E/F is algebraic, and the field fixed by the automorphism group Aut(E/F) is precisely the base field F. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory.
A result of Emil Artin allows one to construct Galois extensions as follows: If E is a given field, and G is a finite group of automorphisms of E with fixed field F, then E/F is a Galois extension.
Characterization of Galois extensions
An important theorem of Emil Artin states that for a finite extension each of the following statements is equivalent to the statement that is Galois:
is a normal extension and a separable extension.
is a splitting field of a separable polynomial with coefficients in
that is, the number of automorphisms equals the degree of the extension.
Other equivalent statements are:
Every irreducible polynomial in with at least one root in splits over and is separable.
that is, the number of automorphisms is at least the degree of the extension.
is the fixed field of a subgroup of
is the fixed field of
There is a one-to-one correspondence between subfields of and subgroups of
Examples
There are two basic ways to construct examples of Galois extensions.
Take any field , any finite subgroup of , and let be the fixed field.
Take any field , any separable polynomial in , and let be its splitting field.
Adjoining to the rational number field the square root of 2 gives a Galois extension, while adjoining the cubic root of 2 gives a non-Galois extension. Both these extensions are separable, because they have characteristic zero. The first of them is the splitting field of ; the second has normal closure that includes the complex cubic roots of unity, and so is not a splitting field. In fact, it has no automorphism other than the identity, because it is contained in the real numbers and has just one real root. For more detailed examples, see the page on the fundamental theorem of Galois theory.
An algebraic closure of an arbitrary field is Galois over if and only if is a perfect field.
Notes
Citations
References
Further reading
(Galois' original paper, with extensive background and commentary.)
(Chapter 4 gives an introduction to the field-theoretic approach to Galois theory.)
(This book introduces the reader to the Galois theory of Grothendieck, and some generalisations, leading to Galois groupoids.)
. English translation (of 2nd revised edition): (Later republished in English by Springer under the title "Algebra".)
Galois theory
Algebraic number theory
Field extensions
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https://en.wikipedia.org/wiki/Field%20trace
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In mathematics, the field trace is a particular function defined with respect to a finite field extension L/K, which is a K-linear map from L onto K.
Definition
Let K be a field and L a finite extension (and hence an algebraic extension) of K. L can be viewed as a vector space over K. Multiplication by α, an element of L,
,
is a K-linear transformation of this vector space into itself. The trace, TrL/K(α), is defined as the trace (in the linear algebra sense) of this linear transformation.
For α in L, let σ(α), ..., σ(α) be the roots (counted with multiplicity) of the minimal polynomial of α over K (in some extension field of K). Then
If L/K is separable then each root appears only once (however this does not mean the coefficient above is one; for example if α is the identity element 1 of K then the trace is [L:K ] times 1).
More particularly, if L/K is a Galois extension and α is in L, then the trace of α is the sum of all the Galois conjugates of α, i.e.,
where Gal(L/K) denotes the Galois group of L/K.
Example
Let be a quadratic extension of . Then a basis of is If then the matrix of is:
,
and so, . The minimal polynomial of α is .
Properties of the trace
Several properties of the trace function hold for any finite extension.
The trace is a K-linear map (a K-linear functional), that is
.
If then
Additionally, trace behaves well in towers of fields: if M is a finite extension of L, then the trace from M to K is just the composition of the trace from M to L with the trace from L to K, i.e.
.
Finite fields
Let L = GF(qn) be a finite extension of a finite field K = GF(q). Since L/K is a Galois extension, if α is in L, then the trace of α is the sum of all the Galois conjugates of α, i.e.
In this setting we have the additional properties:
.
For any , there are exactly elements with .
Theorem. For b ∈ L, let Fb be the map Then if . Moreover, the K-linear transformations from L to K are exactly the maps of the form Fb as b varies over the field L.
When K is the prime subfield of L, the trace is called the absolute trace and otherwise it is a relative trace.
Application
A quadratic equation, with a ≠ 0, and coefficients in the finite field has either 0, 1 or 2 roots in GF(q) (and two roots, counted with multiplicity, in the quadratic extension GF(q2)). If the characteristic of GF(q) is odd, the discriminant indicates the number of roots in GF(q) and the classical quadratic formula gives the roots. However, when GF(q) has even characteristic (i.e., for some positive integer h), these formulas are no longer applicable.
Consider the quadratic equation with coefficients in the finite field GF(2h). If b = 0 then this equation has the unique solution in GF(q). If then the substitution converts the quadratic equation to the form:
This equation has two solutions in GF(q) if and only if the absolute trace In this case, if y = s is one of the solutions, then y = s + 1 is the other. Let k be any element of GF(q) wi
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https://en.wikipedia.org/wiki/Pure%20mathematics
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Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, but pure mathematicians are not primarily motivated by such applications. Instead, the appeal is attributed to the intellectual challenge and aesthetic beauty of working out the logical consequences of basic principles.
While pure mathematics has existed as an activity since at least ancient Greece, the concept was elaborated upon around the year 1900, after the introduction of theories with counter-intuitive properties (such as non-Euclidean geometries and Cantor's theory of infinite sets), and the discovery of apparent paradoxes (such as continuous functions that are nowhere differentiable, and Russell's paradox). This introduced the need to renew the concept of mathematical rigor and rewrite all mathematics accordingly, with a systematic use of axiomatic methods. This led many mathematicians to focus on mathematics for its own sake, that is, pure mathematics.
Nevertheless, almost all mathematical theories remained motivated by problems coming from the real world or from less abstract mathematical theories. Also, many mathematical theories, which had seemed to be totally pure mathematics, were eventually used in applied areas, mainly physics and computer science. A famous early example is Isaac Newton's demonstration that his law of universal gravitation implied that planets move in orbits that are conic sections, geometrical curves that had been studied in antiquity by Apollonius. Another example is the problem of factoring large integers, which is the basis of the RSA cryptosystem, widely used to secure internet communications.
It follows that, presently, the distinction between pure and applied mathematics is more a philosophical point of view or a mathematician's preference rather than a rigid subdivision of mathematics. In particular, it is not uncommon that some members of a department of applied mathematics describe themselves as pure mathematicians.
History
Ancient Greece
Ancient Greek mathematicians were among the earliest to make a distinction between pure and applied mathematics. Plato helped to create the gap between "arithmetic", now called number theory, and "logistic", now called arithmetic. Plato regarded logistic (arithmetic) as appropriate for businessmen and men of war who "must learn the art of numbers or [they] will not know how to array [their] troops" and arithmetic (number theory) as appropriate for philosophers "because [they have] to arise out of the sea of change and lay hold of true being." Euclid of Alexandria, when asked by one of his students of what use was the study of geometry, asked his slave to give the student threepence, "since he must make gain of what he learns." The Greek mathematician Apollonius of Perga was asked about the usefulness of some of his theore
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https://en.wikipedia.org/wiki/Block%20matrix
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In mathematics, a block matrix or a partitioned matrix is a matrix that is interpreted as having been broken into sections called blocks or submatrices. Intuitively, a matrix interpreted as a block matrix can be visualized as the original matrix with a collection of horizontal and vertical lines, which break it up, or partition it, into a collection of smaller matrices. Any matrix may be interpreted as a block matrix in one or more ways, with each interpretation defined by how its rows and columns are partitioned.
This notion can be made more precise for an by matrix by partitioning into a collection , and then partitioning into a collection . The original matrix is then considered as the "total" of these groups, in the sense that the entry of the original matrix corresponds in a 1-to-1 way with some offset entry of some , where and .
Block matrix algebra arises in general from biproducts in categories of matrices.
Example
The matrix
can be partitioned into four 2×2 blocks
The partitioned matrix can then be written as
Block matrix multiplication
It is possible to use a block partitioned matrix product that involves only algebra on submatrices of the factors. The partitioning of the factors is not arbitrary, however, and requires "conformable partitions" between two matrices and such that all submatrix products that will be used are defined. Given an matrix with row partitions and column partitions
and a matrix with row partitions and column partitions
that are compatible with the partitions of , the matrix product
can be performed blockwise, yielding as an matrix with row partitions and column partitions. The matrices in the resulting matrix are calculated by multiplying:
Or, using the Einstein notation that implicitly sums over repeated indices:
Block matrix inversion
If a matrix is partitioned into four blocks, it can be inverted blockwise as follows:
where A and D are square blocks of arbitrary size, and B and C are conformable with them for partitioning. Furthermore, A and the Schur complement of A in P: must be invertible.
Equivalently, by permuting the blocks:
Here, D and the Schur complement of D in P: must be invertible.
If A and D are both invertible, then:
By the Weinstein–Aronszajn identity, one of the two matrices in the block-diagonal matrix is invertible exactly when the other is.
Block matrix determinant
The formula for the determinant of a -matrix above continues to hold, under appropriate further assumptions, for a matrix composed of four submatrices . The easiest such formula, which can be proven using either the Leibniz formula or a factorization involving the Schur complement, is
Using this formula, we can derive that characteristic polynomials of and are same and equal to the product of characteristic polynomials of and . Furthermore, If or is diagonalizable, then and are diagonalizable too. The converse is false; simply check .
If is invertible (and similarly if is in
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https://en.wikipedia.org/wiki/Spline%20%28mathematics%29
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In mathematics, a spline is a special function defined piecewise by polynomials.
In interpolating problems, spline interpolation is often preferred to polynomial interpolation because it yields similar results, even when using low degree polynomials, while avoiding Runge's phenomenon for higher degrees.
In the computer science subfields of computer-aided design and computer graphics, the term spline more frequently refers to a piecewise polynomial (parametric) curve. Splines are popular curves in these subfields because of the simplicity of their construction, their ease and accuracy of evaluation, and their capacity to approximate complex shapes through curve fitting and interactive curve design.
The term spline comes from the flexible spline devices used by shipbuilders and draftsmen to draw smooth shapes.
Introduction
The term "spline" is used to refer to a wide class of functions that are used in applications requiring data interpolation and/or smoothing. The data may be either one-dimensional or multi-dimensional. Spline functions for interpolation are normally determined as the minimizers of suitable measures of roughness (for example integral squared curvature) subject to the interpolation constraints. Smoothing splines may be viewed as generalizations of interpolation splines where the functions are determined to minimize a weighted combination of the average squared approximation error over observed data and the roughness measure. For a number of meaningful definitions of the roughness measure, the spline functions are found to be finite dimensional in nature, which is the primary reason for their utility in computations and representation. For the rest of this section, we focus entirely on one-dimensional, polynomial splines and use the term "spline" in this restricted sense.
Definition
We begin by limiting our discussion to polynomials in one variable. In this case, a spline is a piecewise polynomial function.
This function, call it S, takes values from an interval [a,b] and maps them to , the set of real numbers,
We want S to be piecewise defined. To accomplish this, let the interval [a,b] be covered by k ordered, disjoint subintervals,
On each of these k "pieces" of [a,b], we want to define a polynomial, call it Pi.
.
On the ith subinterval of [a,b], S is defined by Pi,
The given k+1 points ti are called knots. The vector
is called a knot vector for the spline.
If the knots are equidistantly distributed in the interval [a,b] we say the spline is uniform, otherwise we say it is non-uniform.
If the polynomial pieces Pi each have degree at most n, then the spline is said to be of degree (or of
order n+1).
If in a neighborhood of ti, then the spline is said to be
of smoothness (at least) at ti. That is,
at ti the two polynomial pieces Pi-1 and Pi share common
derivative values from the derivative of order 0 (the function value)
up through the derivative of order ri (in other words, the two adjacent polynomial pieces c
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https://en.wikipedia.org/wiki/153%20%28number%29
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153 (one hundred [and] fifty-three) is the natural number following 152 and preceding 154.
In mathematics
The number 153 is associated with the geometric shape known as the Vesica Piscis or Mandorla. Archimedes, in his Measurement of a Circle, referred to this ratio (153/265), as constituting the "measure of the fish", this ratio being an imperfect representation of .
