source stringlengths 31 168 | text stringlengths 51 3k |
|---|---|
https://en.wikipedia.org/wiki/Ivan%20Vinogradov | Ivan Matveevich Vinogradov (; 14 September 1891 – 20 March 1983) was a Soviet mathematician, who was one of the creators of modern analytic number theory, and also a dominant figure in mathematics in the USSR. He was born in the Velikiye Luki district, Pskov Oblast. He graduated from the University of St. Petersburg, where in 1920 he became a Professor. From 1934 he was a Director of the Steklov Institute of Mathematics, a position he held for the rest of his life, except for the five-year period (1941–1946) when the institute was directed by Academician Sergei Sobolev. In 1941 he was awarded the Stalin Prize. He was elected to the American Philosophical Society in 1942. In 1951 he became a foreign member of the Polish Academy of Sciences and Letters in Kraków.
Mathematical contributions
In analytic number theory, Vinogradov's method refers to his main problem-solving technique, applied to central questions involving the estimation of exponential sums. In its most basic form, it is used to estimate sums over prime numbers, or Weyl sums. It is a reduction from a complicated sum to a number of smaller sums which are then simplified. The canonical form for prime number sums is
With the help of this method, Vinogradov tackled questions such as the ternary Goldbach problem in 1937 (using Vinogradov's theorem), and the zero-free region for the Riemann zeta function. His own use of it was inimitable; in terms of later techniques, it is recognised as a prototype of the large sieve method in its application of bilinear forms, and also as an exploitation of combinatorial structure. In some cases his results resisted improvement for decades.
He also used this technique on the Dirichlet divisor problem, allowing him to estimate the number of integer points under an arbitrary curve. This was an improvement on the work of Georgy Voronoy.
In 1918 Vinogradov proved the Pólya–Vinogradov inequality for character sums.
Personality and career
Vinogradov served as director of the Mathematical Institute for 49 years. For his long service he was twice awarded the order of The Hero of the Socialist Labour. The house where he was born was converted into his memorial – a unique honour among Russian mathematicians. As the head of a leading mathematical institute, Vinogradov enjoyed significant influence in the Academy of Sciences and was regarded as an informal leader of Soviet mathematicians, not always in a positive way: his anti-Semitic feelings led him to hinder the careers of many prominent Soviet mathematicians.
Although he was always faithful to the official line, he was never a member of the Communist Party and his overall mindset was nationalistic rather than communist. This can at least partly be attributed to his origins: his father was a priest of the Russian Orthodox Church. Vinogradov was enormously strong: in some recollections it is stated that he could lift a chair with a person sitting on it by holding the leg of the chair in his hands. He w |
https://en.wikipedia.org/wiki/1692%20in%20science | The year 1692 in science and technology:
Events
In the American colonies, the Salem witch trials develop, following 250 years of witch-hunts in Europe.
Mathematics
The tractrix, sometimes called a tractory or equitangential curve, is first studied by Christiaan Huygens, who gives it its name.
John Arbuthnot publishes Of the Laws of Chance (translated from Huygens' De ratiociniis in ludo aleae), the first work on probability theory in English.
Medicine
Thomas Sydenham's Processus integri ("The Process of Healing") is published posthumously.
Births
April 22 – James Stirling, Scottish mathematician (died 1770)
Deaths
May – John Banister, English missionary and botanist, accidentally shot (born 1654)
References
17th century in science
1690s in science |
https://en.wikipedia.org/wiki/1640%20in%20science | The year 1640 in science and technology involved some significant events.
Botany
John Parkinson publishes Theatrum Botanicum:The Theater of Plants, or, An Herbal of a Large Extent.
Mathematics
The 16-year-old Blaise Pascal demonstrates the properties of the hexagrammum mysticum in his Essai pour les coniques which he sends to Mersenne.
October 18 – Fermat states his "little theorem" in a letter to Frénicle de Bessy: if p is a prime number, then for any integer a, a p − a will be divisible by p.
December 25 – Fermat claims a proof of the theorem on sums of two squares in a letter to Mersenne ("Fermat's Christmas Theorem"): an odd prime p is expressible as the sum of two squares.
Technology
The micrometer is developed.
A form of bayonet is invented; in later years it will gradually replace the pike.
The reticle telescope is developed and initiates the birth of sharpshooting.
Births
April 1 – Georg Mohr, Danish mathematician (died 1697)
December 13 (bapt.) – Robert Plot, English naturalist and chemist (died 1696)
Elias Tillandz, Swedish physician and botanist in Finland (died 1693)
Deaths
December 22 – Jean de Beaugrand, French mathematician (born c. 1584)
References
17th century in science
1640s in science |
https://en.wikipedia.org/wiki/1659%20in%20science | The year 1659 in science and technology involved some significant events.
Astronomy
Christiaan Huygens publishes Systema Saturnium, including the first illustration of the Orion Nebula.
Mathematics
First known use of the term Abscissa, by Stefano degli Angeli.
Swiss mathematician Johann Rahn publishes Teutsche Algebra, containing the first printed use of the 'division sign' (÷, a repurposed obelus variant) as a mathematical symbol for division and of the 'therefore sign' (∴).
Medicine
Thomas Willis publishes De Febribus.
Physics
Christiaan Huygens derives the formula for centripedal force.
Births
February 27 – William Sherard, English botanist (died 1728)
June 3 – David Gregory, Scottish astronomer (died 1708)
Deaths
October 10 – Abel Tasman, Dutch explorer (born 1603)
References
17th century in science
1650s in science |
https://en.wikipedia.org/wiki/Shift%20operator | In mathematics, and in particular functional analysis, the shift operator, also known as the translation operator, is an operator that takes a function
to its translation . In time series analysis, the shift operator is called the lag operator.
Shift operators are examples of linear operators, important for their simplicity and natural occurrence. The shift operator action on functions of a real variable plays an important role in harmonic analysis, for example, it appears in the definitions of almost periodic functions, positive-definite functions, derivatives, and convolution. Shifts of sequences (functions of an integer variable) appear in diverse areas such as Hardy spaces, the theory of abelian varieties, and the theory of symbolic dynamics, for which the baker's map is an explicit representation. The notion of triangulated category is a categorified analogue of the shift operator.
Definition
Functions of a real variable
The shift operator (where ) takes a function on to its translation ,
A practical operational calculus representation of the linear operator in terms of the plain derivative was introduced by Lagrange,
which may be interpreted operationally through its formal Taylor expansion in ; and whose action on the monomial is evident by the binomial theorem, and hence on all series in , and so all functions as above. This, then, is a formal encoding of the Taylor expansion in Heaviside's calculus.
The operator thus provides the prototype
for Lie's celebrated advective flow for Abelian groups,
where the canonical coordinates (Abel functions) are defined such that
For example, it easily follows that yields scaling,
hence (parity); likewise,
yields
yields
yields
etc.
The initial condition of the flow and the group property completely determine the entire Lie flow, providing a solution to the translation functional equation
Sequences
The left shift operator acts on one-sided infinite sequence of numbers by
and on two-sided infinite sequences by
The right shift operator acts on one-sided infinite sequence of numbers by
and on two-sided infinite sequences by
The right and left shift operators acting on two-sided infinite sequences are called bilateral shifts.
Abelian groups
In general, as illustrated above, if is a function on an abelian group , and is an element of , the shift operator maps to
Properties of the shift operator
The shift operator acting on real- or complex-valued functions or sequences is a linear operator which preserves most of the standard norms which appear in functional analysis. Therefore, it is usually a continuous operator with norm one.
Action on Hilbert spaces
The shift operator acting on two-sided sequences is a unitary operator on The shift operator acting on functions of a real variable is a unitary operator on
In both cases, the (left) shift operator satisfies the following commutation relation with the Fourier transform:
where is the multiplication |
https://en.wikipedia.org/wiki/1685%20in%20science | The year 1685 in science and technology involved some significant events.
Mathematics
Adam Adamandy Kochański publishes an approximation for squaring the circle.
Physiology and medicine
Charles Allen publishes the first book in English on dentistry, The Operator for the Teeth.
Govert Bidloo publishes an atlas of human anatomy, Ontleding des menschelyken lichaams, with plates by Gerard de Lairesse.
Technology
Menno van Coehoorn publishes his principal treatise on fortification, Nieuwe Vestingbouw op een natte of lage horisont, in Leeuwarden.
Births
August 18 – Brook Taylor, English mathematician (died 1731)
November 17 – Pierre Gaultier de Varennes et de la Vérendrye, French Canadian explorer (died 1749)
Deaths
February 2 – Pierre Bourdelot, French physician, anatomist, freethinker, abbé and libertine (born 1610)
November 23 – Bernard de Gomme, military engineer in England (born 1620)
December 12 – John Pell, English mathematician (born 1610)
References
17th century in science
1680s in science |
https://en.wikipedia.org/wiki/1637%20in%20science | The year 1637 in science and technology involved some significant events.
Mathematics
René Descartes promotes intellectual rigour in Discours de la méthode pour bien conduire sa raison, et chercher la vérité dans les sciences and introduces the Cartesian coordinate system in its appendix La Géométrie (published in Leiden).
Pierre de Fermat conjectures Fermat's Last Theorem.
Publications
May – Chinese encyclopedist Song Yingxing publishes his Tiangong Kaiwu ("Exploitation of the Works of Nature").
Births
February 12 – Jan Swammerdam, Dutch naturalist, pioneer of comparative anatomy and entomology (died 1680)
François Mauriceau, French obstetrician (died 1709)
Deaths
June 24 – Nicolas-Claude Fabri de Peiresc, French astronomer (born 1580)
May 19 – Isaac Beeckman, Dutch philosopher and scientist (born 1588)
Henry Gellibrand, English mathematician (born 1597)
References
17th century in science
1630s in science |
https://en.wikipedia.org/wiki/169%20%28number%29 | 169 (one hundred [and] sixty-nine) is the natural number following 168 and preceding 170.
In mathematics
169 is an odd number, a composite number, and a deficient number.
169 is a square number: 13 × 13 = 169, and if each number is reversed the equation is still true: 31 × 31 = 961. 144 shares this property: 12 × 12 = 144, 21 × 21 = 441.
169 is one of the few squares to also be a centered hexagonal number. Like all odd squares, it is a centered octagonal number. 169 is an odd-indexed Pell number, thus it is also a Markov number, appearing in the solutions (2, 169, 985), (2, 29, 169), (29, 169, 14701), etc. 169 is the sum of seven consecutive primes: 13 + 17 + 19 + 23 + 29 + 31 + 37. 169 is a difference in consecutive cubes, equaling
In astronomy
169 Zelia is a bright main belt asteroid
Gliese 169 is an orange, main sequence (K7 V) star in the constellation Taurus
QSO B0307+169 is a quasar in the constellation Aries
Sayh al Uhaymir 169 is a 206g lunar meteorite found in Sultanate of Oman
In the military
was a United States Navy technical research ship during the 1960s
was a United States Navy during World War II
was a United States Navy during World War II
was a United States Navy following World War I
was a United States Navy during World War II
was a United States Navy submarine during World War II
169th Battalion, CEF unit in the Canadian Expeditionary Force during the World War I
169th Fires Brigade the US Army National Guard artillery brigade, a part of the Colorado Army National Guard
The United States Air Force's 169th Fighter Wing fighter unit at McEntire Joint National Guard Station, South Carolina
169 or 169th Squadrons
169th Airlift Squadron, a unit of the U.S. Air Force
Marine Light Attack Helicopter Squadron 169, United States Marine Corps Light Attack Helicopter Squadron
No. 169 Squadron RAF, a unit of the United Kingdom Royal Air Force
In transportation
Metro Transit Route 169 in Seattle
169th Street station on the IND Queens Boulevard Line of the New York City Subway served by the and trains
169th Street was a station on the demolished IRT Third Avenue Line of the New York City Subway
In TV and radio
The IEC 169-2 connector TV aerial plug
In other fields
169 is also:
The year AD 169 or 169 BC
The atomic number of an element temporarily called Unhexennium
169 is the number of nonequivalent starting hands in the card game Texas hold 'em
169 is known in the computing world as the first number of an automatic IPv4 address assigned by TCP/IP when no external networking device is contactable
Minuscule 169 is a Greek minuscule manuscript of the New Testament, on parchment
See also
List of highways numbered 169
United States Supreme Court cases, Volume 169
United Nations Security Council Resolution 169
St. Joseph Community Consolidated School District 169
References
External links
Number Facts and Trivia: 169
The Positive Integer 169
Prime curiosities: 169
The Number 169
Number Go |
https://en.wikipedia.org/wiki/Kikuchi%20District%2C%20Kumamoto | is a district located in Kumamoto Prefecture, Japan.
As of the Koshi merger (but with 2003 population statistics), the district has an estimated population of 58,300 and a density of 427 persons per square kilometer. The total area is 136.66 km2.
Towns
Kikuyō
Ōzu
Mergers
See merger and dissolution of municipalities of Japan.
On March 22, 2005 the towns of Shichijō and Shisui, and the village of Kyokushi merged into the expanded city of Kikuchi.
On February 27, 2006 the towns of Kōshi and Nishigōshi merged to form the new city of Kōshi.
Districts in Kumamoto Prefecture |
https://en.wikipedia.org/wiki/Kamimashiki%20District%2C%20Kumamoto | is a district located in Kumamoto Prefecture, Japan.
As of the Yamato merger (but with 2003 population statistics), the district had an estimated population of 90,315 and a density of 115.2 persons per square kilometer. The total area is 784.03 km2.
Towns and villages
Kashima
Kōsa
Mashiki
Mifune
Yamato
Mergers
On February 11, 2005, the municipalities of Yabe and Seiwa merged with the town of Soyō from Aso District to form the new town of Yamato.
Districts in Kumamoto Prefecture |
https://en.wikipedia.org/wiki/Nagell%E2%80%93Lutz%20theorem | In mathematics, the Nagell–Lutz theorem is a result in the diophantine geometry of elliptic curves, which describes rational torsion points on elliptic curves over the integers.
It is named for Trygve Nagell and Élisabeth Lutz.
Definition of the terms
Suppose that the equation
defines a non-singular cubic curve with integer coefficients a, b, c, and let D be the discriminant of the cubic polynomial on the right side:
Statement of the theorem
If P = (x,y) is a rational point of finite order on C, for the elliptic curve group law, then:
1) x and y are integers
2) either y = 0, in which case P has order two, or else y divides D, which immediately implies that y2 divides D.
Generalizations
The Nagell–Lutz theorem generalizes to arbitrary number fields and more
general cubic equations.
For curves over the rationals, the
generalization says that, for a nonsingular cubic curve
whose Weierstrass form
has integer coefficients, any rational point P=(x,y) of finite
order must have integer coordinates, or else have order 2 and
coordinates of the form x=m/4, y=n/8, for m and n integers.
History
The result is named for its two independent discoverers, the Norwegian Trygve Nagell (1895–1988) who published it in 1935, and Élisabeth Lutz (1937).
See also
Mordell–Weil theorem
References
Joseph H. Silverman, John Tate (1994), "Rational Points on Elliptic Curves", Springer, .
Elliptic curves
Theorems in number theory |
https://en.wikipedia.org/wiki/Discrete%20group | In mathematics, a topological group G is called a discrete group if there is no limit point in it (i.e., for each element in G, there is a neighborhood which only contains that element). Equivalently, the group G is discrete if and only if its identity is isolated.
A subgroup H of a topological group G is a discrete subgroup if H is discrete when endowed with the subspace topology from G. In other words there is a neighbourhood of the identity in G containing no other element of H. For example, the integers, Z, form a discrete subgroup of the reals, R (with the standard metric topology), but the rational numbers, Q, do not.
Any group can be endowed with the discrete topology, making it a discrete topological group. Since every map from a discrete space is continuous, the topological homomorphisms between discrete groups are exactly the group homomorphisms between the underlying groups. Hence, there is an isomorphism between the category of groups and the category of discrete groups. Discrete groups can therefore be identified with their underlying (non-topological) groups.
There are some occasions when a topological group or Lie group is usefully endowed with the discrete topology, 'against nature'. This happens for example in the theory of the Bohr compactification, and in group cohomology theory of Lie groups.
A discrete isometry group is an isometry group such that for every point of the metric space the set of images of the point under the isometries is a discrete set. A discrete symmetry group is a symmetry group that is a discrete isometry group.
Properties
Since topological groups are homogeneous, one need only look at a single point to determine if the topological group is discrete. In particular, a topological group is discrete only if the singleton containing the identity is an open set.
A discrete group is the same thing as a zero-dimensional Lie group (uncountable discrete groups are not second-countable, so authors who require Lie groups to satisfy this axiom do not regard these groups as Lie groups). The identity component of a discrete group is just the trivial subgroup while the group of components is isomorphic to the group itself.
Since the only Hausdorff topology on a finite set is the discrete one, a finite Hausdorff topological group must necessarily be discrete. It follows that every finite subgroup of a Hausdorff group is discrete.
A discrete subgroup H of G is cocompact if there is a compact subset K of G such that HK = G.
Discrete normal subgroups play an important role in the theory of covering groups and locally isomorphic groups. A discrete normal subgroup of a connected group G necessarily lies in the center of G and is therefore abelian.
Other properties:
every discrete group is totally disconnected
every subgroup of a discrete group is discrete.
every quotient of a discrete group is discrete.
the product of a finite number of discrete groups is discrete.
a discrete group is compact if and only if it is f |
https://en.wikipedia.org/wiki/List%20of%20curves | This is a list of Wikipedia articles about curves used in different fields: mathematics (including geometry, statistics, and applied mathematics), physics, engineering, economics, medicine, biology, psychology, ecology, etc.
Mathematics (Geometry)
Algebraic curves
Rational curves
Rational curves are subdivided according to the degree of the polynomial.
Degree 1
Line
Degree 2
Plane curves of degree 2 are known as conics or conic sections and include
Circle
Unit circle
Ellipse
Parabola
Hyperbola
Unit hyperbola
Degree 3
Cubic plane curves include
Cubic parabola
Folium of Descartes
Cissoid of Diocles
Conchoid of de Sluze
Right strophoid
Semicubical parabola
Serpentine curve
Trident curve
Trisectrix of Maclaurin
Tschirnhausen cubic
Witch of Agnesi
Degree 4
Quartic plane curves include
Ampersand curve
Bean curve
Bicorn
Bow curve
Bullet-nose curve
Cartesian oval
Cruciform curve
Deltoid curve
Devil's curve
Hippopede
Kampyle of Eudoxus
Kappa curve
Lemniscate
Lemniscate of Booth
Lemniscate of Gerono
Lemniscate of Bernoulli
Limaçon
Cardioid
Limaçon trisectrix
Ovals of Cassini
Squircle
Trifolium Curve
Degree 5
Degree 6
Astroid
Atriphtaloid
Nephroid
Quadrifolium
Curve families of variable degree
Epicycloid
Epispiral
Epitrochoid
Hypocycloid
Lissajous curve
Poinsot's spirals
Rational normal curve
Rose curve
Curves with genus 1
Bicuspid curve
Cassinoide
Cubic curve
Elliptic curve
Watt's curve
Curves with genus > 1
Bolza surface (genus 2)
Klein quartic (genus 3)
Bring's curve (genus 4)
Macbeath surface (genus 7)
Butterfly curve (algebraic) (genus 7)
Curve families with variable genus
Polynomial lemniscate
Fermat curve
Sinusoidal spiral
Superellipse
Hurwitz surface
Elkies trinomial curves
Hyperelliptic curve
Classical modular curve
Cassini oval
Transcendental curves
Bowditch curve
Brachistochrone
Butterfly curve (transcendental)
Catenary
Clélies
Cochleoid
Cycloid
Horopter
Isochrone
Isochrone of Huygens (Tautochrone)
Isochrone of Leibniz
Isochrone of Varignon
Lamé curve
Pursuit curve
Rhumb line
Sinusoid
Spirals
Archimedean spiral
Cornu spiral
Cotes' spiral
Fermat's spiral
Galileo's spiral
Hyperbolic spiral
Lituus
Logarithmic spiral
Nielsen's spiral
Syntractrix
Tractrix
Trochoid
Piecewise constructions
Bézier curve
Loess curve
Lowess
Ogee
Polygonal curve
Maurer rose
Reuleaux triangle
Splines
B-spline
Nonuniform rational B-spline
Fractal curves
Blancmange curve
De Rham curve
Dragon curve
Koch curve
Lévy C curve
Sierpiński curve
Space-filling curve (Peano curve)
See also List of fractals by Hausdorff dimension.
Space curves/Skew curves
Conchospiral
Helix
Hemihelix, a quasi-helical shape characterized by multiple tendril perversions
Tendril perversion (a transition between back-to-back helices)
Seiffert's spiral
Slinky spiral
Twisted cubic
Viviani's curve
Curves generated by other curves
Caustic including Catacaustic and Diacaustic
Cissoid
Conchoid
Evolute
Glissette
Inverse curve
Involute
Isoptic including Orthoptic
Negative pedal cu |
https://en.wikipedia.org/wiki/Fisher%20equation | In financial mathematics and economics, the Fisher equation expresses the relationship between nominal interest rates, real interest rates, and inflation. Named after Irving Fisher, an American economist, it can be expressed as real interest rate ≈ nominal interest rate − inflation rate.
In more formal terms, where equals the real interest rate, equals the nominal interest rate, and equals the inflation rate, then . The approximation of is often used instead since the nominal interest rate, real interest rate, and inflation rate are usually close to zero.
Applications
Borrowing, lending and the time value of money
When loans are made, the amount borrowed and the repayments due to the lender are normally stated in nominal terms, before inflation. However, when inflation occurs, a dollar repaid in the future is worth less than a dollar borrowed today. To calculate the true economics of the loan, it is necessary to adjust the nominal cash flows to account for future inflation.
Inflation-indexed bonds
The Fisher equation can be used in the analysis of bonds. The real return on a bond is roughly equivalent to the nominal interest rate minus the expected inflation rate. But if actual inflation exceeds expected inflation during the life of the bond, the bondholder's real return will suffer. This risk is one of the reasons inflation-indexed bonds such as U.S. Treasury Inflation-Protected Securities were created to eliminate inflation uncertainty. Holders of indexed bonds are assured that the real cash flow of the bond (principal plus interest) will not be affected by inflation.
Cost–benefit analysis
As detailed by Steve Hanke, Philip Carver, and Paul Bugg (1975), cost benefit analysis can be greatly distorted if the exact Fisher equation is not applied. Prices and interest rates must both be projected in either real or nominal terms.
Monetary policy
The Fisher equation plays a key role in the Fisher hypothesis, which asserts that the real interest rate is unaffected by monetary policy and hence unaffected by the expected inflation rate. With a fixed real interest rate, a given percent change in the expected inflation rate will, according to the equation, necessarily be met with an equal percent change in the nominal interest rate in the same direction.
See also
Real versus nominal value (economics)
Yield
Yield curve
Interest rate
Inflation
References
Further reading
.
Mathematical finance
Monetary economics
Eponyms in economics
Equations
Fixed income analysis
Inflation
Interest rates |
https://en.wikipedia.org/wiki/Exponential%20sum | In mathematics, an exponential sum may be a finite Fourier series (i.e. a trigonometric polynomial), or other finite sum formed using the exponential function, usually expressed by means of the function
Therefore, a typical exponential sum may take the form
summed over a finite sequence of real numbers xn.
