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https://en.wikipedia.org/wiki/Adolf%20Lindenbaum
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Adolf Lindenbaum (12 June 1904 – August 1941) was a Polish-Jewish logician and mathematician best known for Lindenbaum's lemma and Lindenbaum–Tarski algebras.
He was born and brought up in Warsaw. He earned a Ph.D. in 1928 under Wacław Sierpiński and habilitated at the University of Warsaw in 1934. He published works on mathematical logic, set theory, cardinal and ordinal arithmetic, the axiom of choice, the continuum hypothesis, theory of functions, measure theory, point-set topology, geometry and real analysis. He served as an assistant professor at the University of Warsaw from 1935 until the outbreak of war in September 1939. He was Alfred Tarski's closest collaborator of the inter-war period. Around the end of October or beginning of November 1935 he married Janina Hosiasson, a fellow logician of the Lwow–Warsaw school. He and his wife were adherents of logical empiricism, participated in and contributed to the international unity of science movement, and were members of the original Vienna Circle. Sometime before the middle of August 1941 he and his sister Stefanja were shot to death in Naujoji Vilnia (Nowa Wilejka), 7 km east of Vilnius, by the occupying German forces or Lithuanian collaborators.
References
External links
Adolf Lindenbaum entry at The Internet Encyclopedia Of Philosophy by Jan Woleński (includes a portrait)
An Open Access article on Lindenbaum's life and works in Logica Universalis, Volume 8, Issue 3–4 (December 2014), pp 285–320 [note: the authors revisited the life of Adolf Lindenbaum in light of new research findings in a later non Open Access paper here.
Page on Sierpinski, contains fragments of his memoirs mentioning the murder of Lindenbaum
1904 births
1941 deaths
20th-century Polish philosophers
20th-century Polish mathematicians
Polish logicians
Polish Jews who died in the Holocaust
Polish set theorists
Polish people executed by Nazi Germany
Scientists from Warsaw
Victims of the Ponary massacre
Executed people from Masovian Voivodeship
Vienna Circle
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https://en.wikipedia.org/wiki/Aleksandr%20Khinchin
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Aleksandr Yakovlevich Khinchin (, ; July 19, 1894 – November 18, 1959) was a Soviet mathematician and one of the most significant contributors to the Soviet school of probability theory.
Due to romanization conventions, his name is sometimes written as "Khinchin" and other times as "Khintchine".
Life and career
He was born in the village of Kondrovo, Kaluga Governorate, Russian Empire. While studying at Moscow State University, he became one of the first followers of the famous Luzin school. Khinchin graduated from the university in 1916 and six years later he became a full professor there, retaining that position until his death.
Khinchin's early works focused on real analysis. Later he applied methods from the metric theory of functions to problems in probability theory and number theory. He became one of the founders of modern probability theory, discovering the law of the iterated logarithm in 1924, achieving important results in the field of limit theorems, giving a definition of a stationary process and laying a foundation for the theory of such processes.
Khinchin made significant contributions to the metric theory of Diophantine approximations and established an important result for simple real continued fractions, discovering a property of such numbers that leads to what is now known as Khinchin's constant. He also published several important works on statistical physics, where he used the methods of probability theory, and on information theory, queuing theory and mathematical analysis.
In 1939 Khinchin was elected as a Correspondent Member of the Academy of Sciences of the USSR. He was awarded the Stalin Prize (1941), the Order of Lenin, three other orders, and medals.
See also
Pollaczek–Khinchine formula
Wiener–Khinchin theorem
Khinchin inequality
Equidistribution theorem
Khinchin's constant
Khinchin–Lévy constant
Khinchin's theorem on Diophantine approximations
Law of the iterated logarithm
Palm-Khintchine Theorem
Weak law of large numbers (Khinchin's law)
Lévy–Khintchine formula of characteristic function of Lévy process
Bibliography
Sur la Loi des Grandes Nombres, in Comptes Rendus de l'Académie des Sciences, Paris, 1929
Asymptotische Gesetze der Wahrscheinlichkeitsrechnung, Berlin: Julius Springer, 1933
Continued Fractions, Mineola, N.Y. : Dover Publications, 1997, (first published in Moscow, 1935)
Three Pearls of Number Theory, Mineola, NY : Dover Publications, 1998, (first published in Moscow and Leningrad, 1947)
Mathematical Foundations of Quantum Statistics, Mineola, N.Y. : Dover Publications, 1998, (first published in Moscow and Leningrad, 1951; trans. in 1960 by Irwin Shapiro)
Mathematical Foundations of Information Theory, Dover Publications, 1957,
References
External links
List of books by Khinchin provided by National Library of Australia
A.Ya. Khinchin at Math-Net.Ru.
20th-century Russian mathematicians
Soviet mathematicians
Number theorists
Probability theorists
Queueing theorists
Recipients of the Stalin
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https://en.wikipedia.org/wiki/Equidistributed%20sequence
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In mathematics, a sequence (s1, s2, s3, ...) of real numbers is said to be equidistributed, or uniformly distributed, if the proportion of terms falling in a subinterval is proportional to the length of that subinterval. Such sequences are studied in Diophantine approximation theory and have applications to Monte Carlo integration.
Definition
A sequence (s1, s2, s3, ...) of real numbers is said to be equidistributed on a non-degenerate interval [a, b] if for every subinterval [c, d] of [a, b] we have
(Here, the notation |{s1,...,sn} ∩ [c, d]| denotes the number of elements, out of the first n elements of the sequence, that are between c and d.)
For example, if a sequence is equidistributed in [0, 2], since the interval [0.5, 0.9] occupies 1/5 of the length of the interval [0, 2], as n becomes large, the proportion of the first n members of the sequence which fall between 0.5 and 0.9 must approach 1/5. Loosely speaking, one could say that each member of the sequence is equally likely to fall anywhere in its range. However, this is not to say that (sn) is a sequence of random variables; rather, it is a determinate sequence of real numbers.
Discrepancy
We define the discrepancy DN for a sequence (s1, s2, s3, ...) with respect to the interval [a, b] as
A sequence is thus equidistributed if the discrepancy DN tends to zero as N tends to infinity.
Equidistribution is a rather weak criterion to express the fact that a sequence fills the segment leaving no gaps. For example, the drawings of a random variable uniform over a segment will be equidistributed in the segment, but there will be large gaps compared to a sequence which first enumerates multiples of ε in the segment, for some small ε, in an appropriately chosen way, and then continues to do this for smaller and smaller values of ε. For stronger criteria and for constructions of sequences that are more evenly distributed, see low-discrepancy sequence.
Riemann integral criterion for equidistribution
Recall that if f is a function having a Riemann integral in the interval [a, b], then its integral is the limit of Riemann sums taken by sampling the function f in a set of points chosen from a fine partition of the interval. Therefore, if some sequence is equidistributed in [a, b], it is expected that this sequence can be used to calculate the integral of a Riemann-integrable function. This leads to the following criterion for an equidistributed sequence:
Suppose (s1, s2, s3, ...) is a sequence contained in the interval [a, b]. Then the following conditions are equivalent:
The sequence is equidistributed on [a, b].