As a triangular number, 153 is the sum of the first 17 integers, and is also the sum of the first five positive factorials:.
The number 153 is also a hexagonal number, and a truncated triangle number, meaning that 1, 15, and 153 are all triangle numbers.
The distinct prime factors of 153 add up to 20, and so do the ones of 154, hence the two form a Ruth-Aaron pair.
Since , it is a 3-narcissistic number, and it is also the smallest three-digit number which can be expressed as the sum of cubes of its digits. Only five other numbers can be expressed as the sum of the cubes of their digits: 0, 1, 370, 371 and 407. It is also a Friedman number, since 153 = 3 × 51.
The Biggs–Smith graph is a symmetric graph with 153 edges, all equivalent.
Another feature of the number 153 is that it is the limit of the following algorithm:
Take a random positive integer, divisible by three
Split that number into its base 10 digits
Take the sum of their cubes
Go back to the second step
An example, starting with the number 84:
There are 153 uniform polypeta that are generated from four different fundamental Coxeter groups in six-dimensional space.
In the Bible
The Gospel of John (chapter 21:1–14) includes the narrative of the miraculous catch of 153 fish as the third appearance of Jesus after his resurrection.
The precision of the number of fish in this narrative has long been considered peculiar, and many scholars have argued that 153 has some deeper significance. Jerome, for example, wrote that Oppian's Halieutica listed 153 species of fish, although this could not have been the intended meaning of the Gospel writer because Oppian composed Halieutica after the Gospel text was written, and at any rate never gave a list of fish species that clearly adds up to 153.
The number is clearly an intentional detail, given the lack of precision and detail elsewhere in the story; and theologians have lent much credence to Augustine's numerology simply because it comes from historic rather than contemporary theology.
Jerome reached much the same conclusion as Augustine that the figure is an allegorical representation of totality, but through more straightforward means rather than through numerology.
In his Commentary on Ezekiel he propounded the hypothesis that 153 was meant to represent the whole universe of fish, citing as proof that contemporary poets, giving Oppian as an example, believed that there were 153 species of fish in the world.
However, Robert M. Grant disproved Jerome's hypothesis by noting that Oppian actually enumerated only 149 (as catalogued by Alexander William Mair) fish specie
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https://en.wikipedia.org/wiki/Rotation%20%28disambiguation%29
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Rotation is a circular motion of a body about a center.
Rotation may also refer to:
Science, mathematics and computing
Rotation (anatomy)
Rotation (mathematics)
Rotation (medicine), medical student training
Rotation (physics), ratio between a given angle and a full turn of 2π radians
Bitwise rotation, a mathematical operator on bit patterns
Curl (mathematics), a vector operator
Differential rotation, objects rotating at different speeds
Display rotation, of a computer monitor or display
Earth's rotation
Improper rotation or rotoreflection, a rotation and reflection in one
Internal rotation, a term in anatomy
Optical rotation, rotation acting on polarized light
Rotation around a fixed axis
Rotational spectroscopy, a spectroscopy technique
Tree rotation, a well-known method used in order to make a tree balanced.
Arts, entertainment, and media
Music
Rotation (Cute Is What We Aim For album), 2008
Rotation (Joe McPhee album), 1977
Rotation (music), the repeated airing of a limited playlist of songs on a radio station
"Rotate" (song), a song on the album Channel 10 by Capone-N-Noreaga
"Rotation", a song on the 1979 album Rise by Herb Alpert
"Rotation (LOTUS-2)", a song on the 2000 album Philosopher's Propeller by Susumu Hirasawa
Other uses in arts, entertainment, and media
Rotation (film), a 1949 East German film
Rotation (pool), a type of pocket billiards game
Politics
Rotation government, the practice of a government switching Prime Ministers mid-term from an individual in one political party to a different individual in another political party
Other uses
Rotation (aviation), the act of lifting the nose off the runway during takeoff
Rotation, in baseball pitching; see the glossary of baseball
Crop rotation, a farming practice
Job rotation, a business management technique
Robson Rotation, a method of having ballot papers in elections
Stock rotation, a retail practice
See also
Rotator (disambiguation)
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https://en.wikipedia.org/wiki/Conjugate%20element%20%28field%20theory%29
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In mathematics, in particular field theory, the conjugate elements or algebraic conjugates of an algebraic element , over a field extension , are the roots of the minimal polynomial of over . Conjugate elements are commonly called conjugates in contexts where this is not ambiguous. Normally itself is included in the set of conjugates of .
Equivalently, the conjugates of are the images of under the field automorphisms of that leave fixed the elements of . The equivalence of the two definitions is one of the starting points of Galois theory.
The concept generalizes the complex conjugation, since the algebraic conjugates over of a complex number are the number itself and its complex conjugate.
Example
The cube roots of the number one are:
The latter two roots are conjugate elements in with minimal polynomial
Properties
If K is given inside an algebraically closed field C, then the conjugates can be taken inside C. If no such C is specified, one can take the conjugates in some relatively small field L. The smallest possible choice for L is to take a splitting field over K of pK,α, containing α. If L is any normal extension of K containing α, then by definition it already contains such a splitting field.
Given then a normal extension L of K, with automorphism group Aut(L/K) = G, and containing α, any element g(α) for g in G will be a conjugate of α, since the automorphism g sends roots of p to roots of p. Conversely any conjugate β of α is of this form: in other words, G acts transitively on the conjugates. This follows as K(α) is K-isomorphic to K(β) by irreducibility of the minimal polynomial, and any isomorphism of fields F and F that maps polynomial p to p can be extended to an isomorphism of the splitting fields of p over F and p over F, respectively.
In summary, the conjugate elements of α are found, in any normal extension L of K that contains K(α), as the set of elements g(α) for g in Aut(L/K). The number of repeats in that list of each element is the separable degree [L:K(α)]sep.
A theorem of Kronecker states that if α is a nonzero algebraic integer such that α and all of its conjugates in the complex numbers have absolute value at most 1, then α is a root of unity. There are quantitative forms of this, stating more precisely bounds (depending on degree) on the largest absolute value of a conjugate that imply that an algebraic integer is a root of unity.
References
David S. Dummit, Richard M. Foote, Abstract algebra, 3rd ed., Wiley, 2004.
External links
Field (mathematics)
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https://en.wikipedia.org/wiki/137%20%28number%29
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137 (one hundred [and] thirty-seven) is the natural number following 136 and preceding 138.
Mathematics
the 33rd prime number; the next is 139, with which it comprises a twin prime, and thus 137 is a Chen prime.
an Eisenstein prime with no imaginary part and a real part of the form .
the fourth Stern prime.
a Pythagorean prime: a prime number of the form , where () or the sum of two squares .
a strong prime in the sense that it is more than the arithmetic mean of its two neighboring primes.
a strictly non-palindromic number and a primeval number.
a factor of 10001 (the other being 73) and the repdigit 11111111 (= 10001 × 1111).
using two radii to divide a circle according to the golden ratio yields sectors of approximately 137.51° (the golden angle) and 222° in degree system so 137 is the largest integer before it.
In decimal notation, 1/137 = 0.007299270072992700..., so its period value happens to be palindromic and has a period length of only 8. However, this is only special to decimal, as in pentadecimal it (1/92) has a period length of twenty-four (24) and the period value is not at all palindromic.
Physics
Since the early 1900s, physicists have postulated that the number could lie at the heart of a grand unified theory, relating theories of electromagnetism, quantum mechanics and, especially, gravity.
1/137 was once believed to be the exact value of the fine-structure constant. The fine-structure constant, a dimensionless physical constant, is approximately 1/137, and the astronomer Arthur Eddington conjectured in 1929 that its reciprocal was in fact precisely the integer 137, which he claimed could be "obtained by pure deduction". This conjecture was not widely adopted, and by the 1940s, the experimental values for the constant were clearly inconsistent with the conjecture, being roughly 1/137.036. Recent work at the Kastler Brossel Laboratory in Paris reported the most precise measurement yet taking the value of this constant to the 11th decimal place, nearly three times more precise than the 2018 results by a group led by Holger Müller at University of California, Berkeley, with a margin of error of just 81 parts per trillion.
Physicist Leon M. Lederman numbered his home near Fermilab 137 based on the significance of the number to those in his profession. Lederman expounded on the significance of the number in his 1993 book The God Particle: If the Universe Is the Answer, What Is the Question?, noting that not only was it the inverse of the fine-structure constant, but was also related to the probability that an electron will emit or absorb a photon—i.e., Feynman's conjecture. He added that it also "contains the crux of electromagnetism (the electron), relativity (the velocity of light), and quantum theory (Planck's constant). It would be less unsettling if the relationship between all these important concepts turned out to be one or three or maybe a multiple of pi. But 137?" The number 137, according to Lederman, "sh
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https://en.wikipedia.org/wiki/1734%20in%20science
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The year 1734 in science and technology involved some significant events.
Mathematics
George Berkeley publishes The Analyst, an empiricist critique of the foundations of infinitesimal calculus, influential in the development of mathematics.
Leonhard Euler introduces the integrating factor technique for solving first-order ordinary differential equations.
Technology
James Short constructs a Gregorian reflecting telescope with an aperture of .
Zoology
René Antoine Ferchault de Réaumur begins publication of Mémoires pour servir à l'histoire des insectes in Amsterdam.
Awards
Copley Medal: John Theophilus Desaguliers
Births
January 23 – Wolfgang von Kempelen, Hungarian inventor (died 1804)
April 18 – Elsa Beata Bunge, Swedish botanist (died 1819)
May 23 – Franz Mesmer, German physician (died 1815)
September 3 – Joseph Wright, English painter of scientific subjects (died 1797)
Deaths
February 1 – John Floyer, English physician (born 1649)
April 25 – Johann Konrad Dippel, German theologian, alchemist and physician (born 1673)
References
18th century in science
1730s in science
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https://en.wikipedia.org/wiki/1730%20in%20science
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The year 1730 in science and technology involved some significant events.
Astronomy
The analemma is developed by the French astronomer Grandjean de Fouchy.
Mathematics
James Stirling publishes Methodus differentialis, sive tractatus de summatione et interpolatione serierum infinitarum.
Physics
The Reaumur scale is developed by French naturalist René Antoine Ferchault de Réaumur, with 0° = the freezing point of water and 80° = the boiling point.
Technology
Joseph Foljambe of Rotherham, England, produces the iron-clad Rotherham swing plough.
Births
April 15 – Moses Harris, English entomologist and engraver (died c. 1788)
July 12 – Anna Barbara Reinhart, Swiss mathematician (died 1796)
June 26 – Charles Messier, French astronomer (died 1817)
August 12 – Edmé-Louis Daubenton, French naturalist (died 1785)
December 8
Johann Hedwig, Transylvanian-born German botanist (died 1799)
Jan Ingenhousz, Dutch physiologist (died 1799)
Maria Angela Ardinghelli, Italian scientific translator (died 1825)
between 1730 and 1732 – William Hudson, English botanist (died 1793)
Deaths
January 18 – Antonio Vallisneri, Italian physician and natural scientist (born 1661)
April 21 - Jan Palfijn, Flemish surgeon and obstetrician (born 1650)
December 5 (bur.) – Alida Withoos, Dutch botanical artist (born c. 1661/1662)
References
18th century in science
1730s in science
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https://en.wikipedia.org/wiki/Weak%20operator%20topology
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In functional analysis, the weak operator topology, often abbreviated WOT, is the weakest topology on the set of bounded operators on a Hilbert space , such that the functional sending an operator to the complex number is continuous for any vectors and in the Hilbert space.
Explicitly, for an operator there is base of neighborhoods of the following type: choose a finite number of vectors , continuous functionals , and positive real constants indexed by the same finite set . An operator lies in the neighborhood if and only if for all .
Equivalently, a net of bounded operators converges to in WOT if for all and , the net converges to .
Relationship with other topologies on B(H)
The WOT is the weakest among all common topologies on , the bounded operators on a Hilbert space .