Formulation
If we allow some real coefficients an, to get the form
it is the same as allowing exponents that are complex numbers. Both forms are certainly useful in applications. A large part of twentieth century analytic number theory was devoted to finding good estimates for these sums, a trend started by basic work of Hermann Weyl in diophantine approximation.
Estimates
The main thrust of the subject is that a sum
is trivially estimated by the number N of terms. That is, the absolute value
by the triangle inequality, since each summand has absolute value 1. In applications one would like to do better. That involves proving some cancellation takes place, or in other words that this sum of complex numbers on the unit circle is not of numbers all with the same argument. The best that is reasonable to hope for is an estimate of the form
which signifies, up to the implied constant in the big O notation, that the sum resembles a random walk in two dimensions.
Such an estimate can be considered ideal; it is unattainable in many of the major problems, and estimates
have to be used, where the o(N) function represents only a small saving on the trivial estimate. A typical 'small saving' may be a factor of log(N), for example. Even such a minor-seeming result in the right direction has to be referred all the way back to the structure of the initial sequence xn, to show a degree of randomness. The techniques involved are ingenious and subtle.
A variant of 'Weyl differencing' investigated by Weyl involving a generating exponential sum
was previously studied by Weyl himself, he developed a method to express the sum as the value , where 'G' can be defined via a linear differential equation similar to Dyson equation obtained via summation by parts.
History
If the sum is of the form
where ƒ is a smooth function, we could use the Euler–Maclaurin formula to convert the series into an integral, plus some corrections involving derivatives of S(x), then for large values of a you could use "stationary phase" method to calculate the integral and give an approximate evaluation of the sum. Major advances in the subject were Van der Corput's method (c. 1920), related to the principle of stationary phase, and the later Vinogradov method (c.1930).
The large sieve method (c.1960), the work of many researchers, is a relatively transparent general principle; but no one method has general application.
Types of exponential sum
Many types of sums are used in formulating particular problems; applications require usually a reduction to some known type, often by ingenious manipulations. Partial summation can be used to remove coefficients an, in many cases.
|
https://en.wikipedia.org/wiki/Lattice%20%28group%29 | In geometry and group theory, a lattice in the real coordinate space is an infinite set of points in this space with the properties that coordinate-wise addition or subtraction of two points in the lattice produces another lattice point, that the lattice points are all separated by some minimum distance, and that every point in the space is within some maximum distance of a lattice point. Closure under addition and subtraction means that a lattice must be a subgroup of the additive group of the points in the space, and the requirements of minimum and maximum distance can be summarized by saying that a lattice is a Delone set. More abstractly, a lattice can be described as a free abelian group of dimension which spans the vector space . For any basis of , the subgroup of all linear combinations with integer coefficients of the basis vectors forms a lattice, and every lattice can be formed from a basis in this way. A lattice may be viewed as a regular tiling of a space by a primitive cell.
Lattices have many significant applications in pure mathematics, particularly in connection to Lie algebras, number theory and group theory. They also arise in applied mathematics in connection with coding theory, in percolation theory to study connectivity arising from small-scale interactions, cryptography because of conjectured computational hardness of several lattice problems, and are used in various ways in the physical sciences. For instance, in materials science and solid-state physics, a lattice is a synonym for the framework of a crystalline structure, a 3-dimensional array of regularly spaced points coinciding in special cases with the atom or molecule positions in a crystal. More generally, lattice models are studied in physics, often by the techniques of computational physics.
Symmetry considerations and examples
A lattice is the symmetry group of discrete translational symmetry in n directions. A pattern with this lattice of translational symmetry cannot have more, but may have less symmetry than the lattice itself. As a group (dropping its geometric structure) a lattice is a finitely-generated free abelian group, and thus isomorphic to .
A lattice in the sense of a 3-dimensional array of regularly spaced points coinciding with e.g. the atom or molecule positions in a crystal, or more generally, the orbit of a group action under translational symmetry, is a translation of the translation lattice: a coset, which need not contain the origin, and therefore need not be a lattice in the previous sense.
A simple example of a lattice in is the subgroup . More complicated examples include the E8 lattice, which is a lattice in , and the Leech lattice in . The period lattice in is central to the study of elliptic functions, developed in nineteenth century mathematics; it generalizes to higher dimensions in the theory of abelian functions. Lattices called root lattices are important in the theory of simple Lie algebras; for example, the E8 lattice i |
https://en.wikipedia.org/wiki/Lattice%20%28order%29 | A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet). An example is given by the power set of a set, partially ordered by inclusion, for which the supremum is the union and the infimum is the intersection. Another example is given by the natural numbers, partially ordered by divisibility, for which the supremum is the least common multiple and the infimum is the greatest common divisor.
Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities. Since the two definitions are equivalent, lattice theory draws on both order theory and universal algebra. Semilattices include lattices, which in turn include Heyting and Boolean algebras. These lattice-like structures all admit order-theoretic as well as algebraic descriptions.
The sub-field of abstract algebra that studies lattices is called lattice theory.
Definition
A lattice can be defined either order-theoretically as a partially ordered set, or as an algebraic structure.
As partially ordered set
A partially ordered set (poset) is called a lattice if it is both a join- and a meet-semilattice, i.e. each two-element subset has a join (i.e. least upper bound, denoted by ) and dually a meet (i.e. greatest lower bound, denoted by ). This definition makes and binary operations. Both operations are monotone with respect to the given order: and implies that and
It follows by an induction argument that every non-empty finite subset of a lattice has a least upper bound and a greatest lower bound. With additional assumptions, further conclusions may be possible; see Completeness (order theory) for more discussion of this subject. That article also discusses how one may rephrase the above definition in terms of the existence of suitable Galois connections between related partially ordered sets—an approach of special interest for the category theoretic approach to lattices, and for formal concept analysis.
Given a subset of a lattice, meet and join restrict to partial functions – they are undefined if their value is not in the subset The resulting structure on is called a . In addition to this extrinsic definition as a subset of some other algebraic structure (a lattice), a partial lattice can also be intrinsically defined as a set with two partial binary operations satisfying certain axioms.
As algebraic structure
A lattice is an algebraic structure , consisting of a set and two binary, commutative and associative operations and on satisfying the following axiomatic identities for all elements (sometimes called ):
The following two identities are also usually regarded as axioms, even though they follow from the two absorption laws taken together. These are called .
These axioms asse |
https://en.wikipedia.org/wiki/List%20of%20curves%20topics | This is an alphabetical index of articles related to curves used in mathematics.
Acnode
Algebraic curve
Arc
Asymptote
Asymptotic curve
Barbier's theorem
Bézier curve
Bézout's theorem
Birch and Swinnerton-Dyer conjecture
Bitangent
Bitangents of a quartic
Cartesian coordinate system
Caustic
Cesàro equation
Chord (geometry)
Cissoid
Circumference
Closed timelike curve
concavity
Conchoid (mathematics)
Confocal
Contact (mathematics)
Contour line
Crunode
Cubic Hermite curve
Curvature
Curve orientation
Curve fitting
Curve-fitting compaction
Curve of constant width
Curve of pursuit
Curves in differential geometry
Cusp
Cyclogon
De Boor algorithm
Differential geometry of curves
Eccentricity (mathematics)
Elliptic curve cryptography
Envelope (mathematics)
Fenchel's theorem
Genus (mathematics)
Geodesic
Geometric genus
Great-circle distance
Harmonograph
Hedgehog (curve)
Hilbert's sixteenth problem
Hyperelliptic curve cryptography
Inflection point
Inscribed square problem
intercept, y-intercept, x-intercept
Intersection number
Intrinsic equation
Isoperimetric inequality
Jordan curve
Jordan curve theorem
Knot
Limit cycle
Linking coefficient
List of circle topics
Loop (knot)
M-curve
Mannheim curve
Meander (mathematics)
Mordell conjecture
Natural representation
Opisometer
Orbital elements
Osculating circle
Osculating plane
Osgood curve
Parallel (curve)
Parallel transport
Parametric curve
Bézier curve
Spline (mathematics)
Hermite spline
Beta spline
B-spline
Higher-order spline
NURBS
Perimeter
Pi
Plane curve
Pochhammer contour
Polar coordinate system
Prime geodesic
Projective line
Ray
Regular parametric representation
Reuleaux triangle
Ribaucour curve
Riemann–Hurwitz formula
Riemann–Roch theorem
Riemann surface
Road curve
Sato–Tate conjecture
secant
Singular solution
Sinuosity
Slope
Space curve
Spinode
Square wheel
Subtangent
Tacnode
Tangent
Tangent space
Tangential angle
Torsion of curves
Trajectory
Transcendental curve
W-curve
Whewell equation
World line
See also
Curve
List of curves
List of differential geometry topics
Curve |
https://en.wikipedia.org/wiki/Superalgebra | In mathematics and theoretical physics, a superalgebra is a Z2-graded algebra. That is, it is an algebra over a commutative ring or field with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading.
The prefix super- comes from the theory of supersymmetry in theoretical physics. Superalgebras and their representations, supermodules, provide an algebraic framework for formulating supersymmetry. The study of such objects is sometimes called super linear algebra. Superalgebras also play an important role in related field of supergeometry where they enter into the definitions of graded manifolds, supermanifolds and superschemes.
Formal definition
Let K be a commutative ring. In most applications, K is a field of characteristic 0, such as R or C.
A superalgebra over K is a K-module A with a direct sum decomposition
together with a bilinear multiplication A × A → A such that
where the subscripts are read modulo 2, i.e. they are thought of as elements of Z2.
A superring, or Z2-graded ring, is a superalgebra over the ring of integers Z.
The elements of each of the Ai are said to be homogeneous. The parity of a homogeneous element x, denoted by , is 0 or 1 according to whether it is in A0 or A1. Elements of parity 0 are said to be even and those of parity 1 to be odd. If x and y are both homogeneous then so is the product xy and .
An associative superalgebra is one whose multiplication is associative and a unital superalgebra is one with a multiplicative identity element. The identity element in a unital superalgebra is necessarily even. Unless otherwise specified, all superalgebras in this article are assumed to be associative and unital.
A commutative superalgebra (or supercommutative algebra) is one which satisfies a graded version of commutativity. Specifically, A is commutative if
for all homogeneous elements x and y of A. There are superalgebras that are commutative in the ordinary sense, but not in the superalgebra sense. For this reason, commutative superalgebras are often called supercommutative in order to avoid confusion.
Examples
Any algebra over a commutative ring K may be regarded as a purely even superalgebra over K; that is, by taking A1 to be trivial.
Any Z- or N-graded algebra may be regarded as superalgebra by reading the grading modulo 2. This includes examples such as tensor algebras and polynomial rings over K.
In particular, any exterior algebra over K is a superalgebra. The exterior algebra is the standard example of a supercommutative algebra.
The symmetric polynomials and alternating polynomials together form a superalgebra, being the even and odd parts, respectively. Note that this is a different grading from the grading by degree.
Clifford algebras are superalgebras. They are generally noncommutative.
The set of all endomorphisms (denoted , where the boldface is referred to as internal , composed of all linear maps) of a super vector space forms a superalgebra under com |
https://en.wikipedia.org/wiki/California%20unemployment%20statistics | The following is a list of California unemployment statistics.
Many of the counties with the lowest unemployment rates had relatively high levels of income. They were also located in Northern California, with two exceptions: Orange and San Luis Obispo counties. The counties with the highest unemployment rates were generally located in inland areas and had lower levels of income.
Unemployment rate has reached 12.4 percent in 2010 which is highest recorded from 1976. Unemployment rates in California reached historic lows in 2000 and 2006. Unemployment rates in California were relatively low during the early 2000s but increased drastically in late 2000s
Statewide unemployment
Data released December 13, 2017 for November 2017.
Unemployment by county
Data released March 7, 2014 for January 2014, except population data (released 2012).
Historical statewide unemployment rates
The unemployment rates in this table are annual averages without seasonal adjustment. The 1976-1989 rates are based on the March 2004 benchmark and were last updated April 26, 2005. The 1990-2012 rates are based on the March 2006 benchmark.
See also
Economy of California
List of U.S. states by unemployment rate
Unemployment in the United States
References
Unemployment in the United States
unemployment statistics |
https://en.wikipedia.org/wiki/Logarithmic%20derivative | In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function f is defined by the formula
where is the derivative of f. Intuitively, this is the infinitesimal relative change in f; that is, the infinitesimal absolute change in f, namely scaled by the current value of f.
When f is a function f(x) of a real variable x, and takes real, strictly positive values, this is equal to the derivative of ln(f), or the natural logarithm of f. This follows directly from the chain rule:
Basic properties
Many properties of the real logarithm also apply to the logarithmic derivative, even when the function does not take values in the positive reals. For example, since the logarithm of a product is the sum of the logarithms of the factors, we have
So for positive-real-valued functions, the logarithmic derivative of a product is the sum of the logarithmic derivatives of the factors. But we can also use the Leibniz law for the derivative of a product to get
Thus, it is true for any function that the logarithmic derivative of a product is the sum of the logarithmic derivatives of the factors (when they are defined).
A corollary to this is that the logarithmic derivative of the reciprocal of a function is the negation of the logarithmic derivative of the function:
just as the logarithm of the reciprocal of a positive real number is the negation of the logarithm of the number.
More generally, the logarithmic derivative of a quotient is the difference of the logarithmic derivatives of the dividend and the divisor:
just as the logarithm of a quotient is the difference of the logarithms of the dividend and the divisor.
Generalising in another direction, the logarithmic derivative of a power (with constant real exponent) is the product of the exponent and the logarithmic derivative of the base:
just as the logarithm of a power is the product of the exponent and the logarithm of the base.
In summary, both derivatives and logarithms have a product rule, a reciprocal rule, a quotient rule, and a power rule (compare the list of logarithmic identities); each pair of rules is related through the logarithmic derivative.
Computing ordinary derivatives using logarithmic derivatives
Logarithmic derivatives can simplify the computation of derivatives requiring the product rule while producing the same result. The procedure is as follows: Suppose that and that we wish to compute . Instead of computing it directly as , we compute its logarithmic derivative. That is, we compute:
Multiplying through by ƒ computes :
This technique is most useful when ƒ is a product of a large number of factors. This technique makes it possible to compute by computing the logarithmic derivative of each factor, summing, and multiplying by .
For example, we can compute the logarithmic derivative of to be .
Integrating factors
The logarithmic derivative idea is closely connected to the integrating factor method for first-order differential e |
https://en.wikipedia.org/wiki/Connection%20%28principal%20bundle%29 | In mathematics, and especially differential geometry and gauge theory, a connection is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. A principal G-connection on a principal G-bundle P over a smooth manifold M is a particular type of connection which is compatible with the action of the group G.
A principal connection can be viewed as a special case of the notion of an Ehresmann connection, and is sometimes called a principal Ehresmann connection. It gives rise to (Ehresmann) connections on any fiber bundle associated to P via the associated bundle construction. In particular, on any associated vector bundle the principal connection induces a covariant derivative, an operator that can differentiate sections of that bundle along tangent directions in the base manifold. Principal connections generalize to arbitrary principal bundles the concept of a linear connection on the frame bundle of a smooth manifold.
Formal definition
Let be a smooth principal G-bundle over a smooth manifold . Then a principal -connection on is a differential 1-form on with values in the Lie algebra of which is -equivariant and reproduces the Lie algebra generators of the fundamental vector fields on .
In other words, it is an element ω of such that
where denotes right multiplication by , and is the adjoint representation on (explicitly, );
if and is the vector field on P associated to ξ by differentiating the G action on P, then (identically on ).
Sometimes the term principal G-connection refers to the pair and itself is called the connection form or connection 1-form of the principal connection.
Computational remarks
Most known non-trivial computations of principal G-connections are done with homogeneous spaces because of the triviality of the (co)tangent bundle. (For example, let , be a principal G-bundle over ) This means that 1-forms on the total space are canonically isomorphic to , where is the dual lie algebra, hence G-connections are in bijection with .
Relation to Ehresmann connections
A principal G-connection on determines an Ehresmann connection on in the following way. First note that the fundamental vector fields generating the action on provide a bundle isomorphism (covering the identity of ) from the bundle to , where is the kernel of the tangent mapping which is called the vertical bundle of . It follows that determines uniquely a bundle map which is the identity on . Such a projection is uniquely determined by its kernel, which is a smooth subbundle of (called the horizontal bundle) such that . This is an Ehresmann connection.
Conversely, an Ehresmann connection (or ) on defines a principal -connection if and only if it is -equivariant in the sense that .
Pull back via trivializing section
A trivializing section of a principal bundle P is given by a section s of P over an open subset U of M. Then the pullback s*ω of a principa |
https://en.wikipedia.org/wiki/Expectation%E2%80%93maximization%20algorithm | In statistics, an expectation–maximization (EM) algorithm is an iterative method to find (local) maximum likelihood or maximum a posteriori (MAP) estimates of parameters in statistical models, where the model depends on unobserved latent variables. The EM iteration alternates between performing an expectation (E) step, which creates a function for the expectation of the log-likelihood evaluated using the current estimate for the parameters, and a maximization (M) step, which computes parameters maximizing the expected log-likelihood found on the E step. These parameter-estimates are then used to determine the distribution of the latent variables in the next E step.
History
The EM algorithm was explained and given its name in a classic 1977 paper by Arthur Dempster, Nan Laird, and Donald Rubin. They pointed out that the method had been "proposed many times in special circumstances" by earlier authors. One of the earliest is the gene-counting method for estimating allele frequencies by Cedric Smith. Another was proposed by H.O. Hartley in 1958, and Hartley and Hocking in 1977, from which many of the ideas in the Dempster–Laird–Rubin paper originated. Another one by S.K Ng, Thriyambakam Krishnan and G.J McLachlan in 1977. Hartley’s ideas can be broadened to any grouped discrete distribution. A very detailed treatment of the EM method for exponential families was published by Rolf Sundberg in his thesis and several papers, following his collaboration with Per Martin-Löf and Anders Martin-Löf. The Dempster–Laird–Rubin paper in 1977 generalized the method and sketched a convergence analysis for a wider class of problems. The Dempster–Laird–Rubin paper established the EM method as an important tool of statistical analysis. See also Meng and van Dyk (1997).
The convergence analysis of the Dempster–Laird–Rubin algorithm was flawed and a correct convergence analysis was published by C. F. Jeff Wu in 1983.
Wu's proof established the EM method's convergence also outside of the exponential family, as claimed by Dempster–Laird–Rubin.
Introduction
The EM algorithm is used to find (local) maximum likelihood parameters of a statistical model in cases where the equations cannot be solved directly. Typically these models involve latent variables in addition to unknown parameters and known data observations. That is, either missing values exist among the data, or the model can be formulated more simply by assuming the existence of further unobserved data points. For example, a mixture model can be described more simply by assuming that each observed data point has a corresponding unobserved data point, or latent variable, specifying the mixture component to which each data point belongs.
Finding a maximum likelihood solution typically requires taking the derivatives of the likelihood function with respect to all the unknown values, the parameters and the latent variables, and simultaneously solving the resulting equations. In statistical models with latent |
https://en.wikipedia.org/wiki/Gona%C3%AFves | Gonaïves (; , ) is a commune in northern Haiti, and the capital of the Artibonite department of Haiti. It has a population of about 300,000 people, but current statistics are unclear, as there has been no census since 2003.
History
The city of Gonaïves was founded around 1422 by a group of Taíno, who named it Gonaibo (to designate a locality of cacicat of the Jaragua). The Gulf of Gonâve is named after the town.
In 1802, an important battle of the Haitian Revolution, the Battle of Ravine-à-Couleuvres was fought near Gonaïves.
Gonaïves is also known as Haiti's city of independence, because it was the location of Jean-Jacques Dessalines declaring Haiti independent from France on January 1, 1804, by reading the Act of Independence, drafted by Boisrond Tonnerre, on the Place d'Armes of the town.
Marie-Claire Heureuse Félicité, the wife of Jean-Jacques Dessalines, died here in August 1858.
In the early 2000s, Gonaïves was the scene of substantial rioting and violence motivated primarily by opposition to President Jean-Bertrand Aristide, and on February 5, 2004, a group calling itself the Revolutionary Artibonite Resistance Front seized control of the city, starting the 2004 Haïtian rebellion. But in recent years, the city has seen a complete return to order.
Even compared with other Haitian port cities, Gonaïves has long been vulnerable to hurricanes due to its location in a flood plain and due to the surrounding unforested mountains. In September 2004, Hurricane Jeanne caused major flooding and mudslides in the city. Four years later, the city was again devastated by another storm, Hurricane Hanna, which killed 529 people, mostly in flooded sections of Gonaïves, where the destruction was described as "catastrophic" and 495 bodies were discovered as late as September 5. Haitian authorities said the tally would grow once officials were able to make their way through the city. "The assessment was only partial, because it was impossible to enter the city at that moment". Gonaïves Mayor Stephen Moise said at least 48,000 people from the Gonaïves area were forced into shelters.
In 2020, President Jovenel Moïse skipped a traditional visit to Gonaïves during a climate of violence. According to local media, an armed group targeted Prime Minister Ariel Henry's visit on 1 January 2022, resulting in one death and two injuries.
Education
University
Gonaives has some training centers including the Université Publique de l'Artibonite aux Gonaïves (UPAG); and
The Law and Economics School of Gonaives.
Health
Gonaives is home to the recently renovated La Providence Hospital.
Sport
Gonaives has some major league teams including Eclair AC and Racing FC (Gonaives).
Media
Radio SuperXtraL'unique 94.1 FM
Radio Motivation Fm Listen Live - 95.5 MHz FM, Gonaïves, Haiti
Radio Boss Haiti 103.9 FM - The Radio of the moment.
Radio Redemption 100.9 FM
Radio Xplosion 96.5 FM
Radio Tele Satellite FM - 100.5 FM
Tele Radio new star fm 99.9 Chaine 13
Radio Continentale 99 |
https://en.wikipedia.org/wiki/1673%20in%20science | The year 1673 in science and technology involved some significant events.
Mathematics
John Kersey begins publication of The Elements of that Mathematical Art Commonly Called Algebra.
Samuel Morland publishes A Perpetual Almanack and Several Useful Tables.
Microbiology
Antonie van Leeuwenhoek's observations with the microscope are first published in Philosophical Transactions of the Royal Society.
Physics
Christiaan Huygens publishes his mathematical analysis of the pendulum, Horologium Oscillatorium sive de motu pendulorum.
Births
August 10 – Johann Konrad Dippel, German theologian, alchemist and physician (died 1734)
Deaths
May 6 – Werner Rolfinck, German scientist (born 1599)
August 17 – Regnier de Graaf, Dutch physician and anatomist who discovered the ovarian follicles (born 1641)
December 15 – Margaret Cavendish, Duchess of Newcastle-upon-Tyne, English natural philosopher (born 1623)
References
17th century in science
1670s in science |
https://en.wikipedia.org/wiki/1622%20in%20science | The year 1622 in science and technology involved some significant events.