For every Riemann-integrable (complex-valued) function , the following limit holds:
{| class="toccolours collapsible collapsed" width="90%" style="text-align:left"
!Proof
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|First note that the definition of an equidistributed sequence is equivalent to the integral criterion whenever f is the indicator function of an interval: If f = 1[c, d], then the left hand side is the p
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https://en.wikipedia.org/wiki/Minimum%20polynomial
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Minimum polynomial can refer to:
Minimal polynomial (field theory)
Minimal polynomial (linear algebra)
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https://en.wikipedia.org/wiki/Surface%20%28disambiguation%29
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A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space.
Surface or surfaces may also refer to:
Mathematics
Surface (mathematics), a generalization of a plane which needs not be flat
Surface (differential geometry), a differentiable two-dimensional manifold
Surface (topology), a two-dimensional manifold
Algebraic surface, an algebraic variety of dimension two
Coordinate surfaces
Fractal surface, generated using a stochastic algorithm
Polyhedral surface
Surface area
Surface integral
Arts and entertainment
Surface (band), an American R&B and pop trio
Surface (Surface album), 1986
Surfaces (band), American musical duo
Surface (Circle album), 1998
"Surface" (Aero Chord song), 2014
Surface (2005 TV series), an American science fiction show, 2005–2006
Surface (2022 TV series), an American psychological thriller miniseries that began streaming in 2022
The Surface, an American film, 2014
"Surface", a song by Your Memorial from the 2010 album Atonement
Physical sciences
Surface finishing, a range of industrial processes that alter the surface of a manufactured item to achieve a certain property
Surface science, the study of physical and chemical phenomena that occur at the interface of two phases
Surface wave, a mechanical wave, in physics
Interface (matter), common boundary among two different phases of matter
Planetary surface
Surface of the Earth
Sea surface
Transportation
Surface mail, transportation of mail that travel on land and sea but not air
Surface transport, transportation of goods and people on land and sea
People
Harvey A. Surface (1867–1941), American zoologist and Pennsylvania legislator
Mary Hall Surface (born 1958), American playwright and theater director
Technology
Microsoft Surface, the brand for a line of computers and related accessories by Microsoft
Microsoft PixelSense (formerly known as Surface), a commercial computing platform
Computer representation of surfaces, a way of representing objects, in technical applications of 3D computer graphics
Deep structure and surface structure, concepts in Chomskyan linguistics
Publications
Surface (magazine), an American architecture magazine since 1993
Surfaces (Université de Montréal journal), published from 1991 to 1999
Surfaces (MDPI journal), published from 2018 onwards
Surface Science (journal)
See also
Surfacing (disambiguation)
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https://en.wikipedia.org/wiki/All%20one%20polynomial
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In mathematics, an all one polynomial (AOP) is a polynomial in which all coefficients are one. Over the finite field of order two, conditions for the AOP to be irreducible are known, which allow this polynomial to be used to define efficient algorithms and circuits for multiplication in finite fields of characteristic two. The AOP is a 1-equally spaced polynomial.
Definition
An AOP of degree m has all terms from xm to x0 with coefficients of 1, and can be written as
or
or
Thus the roots of the all one polynomial of degree m are all (m+1)th roots of unity other than unity itself.
Properties
Over GF(2) the AOP has many interesting properties, including:
The Hamming weight of the AOP is m + 1, the maximum possible for its degree
The AOP is irreducible if and only if m + 1 is prime and 2 is a primitive root modulo m + 1 (over GF(p) with prime p, it is irreducible if and only if m + 1 is prime and p is a primitive root modulo m + 1)
The only AOP that is a primitive polynomial is x2 + x + 1.
Despite the fact that the Hamming weight is large, because of the ease of representation and other improvements there are efficient implementations in areas such as coding theory and cryptography.
Over , the AOP is irreducible whenever m + 1 is a prime p, and therefore in these cases, the pth cyclotomic polynomial.
References
External links
Field (mathematics)
Polynomials
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https://en.wikipedia.org/wiki/Focus%20%28geometry%29
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In geometry, focuses or foci (; : focus) are special points with reference to which any of a variety of curves is constructed. For example, one or two foci can be used in defining conic sections, the four types of which are the circle, ellipse, parabola, and hyperbola. In addition, two foci are used to define the Cassini oval and the Cartesian oval, and more than two foci are used in defining an n-ellipse.
Conic sections
Defining conics in terms of two foci
An ellipse can be defined as the locus of points for which the sum of the distances to two given foci is constant.
A circle is the special case of an ellipse in which the two foci coincide with each other. Thus, a circle can be more simply defined as the locus of points each of which is a fixed distance from a single given focus. A circle can also be defined as the circle of Apollonius, in terms of two different foci, as the locus of points having a fixed ratio of distances to the two foci.
A parabola is a limiting case of an ellipse in which one of the foci is a point at infinity.
A hyperbola can be defined as the locus of points for which the absolute value of the difference between the distances to two given foci is constant.
Defining conics in terms of a focus and a directrix
It is also possible to describe all conic sections in terms of a single focus and a single directrix, which is a given line not containing the focus. A conic is defined as the locus of points for each of which the distance to the focus divided by the distance to the directrix is a fixed positive constant, called the eccentricity . If the conic is an ellipse, if the conic is a parabola, and if the conic is a hyperbola. If the distance to the focus is fixed and the directrix is a line at infinity, so the eccentricity is zero, then the conic is a circle.
Defining conics in terms of a focus and a directrix circle
It is also possible to describe all the conic sections as loci of points that are equidistant from a single focus and a single, circular directrix. For the ellipse, both the focus and the center of the directrix circle have finite coordinates and the radius of the directrix circle is greater than the distance between the center of this circle and the focus; thus, the focus is inside the directrix circle. The ellipse thus generated has its second focus at the center of the directrix circle, and the ellipse lies entirely within the circle.
For the parabola, the center of the directrix moves to the point at infinity (see Projective geometry). The directrix "circle" becomes a curve with zero curvature, indistinguishable from a straight line. The two arms of the parabola become increasingly parallel as they extend, and "at infinity" become parallel; using the principles of projective geometry, the two parallels intersect at the point at infinity and the parabola becomes a closed curve (elliptical projection).
To generate a hyperbola, the radius of the directrix circle is chosen to be less than the distan
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https://en.wikipedia.org/wiki/Kitaro%20Nishida
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was a Japanese moral
philosopher, philosopher of mathematics and science, and religious scholar. He was the founder of what has been called the Kyoto School of philosophy. He graduated from the University of Tokyo during the Meiji period in 1894 with a degree in philosophy. He was named professor of the Fourth Higher School in Ishikawa Prefecture in 1899 and later became professor of philosophy at Kyoto University. Nishida retired in 1927. In 1940, he was awarded the Order of Culture (文化勲章, bunka kunshō). He participated in establishing the Chiba Institute of Technology (千葉工業大学) from 1940.