Strong operator topology
The strong operator topology, or SOT, on is the topology of pointwise convergence. Because the inner product is a continuous function, the SOT is stronger than WOT. The following example shows that this inclusion is strict. Let and consider the sequence of right shifts. An application of Cauchy-Schwarz shows that in WOT. But clearly does not converge to in SOT.
The linear functionals on the set of bounded operators on a Hilbert space that are continuous in the strong operator topology are precisely those that are continuous in the WOT (actually, the WOT is the weakest operator topology that leaves continuous all strongly continuous linear functionals on the set of bounded operators on the Hilbert space H). Because of this fact, the closure of a convex set of operators in the WOT is the same as the closure of that set in the SOT.
It follows from the polarization identity that a net converges to in SOT if and only if in WOT.
Weak-star operator topology
The predual of B(H) is the trace class operators C1(H), and it generates the w*-topology on B(H), called the weak-star operator topology or σ-weak topology. The weak-operator and σ-weak topologies agree on norm-bounded sets in B(H).
A net {Tα} ⊂ B(H) converges to T in WOT if and only Tr(TαF) converges to Tr(TF) for all finite-rank operator F. Since every finite-rank operator is trace-class, this implies that WOT is weaker than the σ-weak topology. To see why the claim is true, recall that every finite-rank operator F is a finite sum
So {Tα} converges to T in WOT means
Extending slightly, one can say that the weak-operator and σ-weak topologies agree on norm-bounded sets in B(H): Every trace-class operator is of the form
where the series converges. Suppose and in WOT. For every trace-class S,
by invoking, for instance, the dominated convergence theorem.
Therefore every norm-bounded set is compact in WOT, by the Banach–Alaoglu theorem.
Other properties
The adjoint operation T → T*, as an immediate consequence of its definition, is continuous in WOT.
Multiplication is not jointly continuous in WOT: again let be the unilateral shift. Appealing to Cauchy-Schwarz, one
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https://en.wikipedia.org/wiki/Strong%20operator%20topology
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In functional analysis, a branch of mathematics, the strong operator topology, often abbreviated SOT, is the locally convex topology on the set of bounded operators on a Hilbert space H induced by the seminorms of the form , as x varies in H.
Equivalently, it is the coarsest topology such that, for each fixed x in H, the evaluation map (taking values in H) is continuous in T. The equivalence of these two definitions can be seen by observing that a subbase for both topologies is given by the sets (where T0 is any bounded operator on H, x is any vector and ε is any positive real number).
In concrete terms, this means that in the strong operator topology if and only if for each x in H.
The SOT is stronger than the weak operator topology and weaker than the norm topology.
The SOT lacks some of the nicer properties that the weak operator topology has, but being stronger, things are sometimes easier to prove in this topology. It can be viewed as more natural, too, since it is simply the topology of pointwise convergence.
The SOT topology also provides the framework for the measurable functional calculus, just as the norm topology does for the continuous functional calculus.
The linear functionals on the set of bounded operators on a Hilbert space that are continuous in the SOT are precisely those continuous in the weak operator topology (WOT). Because of this, the closure of a convex set of operators in the WOT is the same as the closure of that set in the SOT.
This language translates into convergence properties of Hilbert space operators. For a complex Hilbert space, it is easy to verify by the polarization identity, that Strong Operator convergence implies Weak Operator convergence.
See also
Strongly continuous semigroup
Topologies on the set of operators on a Hilbert space
References
Banach spaces
Topology of function spaces
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https://en.wikipedia.org/wiki/Predual
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In mathematics, the predual of an object D is an object P whose dual space is D.
For example, the predual of the space of bounded operators is the space of trace class operators, and the predual of the space L∞(R) of essentially bounded functions on R is the Banach space L1(R) of integrable functions.
Abstract algebra
Functional analysis
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https://en.wikipedia.org/wiki/Kimotsuki%20District%2C%20Kagoshima
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is a district located in Kagoshima Prefecture, Japan.
As of the January 1, 2006 merger but with 2003 population statistics, the district has an estimated population of 46,943 and a density of 65.9 persons per km2. The total area is 712.55 km2.
Towns and villages
Higashikushira
Kimotsuki
Kinkō
Minamiōsumi
Mergers
On March 22, 2005 the towns of Ōnejime and Tashiro merged into the town of Kinkō.
On March 31, 2005 the towns of Nejime and Sata merged into the town of Minamiōsumi.
On July 1, 2005 the towns of Kōyama and Uchinoura merged into the town of Kimotsuki.
On January 1, 2006 the towns of Aira and Kushira, and the town of Kihoku, from Soo District, merged into the expanded city of Kanoya.
Districts in Kagoshima Prefecture
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https://en.wikipedia.org/wiki/%C5%8Cshima%20District%2C%20Kagoshima
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is a district located in Kagoshima Prefecture, Japan.
As of the March 20, 2006 merger but with 2003 population statistics, the district has an estimated population of 78,882 and a density of 84.4 persons per km2. The total area is 934.10 km2.
Towns and villages
Amagi
China
Isen
Kikai
Setouchi
Tatsugō
Tokunoshima
Wadomari
Yoron
Uken
Yamato
District timeline (after WWII)
February 28, 1946 – The district fell under United States Army control except for the current village of Mishima areas.
July 1, 1946 – The town of Naze gained city status.
September 1, 1946 – The village of China gained town status.
February 4, 1952 – Japan regains the current village of Toshima areas.
February 10, 1952 – The village of Mishima broke off from the village of Toshima.
December 25, 1953 – The remaining parts of the district returned to Japan.
February 1, 1955 – The village of Mikata merged into the city of Naze.
September 1, 1956 – The town of Koniya, and the villages of Chinzei, Saneku, and Nishikata merged to form the town of Setouchi.
September 10, 1956 – The town of Kikai and the village of Sōmachi merged to form the town of Kikai.
April 1, 1958 – The town of Kametsu and the village of Higashiamagi merged to form the town of Tokunoshima.
January 1, 1961
The village of Kasari gained town status.
The village of Amagi gained town status.
January 1, 1962 – The village of Isen gained town status.
January 1, 1963 – The village of Yoron gained town status.
April 1, 1973 – The district transferred the villages of Mishima and Toshima to Kagoshima District.
February 10, 1975 – The village of Tatsugō gained town status.
March 20, 2006 – The village of Sumiyō and town of Kasari merged with the city of Naze to form the new city of Amami.
Transportation
Kikai Airport is located in the district.
Amami Reversion Movement
The "restoration of Ōshima District of Kagoshima Prefecture" was a slogan of the Amami reversion movement during the United States military occupation of the Amami Islands from 1945/6 to 1953. An overwhelming majority of people of Amami, including those in mainland Japan, urged the immediate return of the islands to Japan. The reversion movements except those by leftist minorities tried to differentiate Amami from Okinawa because the U.S. seemingly intended permanent control of Okinawa. They opposed the name "Northern Ryukyu" occasionally labeled by the U.S. occupiers. Instead they used "Ōshima District, Kagoshima Prefecture" as a symbol of national belongingness.
References
Districts in Kagoshima Prefecture
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https://en.wikipedia.org/wiki/Ternary%20operation
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In mathematics, a ternary operation is an n-ary operation with n = 3. A ternary operation on a set A takes any given three elements of A and combines them to form a single element of A.
In computer science, a ternary operator is an operator that takes three arguments as input and returns one output.
Examples
The function is an example of a ternary operation on the integers (or on any structure where and are both defined). Properties of this ternary operation have been used to define planar ternary rings in the foundations of projective geometry.
In the Euclidean plane with points a, b, c referred to an origin, the ternary operation has been used to define free vectors. Since (abc) = d implies a – b = c – d, these directed segments are equipollent and are associated with the same free vector. Any three points in the plane a, b, c thus determine a parallelogram with d at the fourth vertex.
In projective geometry, the process of finding a projective harmonic conjugate is a ternary operation on three points. In the diagram, points A, B and P determine point V, the harmonic conjugate of P with respect to A and B. Point R and the line through P can be selected arbitrarily, determining C and D. Drawing AC and BD produces the intersection Q, and RQ then yields V.
Suppose A and B are given sets and is the collection of binary relations between A and B. Composition of relations is always defined when A = B, but otherwise a ternary composition can be defined by where is the converse relation of q. Properties of this ternary relation have been used to set the axioms for a heap.
In Boolean algebra, defines the formula .
Computer science
In computer science, a ternary operator is an operator that takes three arguments (or operands). The arguments and result can be of different types. Many programming languages that use C-like syntax feature a ternary operator, ?:, which defines a conditional expression. In some languages, this operator is referred to as the conditional operator.
In Python, the ternary conditional operator reads x if C else y. Python also supports ternary operations called array slicing, e.g. a[b:c] return an array where the first element is a[b] and last element is a[c-1]. OCaml expressions provide ternary operations against records, arrays, and strings: a.[b]<-c would mean the string a where index b has value c.
The multiply–accumulate operation is another ternary operator.
Another example of a ternary operator is between, as used in SQL.
The Icon programming language has a "to-by" ternary operator: the expression 1 to 10 by 2 generates the odd integers from 1 through 9.
In Excel formulae, the form is =if(C, x, y).
See also
Median algebra or Majority function
Ternary conditional operator for a list of ternary operators in computer programming languages
Ternary Exclusive or
Ternary equivalence relation
References
External links
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https://en.wikipedia.org/wiki/Miyazaki%20District%2C%20Miyazaki
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was a district located in Miyazaki Prefecture, Japan.
As of 2003 population statistics (but following the January 1, 2006 merger of the towns of Sadowara and Tano), the district had an estimated population of 28,937 and the density of 605.25 persons per km2. The total area was 47.81 km2.
Former towns and villages
Kiyotake
Sadowara
Tano
Mergers
On January 1, 2006 - the towns of Sadowara and Tano, along with the town of Takaoka (from Higashimorokata District), were merged into the expanded city of Miyazaki.
On March 23, 2010 - the town of Kiyotake was merged into the expanded city of Miyazaki. Miyazaki District was dissolved as a result of this merger.
References
Former districts of Miyazaki Prefecture
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https://en.wikipedia.org/wiki/Kitamorokata%20District%2C%20Miyazaki
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is a district located in Miyazaki Prefecture, Japan.
As of the Miyakonojō merger (but with the population statistics as of October 2020), the district has an estimated population of 25,591 and a density of 232.6 persons per km2. The total area is 110.0 km2.
Towns and villages
Mimata
Mergers
On January 1, 2006 the towns of Takajō, Takazaki, Yamada, and Yamanokuchi merged into the expanded city of Miyakonojō.
References
Districts in Miyazaki Prefecture
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https://en.wikipedia.org/wiki/Ramanujan%E2%80%93Soldner%20constant
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In mathematics, the Ramanujan–Soldner constant (also called the Soldner constant) is a mathematical constant defined as the unique positive zero of the logarithmic integral function. It is named after Srinivasa Ramanujan and Johann Georg von Soldner.
Its value is approximately μ ≈ 1.45136923488338105028396848589202744949303228…
Since the logarithmic integral is defined by
then using we have
thus easing calculation for numbers greater than μ. Also, since the exponential integral function satisfies the equation
the only positive zero of the exponential integral occurs at the natural logarithm of the Ramanujan–Soldner constant, whose value is approximately ln(μ) ≈ 0.372507410781366634461991866…
External links
Mathematical constants
Srinivasa Ramanujan
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https://en.wikipedia.org/wiki/Dirichlet%27s%20unit%20theorem
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In mathematics, Dirichlet's unit theorem is a basic result in algebraic number theory due to Peter Gustav Lejeune Dirichlet. It determines the rank of the group of units in the ring of algebraic integers of a number field . The regulator is a positive real number that determines how "dense" the units are.