Mathematics
The slide rule is invented by William Oughtred (1574–1660), an English mathematician, and later becomes the calculating tool of choice until the electronic calculator takes over in the early 1970s.
Physiology and medicine
Gaspare Aselli discovers the lacteal vessels of the lymphatic system.
Flemish anatomist Giulio Casserio publishes Nova anatomia in Frankfurt, containing clear copperplate engravings of the human anatomy.
Technology
February 22 – An English patent is granted for Dud Dudley's process for smelting iron ore with coke.
Births
January 28 – Adrien Auzout, French astronomer (died 1691)
March 10 – Johann Rahn, Swiss mathematician (died 1676)
April 5 – Vincenzo Viviani, Italian mathematician and scientist (died 1703)
undated – Jean Pecquet, French anatomist (died 1674)
Deaths
January 23 – William Baffin, English explorer and navigator (born 1584)
February 19 – Sir Henry Savile, English polymath and benefactor (born 1549)
April 13 – Katharina Kepler, German healer and mother of Johannes Kepler (born 1546)
May 15 – Petrus Plancius, Flemish cartographer and cosmographer (born 1552)
References
17th century in science
1620s in science |
https://en.wikipedia.org/wiki/Partition%20of%20an%20interval | In mathematics, a partition of an interval on the real line is a finite sequence of real numbers such that
.
In other terms, a partition of a compact interval is a strictly increasing sequence of numbers (belonging to the interval itself) starting from the initial point of and arriving at the final point of .
Every interval of the form is referred to as a subinterval of the partition x.
Refinement of a partition
Another partition of the given interval [a, b] is defined as a refinement of the partition , if contains all the points of and possibly some other points as well; the partition is said to be “finer” than . Given two partitions, and , one can always form their common refinement, denoted , which consists of all the points of and , in increasing order.
Norm of a partition
The norm (or mesh) of the partition
is the length of the longest of these subintervals
{{math|maxxi − xi−1}} : i 1, … , n .
Applications
Partitions are used in the theory of the Riemann integral, the Riemann–Stieltjes integral and the regulated integral. Specifically, as finer partitions of a given interval are considered, their mesh approaches zero and the Riemann sum based on a given partition approaches the Riemann integral.
Tagged partitions
A tagged partition is a partition of a given interval together with a finite sequence of numbers subject to the conditions that for each ,
.
In other words, a tagged partition is a partition together with a distinguished point of every subinterval: its mesh is defined in the same way as for an ordinary partition. It is possible to define a partial order on the set of all tagged partitions by saying that one tagged partition is bigger than another if the bigger one is a refinement of the smaller one.
Suppose that together with is a tagged partition of , and that together with is another tagged partition of . We say that together with is a refinement of a tagged partition together with if for each integer with , there is an integer such that and such that for some with . Said more simply, a refinement of a tagged partition takes the starting partition and adds more tags, but does not take any away.
See also
Regulated integral
Riemann integral
Riemann–Stieltjes integral
References
Further reading
Mathematical analysis |
https://en.wikipedia.org/wiki/Fuchsian%20group | In mathematics, a Fuchsian group is a discrete subgroup of PSL(2,R). The group PSL(2,R) can be regarded equivalently as a group of orientation-preserving isometries of the hyperbolic plane, or conformal transformations of the unit disc, or conformal transformations of the upper half plane, so a Fuchsian group can be regarded as a group acting on any of these spaces. There are some variations of the definition: sometimes the Fuchsian group is assumed to be finitely generated, sometimes it is allowed to be a subgroup of PGL(2,R) (so that it contains orientation-reversing elements), and sometimes it is allowed to be a Kleinian group (a discrete subgroup of PSL(2,C)) which is conjugate to a subgroup of PSL(2,R).
Fuchsian groups are used to create Fuchsian models of Riemann surfaces. In this case, the group may be called the Fuchsian group of the surface. In some sense, Fuchsian groups do for non-Euclidean geometry what crystallographic groups do for Euclidean geometry. Some Escher graphics are based on them (for the disc model of hyperbolic geometry).
General Fuchsian groups were first studied by , who was motivated by the paper , and therefore named them after Lazarus Fuchs.
Fuchsian groups on the upper half-plane
Let H = {z in C : Im(z) > 0} be the upper half-plane. Then H is a model of the hyperbolic plane when endowed with the metric
The group PSL(2,R) acts on H by linear fractional transformations (also known as Möbius transformations):
This action is faithful, and in fact PSL(2,R) is isomorphic to the group of all orientation-preserving isometries of H.
A Fuchsian group Γ may be defined to be a subgroup of PSL(2,R), which acts discontinuously on H. That is,
For every z in H, the orbit Γz = {γz : γ in Γ} has no accumulation point in H.
An equivalent definition for Γ to be Fuchsian is that Γ be a discrete group, which means that:
Every sequence {γn} of elements of Γ converging to the identity in the usual topology of point-wise convergence is eventually constant, i.e. there exists an integer N such that for all n > N, γn = I, where I is the identity matrix.
Although discontinuity and discreteness are equivalent in this case, this is not generally true for the case of an arbitrary group of conformal homeomorphisms acting on the full Riemann sphere (as opposed to H). Indeed, the Fuchsian group PSL(2,Z) is discrete but has accumulation points on the real number line Im z = 0: elements of PSL(2,Z) will carry z = 0 to every rational number, and the rationals Q are dense in R.
General definition
A linear fractional transformation defined by a matrix from PSL(2,C) will preserve the Riemann sphere P1(C) = C ∪ ∞, but will send the upper-half plane H to some open disk Δ. Conjugating by such a transformation will send a discrete subgroup of PSL(2,R) to a discrete subgroup of PSL(2,C) preserving Δ.
This motivates the following definition of a Fuchsian group. Let Γ ⊂ PSL(2,C) act invariantly on a proper, open disk Δ ⊂ C ∪ ∞, that is, Γ(Δ) = |
https://en.wikipedia.org/wiki/%C3%89l%C3%A9ments%20de%20g%C3%A9om%C3%A9trie%20alg%C3%A9brique | The Éléments de géométrie algébrique ("Elements of Algebraic Geometry") by Alexander Grothendieck (assisted by Jean Dieudonné), or EGA for short, is a rigorous treatise, in French, on algebraic geometry that was published (in eight parts or fascicles) from 1960 through 1967 by the Institut des Hautes Études Scientifiques. In it, Grothendieck established systematic foundations of algebraic geometry, building upon the concept of schemes, which he defined. The work is now considered the foundation stone and basic reference of modern algebraic geometry.
Editions
Initially thirteen chapters were planned, but only the first four (making a total of approximately 1500 pages) were published. Much of the material which would have been found in the following chapters can be found, in a less polished form, in the Séminaire de géométrie algébrique (known as SGA). Indeed, as explained by Grothendieck in the preface of the published version of SGA, by 1970 it had become clear that incorporating all of the planned material in EGA would require significant changes in the earlier chapters already published, and that therefore the prospects of completing EGA in the near term were limited. An obvious example is provided by derived categories, which became an indispensable tool in the later SGA volumes, but was not yet used in EGA III as the theory was not yet developed at the time. Considerable effort was therefore spent to bring the published SGA volumes to a high degree of completeness and rigour. Before work on the treatise was abandoned, there were plans in 1966–67 to expand the group of authors to include Grothendieck's students Pierre Deligne and Michel Raynaud, as evidenced by published correspondence between Grothendieck and David Mumford. Grothendieck's letter of 4 November 1966 to Mumford also indicates that the second-edition revised structure was in place by that time, with Chapter VIII already intended to cover the Picard scheme. In that letter he estimated that at the pace of writing up to that point, the following four chapters (V to VIII) would have taken eight years to complete, indicating an intended length comparable to the first four chapters, which had been in preparation for about eight years at the time.
Grothendieck nevertheless wrote a revised version of EGA I which was published by Springer-Verlag. It updates the terminology, replacing "prescheme" by "scheme" and "scheme" by "separated scheme", and heavily emphasizes the use of representable functors. The new preface of the second edition also includes a slightly revised plan of the complete treatise, now divided into twelve chapters.
Grothendieck's EGA V which deals with Bertini type theorems is to some extent available from the Grothendieck Circle website. Monografie Matematyczne in Poland has accepted this volume for publication, but the editing process is quite slow (as of 2010).
James Milne has preserved some of the original Grothendieck notes and a translation of them into Engli |
https://en.wikipedia.org/wiki/List%20of%20census%20divisions%20of%20Alberta | Statistics Canada divides the province of Alberta into nineteen census divisions. Unlike in some other provinces, census divisions do not reflect the organization of local government in Alberta. These areas exist solely for the purposes of statistical analysis and presentation; they have no government of their own.
Alberta's census divisions consist of numerous census subdivisions. The types of census subdivisions within an Alberta census division may include:
cities, towns, villages, and summer villages (urban municipalities);
specialized municipalities;
municipal districts, special areas, and improvement districts (rural municipalities);
Indian reserves; and
Indian settlements.
List of census divisions
The following is a list of Alberta's census divisions. Population, area, and density figures are from the 2016 Census.
See also
List of cities in Alberta
List of communities in Alberta
List of designated places in Alberta
List of hamlets in Alberta
List of Indian reserves in Alberta
List of municipal districts in Alberta
List of municipalities in Alberta
List of population centres in Alberta
List of summer villages in Alberta
List of towns in Alberta
List of villages in Alberta
Specialized municipalities of Alberta
Subdivisions of Canada
References
Census divisions |
https://en.wikipedia.org/wiki/Kaprekar%20number | In mathematics, a natural number in a given number base is a -Kaprekar number if the representation of its square in that base can be split into two parts, where the second part has digits, that add up to the original number. The numbers are named after D. R. Kaprekar.
Definition and properties
Let be a natural number. We define the Kaprekar function for base and power to be the following:
,
where and
A natural number is a -Kaprekar number if it is a fixed point for , which occurs if . and are trivial Kaprekar numbers for all and , all other Kaprekar numbers are nontrivial Kaprekar numbers.
For example, in base 10, 45 is a 2-Kaprekar number, because
A natural number is a sociable Kaprekar number if it is a periodic point for , where for a positive integer (where is the th iterate of ), and forms a cycle of period . A Kaprekar number is a sociable Kaprekar number with , and a amicable Kaprekar number is a sociable Kaprekar number with .
The number of iterations needed for to reach a fixed point is the Kaprekar function's persistence of , and undefined if it never reaches a fixed point.
There are only a finite number of -Kaprekar numbers and cycles for a given base , because if , where then
and , , and . Only when do Kaprekar numbers and cycles exist.
If is any divisor of , then is also a -Kaprekar number for base .
In base , all even perfect numbers are Kaprekar numbers. More generally, any numbers of the form or for natural number are Kaprekar numbers in base 2.
Set-theoretic definition and unitary divisors
We can define the set for a given integer as the set of integers for which there exist natural numbers and satisfying the Diophantine equation
, where
An -Kaprekar number for base is then one which lies in the set .
It was shown in 2000 that there is a bijection between the unitary divisors of and the set defined above. Let denote the multiplicative inverse of modulo , namely the least positive integer such that , and for each unitary divisor of let and . Then the function is a bijection from the set of unitary divisors of onto the set . In particular, a number is in the set if and only if for some unitary divisor of .
The numbers in occur in complementary pairs, and . If is a unitary divisor of then so is , and if then .
Kaprekar numbers for
b = 4k + 3 and p = 2n + 1
Let and be natural numbers, the number base , and . Then:
is a Kaprekar number.
is a Kaprekar number for all natural numbers .
b = m2k + m + 1 and p = mn + 1
Let , , and be natural numbers, the number base , and the power . Then:
is a Kaprekar number.
is a Kaprekar number.
b = m2k + m + 1 and p = mn + m − 1
Let , , and be natural numbers, the number base , and the power . Then:
is a Kaprekar number.
is a Kaprekar number.
b = m2k + m2 − m + 1 and p = mn + 1
Let , , and be natural numbers, the number base , and the power . Then:
is a Kaprekar number.
is a Kaprekar num |
https://en.wikipedia.org/wiki/Divisible%20group | In mathematics, especially in the field of group theory, a divisible group is an abelian group in which every element can, in some sense, be divided by positive integers, or more accurately, every element is an nth multiple for each positive integer n. Divisible groups are important in understanding the structure of abelian groups, especially because they are the injective abelian groups.
Definition
An abelian group is divisible if, for every positive integer and every , there exists such that . An equivalent condition is: for any positive integer , , since the existence of for every and implies that , and the other direction is true for every group. A third equivalent condition is that an abelian group is divisible if and only if is an injective object in the category of abelian groups; for this reason, a divisible group is sometimes called an injective group.
An abelian group is -divisible for a prime if for every , there exists such that . Equivalently, an abelian group is -divisible if and only if .
Examples
The rational numbers form a divisible group under addition.
More generally, the underlying additive group of any vector space over is divisible.
Every quotient of a divisible group is divisible. Thus, is divisible.
The p-primary component of , which is isomorphic to the p-quasicyclic group , is divisible.
The multiplicative group of the complex numbers is divisible.
Every existentially closed abelian group (in the model theoretic sense) is divisible.
Properties
If a divisible group is a subgroup of an abelian group then it is a direct summand of that abelian group.
Every abelian group can be embedded in a divisible group.
Non-trivial divisible groups are not finitely generated.
Further, every abelian group can be embedded in a divisible group as an essential subgroup in a unique way.
An abelian group is divisible if and only if it is p-divisible for every prime p.
Let be a ring. If is a divisible group, then is injective in the category of -modules.
Structure theorem of divisible groups
Let G be a divisible group. Then the torsion subgroup Tor(G) of G is divisible. Since a divisible group is an injective module, Tor(G) is a direct summand of G. So
As a quotient of a divisible group, G/Tor(G) is divisible. Moreover, it is torsion-free. Thus, it is a vector space over Q and so there exists a set I such that
The structure of the torsion subgroup is harder to determine, but one can show that for all prime numbers p there exists such that
where is the p-primary component of Tor(G).
Thus, if P is the set of prime numbers,
The cardinalities of the sets I and Ip for p ∈ P are uniquely determined by the group G.
Injective envelope
As stated above, any abelian group A can be uniquely embedded in a divisible group D as an essential subgroup. This divisible group D is the injective envelope of A, and this concept is the injective hull in the category of abelian groups.
Reduced abelian groups
An |
https://en.wikipedia.org/wiki/Pierre%20Boutroux | Pierre Léon Boutroux (; 6 December 1880 – 15 August 1922) was a French mathematician and historian of science. Boutroux is chiefly known for his work in the history and philosophy of mathematics.
Biography
He was born in Paris on 6 December 1880 into a well connected family of the French intelligentsia. His father was the philosopher Émile Boutroux. His mother was Aline Catherine Eugénie Poincaré, sister of the scientist and mathematician Henri Poincaré. A cousin of Aline, Raymond Poincaré was to be President of France.
He occupied the mathematics chair at Princeton University from 1913 until 1914. He occupied the History of sciences chair from 1920 to 1922.
Boutroux published his major work Les principes de l'analyse mathématique in two volumes; Volume 1 in 1914 and Volume 2 in 1919. This is a comprehensive view of the whole field of mathematics at the time.
He was an Invited Speaker of the ICM in 1904 at Heidelberg, in 1908 at Rome, and in 1920 at Strasbourg.
He died on 15 August 1922, aged 41 years.
Works
L'Imagination et les mathématiques selon Descartes (1900)
Sur quelques propriétés des fonctions entières (1903)
Œuvres de Blaise Pascal, publiées suivant l'ordre chronologique, avec documents complémentaires, introductions et notes, par Léon Brunschvicg et Pierre Boutroux (1908)
Leçons sur les fonctions définies par les équations différentielles du premier ordre, professées au Collège de France (1908)
Les Principes de l'analyse mathématique, exposé historique et critique (2 volumes, 1914-1919) Texte en ligne 1 2
Contient : (I) Les nombres, les grandeurs, les figures, le calcul combinatoire, le calcul algébrique, calcul des fonctions, l'algèbre géométrique. (2) La géométrie algébrique. Extensions de l'algèbre et constructions logiques. Extensions de l'algèbre ; les développements en séries. La méthode analytique en mathématiques. Analyse infinitésimale. Analyse des principes mathématiques. Analyse de la notion de fonction.
L'Idéal scientifique des mathématiciens dans l'antiquité et dans les temps modernes (1920)<ref>{{cite journal|author=Young, J. W.|author-link=John Wesley Young|title=Review: L'Idéal Scientifique des Mathématiciens dans l'Antiquité et dans les Temps Modernes. By Pierre Boutroux|year=1923|volume=29|issue=10|pages=470–473|url=https://www.ams.org/journals/bull/1923-29-10/S0002-9904-1923-03800-7/S0002-9904-1923-03800-7.pdf|doi=10.1090/S0002-9904-1923-03800-7|journal=Bulletin of the American Mathematical Society}}</ref> Texte en ligne
Les Mathématiques'' (1922)
References
Further reading
R S Calinger, Biography in Dictionary of Scientific Biography (New York 1970-1990).
L Brunschvicg, L'oeuvre de Pierre Boutroux, Revue de métaphysique et de morale 29-30 (1922), 285-289.
Lettre de M Pierre Boutroux a M Mittag-Leffler, in The mathematical heritage of Henri Poincaré 2 (Providence, R.I., 1983), 441-445.
External links
Boutroux summary at www-gap.dcs.st-and.ac.uk
1880 births
1922 deaths
Academic staff of the Collè |
https://en.wikipedia.org/wiki/Subtangent | In geometry, the subtangent and related terms are certain line segments defined using the line tangent to a curve at a given point and the coordinate axes. The terms are somewhat archaic today but were in common use until the early part of the 20th century.
Definitions
Let P = (x, y) be a point on a given curve with A = (x, 0) its projection onto the x-axis. Draw the tangent to the curve at P and let T be the point where this line intersects the x-axis. Then TA is defined to be the subtangent at P. Similarly, if normal to the curve at P intersects the x-axis at N then AN is called the subnormal. In this context, the lengths PT and PN are called the tangent and normal, not to be confused with the tangent line and the normal line which are also called the tangent and normal.
Equations
Let φ be the angle of inclination of the tangent with respect to the x-axis; this is also known as the tangential angle. Then
So the subtangent is
and the subnormal is
The normal is given by
and the tangent is given by
Polar definitions
Let P = (r, θ) be a point on a given curve defined by polar coordinates and let O denote the origin. Draw a line through O which is perpendicular to OP and let T now be the point where this line intersects the tangent to the curve at P. Similarly, let N now be the point where the normal to the curve intersects the line. Then OT and ON are, respectively, called the polar subtangent and polar subnormal of the curve at P.
Polar equations
Let ψ be the angle between the tangent and the ray OP; this is also known as the polar tangential angle. Then
So the polar subtangent is
and the subnormal is
References
B. Williamson "Subtangent and Subnormal" and "Polar Subtangent and Polar Subnormal" in An elementary treatise on the differential calculus (1899) p 215, 223 Internet Archive
Curves |
https://en.wikipedia.org/wiki/Serre%27s%20multiplicity%20conjectures | In mathematics, Serre's multiplicity conjectures, named after Jean-Pierre Serre, are certain purely algebraic problems, in commutative algebra, motivated by the needs of algebraic geometry. Since André Weil's initial definition of intersection numbers, around 1949, there had been a question of how to provide a more flexible and computable theory.
Let R be a (Noetherian, commutative) regular local ring and P and Q be prime ideals of R. In 1958, Serre realized that classical algebraic-geometric ideas of multiplicity could be generalized using the concepts of homological algebra. Serre defined the intersection multiplicity of R/P and R/Q by means of the Tor functors of homological algebra, as
This requires the concept of the length of a module, denoted here by , and the assumption that
If this idea were to work, however, certain classical relationships would presumably have to continue to hold. Serre singled out four important properties. These then became conjectures, challenging in the general case. (There are more general statements of these conjectures where R/P and R/Q are replaced by finitely generated modules: see Serre's Local Algebra for more details.)
Dimension inequality
Serre proved this for all regular local rings. He established the following three properties when R is either of equal characteristic or of mixed characteristic and unramified (which in this case means that characteristic of the residue field is not an element of the square of the maximal ideal of the local ring), and conjectured that they hold in general.
Nonnegativity
This was proven by Ofer Gabber in 1995.
Vanishing
If
then
This was proven in 1985 by Paul C. Roberts, and independently by Henri Gillet and Christophe Soulé.
Positivity
If
then
This remains open.
See also
Homological conjectures in commutative algebra
References
Commutative algebra
Intersection theory
Conjectures
Unsolved problems in mathematics |
https://en.wikipedia.org/wiki/Prior%20probability | A prior probability distribution of an uncertain quantity, often simply called the prior, is its assumed probability distribution before some evidence is taken into account. For example, the prior could be the probability distribution representing the relative proportions of voters who will vote for a particular politician in a future election. The unknown quantity may be a parameter of the model or a latent variable rather than an observable variable.
In Bayesian statistics, Bayes' rule prescribes how to update the prior with new information to obtain the posterior probability distribution, which is the conditional distribution of the uncertain quantity given new data. Historically, the choice of priors was often constrained to a conjugate family of a given likelihood function, for that it would result in a tractable posterior of the same family. The widespread availability of Markov chain Monte Carlo methods, however, has made this less of a concern.
There are many ways to construct a prior distribution. In some cases, a prior may be determined from past information, such as previous experiments. A prior can also be elicited from the purely subjective assessment of an experienced expert. When no information is available, an uninformative prior may be adopted as justified by the principle of indifference. In modern applications, priors are also often chosen for their mechanical properties, such as regularization and feature selection.
The prior distributions of model parameters will often depend on parameters of their own. Uncertainty about these hyperparameters can, in turn, be expressed as hyperprior probability distributions. For example, if one uses a beta distribution to model the distribution of the parameter p of a Bernoulli distribution, then:
p is a parameter of the underlying system (Bernoulli distribution), and
α and β are parameters of the prior distribution (beta distribution); hence hyperparameters.
In principle, priors can be decomposed into many conditional levels of distributions, so-called hierarchical priors.
Informative priors
An informative prior expresses specific, definite information about a variable.
An example is a prior distribution for the temperature at noon tomorrow.
A reasonable approach is to make the prior a normal distribution with expected value equal to today's noontime temperature, with variance equal to the day-to-day variance of atmospheric temperature,
or a distribution of the temperature for that day of the year.
This example has a property in common with many priors,
namely, that the posterior from one problem (today's temperature) becomes the prior for another problem (tomorrow's temperature); pre-existing evidence which has already been taken into account is part of the prior and, as more evidence accumulates, the posterior is determined largely by the evidence rather than any original assumption, provided that the original assumption admitted the possibility of what the evidence is suggesting. |
https://en.wikipedia.org/wiki/Kirszbraun%20theorem | In mathematics, specifically real analysis and functional analysis, the Kirszbraun theorem states that if is a subset of some Hilbert space , and is another Hilbert space, and
is a Lipschitz-continuous map, then there is a Lipschitz-continuous map
that extends and has the same Lipschitz constant as .