Nishida Kitarō died at the age of 75 of a renal infection. His cremated remains were divided in three and buried at different locations. Part of his remains were buried in the Nishida family grave in his birthplace Unoke, Ishikawa. A second grave can be found at Tōkei-ji Temple in Kamakura, where his friend D. T. Suzuki organized Nishida's funeral and was later also buried in the adjacent plot. Nishida's third grave is at Reiun'in (霊雲院, Reiun'in), a temple in the Myōshin-ji compound in Kyoto.
Philosophy
Being born in the third year of the Meiji period, Nishida was presented with a new, unique opportunity to contemplate Eastern philosophical issues in the fresh light that Western philosophy shone on them. Nishida's original and creative philosophy, incorporating ideas of Zen and Western philosophy, was aimed at bringing the East and West closer. Throughout his lifetime, Nishida published a number of books and essays including An Inquiry into the Good and The Logic of the Place of Nothingness and the Religious Worldview. Taken as a whole, Nishida's life work was the foundation for the Kyoto School of philosophy and the inspiration for the original thinking of his disciples.
One of the most famous concepts in Nishida's philosophy is the logic of basho (Japanese: 場所; usually translated as "place" or "topos"), a non-dualistic concrete logic, meant to overcome the inadequacy of the subject-object distinction essential to the subject logic of Aristotle and the predicate logic of Immanuel Kant, through the affirmation of what he calls the "absolutely contradictory self-identity", a dynamic tension of opposites that, unlike the dialectical logic of G.W.F. Hegel, does not resolve in a synthesis. Rather, it defines its proper subject by maintaining the tension between affirmation and negation as opposite poles or perspectives.
In David A. Dilworth's survey of Nishida's works, he did not mention the debut book, An Inquiry into the Good. There, Nishida writes about the experience, reality, good and religion. He argues that the most profound form of experience is the pure experience. Nishida analyzes the thought, the will, the intellectual intuition, and the pure experience among them. According to Nishida's vision as well as to the essence of Asian wisdom, one craves harmony in experience, for unity.
Legacy
According to Masao Abe, "During World War II right-wing thin
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https://en.wikipedia.org/wiki/Marcinkiewicz%20interpolation%20theorem
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In mathematics, the Marcinkiewicz interpolation theorem, discovered by , is a result bounding the norms of non-linear operators acting on Lp spaces.
Marcinkiewicz' theorem is similar to the Riesz–Thorin theorem about linear operators, but also applies to non-linear operators.
Preliminaries
Let f be a measurable function with real or complex values, defined on a measure space (X, F, ω). The distribution function of f is defined by
Then f is called weak if there exists a constant C such that the distribution function of f satisfies the following inequality for all t > 0:
The smallest constant C in the inequality above is called the weak norm and is usually denoted by or Similarly the space is usually denoted by L1,w or L1,∞.
(Note: This terminology is a bit misleading since the weak norm does not satisfy the triangle inequality as one can see by considering the sum of the functions on given by and , which has norm 4 not 2.)
Any function belongs to L1,w and in addition one has the inequality
This is nothing but Markov's inequality (aka Chebyshev's Inequality). The converse is not true. For example, the function 1/x belongs to L1,w but not to L1.
Similarly, one may define the weak space as the space of all functions f such that belong to L1,w, and the weak norm using
More directly, the Lp,w norm is defined as the best constant C in the inequality
for all t > 0.
Formulation
Informally, Marcinkiewicz's theorem is
Theorem. Let T be a bounded linear operator from to and at the same time from to . Then T is also a bounded operator from to for any r between p and q.
In other words, even if one only requires weak boundedness on the extremes p and q, regular boundedness still holds. To make this more formal, one has to explain that T is bounded only on a dense subset and can be completed. See Riesz-Thorin theorem for these details.
Where Marcinkiewicz's theorem is weaker than the Riesz-Thorin theorem is in the estimates of the norm. The theorem gives bounds for the norm of T but this bound increases to infinity as r converges to either p or q. Specifically , suppose that
so that the operator norm of T from Lp to Lp,w is at most Np, and the operator norm of T from Lq to Lq,w is at most Nq. Then the following interpolation inequality holds for all r between p and q and all f ∈ Lr:
where
and
The constants δ and γ can also be given for q = ∞ by passing to the limit.
A version of the theorem also holds more generally if T is only assumed to be a quasilinear operator in the following sense: there exists a constant C > 0 such that T satisfies
for almost every x. The theorem holds precisely as stated, except with γ replaced by
An operator T (possibly quasilinear) satisfying an estimate of the form
is said to be of weak type (p,q). An operator is simply of type (p,q) if T is a bounded transformation from Lp to Lq:
A more general formulation of the interpolation theorem is as follows:
If T is a quasilinear operator of w
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https://en.wikipedia.org/wiki/Joseph%20Betts
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Joseph Betts was an English mathematician. He held the Savilian Chair of Geometry at the University of Oxford in 1765.
Betts was an undergraduate and Fellow of University College, Oxford, where he was a tutor of William Jones. He had previously sought election as Savilian Professor of Astronomy with the support of the Earl of Lichfield, the Earl of Halifax, and the Earl of Bute. He thanked his patrons for that failed attempt in the dedication to an engraving of the annular solar eclipse of 1 April 1764.
References
18th-century births
18th-century deaths
Alumni of University College, Oxford
18th-century English mathematicians
Savilian Professors of Geometry
Fellows of University College, Oxford
Year of birth unknown
Year of death unknown
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https://en.wikipedia.org/wiki/Baden%20Powell%20%28mathematician%29
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Baden Powell, MA FRS FRGS (22 August 1796 – 11 June 1860) was an English mathematician and Church of England priest. He held the Savilian Chair of Geometry at the University of Oxford from 1827 to 1860. Powell was a prominent liberal theologian who put forward advanced ideas about evolution.
Origins
Baden Powell II was born at Stamford Hill, Hackney in London. His father, Baden Powell I (1767-1841), of Langton and Speldhurst in Kent, was a wine merchant, who served as High Sheriff of Kent in 1831, and as Master of the Worshipful Company of Mercers in 1822. The mother of Baden Powell II was Hester Powell (1776-1848), his father's paternal first cousin, a daughter of James Powell (1737-1824) of Clapton, Hackney, Middlesex, Master of the Worshipful Company of Salters in 1818.
The Powell family can be traced back to the early 16th century, where they were yeomen farmers at Mildenhall in Suffolk. Baden Powell II's great-grandfather, David Powell (1725-1810) of Homerton, Middlesex, a second son, migrated to the City of London aged 17 in 1712, subsequently going into business as a merchant at Old Broad Street and buying the manor of Wattisfield in Suffolk. In 1740 a branch of the family bought the Whitefriars Glass works.