The statement is that the group of units is finitely generated and has rank (maximal number of multiplicatively independent elements) equal to
where is the number of real embeddings and the number of conjugate pairs of complex embeddings of . This characterisation of and is based on the idea that there will be as many ways to embed in the complex number field as the degree ; these will either be into the real numbers, or pairs of embeddings related by complex conjugation, so that
Note that if is Galois over then either or .
Other ways of determining and are
use the primitive element theorem to write , and then is the number of conjugates of that are real, the number that are complex; in other words, if is the minimal polynomial of over , then is the number of real roots and is the number of non-real complex roots of (which come in complex conjugate pairs);
write the tensor product of fields as a product of fields, there being copies of and copies of .
As an example, if is a quadratic field, the rank is 1 if it is a real quadratic field, and 0 if an imaginary quadratic field. The theory for real quadratic fields is essentially the theory of Pell's equation.
The rank is positive for all number fields besides and imaginary quadratic fields, which have rank 0. The 'size' of the units is measured in general by a determinant called the regulator. In principle a basis for the units can be effectively computed; in practice the calculations are quite involved when is large.
The torsion in the group of units is the set of all roots of unity of , which form a finite cyclic group. For a number field with at least one real embedding the torsion must therefore be only . There are number fields, for example most imaginary quadratic fields, having no real embeddings which also have for the torsion of its unit group.
Totally real fields are special with respect to units. If is a finite extension of number fields with degree greater than 1 and
the units groups for the integers of and have the same rank then is totally real and is a totally complex quadratic extension. The converse holds too. (An example is equal to the rationals and equal to an imaginary quadratic field; both have unit rank 0.)
The theorem not only applies to the maximal order but to any order {{math|O ⊂ OK}}.
There is a generalisation of the unit theorem by Helmut Hasse (and later Claude Chevalley) to describe the structure of the group of -units, determining the rank of the unit group in localizations of rings of integers. Also, the Galois module structure of has been determined.
The regulator
Suppose that K is a number field and are a set of
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https://en.wikipedia.org/wiki/Nest%20algebra
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In functional analysis, a branch of mathematics, nest algebras are a class of operator algebras that generalise the upper-triangular matrix algebras to a Hilbert space context. They were introduced by and have many interesting properties. They are non-selfadjoint algebras, are closed in the weak operator topology and are reflexive.
Nest algebras are among the simplest examples of commutative subspace lattice algebras. Indeed, they are formally defined as the algebra of bounded operators leaving invariant each subspace contained in a subspace nest, that is, a set of subspaces which is totally ordered by inclusion and is also a complete lattice. Since the orthogonal projections corresponding to the subspaces in a nest commute, nests are commutative subspace lattices.
By way of an example, let us apply this definition to recover the finite-dimensional upper-triangular matrices. Let us work in the -dimensional complex vector space , and let be the standard basis. For , let be the -dimensional subspace of spanned by the first basis vectors . Let
then N is a subspace nest, and the corresponding nest algebra of n × n complex matrices M leaving each subspace in N invariant that is, satisfying for each S in N – is precisely the set of upper-triangular matrices.
If we omit one or more of the subspaces Sj from N then the corresponding nest algebra consists of block upper-triangular matrices.
Properties
Nest algebras are hyperreflexive with distance constant 1.
See also
flag manifold
References
Operator theory
Operator algebras
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https://en.wikipedia.org/wiki/Reflexive%20operator%20algebra
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In functional analysis, a reflexive operator algebra A is an operator algebra that has enough invariant subspaces to characterize it. Formally, A is reflexive if it is equal to the algebra of bounded operators which leave invariant each subspace left invariant by every operator in A.
This should not be confused with a reflexive space.
Examples
Nest algebras are examples of reflexive operator algebras. In finite dimensions, these are simply algebras of all matrices of a given size whose nonzero entries lie in an upper-triangular pattern.
In fact if we fix any pattern of entries in an n by n matrix containing the diagonal, then the set of all n by n matrices whose nonzero entries lie in this pattern forms a reflexive algebra.
An example of an algebra which is not reflexive is the set of 2 × 2 matrices
This algebra is smaller than the Nest algebra
but has the same invariant subspaces, so it is not reflexive.
If T is a fixed n by n matrix then the set of all polynomials in T and the identity operator forms a unital operator algebra. A theorem of Deddens and Fillmore states that this algebra is reflexive if and only if the largest two blocks in the Jordan normal form of T differ in size by at most one. For example, the algebra
which is equal to the set of all polynomials in
and the identity is reflexive.
Hyper-reflexivity
Let be a weak*-closed operator algebra contained in B(H), the set of all bounded operators on a Hilbert space H and for T any operator in B(H), let
Observe that P is a projection involved in this supremum precisely if the range of P is an invariant subspace of .
The algebra is reflexive if and only if for every T in B(H):
We note that for any T in B(H) the following inequality is satisfied:
Here is the distance of T from the algebra, namely the smallest norm of an operator T-A where A runs over the algebra. We call hyperreflexive if there is a constant K such that for every operator T in B(H),
The smallest such K is called the distance constant for . A hyper-reflexive operator algebra is automatically reflexive.
In the case of a reflexive algebra of matrices with nonzero entries specified by a given pattern, the problem of finding the distance constant can be rephrased as a matrix-filling problem: if we fill the entries in the complement of the pattern with arbitrary entries, what choice of entries in the pattern gives the smallest operator norm?
Examples
Every finite-dimensional reflexive algebra is hyper-reflexive. However, there are examples of infinite-dimensional reflexive operator algebras which are not hyper-reflexive.
The distance constant for a one-dimensional algebra is 1.
Nest algebras are hyper-reflexive with distance constant 1.
Many von Neumann algebras are hyper-reflexive, but it is not known if they all are.
A type I von Neumann algebra is hyper-reflexive with distance constant at most 2.
See also
Invariant subspace
subspace lattice
reflexive subspace lattice
nest algebra
References
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https://en.wikipedia.org/wiki/Ring%20of%20integers
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In mathematics, the ring of integers of an algebraic number field is the ring of all algebraic integers contained in . An algebraic integer is a root of a monic polynomial with integer coefficients: . This ring is often denoted by or . Since any integer belongs to and is an integral element of , the ring is always a subring of .
The ring of integers is the simplest possible ring of integers. Namely, where is the field of rational numbers. And indeed, in algebraic number theory the elements of are often called the "rational integers" because of this.
The next simplest example is the ring of Gaussian integers , consisting of complex numbers whose real and imaginary parts are integers. It is the ring of integers in the number field of Gaussian rationals, consisting of complex numbers whose real and imaginary parts are rational numbers. Like the rational integers, is a Euclidean domain.
The ring of integers of an algebraic number field is the unique maximal order in the field. It is always a Dedekind domain.
Properties
The ring of integers is a finitely-generated -module. Indeed, it is a free -module, and thus has an integral basis, that is a basis of the -vector space such that each element in can be uniquely represented as
with . The rank of as a free -module is equal to the degree of over .
Examples
Computational tool
A useful tool for computing the integral closure of the ring of integers in an algebraic field is the discriminant. If is of degree over , and form a basis of over , set . Then, is a submodule of the spanned by . pg. 33 In fact, if is square-free, then forms an integral basis for . pg. 35
Cyclotomic extensions
If is a prime, is a th root of unity and is the corresponding cyclotomic field, then an integral basis of is given by .
Quadratic extensions
If is a square-free integer and is the corresponding quadratic field, then is a ring of quadratic integers and its integral basis is given by if and by if . This can be found by computing the minimal polynomial of an arbitrary element where .
Multiplicative structure
In a ring of integers, every element has a factorization into irreducible elements, but the ring need not have the property of unique factorization: for example, in the ring of integers , the element 6 has two essentially different factorizations into irreducibles:
A ring of integers is always a Dedekind domain, and so has unique factorization of ideals into prime ideals.
The units of a ring of integers is a finitely generated abelian group by Dirichlet's unit theorem. The torsion subgroup consists of the roots of unity of . A set of torsion-free generators is called a set of fundamental units.
Generalization
One defines the ring of integers of a non-archimedean local field as the set of all elements of with absolute value ; this is a ring because of the strong triangle inequality. If is the completion of an algebraic number field, its ring of integers is the co
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https://en.wikipedia.org/wiki/Invertible%20%28disambiguation%29
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Invertible may refer to
Mathematics
Invertible element
Invertible function
Invertible ideal
Invertible knot
Invertible jet
Invertible matrix
Invertible module
Invertible sheaf
Others
Invertible counterpoint
See also
Inverse (disambiguation)
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https://en.wikipedia.org/wiki/Ideal%20norm
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In commutative algebra, the norm of an ideal is a generalization of a norm of an element in the field extension. It is particularly important in number theory since it measures the size of an ideal of a complicated number ring in terms of an ideal in a less complicated ring. When the less complicated number ring is taken to be the ring of integers, Z, then the norm of a nonzero ideal I of a number ring R is simply the size of the finite quotient ring R/I.
Relative norm
Let A be a Dedekind domain with field of fractions K and integral closure of B in a finite separable extension L of K. (this implies that B is also a Dedekind domain.) Let and be the ideal groups of A and B, respectively (i.e., the sets of nonzero fractional ideals.) Following the technique developed by Jean-Pierre Serre, the norm map
is the unique group homomorphism that satisfies
for all nonzero prime ideals of B, where is the prime ideal of A lying below .
Alternatively, for any one can equivalently define to be the fractional ideal of A generated by the set of field norms of elements of B.
For , one has , where .
The ideal norm of a principal ideal is thus compatible with the field norm of an element:
Let be a Galois extension of number fields with rings of integers .
Then the preceding applies with , and for any we have
which is an element of .
The notation is sometimes shortened to , an abuse of notation that is compatible with also writing for the field norm, as noted above.
In the case , it is reasonable to use positive rational numbers as the range for since has trivial ideal class group and unit group , thus each nonzero fractional ideal of is generated by a uniquely determined positive rational number.
Under this convention the relative norm from down to coincides with the absolute norm defined below.
Absolute norm
Let be a number field with ring of integers , and a nonzero (integral) ideal of .
The absolute norm of is
By convention, the norm of the zero ideal is taken to be zero.
If is a principal ideal, then
.
The norm is completely multiplicative: if and are ideals of , then
.
Thus the absolute norm extends uniquely to a group homomorphism
defined for all nonzero fractional ideals of .
The norm of an ideal can be used to give an upper bound on the field norm of the smallest nonzero element it contains:
there always exists a nonzero for which
where
is the discriminant of and
is the number of pairs of (non-real) complex embeddings of into (the number of complex places of ).
See also
Field norm
Dedekind zeta function
References
Algebraic number theory
Commutative algebra
Ideals (ring theory)
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https://en.wikipedia.org/wiki/Fractional%20ideal
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In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral domain are like ideals where denominators are allowed. In contexts where fractional ideals and ordinary ring ideals are both under discussion, the latter are sometimes termed integral ideals for clarity.
Definition and basic results
Let be an integral domain, and let be its field of fractions.
A fractional ideal of is an -submodule of such that there exists a non-zero such that . The element can be thought of as clearing out the denominators in , hence the name fractional ideal.
The principal fractional ideals are those -submodules of generated by a single nonzero element of . A fractional ideal is contained in if and only if it is an (integral) ideal of .
A fractional ideal is called invertible if there is another fractional ideal such that
where
is the product of the two fractional ideals.
In this case, the fractional ideal is uniquely determined and equal to the generalized ideal quotient
The set of invertible fractional ideals form an abelian group with respect to the above product, where the identity is the unit ideal itself. This group is called the group of fractional ideals of . The principal fractional ideals form a subgroup. A (nonzero) fractional ideal is invertible if and only if it is projective as an -module. Geometrically, this means an invertible fractional ideal can be interpreted as rank 1 vector bundle over the affine scheme .
Every finitely generated R-submodule of K is a fractional ideal and if is noetherian these are all the fractional ideals of .