Note that this result in particular applies to Euclidean spaces and , and it was in this form that Kirszbraun originally formulated and proved the theorem. The version for Hilbert spaces can for example be found in (Schwartz 1969, p. 21). If is a separable space (in particular, if it is a Euclidean space) the result is true in Zermelo–Fraenkel set theory; for the fully general case, it appears to need some form of the axiom of choice; the Boolean prime ideal theorem is known to be sufficient.
The proof of the theorem uses geometric features of Hilbert spaces; the corresponding statement for Banach spaces is not true in general, not even for finite-dimensional Banach spaces. It is for instance possible to construct counterexamples where the domain is a subset of with the maximum norm and carries the Euclidean norm. More generally, the theorem fails for equipped with any norm () (Schwartz 1969, p. 20).
Explicit formulas
For an -valued function the extension is provided by where is the Lipschitz constant of on .
In general, an extension can also be written for -valued functions as where and conv(g) is the lower convex envelope of g.
History
The theorem was proved by Mojżesz David Kirszbraun, and later it was reproved by Frederick Valentine, who first proved it for the Euclidean plane. Sometimes this theorem is also called Kirszbraun–Valentine theorem.
References
External links
Kirszbraun theorem at Encyclopedia of Mathematics.
Lipschitz maps
Metric geometry
Theorems in real analysis
Theorems in functional analysis
Hilbert spaces |
https://en.wikipedia.org/wiki/1614%20in%20science | The year 1614 in science and technology involved some significant events.
Mathematics
Scottish mathematician John Napier publishes Mirifici Logarithmorum Canonis Descriptio ("Description of the Admirable Table of Logarithms"), outlining his discovery of logarithms and incorporating the decimal mark. Astronomer Johannes Kepler soon begins to employ logarithms in his description of the Solar System.
Medicine
Felix Plater gives a description of Dupuytren's contracture.
Sanctorius publishes De statica medicina, which will go through five editions in the following century.
Births
February 14 – Bishop John Wilkins, English natural philosopher, co-founder of the Royal Society (died 1672)
Deaths
July 28 – Felix Plater, Swiss physician (born 1536)
Pedro Fernandes de Queirós, Portuguese-born navigator (born 1565)
William Lee, English-born inventor (born c. 1563)
References
17th century in science
1610s in science |
https://en.wikipedia.org/wiki/List%20of%20variational%20topics | This is a list of variational topics in from mathematics and physics. See calculus of variations for a general introduction.
Action (physics)
Averaged Lagrangian
Brachistochrone curve
Calculus of variations
Catenoid
Cycloid
Dirichlet principle
Euler–Lagrange equation cf. Action (physics)
Fermat's principle
Functional (mathematics)
Functional derivative
Functional integral
Geodesic
Isoperimetry
Lagrangian
Lagrangian mechanics
Legendre transformation
Luke's variational principle
Minimal surface
Morse theory
Noether's theorem
Path integral formulation
Plateau's problem
Prime geodesic
Principle of least action
Soap bubble
Soap film
Tautochrone curve
Variations |
https://en.wikipedia.org/wiki/Transformation%20geometry | In mathematics, transformation geometry (or transformational geometry) is the name of a mathematical and pedagogic take on the study of geometry by focusing on groups of geometric transformations, and properties that are invariant under them. It is opposed to the classical synthetic geometry approach of Euclidean geometry, that focuses on proving theorems.
For example, within transformation geometry, the properties of an isosceles triangle are deduced from the fact that it is mapped to itself by a reflection about a certain line. This contrasts with the classical proofs by the criteria for congruence of triangles.
The first systematic effort to use transformations as the foundation of geometry was made by Felix Klein in the 19th century, under the name Erlangen programme. For nearly a century this approach remained confined to mathematics research circles. In the 20th century efforts were made to exploit it for mathematical education. Andrei Kolmogorov included this approach (together with set theory) as part of a proposal for geometry teaching reform in Russia. These efforts culminated in the 1960s with the general reform of mathematics teaching known as the New Math movement.
Pedagogy
An exploration of transformation geometry often begins with a study of reflection symmetry as found in daily life. The first real transformation is reflection in a line or reflection against an axis. The composition of two reflections results in a rotation when the lines intersect, or a translation when they are parallel. Thus through transformations students learn about Euclidean plane isometry. For instance, consider reflection in a vertical line and a line inclined at 45° to the horizontal. One can observe that one composition yields a counter-clockwise quarter-turn (90°) while the reverse composition yields a clockwise quarter-turn. Such results show that transformation geometry includes non-commutative processes.
An entertaining application of reflection in a line occurs in a proof of the one-seventh area triangle found in any triangle.
Another transformation introduced to young students is the dilation. However, the reflection in a circle transformation seems inappropriate for lower grades. Thus inversive geometry, a larger study than grade school transformation geometry, is usually reserved for college students.
Experiments with concrete symmetry groups make way for abstract group theory. Other concrete activities use computations with complex numbers, hypercomplex numbers, or matrices to express transformation geometry.
Such transformation geometry lessons present an alternate view that contrasts with classical synthetic geometry. When students then encounter analytic geometry, the ideas of coordinate rotations and reflections follow easily. All these concepts prepare for linear algebra where the reflection concept is expanded.
Educators have shown some interest and described projects and experiences with transformation geometry for children |
https://en.wikipedia.org/wiki/Injective%20module | In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers. Specifically, if Q is a submodule of some other module, then it is already a direct summand of that module; also, given a submodule of a module Y, any module homomorphism from this submodule to Q can be extended to a homomorphism from all of Y to Q. This concept is dual to that of projective modules. Injective modules were introduced in and are discussed in some detail in the textbook .
Injective modules have been heavily studied, and a variety of additional notions are defined in terms of them: Injective cogenerators are injective modules that faithfully represent the entire category of modules. Injective resolutions measure how far from injective a module is in terms of the injective dimension and represent modules in the derived category. Injective hulls are maximal essential extensions, and turn out to be minimal injective extensions. Over a Noetherian ring, every injective module is uniquely a direct sum of indecomposable modules, and their structure is well understood. An injective module over one ring, may not be injective over another, but there are well-understood methods of changing rings which handle special cases. Rings which are themselves injective modules have a number of interesting properties and include rings such as group rings of finite groups over fields. Injective modules include divisible groups and are generalized by the notion of injective objects in category theory.
Definition
A left module Q over the ring R is injective if it satisfies one (and therefore all) of the following equivalent conditions:
If Q is a submodule of some other left R-module M, then there exists another submodule K of M such that M is the internal direct sum of Q and K, i.e. Q + K = M and Q ∩ K = {0}.
Any short exact sequence 0 →Q → M → K → 0 of left R-modules splits.
If X and Y are left R-modules, f : X → Y is an injective module homomorphism and g : X → Q is an arbitrary module homomorphism, then there exists a module homomorphism h : Y → Q such that hf = g, i.e. such that the following diagram commutes:
The contravariant Hom functor Hom(-,Q) from the category of left R-modules to the category of abelian groups is exact.
Injective right R-modules are defined in complete analogy.
Examples
First examples
Trivially, the zero module {0} is injective.
Given a field k, every k-vector space Q is an injective k-module. Reason: if Q is a subspace of V, we can find a basis of Q and extend it to a basis of V. The new extending basis vectors span a subspace K of V and V is the internal direct sum of Q and K. Note that the direct complement K of Q is not uniquely determined by Q, and likewise the extending map h in the above definition is typically not unique.
The rationals Q (with addition) form an injective abelian group (i.e. an injective Z- |
https://en.wikipedia.org/wiki/Injective%20object | In mathematics, especially in the field of category theory, the concept of injective object is a generalization of the concept of injective module. This concept is important in cohomology, in homotopy theory and in the theory of model categories. The dual notion is that of a projective object.
Definition
An object in a category is said to be injective if for every monomorphism and every morphism there exists a morphism extending to , i.e. such that .
That is, every morphism factors through every monomorphism .
The morphism in the above definition is not required to be uniquely determined by and .
In a locally small category, it is equivalent to require that the hom functor carries monomorphisms in to surjective set maps.
In Abelian categories
The notion of injectivity was first formulated for abelian categories, and this is still one of its primary areas of application. When is an abelian category, an object Q of is injective if and only if its hom functor HomC(–,Q) is exact.
If is an exact sequence in such that Q is injective, then the sequence splits.
Enough injectives and injective hulls
The category is said to have enough injectives if for every object X of , there exists a monomorphism from X to an injective object.
A monomorphism g in is called an essential monomorphism if for any morphism f, the composite fg is a monomorphism only if f is a monomorphism.
If g is an essential monomorphism with domain X and an injective codomain G, then G is called an injective hull of X. The injective hull is then uniquely determined by X up to a non-canonical isomorphism.
Examples
In the category of abelian groups and group homomorphisms, Ab, an injective object is necessarily a divisible group. Assuming the axiom of choice, the notions are equivalent.
In the category of (left) modules and module homomorphisms, R-Mod, an injective object is an injective module. R-Mod has injective hulls (as a consequence, R-Mod has enough injectives).
In the category of metric spaces, Met, an injective object is an injective metric space, and the injective hull of a metric space is its tight span.
In the category of T0 spaces and continuous mappings, an injective object is always a Scott topology on a continuous lattice, and therefore it is always sober and locally compact.
Uses
If an abelian category has enough injectives, we can form injective resolutions, i.e. for a given object X we can form a long exact sequence
and one can then define the derived functors of a given functor F by applying F to this sequence and computing the homology of the resulting (not necessarily exact) sequence. This approach is used to define Ext, and Tor functors and also the various cohomology theories in group theory, algebraic topology and algebraic geometry. The categories being used are typically functor categories or categories of sheaves of OX modules over some ringed space (X, OX) or, more generally, any Grothendieck category.
Generalization
Let be a |
https://en.wikipedia.org/wiki/Mutually%20orthogonal%20Latin%20squares | In combinatorial mathematics, two Latin squares of the same size (order) are said to be orthogonal if when superimposed the ordered paired entries in the positions are all distinct. A set of Latin squares, all of the same order, all pairs of which are orthogonal is called a set of mutually orthogonal Latin squares. This concept of orthogonality in combinatorics is strongly related to the concept of blocking in statistics, which ensures that independent variables are truly independent with no hidden confounding correlations. "Orthogonal" is thus synonymous with "independent" in that knowing one variable's value gives no further information about another variable's likely value.
An outdated term for pair of orthogonal Latin squares is Graeco-Latin square, found in older literature.
Graeco-Latin squares
A Graeco-Latin square or Euler square or pair of orthogonal Latin squares of order over two sets and (which may be the same), each consisting of symbols, is an arrangement of cells, each cell containing an ordered pair , where is in and is in , such that every row and every column contains each element of and each element of exactly once, and that no two cells contain the same ordered pair.
The arrangement of the -coordinates by themselves (which may be thought of as Latin characters) and of the -coordinates (the Greek characters) each forms a Latin square. A Graeco-Latin square can therefore be decomposed into two orthogonal Latin squares. Orthogonality here means that every pair from the Cartesian product occurs exactly once.
Orthogonal Latin squares were studied in detail by Leonhard Euler, who took the two sets to be }, the first upper-case letters from the Latin alphabet, and },
the first lower-case letters from the Greek alphabet—hence the name Graeco-Latin square.
Existence
When a Graeco-Latin square is viewed as a pair of orthogonal Latin squares, each of the Latin squares is said to have an orthogonal mate. In an arbitrary Latin square, a selection of positions, one in each row and one in each column whose entries are all distinct is called a transversal of that square. Consider one symbol in a Graeco-Latin square. The positions containing this symbol must all be in different rows and columns, and furthermore the other symbol in these positions must all be distinct. Hence, when viewed as a pair of Latin squares, the positions containing one symbol in the first square correspond to a transversal in the second square (and vice versa).
A given Latin square of order n possesses an orthogonal mate if and only if it has n disjoint transversals.
The Cayley table (without borders) of any group of odd order forms a Latin square which possesses an orthogonal mate.
Thus Graeco-Latin squares exist for all odd orders as there are groups that exist of these orders. Such Graeco-Latin squares are said to be group based.
Euler was able to construct Graeco-Latin squares of orders that are multiples of four, and seemed to be aware of the |
https://en.wikipedia.org/wiki/Gaston%20Tarry | Gaston Tarry (27 September 1843 – 21 June 1913) was a French mathematician. Born in Villefranche de Rouergue, Aveyron, he studied mathematics at high school before joining the civil service in Algeria. He pursued mathematics as an amateur.
In 1901 Tarry confirmed Leonhard Euler's conjecture that no 6×6 Graeco-Latin square was possible (the 36 officers problem).
See also
List of amateur mathematicians
Prouhet-Tarry-Escott problem
Tarry point
Tetramagic square
References
External links
People from Villefranche-de-Rouergue
1843 births
1913 deaths
Combinatorialists
19th-century French mathematicians
20th-century French mathematicians |
https://en.wikipedia.org/wiki/Chemical%20structure | A chemical structure of a molecule is a spatial arrangement of its atoms and their chemical bonds. Its determination includes a chemist's specifying the molecular geometry and, when feasible and necessary, the electronic structure of the target molecule or other solid. Molecular geometry refers to the spatial arrangement of atoms in a molecule and the chemical bonds that hold the atoms together and can be represented using structural formulae and by molecular models; complete electronic structure descriptions include specifying the occupation of a molecule's molecular orbitals. Structure determination can be applied to a range of targets from very simple molecules (e.g., diatomic oxygen or nitrogen) to very complex ones (e.g., such as protein or DNA).
Background
Theories of chemical structure were first developed by August Kekulé, Archibald Scott Couper, and Aleksandr Butlerov, among others, from about 1858. These theories were first to state that chemical compounds are not a random cluster of atoms and functional groups, but rather had a definite order defined by the valency of the atoms composing the molecule, giving the molecules a three dimensional structure that could be determined or solved.
Concerning chemical structure, one has to distinguish between pure connectivity of the atoms within a molecule (chemical constitution), a description of a three-dimensional arrangement (molecular configuration, includes e.g. information on chirality) and the precise determination of bond lengths, angles and torsion angles, i.e. a full representation of the (relative) atomic coordinates.
In determining structures of chemical compounds, one generally aims to obtain, first and minimally, the pattern and degree of bonding between all atoms in the molecule; when possible, one seeks the three dimensional spatial coordinates of the atoms in the molecule (or other solid).
Structural elucidation
The methods by which one can determine the structure of a molecule is called structural elucidation. These methods include:
concerning only connectivity of the atoms: spectroscopies such as nuclear magnetic resonance (proton and carbon-13 NMR), various methods of mass spectrometry (to give overall molecular mass, as well as fragment masses).Techniques such as absorption spectroscopy and the vibrational spectroscopies, infrared and Raman, provide, respectively, important supporting information about the numbers and adjacencies of multiple bonds, and about the types of functional groups (whose internal bonding gives vibrational signatures); further inferential studies that give insight into the contributing electronic structure of molecules include cyclic voltammetry and X-ray photoelectron spectroscopy.
concerning precise metric three-dimensional information: can be obtained for gases by gas electron diffraction and microwave (rotational) spectroscopy (and other rotationally resolved spectroscopy) and for the crystalline solid state by X-ray crystallography or n |
https://en.wikipedia.org/wiki/Inerting%20system | An inerting system decreases the probability of combustion of flammable materials stored in a confined space. The most common such system is a fuel tank containing a combustible liquid, such as gasoline, diesel fuel, aviation fuel, jet fuel, or rocket propellant. After being fully filled, and during use, there is a space above the fuel, called the ullage, that contains evaporated fuel mixed with air, which contains the oxygen necessary for combustion. Under the right conditions this mixture can ignite. An inerting system replaces the air with a gas that cannot support combustion, such as nitrogen.
Principle of operation
Three elements are required to initiate and sustain combustion in the ullage: an ignition source (heat), fuel, and oxygen. Combustion may be prevented by reducing any one of these three elements. In many cases there is no ignition source, e.g. storage tanks. If the presence of an ignition source can not be prevented, as is the case with most tanks that feed fuel to internal combustion engines, then the tank may be made non-ignitable by filling the ullage with an inert gas as the fuel is consumed. At present carbon dioxide or nitrogen are used almost exclusively, although some systems use nitrogen-enriched air, or steam. Using these inert gases reduces the oxygen concentration of the ullage to below the combustion threshold.
Oil tankers
Oil tankers fill the empty space above the oil cargo with inert gas to prevent fire or explosion of hydrocarbon vapors. Oil vapors cannot burn in air with less than 11% oxygen content. The inert gas may be supplied by cooling and scrubbing the flue gas produced by the ship's boilers. Where diesel engines are used, the exhaust gas may contain too much oxygen so fuel-burning inert gas generators may be installed. One-way valves are installed in process piping to the tanker spaces to prevent volatile hydrocarbon vapors or mist from entering other equipment. Inert gas systems have been required on oil tankers since the SOLAS regulations of 1974. The International Maritime Organization (IMO) publishes technical standard IMO-860 describing the requirements for inert gas systems. Other types of cargo such as bulk chemicals may also be carried in inerted tanks, but the inerting gas must be compatible with the chemicals used.
Aircraft
Fuel tanks for combat aircraft have long been inerted, as well as being self-sealing, but those for military cargo aircraft and civilian transport category aircraft have not, largely due to cost and weight considerations. Early applications using nitrogen were on the Handley Page Halifax III and VIII, Short Stirling, and Avro Lincoln B.II, which incorporated inerting systems from around 1944.
Cleve Kimmel first proposed an inerting system to passenger airlines in the early 1960s. His proposed system for passenger aircraft would have used nitrogen. However, the US Federal Aviation Administration (FAA) refused to consider Kimmel's system after the airlines complained it was |
https://en.wikipedia.org/wiki/Module%20homomorphism | In algebra, a module homomorphism is a function between modules that preserves the module structures. Explicitly, if M and N are left modules over a ring R, then a function is called an R-module homomorphism or an R-linear map if for any x, y in M and r in R,
In other words, f is a group homomorphism (for the underlying additive groups) that commutes with scalar multiplication. If M, N are right R-modules, then the second condition is replaced with
The preimage of the zero element under f is called the kernel of f. The set of all module homomorphisms from M to N is denoted by . It is an abelian group (under pointwise addition) but is not necessarily a module unless R is commutative.
The composition of module homomorphisms is again a module homomorphism, and the identity map on a module is a module homomorphism. Thus, all the (say left) modules together with all the module homomorphisms between them form the category of modules.
Terminology
A module homomorphism is called a module isomorphism if it admits an inverse homomorphism; in particular, it is a bijection. Conversely, one can show a bijective module homomorphism is an isomorphism; i.e., the inverse is a module homomorphism. In particular, a module homomorphism is an isomorphism if and only if it is an isomorphism between the underlying abelian groups.
The isomorphism theorems hold for module homomorphisms.
A module homomorphism from a module M to itself is called an endomorphism and an isomorphism from M to itself an automorphism. One writes for the set of all endomorphisms of a module M. It is not only an abelian group but is also a ring with multiplication given by function composition, called the endomorphism ring of M. The group of units of this ring is the automorphism group of M.
Schur's lemma says that a homomorphism between simple modules (modules with no non-trivial submodules) must be either zero or an isomorphism. In particular, the endomorphism ring of a simple module is a division ring.
In the language of the category theory, an injective homomorphism is also called a monomorphism and a surjective homomorphism an epimorphism.
Examples
The zero map M → N that maps every element to zero.
A linear transformation between vector spaces.
.
For a commutative ring R and ideals I, J, there is the canonical identification
given by . In particular, is the annihilator of I.
Given a ring R and an element r, let denote the left multiplication by r. Then for any s, t in R,
.
That is, is right R-linear.
For any ring R,
as rings when R is viewed as a right module over itself. Explicitly, this isomorphism is given by the left regular representation .
Similarly, as rings when R is viewed as a left module over itself. Textbooks or other references usually specify which convention is used.
through for any left module M. (The module structure on Hom here comes from the right R-action on R; see #Module structures on Hom below.)
is called the dual module of M; it is a left (resp |
https://en.wikipedia.org/wiki/Integral%20equation | In mathematics, integral equations are equations in which an unknown function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form: where is an integral operator acting on u. Hence, integral equations may be viewed as the analog to differential equations where instead of the equation involving derivatives, the equation contains integrals. A direct comparison can be seen with the mathematical form of the general integral equation above with the general form of a differential equation which may be expressed as follows:where may be viewed as a differential operator of order i. Due to this close connection between differential and integral equations, one can often convert between the two. For example, one method of solving a boundary value problem is by converting the differential equation with its boundary conditions into an integral equation and solving the integral equation. In addition, because one can convert between the two, differential equations in physics such as Maxwell's equations often have an analog integral and differential form. See also, for example, Green's function and Fredholm theory.
Classification and overview
Various classification methods for integral equations exist. A few standard classifications include distinctions between linear and nonlinear; homogenous and inhomogeneous; Fredholm and Volterra; first order, second order, and third order; and singular and regular integral equations. These distinctions usually rest on some fundamental property such as the consideration of the linearity of the equation or the homogeneity of the equation. These comments are made concrete through the following definitions and examples:
Linearity
: An integral equation is linear if the unknown function u(x) and its integrals appear linear in the equation. Hence, an example of a linear equation would be:As a note on naming convention: i) u(x) is called the unknown function, ii) f(x) is called a known function, iii) K(x,t) is a function of two variables and often called the Kernel function, and iv) λ is an unknown factor or parameter, which plays the same role as the eigenvalue in linear algebra.
: An integral equation is nonlinear if the unknown function u(x) or any of its integrals appear nonlinear in the equation. Hence, examples of nonlinear equations would be the equation above if we replaced u(t) with , such as:Certain kinds of nonlinear integral equations have specific names. A selection of such equations are:
Nonlinear Volterra integral equations of the second kind which have the general form: where is a known function.
Nonlinear Fredholm integral equations of the second kind which have the general form: .
A special type of nonlinear Fredholm integral equations of the second kind is given by the form: , which has the two special subclasses:
Urysohn equation: .
Hammerstein equation: .
More information on the Hammerstein equation and different versions of the |
https://en.wikipedia.org/wiki/Pseudo-spectral%20method | Pseudo-spectral methods, also known as discrete variable representation (DVR) methods, are a class of numerical methods used in applied mathematics and scientific computing for the solution of partial differential equations. They are closely related to spectral methods, but complement the basis by an additional pseudo-spectral basis, which allows representation of functions on a quadrature grid. This simplifies the evaluation of certain operators, and can considerably speed up the calculation when using fast algorithms such as the fast Fourier transform.
Motivation with a concrete example
Take the initial-value problem
with periodic conditions . This specific example is the Schrödinger equation for a particle in a potential , but the structure is more general. In many practical partial differential equations, one has a term that involves derivatives (such as a kinetic energy contribution), and a multiplication with a function (for example, a potential).
In the spectral method, the solution is expanded in a suitable set of basis functions, for example plane waves,
Insertion and equating identical coefficients yields a set of ordinary differential equations for the coefficients,
where the elements are calculated through the explicit Fourier-transform
The solution would then be obtained by truncating the expansion to basis functions, and finding a solution for the . In general, this is done by numerical methods, such as Runge–Kutta methods. For the numerical solutions, the right-hand side of the ordinary differential equation has to be evaluated repeatedly at different time steps. At this point, the spectral method has a major problem with the potential term .