The name Baden originated in Susanna Baden (1663-1737), the maternal grandmother of David Powell (1725-1810) of Homerton, Middlesex, and one of the ten children of Andrew Baden (1637-1716), a Mercer who served as Mayor of Salisbury in 1682.
Education
Powell was admitted as an undergraduate at Oriel College, Oxford in 1814, and graduated with a first-class honours degree in mathematics in 1817.
Ordination
Powell was ordained as a priest of the Church of England in 1821, having served as curate of Midhurst, Sussex. His first living was as Vicar of Plumstead, Kent, of which the advowson was owned by his family. He immediately began his scientific work there, starting with experiments on radiant heat.
Marriages and children
Powell married three times, and had fourteen children in total. His widow changed the last name of the surviving children of his third marriage to "Baden-Powell".
Powell's first marriage on 21 July 1821 to Eliza Rivaz (died 13 March 1836) was childless.
His second marriage on 27 September 1837 to Charlotte Pope (died 14 October 1844) produced one son and three daughters:
Charlotte Elizabeth Powell, (14 September 1838–20 October 1917)
Baden Henry Baden-Powell, FRSE (23 August 1841–2 January 1901)
Louisa Ann Powell, (18 March 1843–1 August 1896)
Laetitia Mary Powell, (4 June 1844–2 September 1865)
His third marriage on 10 March 1846 (at St Luke's Church, Chelsea) to Henrietta Grace Smyth (3 September 1824–13 October 1914), a daughter of Admiral Smyth, produced seven sons and three daughters:
Henry Warington Baden-Powell, (3 February 1847–24 April 1921), a naval officer, a fellow of the Royal Geographical Society and a King's Counsel (K.C.)
Sir George Smyth Baden-Powell, (24 December 1847–20 November 1898)
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https://en.wikipedia.org/wiki/Morava%20K-theory
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In stable homotopy theory, a branch of mathematics, Morava K-theory is one of a collection of cohomology theories introduced in algebraic topology by Jack Morava in unpublished preprints in the early 1970s. For every prime number p (which is suppressed in the notation), it consists of theories K(n) for each nonnegative integer n, each a ring spectrum in the sense of homotopy theory. published the first account of the theories.
Details
The theory K(0) agrees with singular homology with rational coefficients, whereas K(1) is a summand of mod-p complex K-theory. The theory K(n) has coefficient ring
Fp[vn,vn−1]
where vn has degree 2(pn − 1). In particular, Morava K-theory is periodic with this period, in much the same way that complex K-theory has period 2.
These theories have several remarkable properties.
They have Künneth isomorphisms for arbitrary pairs of spaces: that is, for X and Y CW complexes, we have
They are "fields" in the category of ring spectra. In other words every module spectrum over K(n) is free, i.e. a wedge of suspensions of K(n).
They are complex oriented (at least after being periodified by taking the wedge sum of (pn − 1) shifted copies), and the formal group they define has height n.
Every finite p-local spectrum X has the property that K(n)∗(X) = 0 if and only if n is less than a certain number N, called the type of the spectrum X. By a theorem of Devinatz–Hopkins–Smith, every thick subcategory of the category of finite p-local spectra is the subcategory of type-n spectra for some n.
See also
Chromatic homotopy theory
Morava E-theory
References
Hovey-Strickland, "Morava K-theory and localisation"
Algebraic topology
Cohomology theories
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https://en.wikipedia.org/wiki/Complex%20cobordism
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In mathematics, complex cobordism is a generalized cohomology theory related to cobordism of manifolds. Its spectrum is denoted by MU. It is an exceptionally powerful cohomology theory, but can be quite hard to compute, so often instead of using it directly one uses some slightly weaker theories derived from it, such as Brown–Peterson cohomology or Morava K-theory, that are easier to compute.
The generalized homology and cohomology complex cobordism theories were introduced by using the Thom spectrum.
Spectrum of complex cobordism
The complex bordism of a space is roughly the group of bordism classes of manifolds over with a complex linear structure on the stable normal bundle. Complex bordism is a generalized homology theory, corresponding to a spectrum MU that can be described explicitly in terms of Thom spaces as follows.
The space is the Thom space of the universal -plane bundle over the classifying space of the unitary group . The natural inclusion from into induces a map from the double suspension to . Together these maps give the spectrum ; namely, it is the homotopy colimit of .
Examples: is the sphere spectrum. is the desuspension of .
The nilpotence theorem states that, for any ring spectrum , the kernel of consists of nilpotent elements. The theorem implies in particular that, if is the sphere spectrum, then for any , every element of is nilpotent (a theorem of Goro Nishida). (Proof: if is in , then is a torsion but its image in , the Lazard ring, cannot be torsion since is a polynomial ring. Thus, must be in the kernel.)
Formal group laws
and showed that the coefficient ring (equal to the complex cobordism of a point, or equivalently the ring of cobordism classes of stably complex manifolds) is a polynomial ring on infinitely many generators of positive even degrees.
Write for infinite dimensional complex projective space, which is the classifying space for complex line bundles, so that tensor product of line bundles induces a map A complex orientation on an associative commutative ring spectrum E is an element x in whose restriction to
is 1, if the latter ring is identified with the coefficient ring of E. A spectrum E with such an element x is called a complex oriented ring spectrum.
If E is a complex oriented ring spectrum, then
and is a formal group law over the ring .
Complex cobordism has a natural complex orientation. showed that there is a natural isomorphism from its coefficient ring to Lazard's universal ring, making the formal group law of complex cobordism into the universal formal group law. In other words, for any formal group law F over any commutative ring R, there is a unique ring homomorphism from MU*(point) to R such that F is the pullback of the formal group law of complex cobordism.
Brown–Peterson cohomology
Complex cobordism over the rationals can be reduced to ordinary cohomology over the rationals, so the main interest is in the torsion of complex cobordism. It is often
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https://en.wikipedia.org/wiki/Spectrum%20%28topology%29
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In algebraic topology, a branch of mathematics, a spectrum is an object representing a generalized cohomology theory. Every such cohomology theory is representable, as follows from Brown's representability theorem. This means that, given a cohomology theory,there exist spaces such that evaluating the cohomology theory in degree on a space is equivalent to computing the homotopy classes of maps to the space , that is.Note there are several different categories of spectra leading to many technical difficulties, but they all determine the same homotopy category, known as the stable homotopy category. This is one of the key points for introducing spectra because they form a natural home for stable homotopy theory.
The definition of a spectrum
There are many variations of the definition: in general, a spectrum is any sequence of pointed topological spaces or pointed simplicial sets together with the structure maps , where is the smash product. The smash product of a pointed space with a circle is homeomorphic to the reduced suspension of , denoted .
The following is due to Frank Adams (1974): a spectrum (or CW-spectrum) is a sequence of CW complexes together with inclusions of the suspension as a subcomplex of .
For other definitions, see symmetric spectrum and simplicial spectrum.