Dedekind domains
In Dedekind domains, the situation is much simpler. In particular, every non-zero fractional ideal is invertible. In fact, this property characterizes Dedekind domains:
An integral domain is a Dedekind domain if and only if every non-zero fractional ideal is invertible.
The set of fractional ideals over a Dedekind domain is denoted .
Its quotient group of fractional ideals by the subgroup of principal fractional ideals is an important invariant of a Dedekind domain called the ideal class group.
Number fields
For the special case of number fields (such as ) there is an associated ring denoted called the ring of integers of . For example, for square-free and congruent to . The key property of these rings is they are Dedekind domains. Hence the theory of fractional ideals can be described for the rings of integers of number fields. In fact, class field theory is the study of such groups of class rings.
Associated structures
For the ring of integerspg 2 of a number field, the group of fractional ideals forms a group denoted and the subgroup of principal fractional ideals is denoted . The ideal class group is the group of fractional ideals modulo the principal fractional ideals, so
and its
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https://en.wikipedia.org/wiki/Derived%20set
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A derived set may refer to:
Derived set (mathematics), a construction in point-set topology
Derived row, a concept in musical set theory
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https://en.wikipedia.org/wiki/Conservative%20vector%20field
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In vector calculus, a conservative vector field is a vector field that is the gradient of some function. A conservative vector field has the property that its line integral is path independent; the choice of any path between two points does not change the value of the line integral. Path independence of the line integral is equivalent to the vector field under the line integral being conservative. A conservative vector field is also irrotational; in three dimensions, this means that it has vanishing curl. An irrotational vector field is necessarily conservative provided that the domain is simply connected.
Conservative vector fields appear naturally in mechanics: They are vector fields representing forces of physical systems in which energy is conserved. For a conservative system, the work done in moving along a path in a configuration space depends on only the endpoints of the path, so it is possible to define potential energy that is independent of the actual path taken.
Informal treatment
In a two- and three-dimensional space, there is an ambiguity in taking an integral between two points as there are infinitely many paths between the two points—apart from the straight line formed between the two points, one could choose a curved path of greater length as shown in the figure. Therefore, in general, the value of the integral depends on the path taken. However, in the special case of a conservative vector field, the value of the integral is independent of the path taken, which can be thought of as a large-scale cancellation of all elements that don't have a component along the straight line between the two points. To visualize this, imagine two people climbing a cliff; one decides to scale the cliff by going vertically up it, and the second decides to walk along a winding path that is longer in length than the height of the cliff, but at only a small angle to the horizontal. Although the two hikers have taken different routes to get up to the top of the cliff, at the top, they will have both gained the same amount of gravitational potential energy. This is because a gravitational field is conservative.
Intuitive explanation
M. C. Escher's lithograph print Ascending and Descending illustrates a non-conservative vector field, impossibly made to appear to be the gradient of the varying height above ground (gravitational potential) as one moves along the staircase. The force field experienced by the one moving on the staircase is non-conservative in that one can return to the starting point while ascending more than one descends or vice versa, resulting in nonzero work done by gravity. On a real staircase, the height above the ground is a scalar potential field: one has to go upward exactly as much as one goes downward in order to return to the same place, in which case the work by gravity totals to zero. This suggests path-independence of work done on the staircase; equivalently, the force field experienced is conservative (see the later sect
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https://en.wikipedia.org/wiki/Errors%20and%20residuals
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In statistics and optimization, errors and residuals are two closely related and easily confused measures of the deviation of an observed value of an element of a statistical sample from its "true value" (not necessarily observable). The error of an observation is the deviation of the observed value from the true value of a quantity of interest (for example, a population mean). The residual is the difference between the observed value and the estimated value of the quantity of interest (for example, a sample mean). The distinction is most important in regression analysis, where the concepts are sometimes called the regression errors and regression residuals and where they lead to the concept of studentized residuals.
In econometrics, "errors" are also called disturbances.
Introduction
Suppose there is a series of observations from a univariate distribution and we want to estimate the mean of that distribution (the so-called location model). In this case, the errors are the deviations of the observations from the population mean, while the residuals are the deviations of the observations from the sample mean.
A statistical error (or disturbance) is the amount by which an observation differs from its expected value, the latter being based on the whole population from which the statistical unit was chosen randomly. For example, if the mean height in a population of 21-year-old men is 1.75 meters, and one randomly chosen man is 1.80 meters tall, then the "error" is 0.05 meters; if the randomly chosen man is 1.70 meters tall, then the "error" is −0.05 meters. The expected value, being the mean of the entire population, is typically unobservable, and hence the statistical error cannot be observed either.
A residual (or fitting deviation), on the other hand, is an observable estimate of the unobservable statistical error. Consider the previous example with men's heights and suppose we have a random sample of n people. The sample mean could serve as a good estimator of the population mean. Then we have:
The difference between the height of each man in the sample and the unobservable population mean is a statistical error, whereas
The difference between the height of each man in the sample and the observable sample mean is a residual.
Note that, because of the definition of the sample mean, the sum of the residuals within a random sample is necessarily zero, and thus the residuals are necessarily not independent. The statistical errors, on the other hand, are independent, and their sum within the random sample is almost surely not zero.
One can standardize statistical errors (especially of a normal distribution) in a z-score (or "standard score"), and standardize residuals in a t-statistic, or more generally studentized residuals.
In univariate distributions
If we assume a normally distributed population with mean μ and standard deviation σ, and choose individuals independently, then we have
and the sample mean
is a random variable distributed
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https://en.wikipedia.org/wiki/Laplacian%20vector%20field
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In vector calculus, a Laplacian vector field is a vector field which is both irrotational and incompressible. If the field is denoted as v, then it is described by the following differential equations:
From the vector calculus identity it follows that
that is, that the field v satisfies Laplace's equation.
However, the converse is not true; not every vector field that satisfies Laplace's equation is a Laplacian vector field, which can be a point of confusion. For example, the vector field satisfies Laplace's equation, but it has both nonzero divergence and nonzero curl and is not a Laplacian vector field.
A Laplacian vector field in the plane satisfies the Cauchy–Riemann equations: it is holomorphic.
Since the curl of v is zero, it follows that (when the domain of definition is simply connected) v can be expressed as the gradient of a scalar potential (see irrotational field) φ :
Then, since the divergence of v is also zero, it follows from equation (1) that
which is equivalent to
Therefore, the potential of a Laplacian field satisfies Laplace's equation.
See also
Potential flow
Harmonic function
Vector calculus
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https://en.wikipedia.org/wiki/Poincar%C3%A9%20half-plane%20model
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In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane, denoted below as H , together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry.
Equivalently the Poincaré half-plane model is sometimes described as a complex plane where the imaginary part (the y coordinate mentioned above) is positive.
The Poincaré half-plane model is named after Henri Poincaré, but it originated with Eugenio Beltrami who used it, along with the Klein model and the Poincaré disk model, to show that hyperbolic geometry was equiconsistent with Euclidean geometry.
This model is conformal which means that the angles measured at a point are the same in the model as they are in the actual hyperbolic plane.
The Cayley transform provides an isometry between the half-plane model and the Poincaré disk model.
This model can be generalized to model an dimensional hyperbolic space by replacing the real number x by a vector in an n dimensional Euclidean vector space.
Metric
The metric of the model on the half-plane, is:
where s measures the length along a (possibly curved) line.
The straight lines in the hyperbolic plane (geodesics for this metric tensor, i.e., curves which minimize the distance) are represented in this model by circular arcs perpendicular to the x-axis (half-circles whose centers are on the x-axis) and straight vertical rays perpendicular to the x-axis.
Distance calculation
If and are two points in the half-plane and is the reflection of across the x-axis into the lower half plane, the distance between the two points under the hyperbolic-plane metric is:
where is the Euclidean distance between points and is the inverse hyperbolic sine, and is the inverse hyperbolic tangent. This formula can be thought of as coming from the chord length in the Minkowski metric between points in the hyperboloid model, analogous to finding arclength on a sphere in terms of chord length. This formula can be thought of as coming from Euclidean distance in the Poincaré disk model with one point at the origin, analogous to finding arclength on the sphere by taking a stereographic projection centered on one point and measuring the Euclidean distance in the plane from the origin to the other point.
If the two points and are on a hyperbolic line (Euclidean half-circle) which intersects the x-axis at the ideal points and the distance from to is:
Cf. Cross-ratio.
Some special cases can be simplified. Two points with the same coordinate:
Two points with the same coordinate:
One point at the apex of the semicircle and another point at a central angle of
where is the inverse Gudermannian function, and is the inverse hyperbolic tangent.
Special points and curves
Ideal points (points at infinity) in the Poincaré half-plane model are of two kinds:
the points on the x-axis, and
one imaginary point at which is the ideal point to which all lines orthogonal to the x-axis converge.
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https://en.wikipedia.org/wiki/Vanish%20at%20infinity
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In mathematics, a function is said to vanish at infinity if its values approach 0 as the input grows without bounds. There are two different ways to define this with one definition applying to functions defined on normed vector spaces and the other applying to functions defined on locally compact spaces.
Aside from this difference, both of these notions correspond to the intuitive notion of adding a point at infinity, and requiring the values of the function to get arbitrarily close to zero as one approaches it. This definition can be formalized in many cases by adding an (actual) point at infinity.
Definitions
A function on a normed vector space is said to if the function approaches as the input grows without bounds (that is, as ). Or,
in the specific case of functions on the real line.
For example, the function
defined on the real line vanishes at infinity.
Alternatively, a function on a locally compact space , if given any positive number , there exists a compact subset such that
whenever the point lies outside of In other words, for each positive number the set has compact closure.
For a given locally compact space the set of such functions
valued in which is either or forms a -vector space with respect to pointwise scalar multiplication and addition, which is often denoted
As an example, the function
where and are reals greater or equal 1 and correspond to the point on vanishes at infinity.
A normed space is locally compact if and only if it is finite-dimensional so in this particular case, there are two different definitions of a function "vanishing at infinity".
The two definitions could be inconsistent with each other: if in an infinite dimensional Banach space, then vanishes at infinity by the definition, but not by the compact set definition.
Rapidly decreasing
Refining the concept, one can look more closely to the of functions at infinity. One of the basic intuitions of mathematical analysis is that the Fourier transform interchanges smoothness conditions with rate conditions on vanishing at infinity. The test functions of tempered distribution theory are smooth functions that are
for all , as , and such that all their partial derivatives satisfy the same condition too. This condition is set up so as to be self-dual under Fourier transform, so that the corresponding distribution theory of will have the same property.
See also
Citations
References
Mathematical analysis
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https://en.wikipedia.org/wiki/Standard%20basis
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In mathematics, the standard basis (also called natural basis or canonical basis) of a coordinate vector space (such as or ) is the set of vectors, each of whose components are all zero, except one that equals 1. For example, in the case of the Euclidean plane formed by the pairs of real numbers, the standard basis is formed by the vectors
Similarly, the standard basis for the three-dimensional space is formed by vectors
Here the vector ex points in the x direction, the vector ey points in the y direction, and the vector ez points in the z direction. There are several common notations for standard-basis vectors, including {ex, ey, ez}, {e1, e2, e3}, {i, j, k}, and {x, y, z}. These vectors are sometimes written with a hat to emphasize their status as unit vectors (standard unit vectors).
These vectors are a basis in the sense that any other vector can be expressed uniquely as a linear combination of these. For example, every vector v in three-dimensional space can be written uniquely as
the scalars , , being the scalar components of the vector v.
In the -dimensional Euclidean space , the standard basis consists of n distinct vectors
where ei denotes the vector with a 1 in the th coordinate and 0's elsewhere.
Standard bases can be defined for other vector spaces, whose definition involves coefficients, such as polynomials and matrices. In both cases, the standard basis consists of the elements of the space such that all coefficients but one are 0 and the non-zero one is 1. For polynomials, the standard basis thus consists of the monomials and is commonly called monomial basis. For matrices , the standard basis consists of the m×n-matrices with exactly one non-zero entry, which is 1. For example, the standard basis for 2×2 matrices is formed by the 4 matrices
Properties
By definition, the standard basis is a sequence of orthogonal unit vectors. In other words, it is an ordered and orthonormal basis.