In the spectral representation, the multiplication with the function transforms into a vector-matrix multiplication, which scales as . Also, the matrix elements need to be evaluated explicitly before the differential equation for the coefficients can be solved, which requires an additional step.
In the pseudo-spectral method, this term is evaluated differently. Given the coefficients , an inverse discrete Fourier transform yields the value of the function at discrete grid points . At these grid points, the function is then multiplied, , and the result Fourier-transformed back. This yields a new set of coefficients that are used instead of the matrix product .
It can be shown that both methods have similar accuracy. However, the pseudo-spectral method allows the use of a fast Fourier transform, which scales as , and is therefore significantly more efficient than the matrix multiplication. Also, the function can be used directly without evaluating any additional integrals.
Technical discussion
In a more abstract way, the pseudo-spectral method deals with the multiplication of two functions and as part of a partial differential equation. To simplify the notation, the time-dependence is dropped. Conceptually, it consists of three steps:
are expanded in a finite set of basi |
https://en.wikipedia.org/wiki/Legendre%20transform%20%28integral%20transform%29 | In mathematics, Legendre transform is an integral transform named after the mathematician Adrien-Marie Legendre, which uses Legendre polynomials as kernels of the transform. Legendre transform is a special case of Jacobi transform.
The Legendre transform of a function is
The inverse Legendre transform is given by
Associated Legendre transform
Associated Legendre transform is defined as
The inverse Legendre transform is given by
Some Legendre transform pairs
References
Integral transforms
Mathematical physics |
https://en.wikipedia.org/wiki/Pseudoscalar | In linear algebra, a pseudoscalar is a quantity that behaves like a scalar, except that it changes sign under a parity inversion while a true scalar does not.
A pseudoscalar, when multiplied by an ordinary vector, becomes a pseudovector (or axial vector); a similar construction creates the pseudotensor.
A pseudoscalar also results from any scalar product between a pseudovector and an ordinary vector. The prototypical example of a pseudoscalar is the scalar triple product, which can be written as the scalar product between one of the vectors in the triple product and the cross product between the two other vectors, where the latter is a pseudovector.
In physics
In physics, a pseudoscalar denotes a physical quantity analogous to a scalar. Both are physical quantities which assume a single value which is invariant under proper rotations. However, under the parity transformation, pseudoscalars flip their signs while scalars do not. As reflections through a plane are the combination of a rotation with the parity transformation, pseudoscalars also change signs under reflections.
Motivation
One of the most powerful ideas in physics is that physical laws do not change when one changes the coordinate system used to describe these laws. That a pseudoscalar reverses its sign when the coordinate axes are inverted suggests that it is not the best object to describe a physical quantity. In 3D-space, quantities described by a pseudovector are anti-symmetric tensors of order 2, which are invariant under inversion. The pseudovector may be a simpler representation of that quantity, but suffers from the change of sign under inversion. Similarly, in 3D-space, the Hodge dual of a scalar is equal to a constant times the 3-dimensional Levi-Civita pseudotensor (or "permutation" pseudotensor); whereas the Hodge dual of a pseudoscalar is an anti-symmetric (pure) tensor of order three. The Levi-Civita pseudotensor is a completely anti-symmetric pseudotensor of order 3. Since the dual of the pseudoscalar is the product of two "pseudo-quantities", the resulting tensor is a true tensor, and does not change sign upon an inversion of axes. The situation is similar to the situation for pseudovectors and anti-symmetric tensors of order 2. The dual of a pseudovector is an anti-symmetric tensor of order 2 (and vice versa). The tensor is an invariant physical quantity under a coordinate inversion, while the pseudovector is not invariant.
The situation can be extended to any dimension. Generally in an n-dimensional space the Hodge dual of an order r tensor will be an anti-symmetric pseudotensor of order and vice versa. In particular, in the four-dimensional spacetime of special relativity, a pseudoscalar is the dual of a fourth-order tensor and is proportional to the four-dimensional Levi-Civita pseudotensor.
Examples
The stream function for a two-dimensional, incompressible fluid flow .
Magnetic charge is a pseudoscalar as it is mathematically defined, regardless of whethe |
https://en.wikipedia.org/wiki/Psychohistory%20%28fictional%29 | Psychohistory is a fictional science in Isaac Asimov's Foundation universe which combines history, sociology, and mathematical statistics to make general predictions about the future behavior of very large groups of people, such as the Galactic Empire. It was first introduced in the four short stories (1942–1944) which would later be collected as the 1951 novel Foundation.
In-universe
Axioms
Psychohistory depends on the idea that, while one cannot foresee the actions of a particular individual, the laws of statistics as applied to large groups of people could predict the general flow of future events. Asimov used the analogy of a gas: An observer has great difficulty in predicting the motion of a single molecule in a gas, but with the kinetic theory can predict the mass action of the gas to a high level of accuracy. Asimov applied this concept to the population of his fictional Galactic Empire, which numbered one quintillion. The character responsible for the science's creation, Hari Seldon, established two axioms:
the population whose behaviour was modelled should be sufficiently large to represent the entire society.
the population should remain in ignorance of the results of the application of psychohistorical analyses because if it is aware, the group changes its behaviour.
Ebling Mis added these axioms:
there would be no fundamental change in the society
human reactions to stimuli would remain constant.
Golan Trevize in Foundation and Earth added this axiom:
humans are the only sentient intelligence in the galaxy.
The Prime Radiant
Asimov presents the Prime Radiant, a device designed by Hari Seldon and built by Yugo Amaryl, as storing the psychohistorical equations showing the future development of humanity.
The Prime Radiant projects the equations onto walls in some unexplained manner, but it does not cast shadows, thus allowing workers easy interaction. As a tool of the Second Foundation, control operates through the power of the mind, allowing the user to zoom in to details of the equations, and to change them. One can make annotations, but by convention all amendments remain anonymous.
A student in the Second Foundation destined for Speakerhood has to present an amendment to the plan. Five different boards then check the mathematics rigorously. Students have to defend their proposals against concerted and merciless attacks. After two years, the change gets reviewed again. If after the second examination it still passes muster, the contribution becomes part of the Seldon Plan.
The Radiant, as well as being interactive, employs a type of colour-coding to equations within itself for ready comprehension by Psychohistorians.
Seldon Black are the original Seldon Plan equations developed by Seldon and Amaryl during the first four decades of Seldon's work at the University of Streeling, and define Seldon Crises, the Plan's duration, and the eventuation of the Second Galactic Empire.
Speaker Red are additions to the plan by Spea |
https://en.wikipedia.org/wiki/Oscillation%20%28mathematics%29 | In mathematics, the oscillation of a function or a sequence is a number that quantifies how much that sequence or function varies between its extreme values as it approaches infinity or a point. As is the case with limits, there are several definitions that put the intuitive concept into a form suitable for a mathematical treatment: oscillation of a sequence of real numbers, oscillation of a real-valued function at a point, and oscillation of a function on an interval (or open set).
Definitions
Oscillation of a sequence
Let be a sequence of real numbers. The oscillation of that sequence is defined as the difference (possibly infinite) between the limit superior and limit inferior of :
.
The oscillation is zero if and only if the sequence converges. It is undefined if and are both equal to +∞ or both equal to −∞, that is, if the sequence tends to +∞ or −∞.
Oscillation of a function on an open set
Let be a real-valued function of a real variable. The oscillation of on an interval in its domain is the difference between the supremum and infimum of :
More generally, if is a function on a topological space (such as a metric space), then the oscillation of on an open set is
Oscillation of a function at a point
The oscillation of a function of a real variable at a point is defined as the limit as of the oscillation of on an -neighborhood of :
This is the same as the difference between the limit superior and limit inferior of the function at , provided the point is not excluded from the limits.
More generally, if is a real-valued function on a metric space, then the oscillation is
Examples
has oscillation ∞ at = 0, and oscillation 0 at other finite and at −∞ and +∞.
(the topologist's sine curve) has oscillation 2 at = 0, and 0 elsewhere.
has oscillation 0 at every finite , and 2 at −∞ and +∞.
or 1, -1, 1, -1, 1, -1... has oscillation 2.
In the last example the sequence is periodic, and any sequence that is periodic without being constant will have non-zero oscillation. However, non-zero oscillation does not usually indicate periodicity.
Geometrically, the graph of an oscillating function on the real numbers follows some path in the xy-plane, without settling into ever-smaller regions. In well-behaved cases the path might look like a loop coming back on itself, that is, periodic behaviour; in the worst cases quite irregular movement covering a whole region.
Continuity
Oscillation can be used to define continuity of a function, and is easily equivalent to the usual ε-δ definition (in the case of functions defined everywhere on the real line): a function ƒ is continuous at a point x0 if and only if the oscillation is zero; in symbols, A benefit of this definition is that it quantifies discontinuity: the oscillation gives how much the function is discontinuous at a point.
For example, in the classification of discontinuities:
in a removable discontinuity, the distance that the value of the function is off by is the |
https://en.wikipedia.org/wiki/Contact%20geometry | In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a condition called 'complete non-integrability'. Equivalently, such a distribution may be given (at least locally) as the kernel of a differential one-form, and the non-integrability condition translates into a maximal non-degeneracy condition on the form. These conditions are opposite to two equivalent conditions for 'complete integrability' of a hyperplane distribution, i.e. that it be tangent to a codimension one foliation on the manifold, whose equivalence is the content of the Frobenius theorem.
Contact geometry is in many ways an odd-dimensional counterpart of symplectic geometry, a structure on certain even-dimensional manifolds. Both contact and symplectic geometry are motivated by the mathematical formalism of classical mechanics, where one can consider either the even-dimensional phase space of a mechanical system or constant-energy hypersurface, which, being codimension one, has odd dimension.
Applications
Like symplectic geometry, contact geometry has broad applications in physics, e.g. geometrical optics, classical mechanics, thermodynamics, geometric quantization, integrable systems and to control theory. Contact geometry also has applications to low-dimensional topology; for example, it has been used by Kronheimer and Mrowka to prove the property P conjecture, by Michael Hutchings to define an invariant of smooth three-manifolds, and by Lenhard Ng to define invariants of knots. It was also used by Yakov Eliashberg to derive a topological characterization of Stein manifolds of dimension at least six.
Contact forms and structures
A contact structure on an odd dimensional manifold is a smoothly varying family of codimension one subspaces of each tangent space of the manifold, satisfying a non-integrability condition. The family may be described as a section of a bundle as follows:
Given an n-dimensional smooth manifold M, and a point , a contact element of M with contact point p is an (n − 1)-dimensional linear subspace of the tangent space to M at p. A contact element can be given by the kernel of a linear function on the tangent space to M at p. However, if a subspace is given by the kernel of a linear function ω, then it will also be given by the zeros of λω where is any nonzero real number. Thus, the kernels of all give the same contact element. It follows that the space of all contact elements of M can be identified with a quotient of the cotangent bundle T*M (with the zero section removed), namely:
A contact structure on an odd dimensional manifold M, of dimension , is a smooth distribution of contact elements, denoted by ξ, which is generic at each point. The genericity condition is that ξ is non-integrable.
Assume that we have a smooth distribution of contact elements, ξ, given locally by a differential 1-form α; i.e. a smooth section of the cotangent bun |
https://en.wikipedia.org/wiki/1636%20in%20science | The year 1636 in science and technology involved some significant events.
Mathematics
Pierre de Fermat begins to circulate his work in analytic geometry in manuscript.
Muhammad Baqir Yazdi and René Descartes independently discover the pair of amicable numbers 9,363,584 and 9,437,056.
Physics
Marin Mersenne publishes his Traité de l'harmonie universelle, containing Mersenne's laws describing the frequency of oscillation of a stretched string.
Publications
Daniel Schwenter publishes Delicia Physic-Mathematicae, including a description of a quill pen with an ink reservoir.
Births
Father Jacques Marquette, French explorer (died 1675)
December 26 – Justine Siegemund, German midwife (died 1705)
Deaths
February 22 – Sanctorius, Italian physiologist (born 1561)
Louise Bourgeois Boursier, French Royal midwife (born 1563)
Michal Sedziwój, Polish alchemist (born 1566)
References
17th century in science
1630s in science |
https://en.wikipedia.org/wiki/1653%20in%20science | The year 1653 in science and technology involved some significant events.
Biology
Jan van Kessel paints a series of pictures of insects and fruit.
Mathematics
Blaise Pascal publishes his Traité du triangle arithmétique in which he describes a convenient tabular presentation for binomial coefficients, now called Pascal's triangle.
Physics
Blaise Pascal publishes his Treatise on the Equilibrium of Liquids in which he explains his law of pressure.
Births
January 16 – Johann Conrad Brunner, Swiss anatomist (died 1727)
March 24 – Joseph Sauveur, French mathematician and acoustician (died 1716)
Deaths
Jan Stampioen, Dutch mathematician (born 1610) (gunpowder explosion)
References
17th century in science
1650s in science |
https://en.wikipedia.org/wiki/Kriging | In statistics, originally in geostatistics, kriging or Kriging, (pronounced /ˌˈkɹiːɡɪŋ/) also known as Gaussian process regression, is a method of interpolation based on Gaussian process governed by prior covariances. Under suitable assumptions of the prior, kriging gives the best linear unbiased prediction (BLUP) at unsampled locations. Interpolating methods based on other criteria such as smoothness (e.g., smoothing spline) may not yield the BLUP. The method is widely used in the domain of spatial analysis and computer experiments. The technique is also known as Wiener–Kolmogorov prediction, after Norbert Wiener and Andrey Kolmogorov.
The theoretical basis for the method was developed by the French mathematician Georges Matheron in 1960, based on the master's thesis of Danie G. Krige, the pioneering plotter of distance-weighted average gold grades at the Witwatersrand reef complex in South Africa. Krige sought to estimate the most likely distribution of gold based on samples from a few boreholes. The English verb is to krige, and the most common noun is kriging. The word is sometimes capitalized as Kriging in the literature.
Though computationally intensive in its basic formulation, kriging can be scaled to larger problems using various approximation methods.
Main principles
Related terms and techniques
Kriging predicts the value of a function at a given point by computing a weighted average of the known values of the function in the neighborhood of the point. The method is closely related to regression analysis. Both theories derive a best linear unbiased estimator based on assumptions on covariances, make use of Gauss–Markov theorem to prove independence of the estimate and error, and use very similar formulae. Even so, they are useful in different frameworks: kriging is made for estimation of a single realization of a random field, while regression models are based on multiple observations of a multivariate data set.
The kriging estimation may also be seen as a spline in a reproducing kernel Hilbert space, with the reproducing kernel given by the covariance function. The difference with the classical kriging approach is provided by the interpretation: while the spline is motivated by a minimum-norm interpolation based on a Hilbert-space structure, kriging is motivated by an expected squared prediction error based on a stochastic model.
Kriging with polynomial trend surfaces is mathematically identical to generalized least squares polynomial curve fitting.
Kriging can also be understood as a form of Bayesian optimization. Kriging starts with a prior distribution over functions. This prior takes the form of a Gaussian process: samples from a function will be normally distributed, where the covariance between any two samples is the covariance function (or kernel) of the Gaussian process evaluated at the spatial location of two points. A set of values is then observed, each value associated with a spatial location. Now, a new value ca |
https://en.wikipedia.org/wiki/Eda%20%28surname%29 | Eda is a Japanese surname that may refer to:
, mathematician specializing in set theory and algebraic topology
, independent Japanese politician
, Japanese marathon runner
, Japanese politician
, Japanese rower
, Japanese politician in the New Komeito Party
Japanese-language surnames |
https://en.wikipedia.org/wiki/Hierarchical%20clustering | In data mining and statistics, hierarchical clustering (also called hierarchical cluster analysis or HCA) is a method of cluster analysis that seeks to build a hierarchy of clusters. Strategies for hierarchical clustering generally fall into two categories:
Agglomerative: This is a "bottom-up" approach: Each observation starts in its own cluster, and pairs of clusters are merged as one moves up the hierarchy.
Divisive: This is a "top-down" approach: All observations start in one cluster, and splits are performed recursively as one moves down the hierarchy.
In general, the merges and splits are determined in a greedy manner. The results of hierarchical clustering are usually presented in a dendrogram.
Hierarchical clustering has the distinct advantage that any valid measure of distance can be used. In fact, the observations themselves are not required: all that is used is a matrix of distances. On the other hand, except for the special case of single-linkage distance, none of the algorithms (except exhaustive search in ) can be guaranteed to find the optimum solution.
Complexity
The standard algorithm for hierarchical agglomerative clustering (HAC) has a time complexity of and requires memory, which makes it too slow for even medium data sets. However, for some special cases, optimal efficient agglomerative methods (of complexity ) are known: SLINK for single-linkage and CLINK for complete-linkage clustering. With a heap, the runtime of the general case can be reduced to , an improvement on the aforementioned bound of , at the cost of further increasing the memory requirements. In many cases, the memory overheads of this approach are too large to make it practically usable.
Divisive clustering with an exhaustive search is , but it is common to use faster heuristics to choose splits, such as k-means.
Cluster Linkage
In order to decide which clusters should be combined (for agglomerative), or where a cluster should be split (for divisive), a measure of dissimilarity between sets of observations is required. In most methods of hierarchical clustering, this is achieved by use of an appropriate distance d, such as the Euclidean distance, between single observations of the data set, and a linkage criterion, which specifies the dissimilarity of sets as a function of the pairwise distances of observations in the sets. The choice of metric as well as linkage can have a major impact on the result of the clustering, where the lower level metric determines which objects are most similar, whereas the linkage criterion influences the shape of the clusters. For example, complete-linkage tends to produce more spherical clusters than single-linkage.
The linkage criterion determines the distance between sets of observations as a function of the pairwise distances between observations.
Some commonly used linkage criteria between two sets of observations A and B and a distance d are:
Some of these can only be recomputed recursively (WPGMA, WPGMC), for m |
https://en.wikipedia.org/wiki/Whitney%20embedding%20theorem | In mathematics, particularly in differential topology, there are two Whitney embedding theorems, named after Hassler Whitney:
The strong Whitney embedding theorem states that any smooth real -dimensional manifold (required also to be Hausdorff and second-countable) can be smoothly embedded in the real -space, if . This is the best linear bound on the smallest-dimensional Euclidean space that all -dimensional manifolds embed in, as the real projective spaces of dimension cannot be embedded into real -space if is a power of two (as can be seen from a characteristic class argument, also due to Whitney).
The weak Whitney embedding theorem states that any continuous function from an -dimensional manifold to an -dimensional manifold may be approximated by a smooth embedding provided . Whitney similarly proved that such a map could be approximated by an immersion provided . This last result is sometimes called the Whitney immersion theorem.
A little about the proof
The general outline of the proof is to start with an immersion with transverse self-intersections. These are known to exist from Whitney's earlier work on the weak immersion theorem. Transversality of the double points follows from a general-position argument. The idea is to then somehow remove all the self-intersections. If has boundary, one can remove the self-intersections simply by isotoping into itself (the isotopy being in the domain of ), to a submanifold of that does not contain the double-points. Thus, we are quickly led to the case where has no boundary. Sometimes it is impossible to remove the double-points via an isotopy—consider for example the figure-8 immersion of the circle in the plane. In this case, one needs to introduce a local double point. Once one has two opposite double points, one constructs a closed loop connecting the two, giving a closed path in Since is simply connected, one can assume this path bounds a disc, and provided one can further assume (by the weak Whitney embedding theorem) that the disc is embedded in such that it intersects the image of only in its boundary. Whitney then uses the disc to create a 1-parameter family of immersions, in effect pushing across the disc, removing the two double points in the process. In the case of the figure-8 immersion with its introduced double-point, the push across move is quite simple (pictured). This process of eliminating opposite sign double-points by pushing the manifold along a disc is called the Whitney Trick.
To introduce a local double point, Whitney created immersions which are approximately linear outside of the unit ball, but containing a single double point. For such an immersion is given by
Notice that if is considered as a map to like so:
then the double point can be resolved to an embedding:
Notice and for then as a function of , is an embedding.
For higher dimensions , there are that can be similarly resolved in For an embedding into for example, define
This proc |
https://en.wikipedia.org/wiki/2520%20%28number%29 | 2520 (two thousand five hundred twenty) is the natural number following 2519 and preceding 2521.
In mathematics
2520 is:
the smallest number divisible by all integers from one to ten, i.e., it is their least common multiple.
half of 7! (5040), meaning 7 factorial, or .
the product of five consecutive numbers, namely .
a superior highly composite number.
a colossally abundant number.
the last highly composite number that is half of the next highly composite number.
the last highly composite number that is a divisor of all following highly composite numbers.
palindromic in undecimal (199111) and a repdigit in bases 55, 59, and 62.
a Harshad number in all bases between binary and hexadecimal.
the aliquot sum of 1080.
part of the 53-aliquot tree. The complete aliquot sequence starting at 1080 is 1080, 2520, 6840, 16560, 41472, 82311, 27441, 12209, 451, 53, 1, 0.
Factors
The factors, also called divisors, of 2520 are:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 21, 24, 28, 30, 35, 36, 40, 42, 45, 56, 60, 63, 70, 72, 84, 90, 105, 120, 126, 140, 168, 180, 210, 252, 280, 315, 360, 420, 504, 630, 840, 1260, 2520.
References
Integers |
https://en.wikipedia.org/wiki/Algebraically%20compact%20module | In mathematics, algebraically compact modules, also called pure-injective modules, are modules that have a certain "nice" property which allows the solution of infinite systems of equations in the module by finitary means. The solutions to these systems allow the extension of certain kinds of module homomorphisms. These algebraically compact modules are analogous to injective modules, where one can extend all module homomorphisms. All injective modules are algebraically compact, and the analogy between the two is made quite precise by a category embedding.
Definitions
Let be a ring, and a left -module. Consider a system of infinitely many linear equations
where both sets and may be infinite, and for each the number of nonzero is finite.
The goal is to decide whether such a system has a solution, that is whether there exist elements of such that all the equations of the system are simultaneously satisfied. (It is not required that only finitely many are non-zero.)
The module M is algebraically compact if, for all such systems, if every subsystem formed by a finite number of the equations has a solution, then the whole system has a solution. (The solutions to the various subsystems may be different.)
On the other hand, a module homomorphism is a pure embedding if the induced homomorphism between the tensor products is injective for every right -module . The module is pure-injective if any pure injective homomorphism splits (that is, there exists with ).
It turns out that a module is algebraically compact if and only if it is pure-injective.
Examples
All modules with finitely many elements are algebraically compact.
Every vector space is algebraically compact (since it is pure-injective). More generally, every injective module is algebraically compact, for the same reason.
If R is an associative algebra with 1 over some field k, then every R-module with finite k-dimension is algebraically compact. This, together with the fact that all finite modules are algebraically compact, gives rise to the intuition that algebraically compact modules are those (possibly "large") modules which share the nice properties of "small" modules.