Homotopy groups of a spectrum
One of the most important invariants of spectra are the homotopy groups of the spectrum. These groups mirror the definition of the stable homotopy groups of spaces since the structure of the suspension maps is integral in its definition. Given a spectrum define the homotopy group as the colimitwhere the maps are induced from the composition of the map (that is, given by functoriality of ) and the structure map . A spectrum is said to be connective if its are zero for negative k.
Examples
Eilenberg–Maclane spectrum
Consider singular cohomology with coefficients in an abelian group . For a CW complex , the group can be identified with the set of homotopy classes of maps from to , the Eilenberg–MacLane space with homotopy concentrated in degree . We write this asThen the corresponding spectrum has -th space ; it is called the Eilenberg–MacLane spectrum of . Note this construction can be used to embed any ring into the category of spectra. This embedding forms the basis of spectral geometry, a model for derived algebraic geometry. One of the important properties of this embedding are the isomorphismsshowing the category of spectra keeps track of the derived information of commutative rings, where the smash product acts as the derived tensor product. Moreover, Eilenberg–Maclane spectra can be used to define theories such as topological Hochschild homology for commutative rings, a more refined theory than classical Hochschild homology.
Topological complex K-theory
As a second important example, consider topological K-theory. At least for X compact, is defined to be the Grothendieck group of the monoid of complex vector
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https://en.wikipedia.org/wiki/Reduction%20%28mathematics%29
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In mathematics, reduction refers to the rewriting of an expression into a simpler form. For example, the process of rewriting a fraction into one with the smallest whole-number denominator possible (while keeping the numerator a whole number) is called "reducing a fraction". Rewriting a radical (or "root") expression with the smallest possible whole number under the radical symbol is called "reducing a radical". Minimizing the number of radicals that appear underneath other radicals in an expression is called denesting radicals.
Algebra
In linear algebra, reduction refers to applying simple rules to a series of equations or matrices to change them into a simpler form. In the case of matrices, the process involves manipulating either the rows or the columns of the matrix and so is usually referred to as row-reduction or column-reduction, respectively. Often the aim of reduction is to transform a matrix into its "row-reduced echelon form" or "row-echelon form"; this is the goal of Gaussian elimination.
Calculus
In calculus, reduction refers to using the technique of integration by parts to evaluate integrals by reducing them to simpler forms.
Static (Guyan) reduction
In dynamic analysis, static reduction refers to reducing the number of degrees of freedom. Static reduction can also be used in finite element analysis to refer to simplification of a linear algebraic problem. Since a static reduction requires several inversion steps it is an expensive matrix operation and is prone to some error in the solution. Consider the following system of linear equations in an FEA problem:
where K and F are known and K, x and F are divided into submatrices as shown above. If F2 contains only zeros, and only x1 is desired, K can be reduced to yield the following system of equations
is obtained by writing out the set of equations as follows:
Equation () can be solved for (assuming invertibility of ):
And substituting into () gives
Thus
In a similar fashion, any row or column i of F with a zero value may be eliminated if the corresponding value of xi is not desired. A reduced K may be reduced again. As a note, since each reduction requires an inversion, and each inversion is an operation with computational cost O(n3), most large matrices are pre-processed to reduce calculation time.
History
In the 9th century, Persian mathematician Al-Khwarizmi's Al-Jabr introduced the fundamental concepts of "reduction" and "balancing", referring to the transposition of subtracted terms to the other side of an equation and the cancellation of like terms on opposite sides of the equation. This is the operation which Al-Khwarizmi originally described as al-jabr. The name "algebra" comes from the "al-jabr" in the title of his book.
References
Mathematical terminology
Linear algebra
Calculus
Iranian inventions
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https://en.wikipedia.org/wiki/Dixon%27s%20factorization%20method
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In number theory, Dixon's factorization method (also Dixon's random squares method or Dixon's algorithm) is a general-purpose integer factorization algorithm; it is the prototypical factor base method. Unlike for other factor base methods, its run-time bound comes with a rigorous proof that does not rely on conjectures about the smoothness properties of the values taken by a polynomial.
The algorithm was designed by John D. Dixon, a mathematician at Carleton University, and was published in 1981.
Basic idea
Dixon's method is based on finding a congruence of squares modulo the integer N which is intended to factor. Fermat's factorization method finds such a congruence by selecting random or pseudo-random x values and hoping that the integer x2 mod N is a perfect square (in the integers):
For example, if , (by starting at 292, the first number greater than and counting up) the is 256, the square of 16. So . Computing the greatest common divisor of and N using Euclid's algorithm gives 163, which is a factor of N.
In practice, selecting random x values will take an impractically long time to find a congruence of squares, since there are only squares less than N.
Dixon's method replaces the condition "is the square of an integer" with the much weaker one "has only small prime factors"; for example, there are 292 squares smaller than 84923; 662 numbers smaller than 84923 whose prime factors are only 2,3,5 or 7; and 4767 whose prime factors are all less than 30. (Such numbers are called B-smooth with respect to some bound B.)
If there are many numbers whose squares can be factorized as for a fixed set of small primes, linear algebra modulo 2 on the matrix will give a subset of the whose squares combine to a product of small primes to an even power — that is, a subset of the whose squares multiply to the square of a (hopefully different) number mod N.
Method
Suppose the composite number N is being factored. Bound B is chosen, and the factor base is identified (which is called P), the set of all primes less than or equal to B. Next, positive integers z are sought such that z2 mod N is B-smooth. Therefore we can write, for suitable exponents ai,
When enough of these relations have been generated (it is generally sufficient that the number of relations be a few more than the size of P), the methods of linear algebra, such as Gaussian elimination, can be used to multiply together these various relations in such a way that the exponents of the primes on the right-hand side are all even:
This yields a congruence of squares of the form which can be turned into a factorization of N, This factorization might turn out to be trivial (i.e. ), which can only happen if in which case another try must be made with a different combination of relations; but if a nontrivial pair of factors of N is reached, the algorithm terminates.
Pseudocode
input: positive integer
output: non-trivial factor of
Choose bound
Let be all primes
r
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https://en.wikipedia.org/wiki/Abraham%20Robertson
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Abraham or Abram Robertson FRS (4 November 1751 – 4 December 1826), was a Scottish mathematician and astronomer. He held the Savilian Chair of Geometry at the University of Oxford from 1797 to 1809.
Robertson was born at Duns, Berwickshire, the son of Abraham Robertson, “a man of humble station”. He attended school at Great Ryle in Northumberland, and later at Duns. At age 24, he moved to London, he had hopes of travelling to the East Indies, but his patron died.
He took himself alone to Oxford, where he sought to finance himself by opening an evening school for mechanics. This failed, and he served for a while as an assistant to John Ireland, a local apothecary. He then gained patronage from John Smith (1721—1796), the Savilian professor of geometry. Robertson completed a Bachelor of Arts in 1779 and completed his Master in Arts in 1782.
In 1784, he deputized for Smith, who was then acting as a physician at Cheltenham and then followed Smith as Savilian professor of geometry. His lectures were considered clear, and he was always anxious to encourage his pupils. Thus in 1804 he printed a demonstration of Euclid v, Definition 5, for the benefit of beginners.