However, an ordered orthonormal basis is not necessarily a standard basis. For instance the two vectors representing a 30° rotation of the 2D standard basis described above, i.e.
are also orthogonal unit vectors, but they are not aligned with the axes of the Cartesian coordinate system, so the basis with these vectors does not meet the definition of standard basis.
Generalizations
There is a standard basis also for the ring of polynomials in n indeterminates over a field, namely the monomials.
All of the preceding are special cases of the indexed family
where is any set and is the Kronecker delta, equal to zero whenever and equal to 1 if .
This family is the canonical basis of the R-module (free module)
of all families
from I into a ring R, which are zero except for a finite number of indices, if we interpret 1 as 1R, the unit in R.
Other usages
The existence of other 'standard' bases has become a topic of interest in algebraic geometry, beginning with work of Hodge from 1943 on Grassmannians. It is now a part of re
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https://en.wikipedia.org/wiki/1714%20in%20science
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The year 1714 in science and technology involved some significant events.
Mathematics
March – Roger Cotes publishes Logometrica in the Philosophical Transactions of the Royal Society. He provides the first proof of what is now known as Euler's formula and constructs the logarithmic spiral.
May – Brook Taylor publishes a paper, written in 1708, in the Philosophical Transactions of the Royal Society which describes his solution to the center of oscillation problem.
Gottfried Leibniz discusses the harmonic triangle.
Medicine
April 14 – Anne, Queen of Great Britain, performs the last touching for the "King's evil".
Dominique Anel uses the first fine-pointed syringe in surgery, later known as "Anel's syringe".
Herman Boerhaave introduces a modern system of clinical teaching at the University of Leiden.
The anatomical engravings of Bartolomeo Eustachi (died 1574) are published for the first time as Tabulae anatomicae by Giovanni Maria Lancisi.
Technology
Henry Mill obtains a British patent for a machine resembling a typewriter.
Events
July – The Parliament of Great Britain offers the Longitude prize to anyone who can solve the problem of accurately determining a ship's longitude.
Births
January 21 – Anna Morandi, Bolognese anatomist (died 1774)
January 6 – Percivall Pott, English surgeon (died 1788)
June 17 – César-François Cassini de Thury, French astronomer (died 1784)
September 6 – Robert Whytt, Scottish physician (died 1766)
October 16 – Giovanni Arduino, Italian geologist (died 1795)
October 25 – James Burnett, Lord Monboddo, Scottish philosopher and evolutionary thinker (died 1799)
December 19 – John Winthrop, American astronomer (died 1779)
December 31 – Arima Yoriyuki, Japanese mathematician (died 1783)
Alexander Wilson, Scottish surgeon, type founder, astronomer, meteorologist and mathematician (died 1786)
Deaths
October 5 – Kaibara Ekiken, Japanese philosopher and botanist (born 1630)
November 1 – John Radcliffe, English physician and benefactor (born 1652)
November 5 – Bernardino Ramazzini, Italian physician (born 1633)
References
18th century in science
1710s in science
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https://en.wikipedia.org/wiki/1713%20in%20science
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The year 1713 in science and technology involved some significant events.
Astronomy
John Rowley of London produces an orrery to a commission by Charles Boyle, 4th Earl of Orrery.
Mathematics
September 9 – Nicolas Bernoulli first describes the St. Petersburg paradox in a letter to Pierre Raymond de Montmort.
November 13 – James Waldegrave provides the first known minimax mixed strategy solution to a two-person game, in a letter to de Montmort.
Jacob Bernoulli's best known work, Ars Conjectandi (The Art of Conjecture), is published posthumously by his nephew. It contains a mathematical proof of the law of large numbers, the Bernoulli numbers, and other important research in probability theory and enumeration.
Medicine
William Cheselden publishes Anatomy of the Human Body and it becomes a popular work on anatomy, at least in part due to it being written in English rather than Latin.
Italian Bernardino Ramazzini provides one of the first descriptions of task-specific dystonia in his book of occupational diseases, Morbis Artificum, noting in chapter II of its Supplementum that "Scribes and Notaries" may develop "incessant movement of the hand, always in the same direction … the continuous and almost tonic strain on the muscles... that results in failure of power in the right hand".
Physics
The second edition of Isaac Newton's Principia Mathematica is published with an introduction by Roger Cotes and an essay by Newton titled General Scholium where he famously states "Hypotheses non fingo" ("I feign no hypotheses").
Technology
(c. 1713) Daniel Gabriel Fahrenheit switches from using alcohol to mercury as the thermometric fluid in his thermometers, creating the first mercury-in-glass thermometer.
Andrew Robinson builds the first ship called a schooner in Gloucester, Massachusetts.
Births
March 15 – Nicolas Louis de Lacaille, French astronomer (baptized December 28; died 1762)
May 3 – Alexis Claude Clairaut, French mathematician (died 1765)
May 25 – John Stuart, Lord Mount Stuart, Scottish politician and botanist (died 1792)
September 10 – John Needham, English biologist (died 1781)
Anthony Addington, English physician (died 1790)
Jean Paul de Gua de Malves, French mathematician (died 1785)
Deaths
April 29 (bur.) – Francis Hauksbee, English scientific instrument maker and experimentalist (born 1660)
April (end) – Edmund Dummer, English naval engineer (born 1651)
July 7 – Henry Compton, English bishop and botanist (born 1632)
August 26 (bur.) – Denis Papin, French-born physicist, mathematician and inventor (born 1647)
October 20 – Archibald Pitcairne, Scottish physician (born 1652)
References
18th century in science
1710s in science
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https://en.wikipedia.org/wiki/1711%20in%20science
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The year 1711 in science and technology involved some significant events.
Biology
Luigi Ferdinando Marsigli shows that coral is an animal rather than a plant as previously thought.
Mathematics
Giovanni Ceva publishes De Re Nummeraria (Concerning Money Matters), one of the first books on mathematical economics.
John Keill, writing in the journal of the Royal Society and with Isaac Newton's presumed blessing, accuses Gottfried Leibniz of having plagiarized Newton's calculus, formally starting the Leibniz and Newton calculus controversy.
Technology
John Shore invents the tuning fork
Births
May 18 – Ruđer Bošković, Ragusan polymath (died 1787)
July 22 – Georg Wilhelm Richmann, Russian physicist (died 1753)
September 22 – Thomas Wright, English astronomer, mathematician, instrument maker, architect, garden designer, antiquary and genealogist (died 1786)
October 31 – Laura Bassi, Italian scientist (died 1778)
November 19 – Mikhail Lomonosov, Russian scientist (died 1765)
18th century in science
1710s in science
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https://en.wikipedia.org/wiki/Harmonic%20divisor%20number
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In mathematics, a harmonic divisor number or Ore number is a positive integer whose divisors have a harmonic mean that is an integer. The first few harmonic divisor numbers are
1, 6, 28, 140, 270, 496, 672, 1638, 2970, 6200, 8128, 8190 .
Harmonic divisor numbers were introduced by Øystein Ore, who showed that every perfect number is a harmonic divisor number and conjectured that there are no odd harmonic divisor numbers other than 1.
Examples
The number 6 has the four divisors 1, 2, 3, and 6. Their harmonic mean is an integer:
Thus 6 is a harmonic divisor number. Similarly, the number 140 has divisors 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, and 140. Their harmonic mean is
Since 5 is an integer, 140 a harmonic divisor number.
Factorization of the harmonic mean
The harmonic mean of the divisors of any number can be expressed as the formula
where is the sum of th powers of the divisors of : is the number of divisors, and is the sum of divisors .
All of the terms in this formula are multiplicative, but not completely multiplicative.
Therefore, the harmonic mean is also multiplicative.
This means that, for any positive integer , the harmonic mean can be expressed as the product of the harmonic means of the prime powers in the factorization of .
For instance, we have
and
Harmonic divisor numbers and perfect numbers
For any integer M, as Ore observed, the product of the harmonic mean and arithmetic mean of its divisors equals M itself, as can be seen from the definitions. Therefore, M is harmonic, with harmonic mean of divisors k, if and only if the average of its divisors is the product of M with a unit fraction 1/k.
Ore showed that every perfect number is harmonic. To see this, observe that the sum of the divisors of a perfect number M is exactly 2M; therefore, the average of the divisors is M(2/τ(M)), where τ(M) denotes the number of divisors of M. For any M, τ(M) is odd if and only if M is a square number, for otherwise each divisor d of M can be paired with a different divisor M/d. But, no perfect number can be a square: this follows from the known form of even perfect numbers and from the fact that odd perfect numbers (if they exist) must have a factor of the form qα where α ≡ 1 (mod 4). Therefore, for a perfect number M, τ(M) is even and the average of the divisors is the product of M with the unit fraction 2/τ(M); thus, M is a harmonic divisor number.
Ore conjectured that no odd harmonic divisor numbers exist other than 1. If the conjecture is true, this would imply the nonexistence of odd perfect numbers.
Bounds and computer searches
W. H. Mills (unpublished; see Muskat) showed that any odd harmonic divisor number above 1 must have a prime power factor greater than 107, and Cohen showed that any such number must have at least three different prime factors. showed that there are no odd harmonic divisor numbers smaller than 1024.
Cohen, Goto, and others starting with Ore himself have performed computer searches listing all
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https://en.wikipedia.org/wiki/Inscribed%20angle
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In geometry, an inscribed angle is the angle formed in the interior of a circle when two chords intersect on the circle. It can also be defined as the angle subtended at a point on the circle by two given points on the circle.
Equivalently, an inscribed angle is defined by two chords of the circle sharing an endpoint.
The inscribed angle theorem relates the measure of an inscribed angle to that of the central angle subtending the same arc.
The inscribed angle theorem appears as Proposition 20 on Book 3 of Euclid's Elements.
Theorem
Statement
The inscribed angle theorem states that an angle θ inscribed in a circle is half of the central angle 2θ that subtends the same arc on the circle. Therefore, the angle does not change as its vertex is moved to different positions on the circle.
Proof
Inscribed angles where one chord is a diameter
Let O be the center of a circle, as in the diagram at right. Choose two points on the circle, and call them V and A. Draw line VO and extended past O so that it intersects the circle at point B which is diametrically opposite the point V. Draw an angle whose vertex is point V and whose sides pass through points A and B.
Draw line OA. Angle BOA is a central angle; call it θ. Lines OV and OA are both radii of the circle, so they have equal lengths. Therefore, triangle VOA is isosceles, so angle BVA (the inscribed angle) and angle VAO are equal; let each of them be denoted as ψ.
Angles BOA and AOV add up to 180°, since line VB passing through O is a straight line. Therefore, angle AOV measures 180° − θ.
It is known that the three angles of a triangle add up to 180°, and the three angles of triangle VOA are:
180° − θ
ψ
ψ.
Therefore,
Subtract
from both sides,
where θ is the central angle subtending arc AB and ψ is the inscribed angle subtending arc AB.
Inscribed angles with the center of the circle in their interior
Given a circle whose center is point O, choose three points V, C, and D on the circle. Draw lines VC and VD: angle DVC is an inscribed angle. Now draw line VO and extend it past point O so that it intersects the circle at point E. Angle DVC subtends arc DC on the circle.
Suppose this arc includes point E within it. Point E is diametrically opposite to point V. Angles DVE and EVC are also inscribed angles, but both of these angles have one side which passes through the center of the circle, therefore the theorem from the above Part 1 can be applied to them.
Therefore,
then let
so that
Draw lines OC and OD. Angle DOC is a central angle, but so are angles DOE and EOC, and
Let
so that
From Part One we know that and that . Combining these results with equation (2) yields
therefore, by equation (1),
Inscribed angles with the center of the circle in their exterior
The previous case can be extended to cover the case where the measure of the inscribed angle is the difference between two inscribed angles as discussed in the first part of this proof.