The Prüfer groups are algebraically compact abelian groups (i.e. Z-modules). The ring of p-adic integers for each prime p is algebraically compact as both a module over itself and a module over Z. The rational numbers are algebraically compact as a Z-module. Together with the indecomposable finite modules over Z, this is a complete list of indecomposable algebraically compact modules.
Many algebraically compact modules can be produced using the injective cogenerator Q/Z of abelian groups. If H is a right module over the ring R, one forms the (algebraic) character module H* consisting of all group homomorphisms from H to Q/Z. This is then a left R-module, and the *-operation yields a faithful contravariant functor from right R-modules to left R-modules.
Every module of the form H* is algebraicall |
https://en.wikipedia.org/wiki/Low-discrepancy%20sequence | In mathematics, a low-discrepancy sequence is a sequence with the property that for all values of N, its subsequence x1, ..., xN has a low discrepancy.
Roughly speaking, the discrepancy of a sequence is low if the proportion of points in the sequence falling into an arbitrary set B is close to proportional to the measure of B, as would happen on average (but not for particular samples) in the case of an equidistributed sequence. Specific definitions of discrepancy differ regarding the choice of B (hyperspheres, hypercubes, etc.) and how the discrepancy for every B is computed (usually normalized) and combined (usually by taking the worst value).
Low-discrepancy sequences are also called quasirandom sequences, due to their common use as a replacement of uniformly distributed random numbers.
The "quasi" modifier is used to denote more clearly that the values of a low-discrepancy sequence are neither random nor pseudorandom, but such sequences share some properties of random variables and in certain applications such as the quasi-Monte Carlo method their lower discrepancy is an important advantage.
Applications
Quasirandom numbers have an advantage over pure random numbers in that they cover the domain of interest quickly and evenly.
Two useful applications are in finding the characteristic function of a probability density function, and in finding the derivative function of a deterministic function with a small amount of noise. Quasirandom numbers allow higher-order moments to be calculated to high accuracy very quickly.
Applications that don't involve sorting would be in finding the mean, standard deviation, skewness and kurtosis of a statistical distribution, and in finding the integral and global maxima and minima of difficult deterministic functions. Quasirandom numbers can also be used for providing starting points for deterministic algorithms that only work locally, such as Newton–Raphson iteration.
Quasirandom numbers can also be combined with search algorithms. A binary tree Quicksort-style algorithm ought to work exceptionally well because quasirandom numbers flatten the tree far better than random numbers, and the flatter the tree the faster the sorting. With a search algorithm, quasirandom numbers can be used to find the mode, median, confidence intervals and cumulative distribution of a statistical distribution, and all local minima and all solutions of deterministic functions.
Low-discrepancy sequences in numerical integration
Various methods of numerical integration can be phrased as approximating the integral of a function f in some interval, e.g. [0,1], as the average of the function evaluated at a set {x1, ..., xN} in that interval:
If the points are chosen as xi = i/N, this is the rectangle rule.
If the points are chosen to be randomly (or pseudorandomly) distributed, this is the Monte Carlo method.
If the points are chosen as elements of a low-discrepancy sequence, this is the quasi-Monte Carlo method.
A remarkable res |
https://en.wikipedia.org/wiki/Sphere%20eversion | In differential topology, sphere eversion is the process of turning a sphere inside out in a three-dimensional space (the word eversion means "turning inside out"). Remarkably, it is possible to smoothly and continuously turn a sphere inside out in this way (allowing self-intersections of the sphere's surface) without cutting or tearing it or creating any crease. This is surprising, both to non-mathematicians and to those who understand regular homotopy, and can be regarded as a veridical paradox; that is something that, while being true, on first glance seems false.
More precisely, let
be the standard embedding; then there is a regular homotopy of immersions
such that ƒ0 = ƒ and ƒ1 = −ƒ.
History
An existence proof for crease-free sphere eversion was first created by .
It is difficult to visualize a particular example of such a turning, although some digital animations have been produced that make it somewhat easier. The first example was exhibited through the efforts of several mathematicians, including Arnold S. Shapiro and Bernard Morin, who was blind. On the other hand, it is much easier to prove that such a "turning" exists, and that is what Smale did.
Smale's graduate adviser Raoul Bott at first told Smale that the result was obviously wrong .
His reasoning was that the degree of the Gauss map must be preserved in such "turning"—in particular it follows that there is no such turning of S1 in R2. But the degrees of the Gauss map for the embeddings f and −f in R3 are both equal to 1, and do not have opposite sign as one might incorrectly guess. The degree of the Gauss map of all immersions of S2 in R3 is 1, so there is no obstacle. The term "veridical paradox" applies perhaps more appropriately at this level: until Smale's work, there was no documented attempt to argue for or against the eversion of S2, and later efforts are in hindsight, so there never was a historical paradox associated with sphere eversion, only an appreciation of the subtleties in visualizing it by those confronting the idea for the first time.
See h-principle for further generalizations.
Proof
Smale's original proof was indirect: he identified (regular homotopy) classes of immersions of spheres with a homotopy group of the Stiefel manifold. Since the homotopy group that corresponds to immersions of in vanishes, the standard embedding and the inside-out one must be regular homotopic. In principle the proof can be unwound to produce an explicit regular homotopy, but this is not easy to do.
There are several ways of producing explicit examples and mathematical visualization:
Half-way models: these consist of very special homotopies. This is the original method, first done by Shapiro and Phillips via Boy's surface, later refined by many others. The original half-way model homotopies were constructed by hand, and worked topologically but weren't minimal. The movie created by Nelson Max, over a seven-year period, and based on Charles Pugh's chicken-wire models |
https://en.wikipedia.org/wiki/Mii%20District%2C%20Fukuoka | is a district located in Fukuoka Prefecture, Japan.
As of 2003 statistics (but following the merger of Kitano), the district has an estimated population of 15,378 and a density of 674 persons per km2. The total area is 22.83 km2.
Towns and villages
Tachiarai
Mergers
On February 5, 2005 the former town of Kitano merged with three other towns (from other districts) into the city of Kurume.
Districts in Fukuoka Prefecture |
https://en.wikipedia.org/wiki/Mizuma%20District | is a district located in Fukuoka Prefecture, Japan.
As of 2003 statistics and counting the decrease in size and population due to the Kurume merger, the district has an estimated population of 14,305 and a density of 776 persons per km2. The total area is 18.43 km2.
Towns and villages
Ōki
Mergers
On February 5, 2005 the former towns of Jōjima and Mizuma merged with two towns (from other districts) into the expanded city of Kurume.
References
Districts in Fukuoka Prefecture |
https://en.wikipedia.org/wiki/List%20of%20cities%20and%20towns%20in%20Saudi%20Arabia | The following is a list of cities and towns in Saudi Arabia.
Alphabetical list of cities and towns
References
Central Department of Statistics and Information
Saudi Arabia, List of cities and towns in
Cities |
https://en.wikipedia.org/wiki/Asymmetric | Asymmetric may refer to:
Asymmetry in geometry, chemistry, and physics
Computing
Asymmetric cryptography, in public-key cryptography
Asymmetric digital subscriber line, Internet connectivity
Asymmetric multiprocessing, in computer architecture
Other
Asymmetric relation, in set theory
Asymmetric synthesis, in organic synthesis
Asymmetric warfare, in modern war
Asymmetric Publications, a video game company
Asymmetry (Mallory Knox album), 2014
Asymmetry (Karnivool album)
Asymmetry (population ethics)
Asymmetry (novel), a 2018 novel by Lisa Halliday
See also |
https://en.wikipedia.org/wiki/Formal%20group%20law | In mathematics, a formal group law is (roughly speaking) a formal power series behaving as if it were the product of a Lie group. They were introduced by . The term formal group sometimes means the same as formal group law, and sometimes means one of several generalizations. Formal groups are intermediate between Lie groups (or algebraic groups) and Lie algebras. They are used in algebraic number theory and algebraic topology.
Definitions
A one-dimensional formal group law over a commutative ring R is a power series F(x,y) with coefficients in R, such that
F(x,y) = x + y + terms of higher degree
F(x, F(y,z)) = F(F(x,y), z) (associativity).
The simplest example is the additive formal group law F(x, y) = x + y.
The idea of the definition is that F should be something like the formal power series expansion of the product of a Lie group, where we choose coordinates so that the identity of the Lie group is the origin.
More generally, an n-dimensional formal group law is a collection of n power series
Fi(x1, x2, ..., xn, y1, y2, ..., yn) in 2n variables, such that
F(x,y) = x + y + terms of higher degree
F(x, F(y,z)) = F(F(x,y), z)
where we write F for (F1, ..., Fn), x for (x1, ..., xn), and so on.
The formal group law is called commutative if F(x,y) = F(y,x). If R is torsionfree, then one can embed R into a Q-algebra and use the exponential and logarithm to write any one-dimensional formal group law F as F(x,y) = exp(log(x) + log(y)), so F is necessarily commutative. More generally, we have:
Theorem. Every one-dimensional formal group law over R is commutative if and only if R has no nonzero torsion nilpotents (i.e., no nonzero elements that are both torsion and nilpotent).
There is no need for an axiom analogous to the existence of inverse elements for groups, as this turns out to follow automatically from the definition of a formal group law. In other words we can always find a (unique) power series G such that F(x,G(x)) = 0.
A homomorphism from a formal group law F of dimension m to a formal group law G of dimension n is a collection f of n power series in m variables, such that
G(f(x), f(y)) = f(F(x,y)).
A homomorphism with an inverse is called an isomorphism, and is called a strict isomorphism if in addition f(x) = x + terms of higher degree. Two formal group laws with an isomorphism between them are essentially the same; they differ only by a "change of coordinates".
Examples
The additive formal group law is given by
The multiplicative formal group law is given by
This rule can be understood as follows. The product G in the (multiplicative group of the) ring R is given by G(a,b) = ab. If we "change coordinates" to make 0 the identity by putting a = 1 + x, b = 1 + y, and G = 1 + F, then we find that F(x,y) = x + y + xy.
Over the rational numbers, there is an isomorphism from the additive formal group law to the multiplicative one, given by . Over general commutative rings R there is no such homomorphism as defining it requires no |
https://en.wikipedia.org/wiki/Homotopy%20principle | In mathematics, the homotopy principle (or h-principle) is a very general way to solve partial differential equations (PDEs), and more generally partial differential relations (PDRs). The h-principle is good for underdetermined PDEs or PDRs, such as the immersion problem, isometric immersion problem, fluid dynamics, and other areas.
The theory was started by Yakov Eliashberg, Mikhail Gromov and Anthony V. Phillips. It was based on earlier results that reduced partial differential relations to homotopy, particularly for immersions. The first evidence of h-principle appeared in the Whitney–Graustein theorem. This was followed by the Nash–Kuiper isometric C1 embedding theorem and the Smale–Hirsch immersion theorem.
Rough idea
Assume we want to find a function ƒ on Rm which satisfies a partial differential equation of degree k, in co-ordinates . One can rewrite it as
where stands for all partial derivatives of ƒ up to order k. Let us exchange every variable in for new independent variables
Then our original equation can be thought as a system of
and some number of equations of the following type
A solution of
is called a non-holonomic solution, and a solution of the system which is also solution of our original PDE is called a holonomic solution.
In order to check whether a solution to our original equation exists, one can first check if there is a non-holonomic solution. Usually this is quite easy, and if there is no non-holonomic solution, then our original equation did not have any solutions.
A PDE satisfies the h-principle if any non-holonomic solution can be deformed into a holonomic one in the class of non-holonomic solutions. Thus in the presence of h-principle, a differential topological problem reduces to an algebraic topological problem. More explicitly this means that apart from the topological obstruction there is no other obstruction to the existence of a holonomic solution. The topological problem of finding a non-holonomic solution is much easier to handle and can be addressed with the obstruction theory for topological bundles.
Many underdetermined partial differential equations satisfy the h-principle. However, the falsity of an h-principle is also an interesting statement, intuitively this means the objects being studied have non-trivial geometry that cannot be reduced to topology. As an example, embedded Lagrangians in a symplectic manifold do not satisfy an h-principle, to prove this one needs to find invariants coming from pseudo-holomorphic curves.
Simple examples
Monotone functions
Perhaps the simplest partial differential relation is for the derivative to not vanish: Properly, this is an ordinary differential relation, as this is a function in one variable.
A holonomic solution to this relation is a function whose derivative is nowhere vanishing, i.e. a strictly monotone differentiable function, either increasing or decreasing. The space of such functions consists of two disjoint convex sets: the increasin |
https://en.wikipedia.org/wiki/L7 | L7 or L-7 may refer to:
Music
L7 (band), a grunge/metal band from Los Angeles, California
L7 (album), a 1988 album by the band
L-Seven, a post-punk band from Detroit, Michigan
Mathematics and technology
ISO/IEC 8859-13 (Latin-7), an 8-bit character encoding
L7, the application layer in the OSI model of computer communications
A layer 7 switch or load balancer
The Lp space for p=7 in mathematics
Transportation
Vehicles
D-Lieferwagen L-7, a 1927–1930 German three-wheel truck
IM L7, a 2022–present Chinese full-size luxury electric sedan
Landsat 7, an Earth observation satellite
Other
L7, IATA code for Laoag International Airlines
Other uses
Royal Ordnance L7, british and the NATO standard 105mm tank and light-field cannon
L7 (machine gun), a Belgian 7.62 mm general-purpose machine gun
Motorola SLVR L7, a mobile phone
See also
7L (disambiguation)
Bustin' Out of L Seven, an album by Rick James |
https://en.wikipedia.org/wiki/Gromov%E2%80%93Hausdorff%20convergence | In mathematics, Gromov–Hausdorff convergence, named after Mikhail Gromov and Felix Hausdorff, is a notion for convergence of metric spaces which is a generalization of Hausdorff convergence.
Gromov–Hausdorff distance
The Gromov–Hausdorff distance was introduced by David Edwards in 1975, and it was later rediscovered and generalized by Mikhail Gromov in 1981. This distance measures how far two compact metric spaces are from being isometric. If X and Y are two compact metric spaces, then dGH (X, Y) is defined to be the infimum of all numbers dH(f(X), g(Y)) for all (compact) metric spaces M and all isometric embeddings f : X → M and g : Y → M. Here dH denotes Hausdorff distance between subsets in M and the isometric embedding is understood in the global sense, i.e. it must preserve all distances, not only infinitesimally small ones; for example no compact Riemannian manifold admits such an embedding into Euclidean space of the same dimension.
The Gromov–Hausdorff distance turns the set of all isometry classes of compact metric spaces into a metric space, called Gromov–Hausdorff space, and it therefore defines a notion of convergence for sequences of compact metric spaces, called Gromov–Hausdorff convergence. A metric space to which such a sequence converges is called the Gromov–Hausdorff limit of the sequence.
Some properties of Gromov–Hausdorff space
The Gromov–Hausdorff space is path-connected, complete, and separable. It is also geodesic, i.e., any two of its points are the endpoints of a minimizing geodesic. In the global sense, the Gromov–Hausdorff space is totally heterogeneous, i.e., its isometry group is trivial, but locally there are many nontrivial isometries.
Pointed Gromov–Hausdorff convergence
The pointed Gromov–Hausdorff convergence is an analog of Gromov–Hausdorff convergence appropriate for non-compact spaces. A pointed metric space is a pair (X,p) consisting of a metric space X and point p in X. A sequence (Xn, pn) of pointed metric spaces converges to a pointed metric space (Y, p) if, for each R > 0, the sequence of closed R-balls around pn in Xn converges to the closed R-ball around p in Y in the usual Gromov–Hausdorff sense.
Applications
The notion of Gromov–Hausdorff convergence was used by Gromov to prove that
any discrete group with polynomial growth is virtually nilpotent (i.e. it contains a nilpotent subgroup of finite index). See Gromov's theorem on groups of polynomial growth. (Also see D. Edwards for an earlier work.)
The key ingredient in the proof was the observation that for the
Cayley graph of a group with polynomial growth a sequence of rescalings converges in the pointed Gromov–Hausdorff sense.
Another simple and very useful result in Riemannian geometry is Gromov's compactness theorem, which states that
the set of Riemannian manifolds with Ricci curvature ≥ c and diameter ≤ D is relatively compact in the Gromov–Hausdorff metric. The limit spaces are metric spaces. Additional properties on the lengt |
https://en.wikipedia.org/wiki/Tetration | In mathematics, tetration (or hyper-4) is an operation based on iterated, or repeated, exponentiation. There is no standard notation for tetration, though and the left-exponent xb are common.
Under the definition as repeated exponentiation, means , where copies of are iterated via exponentiation, right-to-left, i.e. the application of exponentiation times. is called the "height" of the function, while is called the "base," analogous to exponentiation. It would be read as "the th tetration of ".
It is the next hyperoperation after exponentiation, but before pentation. The word was coined by Reuben Louis Goodstein from tetra- (four) and iteration.
Tetration is also defined recursively as
allowing for attempts to extend tetration to non-natural numbers such as real and complex numbers.
The two inverses of tetration are called super-root and super-logarithm, analogous to the nth root and the logarithmic functions. None of the three functions are elementary.
Tetration is used for the notation of very large numbers.
Introduction
The first four hyperoperations are shown here, with tetration being considered the fourth in the series. The unary operation succession, defined as , is considered to be the zeroth operation.
Addition copies of 1 added to combined by succession.
Multiplication copies of combined by addition.
Exponentiation copies of combined by multiplication.
Tetration copies of combined by exponentiation, right-to-left.
Note that nested exponents are conventionally interpreted from the top down: means and not
Succession, , is the most basic operation; while addition () is a primary operation, for addition of natural numbers it can be thought of as a chained succession of successors of ; multiplication ) is also a primary operation, though for natural numbers it can analogously be thought of as a chained addition involving numbers of . Exponentiation can be thought of as a chained multiplication involving numbers of and tetration () as a chained power involving numbers . Each of the operations above are defined by iterating the previous one; however, unlike the operations before it, tetration is not an elementary function.
The parameter is referred to as the base, while the parameter may be referred to as the height. In the original definition of tetration, the height parameter must be a natural number; for instance, it would be illogical to say "three raised to itself negative five times" or "four raised to itself one half of a time." However, just as addition, multiplication, and exponentiation can be defined in ways that allow for extensions to real and complex numbers, several attempts have been made to generalize tetration to negative numbers, real numbers, and complex numbers. One such way for doing so is using a recursive definition for tetration; for any positive real and non-negative integer , we can define recursively as:
The recursive definition is equivalent to repeated exponentiation f |
https://en.wikipedia.org/wiki/Polylogarithm | In mathematics, the polylogarithm (also known as Jonquière's function, for Alfred Jonquière) is a special function of order and argument . Only for special values of does the polylogarithm reduce to an elementary function such as the natural logarithm or a rational function. In quantum statistics, the polylogarithm function appears as the closed form of integrals of the Fermi–Dirac distribution and the Bose–Einstein distribution, and is also known as the Fermi–Dirac integral or the Bose–Einstein integral. In quantum electrodynamics, polylogarithms of positive integer order arise in the calculation of processes represented by higher-order Feynman diagrams.
The polylogarithm function is equivalent to the Hurwitz zeta function — either function can be expressed in terms of the other — and both functions are special cases of the Lerch transcendent. Polylogarithms should not be confused with polylogarithmic functions, nor with the offset logarithmic integral , which has the same notation without the subscript.
The polylogarithm function is defined by a power series in , which is also a Dirichlet series in :
This definition is valid for arbitrary complex order and for all complex arguments with ; it can be extended to by the process of analytic continuation. (Here the denominator is understood as ). The special case involves the ordinary natural logarithm, , while the special cases and are called the dilogarithm (also referred to as Spence's function) and trilogarithm respectively. The name of the function comes from the fact that it may also be defined as the repeated integral of itself:
thus the dilogarithm is an integral of a function involving the logarithm, and so on. For nonpositive integer orders , the polylogarithm is a rational function.
Properties
In the case where the order is an integer, it will be represented by (or when negative). It is often convenient to define where is the principal branch of the complex logarithm so that Also, all exponentiation will be assumed to be single-valued:
Depending on the order , the polylogarithm may be multi-valued. The principal branch of is taken to be given for by the above series definition and taken to be continuous except on the positive real axis, where a cut is made from to such that the axis is placed on the lower half plane of In terms of this amounts to . The discontinuity of the polylogarithm in dependence on can sometimes be confusing.
For real argument , the polylogarithm of real order is real if and its imaginary part for is :
Going across the cut, if ε is an infinitesimally small positive real number, then:
Both can be concluded from the series expansion (see below) of about
The derivatives of the polylogarithm follow from the defining power series:
The square relationship is seen from the series definition, and is related to the duplication formula (see also , ):
Kummer's function obeys a very similar duplication formula. This is a special case of |
https://en.wikipedia.org/wiki/Closure%20operator | In mathematics, a closure operator on a set S is a function from the power set of S to itself that satisfies the following conditions for all sets
{| border="0"
|-
|
| (cl is extensive),
|-
|
| (cl is increasing),
|-
|
| (cl is idempotent).
|}
Closure operators are determined by their closed sets, i.e., by the sets of the form cl(X), since the closure cl(X) of a set X is the smallest closed set containing X. Such families of "closed sets" are sometimes called closure systems or "Moore families". A set together with a closure operator on it is sometimes called a closure space. Closure operators are also called "hull operators", which prevents confusion with the "closure operators" studied in topology.
History
E. H. Moore studied closure operators in his 1910 Introduction to a form of general analysis, whereas the concept of the closure of a subset originated in the work of Frigyes Riesz in connection with topological spaces. Though not formalized at the time, the idea of closure originated in the late 19th century with notable contributions by Ernst Schröder, Richard Dedekind and Georg Cantor.
Examples
The usual set closure from topology is a closure operator. Other examples include the linear span of a subset of a vector space, the convex hull or affine hull of a subset of a vector space or the lower semicontinuous hull of a function , where is e.g. a normed space, defined implicitly , where is the epigraph of a function .
The relative interior is not a closure operator: although it is idempotent, it is not increasing and if is a cube in and is one of its faces, then , but and , so it is not increasing.
In topology, the closure operators are topological closure operators, which must satisfy
for all (Note that for this gives ).
In algebra and logic, many closure operators are finitary closure operators, i.e. they satisfy
In the theory of partially ordered sets, which are important in theoretical computer science, closure operators have a more general definition that replaces with . (See .)
Closure operators in topology
The topological closure of a subset X of a topological space consists of all points y of the space, such that every neighbourhood of y contains a point of X. The function that associates to every subset X its closure is a topological closure operator. Conversely, every topological closure operator on a set gives rise to a topological space whose closed sets are exactly the closed sets with respect to the closure operator.
Closure operators in algebra
Finitary closure operators play a relatively prominent role in universal algebra, and in this context they are traditionally called algebraic closure operators. Every subset of an algebra generates a subalgebra: the smallest subalgebra containing the set. This gives rise to a finitary closure operator.
Perhaps the best known example for this is the function that associates to every subset of a given vector space its linear span. Similarly, the func |
https://en.wikipedia.org/wiki/Takaoka%20District%2C%20K%C5%8Dchi | is a district located in Kōchi Prefecture, Japan.
As of the Shimanto merger but with 2003 population statistics, the district has an estimated population of 68,854 and a density of 45.1 persons per km2. The total area is 1,527.65 km2.