In 1789, Robertson was presented by the dean and canons of Christ Church to the vicarage of Ravensthorpe, near Northampton, but his principal residence was still in Oxford. He married, about 1790, Miss Bacon of Drayton in Berkshire, who died a few years after he became professor. They had no children.
In 1795, the Royal Society elected him a fellow in recognition of his work on conic sections.
Robertson died on 4 December 1826 at the Radcliffe Observatory, Oxford, and was buried in the churchyard of St Peter-in-the-East.
Robertson's chief works were the following:
Sectionum conicarum libri septem (1792), dedicated to Dr Cyril Jackson, dean of Christ Church, was with an exhaustive survey of the history of the field.
Calculations for the Earl of Liverpool's Coins of the Realm (1805)
He superintended the publication of the works of Archimedes which were prepared for the press by Torelli (1792), and, with much effort, the second volume of Bradley's Greenwich Royal Observatory Astronomical Observations, commenced by Thomas Hornsby (1st ser., 1798–1805).
He declined to publish the manuscripts of Thomas Harriot. Two of Robertson's five papers in the Philosophical Transactions were fiercely criticized, and he responded by publishing a "Reply to a Critical and Monthly Reviewer" (1808). He contributed several papers to the first series of the British Critic, and two to the Edinburgh Philosophical Journal, in 1822.
1801: Robertson gave evidence before a committee of the House of Commons which reported in 1801 on the expediency of replacing London Bridge by a single arch. In 1807 he graduated BD and DD.
1801: The same year he was in London making calculations for Lord Grenville's system of finance, and in 1808 he drew up the tables for Spencer Perceval's system of increasing the sinking fund by
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https://en.wikipedia.org/wiki/Nigel%20Hitchin
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Nigel James Hitchin FRS (born 2 August 1946) is a British mathematician working in the fields of differential geometry, gauge theory, algebraic geometry, and mathematical physics. He is a Professor Emeritus of Mathematics at the University of Oxford.
Academic career
Hitchin attended Ecclesbourne School, Duffield, and earned his BA in mathematics from Jesus College, Oxford, in 1968. After moving to Wolfson College, he received his D.Phil. in 1972. From 1971 to 1973 he visited the Institute for Advanced Study and 1973/74 the Courant Institute of Mathematical Sciences of New York University. He then was a research fellow in Oxford and starting in 1979 tutor, lecturer and fellow of St Catherine's College. In 1990 he became a professor at the University of Warwick and in 1994 the Rouse Ball Professor of Mathematics at the University of Cambridge. In 1997 he was appointed to the Savilian Chair of Geometry at the University of Oxford, a position he held until his retirement in 2016.
Amongst his notable discoveries are the Hitchin–Thorpe inequality; Hitchin's projectively flat connection over Teichmüller space; the Atiyah–Hitchin monopole metric; the Atiyah–Hitchin–Singer theorem; the ADHM construction of instantons (of Michael Atiyah, Vladimir Drinfeld, Hitchin, and Yuri Manin); the hyperkähler quotient (of Hitchin, Anders Karlhede, Ulf Lindström and Martin Roček); Higgs bundles, which arise as solutions to the Hitchin equations, a 2-dimensional reduction of the self-dual Yang–Mills equations; and the Hitchin system, an algebraically completely integrable Hamiltonian system associated to the data of an algebraic curve and a complex reductive group. He and Shoshichi Kobayashi independently conjectured the Kobayashi–Hitchin correspondence. Higgs bundles, which are also developed in the work of Carlos Simpson, are closely related to the Hitchin system, which has an interpretation as a moduli space of semistable Higgs bundles over a compact Riemann surface or algebraic curve. This moduli space has emerged as a focal point for deep connections between algebraic geometry, differential geometry, hyperkähler geometry, mathematical physics, and representation theory.
In his article on generalized Calabi–Yau manifolds, he introduced the notion of generalized complex manifolds, providing a single structure that incorporates, as examples, Poisson manifolds, symplectic manifolds and complex manifolds. These have found wide applications as the geometries of flux compactifications in string theory and also in topological string theory.
In the span of his career, Hitchin has supervised 37 research students, including Simon Donaldson (part-supervised with Atiyah).
Until 2013 Nigel Hitchin served as the managing editor of the journal Mathematische Annalen.
Honours and awards
In 1991 he was elected a Fellow of the Royal Society.
In 2003 he was awarded an Honorary Degree (Doctor of Science) from the University of Bath.
Hitchin was elected as an Honorary Fell
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https://en.wikipedia.org/wiki/Ioan%20James
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Ioan Mackenzie James FRS (born 23 May 1928) is a British mathematician working in the field of topology, particularly in homotopy theory.
Biography
James was born in Croydon, Surrey, England, and was educated at St Paul's School, London and Queen's College, Oxford. In 1953 he earned a D. Phil. from the University of Oxford for his thesis entitled Some problems in algebraic topology, written under the direction of J. H. C. Whitehead.
In 1957 he was appointed reader in pure mathematics, a post which he held until 1969. From 1959 until 1969 he was a senior research fellow at St John's College. He held the Savilian Chair of Geometry at the University of Oxford from 1970 to 1995. He is now a professor emeritus.
He was elected a Fellow of the Royal Society in 1968. In 1978 the London Mathematical Society awarded him the Senior Whitehead Prize, which was established in honour of his doctoral supervisor, Whitehead. In 1984 he became President of the London Mathematical Society.
He married Rosemary Stewart, a writer and researcher in business management and healthcare management, in 1961. She died in 2015, aged 90.
Books
, Topologies and Uniformities (Springer Undergraduate Mathematics Series), Springer, 1999.
, Remarkable Mathematicians, From Euler to von Neumann, Cambridge University Press, 2002.
, Remarkable Physicists: From Galileo to Yukawa, Cambridge University Press, 2004.
, Asperger's Syndrome And High Achievement: Some Very Remarkable People, Jessica Kingsley Pub, 2005.
, The Mind of the Mathematician, JHU Press, 2007.
, Driven to Innovate: A Century of Jewish Mathematicians and Physicists, Peter Lang Oxford, 2009.
, Remarkable Biologists: From Ray to Hamilton, Cambridge University Press, 2009.
, Remarkable Engineers: From Riquet to Shannon, Cambridge University Press, 2010.
See also
James embedding
James reduced product
References
External links
1928 births
Living people
English mathematicians
Fellows of New College, Oxford
Fellows of St John's College, Oxford
Fellows of the Royal Society
British historians of mathematics
Historians of science
Savilian Professors of Geometry
Place of birth missing (living people)
Topologists
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https://en.wikipedia.org/wiki/Scattering%20amplitude
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In quantum physics, the scattering amplitude is the probability amplitude of the outgoing spherical wave relative to the incoming plane wave in a stationary-state scattering process.