Given a circle whose c
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https://en.wikipedia.org/wiki/144%20%28number%29
|
144 (one hundred [and] forty-four) is the natural number following 143 and preceding 145. 144 is a dozen dozens, or one gross.
In mathematics
144 is the square of 12. It is also the twelfth Fibonacci number, following 89 and preceding 233, and the only Fibonacci number (other than 0, and 1) to also be a square. 144 is the smallest number with exactly 15 divisors, but it is not highly composite since the smaller number 120 contains 16. 144 is also equal to the sum of the eighth twin prime pair, (71 + 73). It is divisible by the value of its φ function, which returns 48 in its case, and there are 21 solutions to the equation φ(x) = 144. This is more than any integer below it, which makes it a highly totient number.
As a square number in decimal notation, 144 = 12 × 12, and if each number is reversed the equation still holds: 21 × 21 = 441. 169 shares this property, 13 × 13 = 169, while 31 × 31 = 961. Also in decimal, 144 is the largest of only four sum-product numbers, and it is a Harshad number, since 1 + 4 + 4 = 9, which divides 144.
144 is the smallest number whose fifth power is a sum of four (smaller) fifth powers. This solution was found in 1966 by L. J. Lander and T. R. Parkin, and disproved Euler's sum of powers conjecture. It was famously published in a paper by both authors, whose body consisted of only two sentences:
A regular ten-sided decagon has an internal angle of 144 degrees, which is equal to four times its own central angle, and equivalently twice the central angle of a regular five-sided pentagon.
The snub 24-cell, one of three semiregular polytopes in the fourth dimension, contains a total of 144 polyhedral cells: 120 regular tetrahedra and 24 regular icosahedra.
The maximum determinant in a 9 by 9 matrix of zeroes and ones is 144.
In particular, 144 is the sum of the divisors of 70: σ(70) = 144, where 70 is part of the only solution to the cannonball problem aside from the trivial solution, in-which the sum of the squares of the first twenty-four integers is equal to the square of another integer, 70 — and meaningful in the context of constructing the Leech lattice in twenty-four dimensions via the Lorentzian even unimodular lattice II25,1. Furthermore, 144 is relevant in testing whether two vectors in the quaternionic Leech lattice are equivalent under its automorphism group, Conway group Co0: modulo 1 + , every vector is congruent to either 0 or a minimal vector that is one of 196,560 ÷ 144 = 1,365 algebraic coordinate-frames, in-which a frame sought can be carried to its standard frame that is then checked for equivalence under a group stabilizing the frame of interest.
In sports
College Hoops Net (CHN) annual ranking of the Top 144 NCAA college basketball teams in 144 days.
The CFL record for career touchdown receptions, held by Milt Stegall of the Winnipeg Blue Bombers.
In other fields
144 is also:
The year AD 144 or 144 BC.
144 Vibilia is a dark, large Main belt asteroid.
The measurement, in cubits, of the
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https://en.wikipedia.org/wiki/Metric%20signature
|
In mathematics, the signature of a metric tensor g (or equivalently, a real quadratic form thought of as a real symmetric bilinear form on a finite-dimensional vector space) is the number (counted with multiplicity) of positive, negative and zero eigenvalues of the real symmetric matrix of the metric tensor with respect to a basis. In relativistic physics, the v represents the time or virtual dimension, and the p for the space and physical dimension. Alternatively, it can be defined as the dimensions of a maximal positive and null subspace. By Sylvester's law of inertia these numbers do not depend on the choice of basis and thus can be used to classify the metric. The signature is often denoted by a pair of integers implying r= 0, or as an explicit list of signs of eigenvalues such as or for the signatures and , respectively.
The signature is said to be indefinite or mixed if both v and p are nonzero, and degenerate if r is nonzero. A Riemannian metric is a metric with a positive definite signature . A Lorentzian metric is a metric with signature , or .
There is another notion of signature of a nondegenerate metric tensor given by a single number s defined as , where v and p are as above, which is equivalent to the above definition when the dimension n = v + p is given or implicit. For example, s = 1 − 3 = −2 for and its mirroring s' = −s = +2 for .
Definition
The signature of a metric tensor is defined as the signature of the corresponding quadratic form. It is the number of positive, negative and zero eigenvalues of any matrix (i.e. in any basis for the underlying vector space) representing the form, counted with their algebraic multiplicities. Usually, is required, which is the same as saying a metric tensor must be nondegenerate, i.e. no nonzero vector is orthogonal to all vectors.
By Sylvester's law of inertia, the numbers are basis independent.
Properties
Signature and dimension
By the spectral theorem a symmetric matrix over the reals is always diagonalizable, and has therefore exactly n real eigenvalues (counted with algebraic multiplicity). Thus .
Sylvester's law of inertia: independence of basis choice and existence of orthonormal basis
According to Sylvester's law of inertia, the signature of the scalar product (a.k.a. real symmetric bilinear form), g does not depend on the choice of basis. Moreover, for every metric g of signature there exists a basis such that
for , for and otherwise. It follows that there exists an isometry if and only if the signatures of g1 and g2 are equal. Likewise the signature is equal for two congruent matrices and classifies a matrix up to congruency. Equivalently, the signature is constant on the orbits of the general linear group GL(V) on the space of symmetric rank 2 contravariant tensors S2V∗ and classifies each orbit.
Geometrical interpretation of the indices
The number v (resp. p) is the maximal dimension of a vector subspace on which the scalar product g is positive
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https://en.wikipedia.org/wiki/Space%20group
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In mathematics, physics and chemistry, a space group is the symmetry group of a repeating pattern in space, usually in three dimensions. The elements of a space group (its symmetry operations) are the rigid transformations of the pattern that leave it unchanged. In three dimensions, space groups are classified into 219 distinct types, or 230 types if chiral copies are considered distinct. Space groups are discrete cocompact groups of isometries of an oriented Euclidean space in any number of dimensions. In dimensions other than 3, they are sometimes called Bieberbach groups.
In crystallography, space groups are also called the crystallographic or Fedorov groups, and represent a description of the symmetry of the crystal. A definitive source regarding 3-dimensional space groups is the International Tables for Crystallography .
History
Space groups in 2 dimensions are the 17 wallpaper groups which have been known for several centuries, though the proof that the list was complete was only given in 1891, after the much more difficult classification of space groups had largely been completed.
In 1879 the German mathematician Leonhard Sohncke listed the 65 space groups (called Sohncke groups) whose elements preserve the chirality. More accurately, he listed 66 groups, but both the Russian mathematician and crystallographer Evgraf Fedorov and the German mathematician Arthur Moritz Schoenflies noticed that two of them were really the same. The space groups in three dimensions were first enumerated in 1891 by Fedorov (whose list had two omissions (I3d and Fdd2) and one duplication (Fmm2)), and shortly afterwards in 1891 were independently enumerated by Schönflies (whose list had four omissions (I3d, Pc, Cc, ?) and one duplication (P21m)). The correct list of 230 space groups was found by 1892 during correspondence between Fedorov and Schönflies. later enumerated the groups with a different method, but omitted four groups (Fdd2, I2d, P21d, and P21c) even though he already had the correct list of 230 groups from Fedorov and Schönflies; the common claim that Barlow was unaware of their work is incorrect. describes the history of the discovery of the space groups in detail.
Elements
The space groups in three dimensions are made from combinations of the 32 crystallographic point groups with the 14 Bravais lattices, each of the latter belonging to one of 7 lattice systems. What this means is that the action of any element of a given space group can be expressed as the action of an element of the appropriate point group followed optionally by a translation. A space group is thus some combination of the translational symmetry of a unit cell (including lattice centering), the point group symmetry operations of reflection, rotation and improper rotation (also called rotoinversion), and the screw axis and glide plane symmetry operations. The combination of all these symmetry operations results in a total of 230 different space groups describing all possible cr
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https://en.wikipedia.org/wiki/555%20%28number%29
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555 (five hundred [and] fifty-five) is the natural number following 554 and preceding 556.
In mathematics
555 is a sphenic number. In base 10, it is a repdigit, and because it is divisible by the sum of its digits, it is a Harshad number. It is also a Harshad number in binary, base 11, base 13 and hexadecimal.
It is the sum of the first triplet of three-digit permutable primes in decimal:
.
It is the twenty-sixth number such that its Euler totient (288) is equal to the totient value of its sum-of-divisors: .
Telephone numbers
The NANP reserves telephone numbers in many dialing areas in the 555 local block for fictional purposes, such as 1-308-555-3485.
References
External links
Integers
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https://en.wikipedia.org/wiki/Antiholomorphic%20function
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In mathematics, antiholomorphic functions (also called antianalytic functions) are a family of functions closely related to but distinct from holomorphic functions.
A function of the complex variable z defined on an open set in the complex plane is said to be antiholomorphic if its derivative with respect to exists in the neighbourhood of each and every point in that set, where is the complex conjugate.
A definition of antiholomorphic function follows: "[a] function of one or more complex variables [is said to be anti-holomorphic if (and only if) it] is the complex conjugate of a holomorphic function ."
One can show that if f(z) is a holomorphic function on an open set D, then f() is an antiholomorphic function on , where is the reflection against the x-axis of D, or in other words, is the set of complex conjugates of elements of D. Moreover, any antiholomorphic function can be obtained in this manner from a holomorphic function. This implies that a function is antiholomorphic if and only if it can be expanded in a power series in in a neighborhood of each point in its domain. Also, a function f(z) is antiholomorphic on an open set D if and only if the function is holomorphic on D.
If a function is both holomorphic and antiholomorphic, then it is constant on any connected component of its domain.
References
Complex analysis
Types of functions
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https://en.wikipedia.org/wiki/Y-intercept
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In analytic geometry, using the common convention that the horizontal axis represents a variable x and the vertical axis represents a variable y, a y-intercept or vertical intercept is a point where the graph of a function or relation intersects the y-axis of the coordinate system. As such, these points satisfy x = 0.
Using equations
If the curve in question is given as the y-coordinate of the y-intercept is found by calculating Functions which are undefined at x = 0 have no y-intercept.
If the function is linear and is expressed in slope-intercept form as , the constant term is the y-coordinate of the y-intercept.
Multiple y-intercepts
Some 2-dimensional mathematical relationships such as circles, ellipses, and hyperbolas can have more than one y-intercept. Because functions associate x values to no more than one y value as part of their definition, they can have at most one y-intercept.
x-intercepts
Analogously, an x-intercept is a point where the graph of a function or relation intersects with the x-axis. As such, these points satisfy y=0. The zeros, or roots, of such a function or relation are the x-coordinates of these x-intercepts.
Unlike y-intercepts, functions of the form y = f(x) may contain multiple x-intercepts. The x-intercepts of functions, if any exist, are often more difficult to locate than the y-intercept, as finding the y intercept involves simply evaluating the function at x=0.
In higher dimensions
The notion may be extended for 3-dimensional space and higher dimensions, as well as for other coordinate axes, possibly with other names. For example, one may speak of the I-intercept of the current–voltage characteristic of, say, a diode. (In electrical engineering, I is the symbol used for electric current.)
See also
Regression intercept
References
Elementary mathematics
Functions and mappings
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https://en.wikipedia.org/wiki/NCD
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NCD may refer to:
Language
Nemine contradicente (or N.C.D.), for 'with no one speaking against'
Non-convergent discourse, an asymmetricly bilingual conversation
Mathematics
Normalized compression distance, in statistics and information theory
Nearly completely decomposable Markov chain, in probability theory
Medicine
Non-communicable disease, that cannot be transmitted
National coverage determination, American public healthcare guidance
Neurocognitive disorder, a class of mental illness
Organisations in government and politics
National Council on Disability, United States
National Center for Digitization, Serbia
Naval Construction Division of the U.S. Navy Seabees
New Centre-Right, Italy
Other uses
National Cleavage Day
National Commission On Disabilities, an organization based in Liberia
Naval Combat Dress, a uniform of the Canadian Forces
Network Computing Devices, a company
No claim discount on insurance policies
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https://en.wikipedia.org/wiki/Sperner%27s%20lemma
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In mathematics, Sperner's lemma is a combinatorial result on colorings of triangulations, analogous to the Brouwer fixed point theorem, which is equivalent to it. It states that every Sperner coloring (described below) of a triangulation of an simplex contains a cell whose vertices all have different colors.