Towns and villages
Nakatosa
Ochi
Sakawa
Shimanto
Tsuno
Yusuhara
Hidaka
Geography
As with the majority of Kochi, the terrain in Takaoka is mostly mountainous.
The Shimanto River, that disperses further west in the Hata district, has its source in Tsuno.
Transport
Like most rural areas in Kochi, transport is limited for residents and visitors without private vehicles.
The JR Dosan line that runs from Kochi passes through Hidaka, Sakawa and Kure (Nakatosa) on the way to Kubokawa (Shimanto Town).
The other.
Local buses do operate within areas of Takaoka, such as a regular but infrequent set of routes servicing Sakawa, Ochi and Niyodogawa.
Mergers
On February 1, 2005 the villages of Hayama and Higashitsuno merged to form the new town of Tsuno.
On August 1, 2005 the village of Niyodo merged with the town of Ikegawa, and the village of Agawa, both from Agawa District, to form the new town of Niyodogawa, in Agawa District.
On January 1, 2006 the village of Ōnomi merged into the town of Nakatosa.
On March 20, 2006 the town of Kubokawa merged with the towns of Taishō and Towa, both from Hata District, to form the new town of Shimanto, in Takaoka District.
Districts in Kōchi Prefecture |
https://en.wikipedia.org/wiki/Hata%20District%2C%20K%C5%8Dchi | is a district located in Kōchi Prefecture, Japan.
As of the Shimanto merger but with 2003 population statistics, the district has an estimated population of 22,402 and a density of 59.4 persons per km2. The total area is 376.77 km2.
Towns and villages
Kuroshio
Ōtsuki
Mihara
Mergers
On April 10, 2005 the old city of Nakamura, and the village of Nishitosa merged to form the new city of Shimanto.
On March 20, 2006 the towns of Taishō and Towa merged with the town of Kubokawa, from Takaoka District, to form the new town of Shimanto, in Takaoka District.
On March 20, 2006 the towns of Ōgata and Saga merged to form the new town of Kuroshio.
External links
Districts in Kōchi Prefecture |
https://en.wikipedia.org/wiki/Geodesic%20curvature | In Riemannian geometry, the geodesic curvature of a curve measures how far the curve is from being a geodesic. For example, for 1D curves on a 2D surface embedded in 3D space, it is the curvature of the curve projected onto the surface's tangent plane. More generally, in a given manifold , the geodesic curvature is just the usual curvature of (see below). However, when the curve is restricted to lie on a submanifold of (e.g. for curves on surfaces), geodesic curvature refers to the curvature of in and it is different in general from the curvature of in the ambient manifold . The (ambient) curvature of depends on two factors: the curvature of the submanifold in the direction of (the normal curvature ), which depends only on the direction of the curve, and the curvature of seen in (the geodesic curvature ), which is a second order quantity. The relation between these is . In particular geodesics on have zero geodesic curvature (they are "straight"), so that , which explains why they appear to be curved in ambient space whenever the submanifold is.
Definition
Consider a curve in a manifold , parametrized by arclength, with unit tangent vector . Its curvature is the norm of the covariant derivative of : . If lies on , the geodesic curvature is the norm of the projection of the covariant derivative on the tangent space to the submanifold. Conversely the normal curvature is the norm of the projection of on the normal bundle to the submanifold at the point considered.
If the ambient manifold is the euclidean space , then the covariant derivative is just the usual derivative .
Example
Let be the unit sphere in three-dimensional Euclidean space. The normal curvature of is identically 1, independently of the direction considered. Great circles have curvature , so they have zero geodesic curvature, and are therefore geodesics. Smaller circles of radius will have curvature and geodesic curvature .
Some results involving geodesic curvature
The geodesic curvature is none other than the usual curvature of the curve when computed intrinsically in the submanifold . It does not depend on the way the submanifold sits in .
Geodesics of have zero geodesic curvature, which is equivalent to saying that is orthogonal to the tangent space to .
On the other hand the normal curvature depends strongly on how the submanifold lies in the ambient space, but marginally on the curve: only depends on the point on the submanifold and the direction , but not on .
In general Riemannian geometry, the derivative is computed using the Levi-Civita connection of the ambient manifold: . It splits into a tangent part and a normal part to the submanifold: . The tangent part is the usual derivative in (it is a particular case of Gauss equation in the Gauss-Codazzi equations), while the normal part is , where denotes the second fundamental form.
The Gauss–Bonnet theorem.
See also
Curvature
Darboux frame
Gauss–Codazzi equations
References
.
.
|
https://en.wikipedia.org/wiki/Boquer%C3%B3n%20department | Boquerón () is a department in the western region of Paraguay. It is the country's largest department, with an area of , but, according to the statistics for 2021 by INE, its population is only 68,080, being the second least populated department. The department includes the Russian Mennonite colonies of Fernheim, Menno and its administrative center Loma Plata and Neuland. The capital is Filadelfia. Other towns are General Eugenio A. Garay, Doctor Pedro P. Peña and Mariscal Estigarribia.
In 1945 Boquerón was split, with the northern portion separated off being renamed "Chaco". The reduced remaining area continued to be called "Boquerón", and the department's capital was moved to Filadelfia (the previous capital had been Doctor Pedro P. Peña). However, in 1992 the previous department Chaco was re-integrated into Boquerón, effectively re-forming the department as of 1945 when it was split, except that after 1992 the enlarged department's capital remained at Filadelfia.
Geography
Boquerón Department is located in the Occidental Region of Paraguay, between the southern parallels 20° 06' and 23° 50' of latitude, and the western meridians 50° 20' and 62° 40' of longitude. It is the largest department of Paraguay, with an area of , slightly larger than Hungary or the province of South Sumatra.
Adjacent territories
North: Alto Paraguay Department, separated by an arbitrary straight line that goes from Hito IV Fort Tte. G. Mendoza to Fort Madrejón; and also by a railway from "km 220" to "km 160".
South: Argentina, separated by the Pilcomayo River, from Misión San Lorenzo to Hito I Esmeralda.
East: Presidente Hayes Department, separated by the road that connects Misión San Lorenzo with Fort Gral. Díaz, Ávalos Sánchez, Zenteno, Dr. Gaspar Rodríguez de Francia, Boquerón, Isla Po'í and Casanillo; and from this point, by an imaginary straight line until "km 160". It also borders Alto Paraguay Department, separated from it by a straight imaginary line from Fort Madrejón to Fort Carlos Antonio López and from there by another line to Fort Montanía.
West: Bolivia, separated by an imaginary line from Hito I Esmeralda until Hito IV Fort Tte. Gabino Mendoza.
Natural environment and climate
This is the most arid region of Paraguay; it has some small streams, but with dry riverbeds. The climate is in the transition between a warm semi-arid climate, a tropical savanna climate and a humid subtropical climate (Köppen climate classification BSh, Aw and Cfa) The rain is scarce during the dry season, but during the wet season it can cause flash floods. The average annual rainfall is about a year in the north of the department and a year in the south. On 14 November 2009, Pratts Gill, a small town in the Boquerón Department, recorded a temperature of . This is the highest temperature to have ever been recorded in Paraguay.
The trees in the area are short and thorny; there are brushwood and cactus, dunes and small hills, especially in the north of the department. Some |
https://en.wikipedia.org/wiki/LINPACK | LINPACK is a software library for performing numerical linear algebra on digital computers.
It was written in Fortran by Jack Dongarra, Jim Bunch, Cleve Moler, and Gilbert Stewart, and was intended for use on supercomputers in the 1970s and early 1980s. It has been largely superseded by LAPACK, which runs more efficiently on modern architectures.
LINPACK makes use of the BLAS (Basic Linear Algebra Subprograms) libraries for performing basic vector and matrix operations.
The LINPACK benchmarks appeared initially as part of the LINPACK user's manual. The parallel LINPACK benchmark implementation called HPL (High Performance Linpack) is used to benchmark and rank supercomputers for the TOP500 list.
World's most powerful computer by year
References
Benchmarks (computing)
Fortran libraries
Numerical linear algebra
Numerical software |
https://en.wikipedia.org/wiki/Fibonacci%20polynomials | In mathematics, the Fibonacci polynomials are a polynomial sequence which can be considered as a generalization of the Fibonacci numbers. The polynomials generated in a similar way from the Lucas numbers are called Lucas polynomials.
Definition
These Fibonacci polynomials are defined by a recurrence relation:
The Lucas polynomials use the same recurrence with different starting values:
They can be defined for negative indices by
The Fibonacci polynomials form a sequence of orthogonal polynomials with and .
Examples
The first few Fibonacci polynomials are:
The first few Lucas polynomials are:
Properties
The degree of Fn is n − 1 and the degree of Ln is n.
The Fibonacci and Lucas numbers are recovered by evaluating the polynomials at x = 1; Pell numbers are recovered by evaluating Fn at x = 2.
The ordinary generating functions for the sequences are:
The polynomials can be expressed in terms of Lucas sequences as
They can also be expressed in terms of Chebyshev polynomials and as
where is the imaginary unit.
Identities
As particular cases of Lucas sequences, Fibonacci polynomials satisfy a number of identities, such as
Closed form expressions, similar to Binet's formula are:
where
are the solutions (in t) of
For Lucas Polynomials n > 0, we have
A relationship between the Fibonacci polynomials and the standard basis polynomials is given by
For example,
Combinatorial interpretation
If F(n,k) is the coefficient of xk in Fn(x), namely
then F(n,k) is the number of ways an n−1 by 1 rectangle can be tiled with 2 by 1 dominoes and 1 by 1 squares so that exactly k squares are used. Equivalently, F(n,k) is the number of ways of writing n−1 as an ordered sum involving only 1 and 2, so that 1 is used exactly k times. For example F(6,3)=4 and 5 can be written in 4 ways, 1+1+1+2, 1+1+2+1, 1+2+1+1, 2+1+1+1, as a sum involving only 1 and 2 with 1 used 3 times. By counting the number of times 1 and 2 are both used in such a sum, it is evident that
This gives a way of reading the coefficients from Pascal's triangle as shown on the right.
References
Jin, Z. On the Lucas polynomials and some of their new identities. Advances in Differential Equations 2018, 126 (2018). https://doi.org/10.1186/s13662-018-1527-9
Further reading
External links
Polynomials
Fibonacci numbers |
https://en.wikipedia.org/wiki/ICMC | ICMC may refer to:
International Catholic Migration Commission
International Computer Music Conference
The Indiana College Mathematics Competition
International Cryptographic Module Conference
Integrated Currency Management Centre
Inter College Music Competition
Integrated Call Management Centre |
https://en.wikipedia.org/wiki/Spurious%20relationship | In statistics, a spurious relationship or spurious correlation is a mathematical relationship in which two or more events or variables are associated but not causally related, due to either coincidence or the presence of a certain third, unseen factor (referred to as a "common response variable", "confounding factor", or "lurking variable").
Examples
An example of a spurious relationship can be found in the time-series literature, where a spurious regression is one that provides misleading statistical evidence of a linear relationship between independent non-stationary variables. In fact, the non-stationarity may be due to the presence of a unit root in both variables. In particular, any two nominal economic variables are likely to be correlated with each other, even when neither has a causal effect on the other, because each equals a real variable times the price level, and the common presence of the price level in the two data series imparts correlation to them. (See also spurious correlation of ratios.)
Another example of a spurious relationship can be seen by examining a city's ice cream sales. The sales might be highest when the rate of drownings in city swimming pools is highest. To allege that ice cream sales cause drowning, or vice versa, would be to imply a spurious relationship between the two. In reality, a heat wave may have caused both. The heat wave is an example of a hidden or unseen variable, also known as a confounding variable.
Another commonly noted example is a series of Dutch statistics showing a positive correlation between the number of storks nesting in a series of springs and the number of human babies born at that time. Of course there was no causal connection; they were correlated with each other only because they were correlated with the weather nine months before the observations.
In rare cases, a spurious relationship can occur between two completely unrelated variables without any confounding variable, as was the case between the success of the Washington Commanders professional football team in a specific game before each presidential election and the success of the incumbent President's political party in said election. For 16 consecutive elections between 1940 and 2000, the Redskins Rule correctly matched whether the incumbent President's political party would retain or lose the Presidency. The rule eventually failed shortly after Elias Sports Bureau discovered the correlation in 2000; in 2004, 2012 and 2016, the results of the Commanders' game and the election did not match. In a similar spurious relationship involving the National Football League, in the 1970s, Leonard Koppett noted a correlation between the direction of the stock market and the winning conference of that year's Super Bowl, the Super Bowl indicator; the relationship maintained itself for most of the 20th century before reverting to more random behavior in the 21st.
Hypothesis testing
Often one tests a null hypothesis of no correlation b |
https://en.wikipedia.org/wiki/Nominal%20group | Nominal group may refer to:
Nominal group, alias for nominal category in statistics
Nominal group (functional grammar)
Nominal group technique, group decision-making technique |
https://en.wikipedia.org/wiki/Descriptive%20geometry | Descriptive geometry is the branch of geometry which allows the representation of three-dimensional objects in two dimensions by using a specific set of procedures. The resulting techniques are important for engineering, architecture, design and in art. The theoretical basis for descriptive geometry is provided by planar geometric projections. The earliest known publication on the technique was "Underweysung der Messung mit dem Zirckel und Richtscheyt" (Observation of the measurement with the compass and spirit level), published in Linien, Nuremberg: 1525, by Albrecht Dürer. Italian architect Guarino Guarini was also a pioneer of projective and descriptive geometry, as is clear from his Placita Philosophica (1665), Euclides Adauctus (1671) and Architettura Civile (1686—not published until 1737), anticipating the work of Gaspard Monge (1746–1818), who is usually credited with the invention of descriptive geometry. Gaspard Monge is usually considered the "father of descriptive geometry" due to his developments in geometric problem solving. His first discoveries were in 1765 while he was working as a draftsman for military fortifications, although his findings were published later on.
Monge's protocols allow an imaginary object to be drawn in such a way that it may be modeled in three dimensions. All geometric aspects of the imaginary object are accounted for in true size/to-scale and shape, and can be imaged as seen from any position in space. All images are represented on a two-dimensional surface.
Descriptive geometry uses the image-creating technique of imaginary, parallel projectors emanating from an imaginary object and intersecting an imaginary plane of projection at right angles. The cumulative points of intersections create the desired image.
Protocols
Project two images of an object into mutually perpendicular, arbitrary directions. Each image view accommodates three dimensions of space, two dimensions displayed as full-scale, mutually-perpendicular axes and one as an invisible (point view) axis receding into the image space (depth). Each of the two adjacent image views shares a full-scale view of one of the three dimensions of space.
Either of these images may serve as the beginning point for a third projected view. The third view may begin a fourth projection, and on ad infinitum. These sequential projections each represent a circuitous, 90° turn in space in order to view the object from a different direction.
Each new projection utilizes a dimension in full scale that appears as point-view dimension in the previous view. To achieve the full-scale view of this dimension and accommodate it within the new view requires one to ignore the previous view and proceed to the second previous view where this dimension appears in full-scale.
Each new view may be created by projecting into any of an infinite number of directions, perpendicular to the previous direction of projection. (Envision the many directions of the spokes of a wagon |
https://en.wikipedia.org/wiki/Statistics%20New%20Zealand | Statistics New Zealand (), branded as Stats NZ, is the public service department of New Zealand charged with the collection of statistics related to the economy, population and society of New Zealand. To this end, Stats NZ produces censuses and surveys.
Organization
Statistics New Zealand employs people with a variety of skills: including statisticians, mathematicians, computer science specialists, accountants, economists, demographers, sociologists, geographers, social psychologists, and marketers.
There are seven organizational subgroups, each managed by a Deputy Government Statistician:
Macro-economic and Environment Statistics studies prices, and national accounts, develops macro-economic statistics, does government and international accounts, and ANZSIC 06 implementation (facilitating changeover to new classification code developed jointly with Australian statistics officials.)
Social and Population Statistics studies population, social conditions, standard of living, and census, and has a census planning manager as well as statisticians helping to develop new social statistics measures.
Standards and Methods studies statistical methods, statistical education, and research, solutions and capabilities, information management, and develops new methodologies.
Collections and Dissemination services clients. There is one general manager in Christchurch and one in Auckland. It develops products and services and manages publishing and customer services.
Organization Direction maintains contacts with key government officials, does internal audits and business planning, manages international relations and the Official Statistics System (OSS), and advises on Māori affairs.
Industry and Labour Statistics studies business indicators, finance and performance, agriculture, energy and work knowledge and skills.
Organization Development focuses on services for the agency itself, including information technology management, quality assurance, application development and support, finances, corporate support, and human resources.
Many of the agency's powers, duties, and responsibilities are governed by acts of the New Zealand Parliament. The agency is a state sector organization of New Zealand operating under the authority of the Statistics Act 1975.
Responsibilities and activities
The department conducts the New Zealand census every five years. The census is officially done on one day. The most recent census was in 2018. The count of usual residents (excluding visitors from overseas) from this census was 4,699,755. they lived in 1,664,313 occupied dwellings; their median age was 37.4 years (half older, half younger); 775,836 identified themselves as "Māori" (16.5% of the population); people had a median income of $31,800. This is a main source of information, and data collected from this census is often used for further purposes within the department as well as serving as benchmark information for numerous reports and surveys. For example, the ce |
https://en.wikipedia.org/wiki/Hairy%20ball%20theorem | The hairy ball theorem of algebraic topology (sometimes called the hedgehog theorem in Europe) states that there is no nonvanishing continuous tangent vector field on even-dimensional n-spheres. For the ordinary sphere, or 2‑sphere, if f is a continuous function that assigns a vector in R3 to every point p on a sphere such that f(p) is always tangent to the sphere at p, then there is at least one pole, a point where the field vanishes (a p such that f(p) = 0).
The theorem was first proved by Henri Poincaré for the 2-sphere in 1885, and extended to higher even dimensions in 1912 by Luitzen Egbertus Jan Brouwer.
The theorem has been expressed colloquially as "you can't comb a hairy ball flat without creating a cowlick" or "you can't comb the hair on a coconut".
Counting zeros
Every zero of a vector field has a (non-zero) "index", and it can be shown that the sum of all of the indices at all of the zeros must be two, because the Euler characteristic of the 2-sphere is two. Therefore, there must be at least one zero. This is a consequence of the Poincaré–Hopf theorem. In the case of the torus, the Euler characteristic is 0; and it is possible to "comb a hairy doughnut flat". In this regard, it follows that for any compact regular 2-dimensional manifold with non-zero Euler characteristic, any continuous tangent vector field has at least one zero.
Application to computer graphics
A common problem in computer graphics is to generate a non-zero vector in R3 that is orthogonal to a given non-zero vector. There is no single continuous function that can do this for all non-zero vector inputs. This is a corollary of the hairy ball theorem. To see this, consider the given vector as the radius of a sphere and note that finding a non-zero vector orthogonal to the given one is equivalent to finding a non-zero vector that is tangent to the surface of that sphere where it touches the radius. However, the hairy ball theorem says there exists no continuous function that can do this for every point on the sphere (equivalently, for every given vector).
Lefschetz connection
There is a closely related argument from algebraic topology, using the Lefschetz fixed-point theorem. Since the Betti numbers of a 2-sphere are 1, 0, 1, 0, 0, ... the Lefschetz number (total trace on homology) of the identity mapping is 2. By integrating a vector field we get (at least a small part of) a one-parameter group of diffeomorphisms on the sphere; and all of the mappings in it are homotopic to the identity. Therefore, they all have Lefschetz number 2, also. Hence they have fixed points (since the Lefschetz number is nonzero). Some more work would be needed to show that this implies there must actually be a zero of the vector field. It does suggest the correct statement of the more general Poincaré-Hopf index theorem.
Corollary
A consequence of the hairy ball theorem is that any continuous function that maps an even-dimensional sphere into itself has either a fixed point or a point th |
https://en.wikipedia.org/wiki/Keith%20number | In recreational mathematics, a Keith number or repfigit number (short for repetitive Fibonacci-like digit) is a natural number in a given number base with digits such that when a sequence is created such that the first terms are the digits of and each subsequent term is the sum of the previous terms, is part of the sequence. Keith numbers were introduced by Mike Keith in 1987.
They are computationally very challenging to find, with only about 100 known.
Definition
Let be a natural number, let be the number of digits of in base , and let
be the value of each digit of .
We define the sequence by a linear recurrence relation. For ,
and for
If there exists an such that , then is said to be a Keith number.
For example, 88 is a Keith number in base 6, as
and the entire sequence
and .
Finding Keith numbers
Whether or not there are infinitely many Keith numbers in a particular base is currently a matter of speculation. Keith numbers are rare and hard to find. They can be found by exhaustive search, and no more efficient algorithm is known.
According to Keith, in base 10, on average Keith numbers are expected between successive powers of 10. Known results seem to support this.
Examples
14, 19, 28, 47, 61, 75, 197, 742, 1104, 1537, 2208, 2580, 3684, 4788, 7385, 7647, 7909, 31331, 34285, 34348, 55604, 62662, 86935, 93993, 120284, 129106, 147640, 156146, 174680, 183186, 298320, 355419, 694280, 925993, 1084051, 7913837, 11436171, 33445755, 44121607, 129572008, 251133297, ...
Other bases
In base 2, there exists a method to construct all Keith numbers.
The Keith numbers in base 12, written in base 12, are
11, 15, 1Ɛ, 22, 2ᘔ, 31, 33, 44, 49, 55, 62, 66, 77, 88, 93, 99, ᘔᘔ, ƐƐ, 125, 215, 24ᘔ, 405, 42ᘔ, 654, 80ᘔ, 8ᘔ3, ᘔ59, 1022, 1662, 2044, 3066, 4088, 4ᘔ1ᘔ, 4ᘔƐ1, 50ᘔᘔ, 8538, Ɛ18Ɛ, 17256, 18671, 24ᘔ78, 4718Ɛ, 517Ɛᘔ, 157617, 1ᘔ265ᘔ, 5ᘔ4074, 5ᘔƐ140, 6Ɛ1449, 6Ɛ8515, ...
where ᘔ represents 10 and Ɛ represents 11.
Keith clusters
A Keith cluster is a related set of Keith numbers such that one is a multiple of another. For example, in base 10, , , and are all Keith clusters. These are possibly the only three examples of a Keith cluster in base 10.
Programming example
The example below implements the sequence defined above in Python to determine if a number in a particular base is a Keith number:
def is_repfigit(x: int, b: int) -> bool:
"""Determine if a number in a particular base is a Keith number."""
if x == 0:
return True
sequence = []
y = x
while y > 0:
sequence.append(y % b)
y = y // b
digit_count = len(sequence)
sequence.reverse()
while sequence[len(sequence) - 1] < x:
n = 0
for i in range(0, digit_count):
n = n + sequence[len(sequence) - digit_count + i]
sequence.append(n)
return (sequence[len(sequence) - 1] == x)
See also
Arithmetic dynamics
Fibonacci number
Linear recurrence relation
References
Arithmetic dynamics
Base |
https://en.wikipedia.org/wiki/Householder%20transformation | In linear algebra, a Householder transformation (also known as a Householder reflection or elementary reflector) is a linear transformation that describes a reflection about a plane or hyperplane containing the origin. The Householder transformation was used in a 1958 paper by Alston Scott Householder.