At large distances from the centrally symmetric scattering center, the plane wave is described by the wavefunction
where is the position vector; ; is the incoming plane wave with the wavenumber along the axis; is the outgoing spherical wave; is the scattering angle (angle between the incident and scattered direction); and is the scattering amplitude. The dimension of the scattering amplitude is length. The scattering amplitude is a probability amplitude; the differential cross-section as a function of scattering angle is given as its modulus squared,
The asymptotic from of the wave function in arbitrary external field takes the from
where is the direction of incidient particles and is the direction of scattered particles.
Unitary condition
When conservation of number of particles holds true during scattering, it leads to a unitary condition for the scattering amplitude. In the general case, we have
Optical theorem follows from here by setting
In the centrally symmetric field, the unitary condition becomes
where and are the angles between and and some direction . This condition puts a constraint on the allowed form for , i.e., the real and imaginary part of the scattering amplitude are not independent in this case. For example, if in is known (say, from the measurement of the cross section), then can be determined such that is uniquely determined within the alternative .
Partial wave expansion
In the partial wave expansion the scattering amplitude is represented as a sum over the partial waves,
,
where is the partial scattering amplitude and are the Legendre polynomials. The partial amplitude can be expressed via the partial wave S-matrix element () and the scattering phase shift as
Then the total cross section
,
can be expanded as
is the partial cross section. The total cross section is also equal to due to optical theorem.
For , we can write
X-rays
The scattering length for X-rays is the Thomson scattering length or classical electron radius, 0.
Neutrons
The nuclear neutron scattering process involves the coherent neutron scattering length, often described by .
Quantum mechanical formalism
A quantum mechanical approach is given by the S matrix formalism.
Measurement
The scattering amplitude can be determined by the scattering length in the low-energy regime.
See also
Veneziano amplitude
Plane wave expansion
References
Neutron
X-rays
Electron
Scattering
Diffraction
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https://en.wikipedia.org/wiki/Smooth%20number
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In number theory, an n-smooth (or n-friable) number is an integer whose prime factors are all less than or equal to n. For example, a 7-smooth number is a number whose every prime factor is at most 7, so 49 = 72 and 15750 = 2 × 32 × 53 × 7 are both 7-smooth, while 11 and 702 = 2 × 33 × 13 are not 7-smooth. The term seems to have been coined by Leonard Adleman. Smooth numbers are especially important in cryptography, which relies on factorization of integers. The 2-smooth numbers are just the powers of 2, while 5-smooth numbers are known as regular numbers.
Definition
A positive integer is called B-smooth if none of its prime factors are greater than B. For example, 1,620 has prime factorization 22 × 34 × 5; therefore 1,620 is 5-smooth because none of its prime factors are greater than 5. This definition includes numbers that lack some of the smaller prime factors; for example, both 10 and 12 are 5-smooth, even though they miss out the prime factors 3 and 5, respectively. All 5-smooth numbers are of the form 2a × 3b × 5c, where a, b and c are non-negative integers.
The 3-smooth numbers have also been called "harmonic numbers", although that name has other more widely used meanings.
5-smooth numbers are also called regular numbers or Hamming numbers; 7-smooth numbers are also called humble numbers, and sometimes called highly composite, although this conflicts with another meaning of highly composite numbers.
Here, note that B itself is not required to appear among the factors of a B-smooth number. If the largest prime factor of a number is p then the number is B-smooth for any B ≥ p. In many scenarios B is prime, but composite numbers are permitted as well. A number is B-smooth if and only if it is p-smooth, where p is the largest prime less than or equal to B.
Applications
An important practical application of smooth numbers is the fast Fourier transform (FFT) algorithms (such as the Cooley–Tukey FFT algorithm), which operates by recursively breaking down a problem of a given size n into problems the size of its factors. By using B-smooth numbers, one ensures that the base cases of this recursion are small primes, for which efficient algorithms exist. (Large prime sizes require less-efficient algorithms such as Bluestein's FFT algorithm.)
5-smooth or regular numbers play a special role in Babylonian mathematics. They are also important in music theory (see Limit (music)), and the problem of generating these numbers efficiently has been used as a test problem for functional programming.
Smooth numbers have a number of applications to cryptography. While most applications center around cryptanalysis (e.g. the fastest known integer factorization algorithms, for example: General number field sieve algorithm), the VSH hash function is another example of a constructive use of smoothness to obtain a provably secure design.
Distribution
Let denote the number of y-smooth integers less than or equal to x (the de Bruijn function).
If the smoothnes
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https://en.wikipedia.org/wiki/Sieve%20theory
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Sieve theory is a set of general techniques in number theory, designed to count, or more realistically to estimate the size of, sifted sets of integers. The prototypical example of a sifted set is the set of prime numbers up to some prescribed limit X. Correspondingly, the prototypical example of a sieve is the sieve of Eratosthenes, or the more general Legendre sieve. The direct attack on prime numbers using these methods soon reaches apparently insuperable obstacles, in the way of the accumulation of error terms. In one of the major strands of number theory in the twentieth century, ways were found of avoiding some of the difficulties of a frontal attack with a naive idea of what sieving should be.
One successful approach is to approximate a specific sifted set of numbers (e.g. the set of
prime numbers) by another, simpler set (e.g. the set of almost prime numbers), which is typically somewhat larger than the original set, and easier to analyze. More sophisticated sieves also do not work directly with sets per se, but instead count them according to carefully chosen weight functions on these sets (options for giving some elements of these sets more "weight" than others). Furthermore, in some modern applications, sieves are used not to estimate the size of a sifted
set, but to produce a function that is large on the set and mostly small outside it, while being easier to analyze than
the characteristic function of the set.
Basic sieve theory
For information on notation see at the end.
We start with some countable sequence of non-negative numbers . In the most basic case this sequence is just the indicator function of some set we want to sieve. However this abstraction allows for more general situations. Next we introduce a general set of prime numbers called the sifting range and their product up to as a function .
The goal of sieve theory is to estimate the sifting function
In the case of this just counts the cardinality of a subset of numbers, that are coprime to the prime factors of .
Legendre's identity
We can rewrite the sifting function with Legendre's identity
by using the Möbius function and some functions induced by the elements of
Example
Let and . The Möbius function is negative for every prime, so we get
Approximation of the congruence sum
One assumes then that can be written as
where is a density, meaning a multiplicative function such that
and is an approximation of and is some remainder term. The sifting function becomes
or in short
One tries then to estimate the sifting function by finding upper and lower bounds for respectively and .
The partial sum of the sifting function alternately over- and undercounts, so the remainder term will be huge. Brun's idea to improve this was to replace in the sifting function with a weight sequence consisting of restricted Möbius functions. Choosing two appropriate sequences and and denoting the sifting functions with and , one can get lower and upper bounds
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https://en.wikipedia.org/wiki/Wittgenstein%27s%20rod
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Wittgenstein's rod is a problem in geometry discussed by 20th-century philosopher Ludwig Wittgenstein.