The initial result of this kind was proved by Emanuel Sperner, in relation with proofs of invariance of domain. Sperner colorings have been used for effective computation of fixed points and in root-finding algorithms, and are applied in fair division (cake cutting) algorithms.
According to the Soviet Mathematical Encyclopaedia (ed. I.M. Vinogradov), a related 1929 theorem (of Knaster, Borsuk and Mazurkiewicz) had also become known as the Sperner lemma – this point is discussed in the English translation (ed. M. Hazewinkel). It is now commonly known as the Knaster–Kuratowski–Mazurkiewicz lemma.
Statement
One-dimensional case
In one dimension, Sperner's Lemma can be regarded as a discrete version of the intermediate value theorem. In this case, it essentially says that if a discrete function takes only the values 0 and 1, begins at the value 0 and ends at the value 1, then it must switch values an odd number of times.
Two-dimensional case
The two-dimensional case is the one referred to most frequently. It is stated as follows:
Subdivide a triangle arbitrarily into a triangulation consisting of smaller triangles meeting edge to edge. Then a Sperner coloring of the triangulation is defined as an assignment of three colors to the vertices of the triangulation such that
Each of the three vertices , , and of the initial triangle has a distinct color
The vertices that lie along any edge of triangle have only two colors, the two colors at the endpoints of the edge. For example, each vertex on must have the same color as or .
Then every Sperner coloring of every triangulation has at least one "rainbow triangle", a smaller triangle in the triangulation that has its vertices colored with all three different colors. More precisely, there must be an odd number of rainbow triangles.
Multidimensional case
In the general case the lemma refers to a -dimensional simplex:
Consider any triangulation , a disjoint division of into smaller -dimensional simplices, again meeting face-to-face. Denote the coloring function as:
where is the set of vertices of . A coloring function defines a Sperner coloring when:
The vertices of the large simplex are colored with different colors, that is, without loss of generality, for .
Vertices of located on any -dimensional subface of the large simplex are colored only with the colors
Then every Sperner coloring of every triangulation of the -dimensional simplex has an odd number of instances of a rainbow simplex, meaning a simplex whose vertices are colored with all colors. In particular, there must be at least one rainbow simplex.
Proof
We shall first address the two-dimensional case. Consider a
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https://en.wikipedia.org/wiki/Probabilistic%20proposition
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A probabilistic proposition is a proposition with a measured probability of being true for an arbitrary person at an arbitrary time. They may be contrasted with deterministic propositions, which assert that something is certain with no element of chance. Probabilistic proportions may be either categorical or conditional.
References
Probability interpretations
Propositions
|
https://en.wikipedia.org/wiki/Sallen%E2%80%93Key%20topology
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The Sallen–Key topology is an electronic filter topology used to implement second-order active filters that is particularly valued for its simplicity. It is a degenerate form of a voltage-controlled voltage-source (VCVS) filter topology. It was introduced by R. P. Sallen and E. L. Key of MIT Lincoln Laboratory in 1955.
Explanation of operation
A VCVS filter uses a voltage amplifier with practically infinite input impedance and zero output impedance to implement a 2-pole low-pass, high-pass, bandpass, bandstop, or allpass response. The VCVS filter allows high Q factor and passband gain without the use of inductors. A VCVS filter also has the advantage of independence: VCVS filters can be cascaded without the stages affecting each others tuning. A Sallen–Key filter is a variation on a VCVS filter that uses a unity-voltage-gain amplifier (i.e., a pure buffer amplifier).
History and implementation
In 1955, Sallen and Key used vacuum tube cathode follower amplifiers; the cathode follower is a reasonable approximation to an amplifier with unity voltage gain. Modern analog filter implementations may use operational amplifiers (also called op amps). Because of its high input impedance and easily selectable gain, an operational amplifier in a conventional non-inverting configuration is often used in VCVS implementations. Implementations of Sallen–Key filters often use an op amp configured as a voltage follower; however, emitter or source followers are other common choices for the buffer amplifier.
Sensitivity to component tolerances
VCVS filters are relatively resilient to component tolerance, but obtaining high Q factor may require extreme component value spread or high amplifier gain. Higher-order filters can be obtained by cascading two or more stages.
Generic Sallen–Key topology
The generic unity-gain Sallen–Key filter topology implemented with a unity-gain operational amplifier is shown in Figure 1. The following analysis is based on the assumption that the operational amplifier is ideal.
Because the op amp is in a negative-feedback configuration, its and inputs must match (i.e., ). However, the inverting input is connected directly to the output , and so
By Kirchhoff's current law (KCL) applied at the node,
By combining equations (1) and (2),
Applying equation (1) and KCL at the op amp's non-inverting input gives
which means that
Combining equations (2) and (3) gives
Rearranging equation (4) gives the transfer function
which typically describes a second-order linear time-invariant (LTI) system.
If the component were connected to ground instead of to , the filter would be a voltage divider composed of the and components cascaded with another voltage divider composed of the and components. The buffer amplifier bootstraps the "bottom" of the component to the output of the filter, which will improve upon the simple two-divider case. This interpretation is the reason why Sallen–Key filters are often drawn with the op amp's non-i
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https://en.wikipedia.org/wiki/Flow
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Flow may refer to:
Science and technology
Fluid flow, the motion of a gas or liquid
Flow (geomorphology), a type of mass wasting or slope movement in geomorphology
Flow (mathematics), a group action of the real numbers on a set
Flow (psychology), a mental state of being fully immersed and focused
Flow, a spacecraft of NASA's GRAIL program
Computing
Flow network, graph-theoretic version of a mathematical flow
Flow analysis
Calligra Flow, free diagramming software
Dataflow, a broad concept in computer systems with many different meanings
Microsoft Flow (renamed to Power Automate in 2019), a workflow toolkit in Microsoft Dynamics
Neos Flow, a free and open source web application framework written in PHP
webMethods Flow, a graphical programming language
FLOW (programming language), an educational programming language from the 1970s
Flow (web browser), a web browser with a proprietary rendering engine
Arts, entertainment and media
Flow (journal), an online journal of television and media studies
The Flow (book), a 2022 non-fiction book by Amy-Jane Beer
Flow (video game)
Flow (comics), a fictional character in the International Ultramarine Corps
Flow 93.5, the Canadian radio station CFXJ-FM
Flow FM (Australia), a radio station
Flow (Argentina), a cable television operator
Film and television
Flow (television), the sequencing of TV material from one element to the next
Flow TV, a network of Ripe Digital Entertainment
Flow: For Love of Water, a 2008 documentary film
Flow (1996 film), a 1996 film by Quentin Lee
Flow (2014 film), also known as Ækte vare, 2014 film by Fenar Ahmad
Music
Flow (rapping), the rhythms and rhymes of a hip-hop song's lyrics and how they interact
Flow (American band), a new age band
Flow G, Filipino rapper and songwriter
Flow (Japanese band), a rock group
Flow (rapper) (Widner DeGruy, born 1991)
Flow (Terence Blanchard album), 2005
Flow (Conception album), 1997
Flow (Foetus album), 2001
The Flow, a 1997 album by Chris Leslie
"Flow", a song by Cage the Elephant from the 2011 album Thank You, Happy Birthday
"Flow", a song by Cloud Wan from The Cloud, 2022
"Flow", a song by Sade from the 2000 album Lovers Rock
"Flow", a song by Transister from the 1998 album Transister
Other uses
Flow (brand), a Caribbean telecommunications provider
Flow (policy debate), a form of note-taking in policy debate and public forum debate
Flow (real estate company), an American residential real estate company
FLOW (Belgium), a national health care network
Football League of West Godavari, an Indian football league
See also
Flo (disambiguation)
Floe (disambiguation)
Floh (disambiguation)
Flou (disambiguation)
Streamflow, or channel runoff, the flow of water in streams, rivers, and other channels
Tide, the rise and fall of sea levels
Phlow, a German webzine
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https://en.wikipedia.org/wiki/1644%20in%20science
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The year 1644 AD in science and technology involved some significant events.
Mathematics
The Basel problem is posed by Pietro Mengoli, and will puzzle mathematicians until solved by Leonhard Euler in 1735.
Technology
Jacob van Eyck collaborates with the bellfounding duo Pieter and François Hemony to create the first tuned carillon in Zutphen.
Publications
Jan Baptist van Helmont publishes Dageraad ofte Nieuwe Opkomst der Geneeskunst ("Daybreak, or the New Rise of Medicine").
Births
25 September – Ole Rømer, Danish astronomer who makes the first quantitative measurements of the speed of light (died 1710)
Deaths
2 July – William Gascoigne, English scientist (born 1610)
30 December – Jan Baptist van Helmont, Flemish chemist (born 1580)
References
17th century in science
1640s in science
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https://en.wikipedia.org/wiki/1679%20in%20science
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The year 1679 in science and technology involved some significant events.
Botany
Establishment of Hortus Botanicus (Amsterdam).
Mathematics
Samuel Morland publishes The Doctrine of Interest, both Simple & Compound, probably the first tables produced with the aid of a calculating machine.
Medicine
Great Plague of Vienna.
Franciscus Sylvius' Opera Medica, published posthumously, recognizes scrofula and phthisis as forms of tuberculosis.
Technology
Pierre-Paul Riquet excavates Malpas Tunnel on the Canal du Midi in Hérault, France, Europe's first navigable canal tunnel (165 m, concrete lined).
Publications
Publication in Paris of the first of Edme Mariotte's Essays de physique: De la végétation des plantes, a pioneering discussion of plant physiology; and De la nature de l'air, a statement of Boyle's law.
Publication by the Paris Observatory of the world's first national ephemeris almanac, the Connaissance des tems, compiled by Jean Picard.
Births
January 2 - Pierre Fauchard, French physician (died 1761).
January 24 – Christian Wolff, German philosopher, mathematician and scientist (died 1754)
Deaths
January 14 – Jacques de Billy, French Jesuit mathematician (born 1602)
References
17th century in science
1670s in science
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https://en.wikipedia.org/wiki/1677%20in%20science
|
The year 1677 in science and technology involved some significant events.
Astronomy
Publication of the first English star atlas, John Seller's Atlas Coelestis.
Mathematics
Publication of Cocker's Arithmetick: Being a Plain and Familiar Method Suitable to the Meanest Capacity for the Full Understanding of That Incomparable Art, As It Is Now Taught by the Ablest School-Masters in City and Country, attributed to Edward Cocker (died 1676). It will remain a standard grammar school textbook in England for more than 150 years.
Medicine
January 21 – A pamphlet on smallpox published in Boston becomes the first medical publication in the British colonies in North America.
Microbiology
Antonie van Leeuwenhoek discovers the spermatozoon.
Paleontology
Robert Plot publishes The Natural History of Oxford-shire, Being an Essay Toward the Natural History of England, in which he describes the fossilised femur of a human giant, now known to be from the dinosaur Megalosaurus.
Births
February 8 – Jacques Cassini, French astronomer (died 1756)
September 17 – Stephen Hales, English physiologist and clergyman (died 1761)
September 27 – Johann Gabriel Doppelmayr, German mathematician, astronomer and cartographer (died 1750)
Deaths
May 4 – Isaac Barrow, English mathematician (born 1630)
May 23 (bur.) – John Kersey, English mathematician (born 1677)
October 11 – Sir Cornelius Vermuyden, Dutch-born drainage engineer (born 1595).
October 14 – Francis Glisson, English physician (born 1599?)
References
17th century in science
1670s in science
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