Its analogue over general inner product spaces is the Householder operator.
Definition
Transformation
The reflection hyperplane can be defined by its normal vector, a unit vector (a vector with length ) that is orthogonal to the hyperplane. The reflection of a point about this hyperplane is the linear transformation:
where is given as a column unit vector with conjugate transpose .
Householder matrix
The matrix constructed from this transformation can be expressed in terms of an outer product as:
is known as the Householder matrix, where is the identity matrix.
Properties
The Householder matrix has the following properties:
it is Hermitian: ,
it is unitary: ,
hence it is involutory: .
A Householder matrix has eigenvalues . To see this, notice that if is orthogonal to the vector which was used to create the reflector, then , i.e., is an eigenvalue of multiplicity , since there are independent vectors orthogonal to . Also, notice , and so is an eigenvalue with multiplicity .
The determinant of a Householder reflector is , since the determinant of a matrix is the product of its eigenvalues, in this case one of which is with the remainder being (as in the previous point).
Applications
Geometric optics
In geometric optics, specular reflection can be expressed in terms of the Householder matrix (see ).
Numerical linear algebra
Householder transformations are widely used in numerical linear algebra, for example, to annihilate the entries below the main diagonal of a matrix, to perform QR decompositions and in the first step of the QR algorithm. They are also widely used for transforming to a Hessenberg form. For symmetric or Hermitian matrices, the symmetry can be preserved, resulting in tridiagonalization.
QR decomposition
Householder reflections can be used to calculate QR decompositions by reflecting first one column of a matrix onto a multiple of a standard basis vector, calculating the transformation matrix, multiplying it with the original matrix and then recursing down the minors of that product.
Tridiagonalization
This procedure is presented in Numerical Analysis by Burden and Faires. It uses a slightly altered function with .
In the first step, to form the Householder matrix in each step we need to determine and , which are:
From and , construct vector :
where , , and
for each
Then compute:
Having found and computed the process is repeated for as follows:
Continuing in this manner, the tridiagonal and symmetric matrix is formed.
Examples
In this example, also from Burden and Faires, the given matrix is transformed to the similar tridiagonal matrix A3 by using the Householder method.
|
https://en.wikipedia.org/wiki/H%C3%A9non%20map | In mathematics, the Hénon map, sometimes called Hénon–Pomeau attractor/map, is a discrete-time dynamical system. It is one of the most studied examples of dynamical systems that exhibit chaotic behavior. The Hénon map takes a point in the plane and maps it to a new point
The map depends on two parameters, and , which for the classical Hénon map have values of and . For the classical values the Hénon map is chaotic. For other values of and the map may be chaotic, intermittent, or converge to a periodic orbit. An overview of the type of behavior of the map at different parameter values may be obtained from its orbit diagram.
The map was introduced by Michel Hénon as a simplified model of the Poincaré section of the Lorenz model. For the classical map, an initial point of the plane will either approach a set of points known as the Hénon strange attractor, or diverge to infinity. The Hénon attractor is a fractal, smooth in one direction and a Cantor set in another. Numerical estimates yield a correlation dimension of 1.21 ± 0.01 or 1.25 ± 0.02 (depending on the dimension of the embedding space) and a Box Counting dimension of 1.261 ± 0.003 for the attractor of the classical map.
Attractor
The Hénon map maps two points into themselves: these are the invariant points. For the classical values of a and b of the Hénon map, one of these points is on the attractor:
This point is unstable. Points close to this fixed point and along the slope 1.924 will approach the fixed point and points along the slope -0.156 will move away from the fixed point. These slopes arise from the linearizations of the stable manifold and unstable manifold of the fixed point. The unstable manifold of the fixed point in the attractor is contained in the strange attractor of the Hénon map.
The Hénon map does not have a strange attractor for all values of the parameters a and b. For example, by keeping b fixed at 0.3 the bifurcation diagram shows that for a = 1.25 the Hénon map has a stable periodic orbit as an attractor.
Cvitanović et al. have shown how the structure of the Hénon strange attractor can be understood in terms of unstable periodic orbits within the attractor.
Relationship to bifurcation diagram
If multiple Hénon maps are plotted, for each map varying the value of b, then stacking all maps together, a Bifurcation diagram is produced. A Bifurcation diagram that is folded like a taco. Hence its boomerang shape when viewed in 2D from the top.
Decomposition
The Hénon map may be decomposed into the composition of three functions acting on the domain one after the other.
1) an area-preserving bend:
,
2) a contraction in the x direction:
,
3) a reflection in the line y = x:
.
One-dimensional decomposition
The Hénon map may also be deconstructed into a one-dimensional map, defined similarly to the Fibonacci Sequence.
Four-dimensional extension
Although the Hénon map can be plotted on the x- and y-axes, by varying a and b, we obtain two |
https://en.wikipedia.org/wiki/Primitive%20ideal | In mathematics, specifically ring theory, a left primitive ideal is the annihilator of a (nonzero) simple left module. A right primitive ideal is defined similarly. Left and right primitive ideals are always two-sided ideals.
Primitive ideals are prime. The quotient of a ring by a left primitive ideal is a left primitive ring. For commutative rings the primitive ideals are maximal, and so commutative primitive rings are all fields.
Primitive spectrum
The primitive spectrum of a ring is a non-commutative analog of the prime spectrum of a commutative ring.
Let A be a ring and the set of all primitive ideals of A. Then there is a topology on , called the Jacobson topology, defined so that the closure of a subset T is the set of primitive ideals of A containing the intersection of elements of T.
Now, suppose A is an associative algebra over a field. Then, by definition, a primitive ideal is the kernel of an irreducible representation of A and thus there is a surjection
Example: the spectrum of a unital C*-algebra.
See also
Dixmier mapping
Notes
References
External links
Ideals (ring theory)
Module theory |
https://en.wikipedia.org/wiki/Semiprimitive%20ring | In algebra, a semiprimitive ring or Jacobson semisimple ring or J-semisimple ring is a ring whose Jacobson radical is zero. This is a type of ring more general than a semisimple ring, but where simple modules still provide enough information about the ring. Rings such as the ring of integers are semiprimitive, and an artinian semiprimitive ring is just a semisimple ring. Semiprimitive rings can be understood as subdirect products of primitive rings, which are described by the Jacobson density theorem.
Definition
A ring is called semiprimitive or Jacobson semisimple if its Jacobson radical is the zero ideal.
A ring is semiprimitive if and only if it has a faithful semisimple left module. The semiprimitive property is left-right symmetric, and so a ring is semiprimitive if and only if it has a faithful semisimple right module.
A ring is semiprimitive if and only if it is a subdirect product of left primitive rings.
A commutative ring is semiprimitive if and only if it is a subdirect product of fields, .
A left artinian ring is semiprimitive if and only if it is semisimple, . Such rings are sometimes called semisimple Artinian, .
Examples
The ring of integers is semiprimitive, but not semisimple.
Every primitive ring is semiprimitive.
The product of two fields is semiprimitive but not primitive.
Every von Neumann regular ring is semiprimitive.
Jacobson himself has defined a ring to be "semisimple" if and only if it is a subdirect product of simple rings, . However, this is a stricter notion, since the endomorphism ring of a countably infinite dimensional vector space is semiprimitive, but not a subdirect product of simple rings, .
References
Algebraic structures
Ring theory |
https://en.wikipedia.org/wiki/Primitive%20ring | In the branch of abstract algebra known as ring theory, a left primitive ring is a ring which has a faithful simple left module. Well known examples include endomorphism rings of vector spaces and Weyl algebras over fields of characteristic zero.
Definition
A ring R is said to be a left primitive ring if it has a faithful simple left R-module. A right primitive ring is defined similarly with right R-modules. There are rings which are primitive on one side but not on the other. The first example was constructed by George M. Bergman in . Another example found by Jategaonkar showing the distinction can be found in .
An internal characterization of left primitive rings is as follows: a ring is left primitive if and only if there is a maximal left ideal containing no nonzero two-sided ideals. The analogous definition for right primitive rings is also valid.
The structure of left primitive rings is completely determined by the Jacobson density theorem: A ring is left primitive if and only if it is isomorphic to a dense subring of the ring of endomorphisms of a left vector space over a division ring.
Another equivalent definition states that a ring is left primitive if and only if it is a prime ring with a faithful left module of finite length (, Ex. 11.19, p. 191).
Properties
One-sided primitive rings are both semiprimitive rings and prime rings. Since the product ring of two or more nonzero rings is not prime, it is clear that the product of primitive rings is never primitive.
For a left Artinian ring, it is known that the conditions "left primitive", "right primitive", "prime", and "simple" are all equivalent, and in this case it is a semisimple ring isomorphic to a square matrix ring over a division ring. More generally, in any ring with a minimal one sided ideal, "left primitive" = "right primitive" = "prime".
A commutative ring is left primitive if and only if it is a field.
Being left primitive is a Morita invariant property.
Examples
Every simple ring R with unity is both left and right primitive. (However, a simple non-unital ring may not be primitive.) This follows from the fact that R has a maximal left ideal M, and the fact that the quotient module R/M is a simple left R-module, and that its annihilator is a proper two-sided ideal in R. Since R is a simple ring, this annihilator is {0} and therefore R/M is a faithful left R-module.
Weyl algebras over fields of characteristic zero are primitive, and since they are domains, they are examples without minimal one-sided ideals.
Full linear rings
A special case of primitive rings is that of full linear rings. A left full linear ring is the ring of all linear transformations of an infinite-dimensional left vector space over a division ring. (A right full linear ring differs by using a right vector space instead.) In symbols, where V is a vector space over a division ring D. It is known that R is a left full linear ring if and only if R is von Neumann regular, left self-inj |
https://en.wikipedia.org/wiki/142%20%28number%29 | 142 (one hundred [and] forty-two) is the natural number following 141 and preceding 143.
In mathematics
There are 142 connected functional graphs on four labeled vertices, 142 planar graphs with 6 unlabeled vertices, and 142 partial involutions on five elements.
See also
The year AD 142 or 142 BC
List of highways numbered 142
References
Integers |
https://en.wikipedia.org/wiki/Aliquot%20sequence | In mathematics, an aliquot sequence is a sequence of positive integers in which each term is the sum of the proper divisors of the previous term. If the sequence reaches the number 1, it ends, since the sum of the proper divisors of 1 is 0.
Definition and overview
The aliquot sequence starting with a positive integer can be defined formally in terms of the sum-of-divisors function or the aliquot sum function in the following way:
If the condition is added, then the terms after 0 are all 0, and all aliquot sequences would be infinite, and we can conjecture that all aliquot sequences are convergent, the limit of these sequences are usually 0 or 6.
For example, the aliquot sequence of 10 is because:
Many aliquot sequences terminate at zero; all such sequences necessarily end with a prime number followed by 1 (since the only proper divisor of a prime is 1), followed by 0 (since 1 has no proper divisors). See for a list of such numbers up to 75. There are a variety of ways in which an aliquot sequence might not terminate:
A perfect number has a repeating aliquot sequence of period 1. The aliquot sequence of 6, for example, is
An amicable number has a repeating aliquot sequence of period 2. For instance, the aliquot sequence of 220 is
A sociable number has a repeating aliquot sequence of period 3 or greater. (Sometimes the term sociable number is used to encompass amicable numbers as well.) For instance, the aliquot sequence of 1264460 is
Some numbers have an aliquot sequence which is eventually periodic, but the number itself is not perfect, amicable, or sociable. For instance, the aliquot sequence of 95 is Numbers like 95 that are not perfect, but have an eventually repeating aliquot sequence of period 1 are called aspiring numbers.
The lengths of the aliquot sequences that start at are
1, 2, 2, 3, 2, 1, 2, 3, 4, 4, 2, 7, 2, 5, 5, 6, 2, 4, 2, 7, 3, 6, 2, 5, 1, 7, 3, 1, 2, 15, 2, 3, 6, 8, 3, 4, 2, 7, 3, 4, 2, 14, 2, 5, 7, 8, 2, 6, 4, 3, ...
The final terms (excluding 1) of the aliquot sequences that start at are
1, 2, 3, 3, 5, 6, 7, 7, 3, 7, 11, 3, 13, 7, 3, 3, 17, 11, 19, 7, 11, 7, 23, 17, 6, 3, 13, 28, 29, 3, 31, 31, 3, 7, 13, 17, 37, 7, 17, 43, 41, 3, 43, 43, 3, 3, 47, 41, 7, 43, ...
Numbers whose aliquot sequence terminates in 1 are
1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, ...
Numbers whose aliquot sequence known to terminate in a perfect number, other than perfect numbers themselves (6, 28, 496, ...), are
25, 95, 119, 143, 417, 445, 565, 608, 650, 652, 675, 685, 783, 790, 909, 913, ...
Numbers whose aliquot sequence terminates in a cycle with length at least 2 are
220, 284, 562, 1064, 1184, 1188, 1210, 1308, 1336, 1380, 1420, 1490, 1604, 1690, 1692, 1772, 1816, 1898, 2008, 2122, 2152, 2172, 2362, ...
Numbers whose aliquot sequence is not known to be finite or eventually periodic |
https://en.wikipedia.org/wiki/Peter%20Lax | Peter David Lax (born Lax Péter Dávid; 1 May 1926) is a Hungarian-born American mathematician and Abel Prize laureate working in the areas of pure and applied mathematics.
Lax has made important contributions to integrable systems, fluid dynamics and shock waves, solitonic physics, hyperbolic conservation laws, and mathematical and scientific computing, among other fields.
In a 1958 paper Lax stated a conjecture about matrix representations for third order hyperbolic polynomials which remained unproven for over four decades. Interest in the "Lax conjecture" grew as mathematicians working in several different areas recognized the importance of its implications in their field, until it was finally proven to be true in 2003.
Life and education
Lax was born in Budapest, Hungary to a Jewish family. Lax began displaying an interest in mathematics at age twelve, and soon his parents hired Rózsa Péter as a tutor for him. His parents Klara Kornfield and Henry Lax were both physicians and his uncle Albert Kornfeld (also known as Albert Korodi) was a mathematician, as well as a friend of Leó Szilárd.
The family left Hungary on 15 November 1941, and traveled via Lisbon to the United States. As a high school student at Stuyvesant High School, Lax took no math classes but did compete on the school math team. During this time, he met with John von Neumann, Richard Courant, and Paul Erdős, who introduced him to Albert Einstein.
As he was still 17 when he finished high school, he could avoid military service, and was able to study for three semesters at New York University. He attended a complex analysis class in the role of a student, but ended up taking over as instructor. He met his future wife, Anneli Cahn (married to her first husband at that time) in this class.
Before being able to complete his studies, Lax was drafted into the U.S. Army. After basic training, the Army sent him to Texas A&M University for more studies. He was then sent to Oak Ridge National Laboratory, and soon afterwards to the Manhattan Project at Los Alamos, New Mexico. At Los Alamos, he began working as a calculator operator, but eventually moved on to higher-level mathematics.
After the war ended, he remained with the Army at Los Alamos for another year, while taking courses at the University of New Mexico, then studied at Stanford University for a semester with Gábor Szegő and George Pólya.
Lax returned to NYU for the 1946–1947 academic year, and by pooling credits from the four universities at which he had studied, he graduated that year. He stayed at NYU for his graduate studies, marrying Anneli in 1948 and earning a PhD in 1949 under the supervision of Kurt O. Friedrichs.
Lax holds a faculty position in the Department of Mathematics, Courant Institute of Mathematical Sciences, New York University.
Awards and honors
He is a member of the Norwegian Academy of Science and Letters and the National Academy of Sciences, USA, the American Academy of Arts and Sciences, and the A |
https://en.wikipedia.org/wiki/Galois%20module | In mathematics, a Galois module is a G-module, with G being the Galois group of some extension of fields. The term Galois representation is frequently used when the G-module is a vector space over a field or a free module over a ring in representation theory, but can also be used as a synonym for G-module. The study of Galois modules for extensions of local or global fields and their group cohomology is an important tool in number theory.
Examples
Given a field K, the multiplicative group (Ks)× of a separable closure of K is a Galois module for the absolute Galois group. Its second cohomology group is isomorphic to the Brauer group of K (by Hilbert's theorem 90, its first cohomology group is zero).
If X is a smooth proper scheme over a field K then the ℓ-adic cohomology groups of its geometric fibre are Galois modules for the absolute Galois group of K.
Ramification theory
Let K be a valued field (with valuation denoted v) and let L/K be a finite Galois extension with Galois group G. For an extension w of v to L, let Iw denote its inertia group. A Galois module ρ : G → Aut(V) is said to be unramified if ρ(Iw) = {1}.
Galois module structure of algebraic integers
In classical algebraic number theory, let L be a Galois extension of a field K, and let G be the corresponding Galois group. Then the ring OL of algebraic integers of L can be considered as an OK[G]-module, and one can ask what its structure is. This is an arithmetic question, in that by the normal basis theorem one knows that L is a free K[G]-module of rank 1. If the same is true for the integers, that is equivalent to the existence of a normal integral basis, i.e. of α in OL such that its conjugate elements under G give a free basis for OL over OK. This is an interesting question even (perhaps especially) when K is the rational number field Q.
For example, if L = Q(), is there a normal integral basis? The answer is yes, as one sees by identifying it with Q(ζ) where
ζ = exp(2i/3).
In fact all the subfields of the cyclotomic fields for p-th roots of unity for p a prime number have normal integral bases (over Z), as can be deduced from the theory of Gaussian periods (the Hilbert–Speiser theorem). On the other hand, the Gaussian field does not. This is an example of a necessary condition found by Emmy Noether (perhaps known earlier?). What matters here is tame ramification. In terms of the discriminant D of L, and taking still K = Q, no prime p must divide D to the power p. Then Noether's theorem states that tame ramification is necessary and sufficient for OL to be a projective module over Z[G]. It is certainly therefore necessary for it to be a free module. It leaves the question of the gap between free and projective, for which a large theory has now been built up.
A classical result, based on a result of David Hilbert, is that a tamely ramified abelian number field has a normal integral basis. This may be seen by using the Kronecker–Weber theorem to embed the abelian field into |
https://en.wikipedia.org/wiki/Gaussian%20field | Gaussian field may refer to:
A field of Gaussian rationals in number theory
Gaussian free field, a concept in statistical mechanics
A Gaussian random field, a field of Gaussian-distributed random variables |
https://en.wikipedia.org/wiki/Analytics | Analytics is the systematic computational analysis of data or statistics. It is used for the discovery, interpretation, and communication of meaningful patterns in data. It also entails applying data patterns toward effective decision-making. It can be valuable in areas rich with recorded information; analytics relies on the simultaneous application of statistics, computer programming, and operations research to quantify performance.
Organizations may apply analytics to business data to describe, predict, and improve business performance. Specifically, areas within analytics include descriptive analytics, diagnostic analytics, predictive analytics, prescriptive analytics, and cognitive analytics. Analytics may apply to a variety of fields such as marketing, management, finance, online systems, information security, and software services. Since analytics can require extensive computation (see big data), the algorithms and software used for analytics harness the most current methods in computer science, statistics, and mathematics. According to International Data Corporation, global spending on big data and business analytics (BDA) solutions is estimated to reach $215.7 billion in 2021. As per Gartner, the overall analytic platforms software market grew by $25.5 billion in 2020.
Analytics vs analysis
Data analysis focuses on the process of examining past data through business understanding, data understanding, data preparation, modeling and evaluation, and deployment. It is a subset of data analytics, which takes multiple data analysis processes to focus on why an event happened and what may happen in the future based on the previous data. Data analytics is used to formulate larger organizational decisions.
Data analytics is a multidisciplinary field. There is extensive use of computer skills, mathematics, statistics, the use of descriptive techniques and predictive models to gain valuable knowledge from data through analytics. There is increasing use of the term advanced analytics, typically used to describe the technical aspects of analytics, especially in the emerging fields such as the use of machine learning techniques like neural networks, decision trees, logistic regression, linear to multiple regression analysis, and classification to do predictive modeling. It also includes unsupervised machine learning techniques like cluster analysis, Principal Component Analysis, segmentation profile analysis and association analysis.
Applications
Marketing optimization
Marketing organizations use analytics to determine the outcomes of campaigns or efforts, and to guide decisions for investment and consumer targeting. Demographic studies, customer segmentation, conjoint analysis and other techniques allow marketers to use large amounts of consumer purchase, survey and panel data to understand and communicate marketing strategy.
Marketing analytics consists of both qualitative and quantitative, structured and unstructured data used to drive stra |
https://en.wikipedia.org/wiki/Information%20geometry | Information geometry is an interdisciplinary field that applies the techniques of differential geometry to study probability theory and statistics. It studies statistical manifolds, which are Riemannian manifolds whose points correspond to probability distributions.
Introduction
Historically, information geometry can be traced back to the work of C. R. Rao, who was the first to treat the Fisher matrix as a Riemannian metric. The modern theory is largely due to Shun'ichi Amari, whose work has been greatly influential on the development of the field.
Classically, information geometry considered a parametrized statistical model as a Riemannian manifold. For such models, there is a natural choice of Riemannian metric, known as the Fisher information metric. In the special case that the statistical model is an exponential family, it is possible to induce the statistical manifold with a Hessian metric (i.e a Riemannian metric given by the potential of a convex function). In this case, the manifold naturally inherits two flat affine connections, as well as a canonical Bregman divergence. Historically, much of the work was devoted to studying the associated geometry of these examples. In the modern setting, information geometry applies to a much wider context, including non-exponential families, nonparametric statistics, and even abstract statistical manifolds not induced from a known statistical model. The results combine techniques from information theory, affine differential geometry, convex analysis and many other fields.
The standard references in the field are Shun’ichi Amari and Hiroshi Nagaoka's book, Methods of Information Geometry, and the more recent book by Nihat Ay and others. A gentle introduction is given in the survey by Frank Nielsen. In 2018, the journal Information Geometry was released, which is devoted to the field.
Contributors
The history of information geometry is associated with the discoveries of at least the following people, and many others.
Ronald Fisher
Harald Cramér
Calyampudi Radhakrishna Rao
Harold Jeffreys
Solomon Kullback
Jean-Louis Koszul
Richard Leibler
Claude Shannon
Imre Csiszár
N. N. Cencov (also written as Chentsov)
Bradley Efron
Shun'ichi Amari
Ole Barndorff-Nielsen
Frank Nielsen
Damiano Brigo
A. W. F. Edwards
Grant Hillier
Kees Jan van Garderen
Applications
As an interdisciplinary field, information geometry has been used in various applications.
Here an incomplete list:
Statistical inference
Time series and linear systems
Filtering problem
Quantum systems
Neural networks
Machine learning
Statistical mechanics
Biology
Statistics
Mathematical finance
See also
Ruppeiner geometry
Kullback–Leibler divergence
Stochastic geometry
Stochastic differential geometry
Projection filters
References
External links
Information Geometry journal by Springer
Information Geometry overview by Cosma Rohilla Shalizi, July 2010
Information Geometry notes by John Baez, Novemb |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.