Description
A ray is drawn with its origin on a circle, through an external point and a point is chosen at some constant distance from the starting end of the ray; what figure does describe when all the initial points on the circle are considered? The answer depends on three parameters: the radius of the circle, the distance from the center to , and the length of the segment . The shape described by can be seen as a 'figure-eight' which in some cases degenerates to a single lobe looking like an inverted cardioid.
If remains on the same side of with respect to the center of the circle, instead of a ray one can consider just a segment or the rod .
Wittgenstein sketched a mechanism and wrote:
This text has been included among the notes selected for publication in Remarks on the Foundations of Mathematics and the editors have dated in the as spring of 1944.
Related mechanism
Wittgenstein's rod is a generalization of Hoeckens linkage.
Animations
See also
Oscillating cylinder steam engine — a steam engine using a Wittgenstein's rod scribing a figure-of-eight shape (it does not make use of the curve scribed)
References
External links
Geometer's Sketchpad applet
Geometry
Rod
Linkages (mechanical)
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https://en.wikipedia.org/wiki/Variation%20of%20parameters
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In mathematics, variation of parameters, also known as variation of constants, is a general method to solve inhomogeneous linear ordinary differential equations.
For first-order inhomogeneous linear differential equations it is usually possible to find solutions via integrating factors or undetermined coefficients with considerably less effort, although those methods leverage heuristics that involve guessing and do not work for all inhomogeneous linear differential equations.
Variation of parameters extends to linear partial differential equations as well, specifically to inhomogeneous problems for linear evolution equations like the heat equation, wave equation, and vibrating plate equation. In this setting, the method is more often known as Duhamel's principle, named after Jean-Marie Duhamel (1797–1872) who first applied the method to solve the inhomogeneous heat equation. Sometimes variation of parameters itself is called Duhamel's principle and vice versa.
History
The method of variation of parameters was first sketched by the Swiss mathematician Leonhard Euler (1707–1783), and later completed by the Italian-French mathematician Joseph-Louis Lagrange (1736–1813).
A forerunner of the method of variation of a celestial body's orbital elements appeared in Euler's work in 1748, while he was studying the mutual perturbations of Jupiter and Saturn. In his 1749 study of the motions of the earth, Euler obtained differential equations for the orbital elements. In 1753, he applied the method to his study of the motions of the moon.
Lagrange first used the method in 1766. Between 1778 and 1783, he further developed the method in two series of memoirs: one on variations in the motions of the planets and another on determining the orbit of a comet from three observations. During 1808–1810, Lagrange gave the method of variation of parameters its final form in a third series of papers.
Description of method
Given an ordinary non-homogeneous linear differential equation of order n
Let be a basis of the vector space of solutions of the corresponding homogeneous equation
Then a particular solution to the non-homogeneous equation is given by
where the are differentiable functions which are assumed to satisfy the conditions
Starting with (), repeated differentiation combined with repeated use of () gives
One last differentiation gives
By substituting () into () and applying () and () it follows that
The linear system ( and ) of n equations can then be solved using Cramer's rule yielding
where is the Wronskian determinant of the basis and is the Wronskian determinant of the basis with the i-th column replaced by
The particular solution to the non-homogeneous equation can then be written as
Intuitive explanation
Consider the equation of the forced dispersionless spring, in suitable units:
Here is the displacement of the spring from the equilibrium , and is an external applied force that depends on time. When the external force i
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https://en.wikipedia.org/wiki/Kappa%20curve
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In geometry, the kappa curve or Gutschoven's curve is a two-dimensional algebraic curve resembling the Greek letter . The kappa curve was first studied by Gérard van Gutschoven around 1662. In the history of mathematics, it is remembered as one of the first examples of Isaac Barrow's application of rudimentary calculus methods to determine the tangent of a curve. Isaac Newton and Johann Bernoulli continued the studies of this curve subsequently.
Using the Cartesian coordinate system it can be expressed as
or, using parametric equations,
In polar coordinates its equation is even simpler:
It has two vertical asymptotes at , shown as dashed blue lines in the figure at right.
The kappa curve's curvature:
Tangential angle:
Tangents via infinitesimals
The tangent lines of the kappa curve can also be determined geometrically using differentials and the elementary rules of infinitesimal arithmetic. Suppose and are variables, while a is taken to be a constant. From the definition of the kappa curve,
Now, an infinitesimal change in our location must also change the value of the left hand side, so
Distributing the differential and applying appropriate rules,
Derivative
If we use the modern concept of a functional relationship and apply implicit differentiation, the slope of a tangent line to the kappa curve at a point is:
References
External links
A Java applet for playing with the curve
Algebraic curves
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https://en.wikipedia.org/wiki/124%20%28number%29
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124 (one hundred [and] twenty-four) is the natural number following 123 and preceding 125.
In mathematics
124 is an untouchable number, meaning that it is not the sum of proper divisors of any positive number.
It is a stella octangula number, the number of spheres packed in the shape of a stellated octahedron. It is also an icosahedral number.
There are 124 different polygons of length 12 formed by edges of the integer lattice, counting two polygons as the same only when one is a translated copy of the other.
124 is a perfectly partitioned number, meaning that it divides the number of partitions of 124. It is the first number to do so after 1, 2, and 3.
See also
The year AD 124 or 124 BC
124th (disambiguation)
List of highways numbered 124
References
Integers
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https://en.wikipedia.org/wiki/Proper%20linear%20model
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In statistics, a proper linear model is a linear regression model in which the weights given to the predictor variables are chosen in such a way as to optimize the relationship between the prediction and the criterion. Simple regression analysis is the most common example of a proper linear model. Unit-weighted regression is the most common example of an improper linear model.
Bibliography
Regression models
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https://en.wikipedia.org/wiki/Cauchy%E2%80%93Euler%20equation
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In mathematics, an Euler–Cauchy equation, or Cauchy–Euler equation, or simply Euler's equation is a linear homogeneous ordinary differential equation with variable coefficients. It is sometimes referred to as an equidimensional equation. Because of its particularly simple equidimensional structure, the differential equation can be solved explicitly.
The equation
Let be the nth derivative of the unknown function . Then a Cauchy–Euler equation of order n has the form
The substitution (that is, ; for , one might replace all instances of by , which extends the solution's domain to ) may be used to reduce this equation to a linear differential equation with constant coefficients. Alternatively, the trial solution may be used to directly solve for the basic solutions.
Second order – solving through trial solution
The most common Cauchy–Euler equation is the second-order equation, appearing in a number of physics and engineering applications, such as when solving Laplace's equation in polar coordinates. The second order Cauchy–Euler equation is
We assume a trial solution
Differentiating gives and
Substituting into the original equation leads to requiring
Rearranging and factoring gives the indicial equation
We then solve for m. There are three particular cases of interest:
Case 1 of two distinct roots, and ;
Case 2 of one real repeated root, ;
Case 3 of complex roots, .
In case 1, the solution is
In case 2, the solution is
To get to this solution, the method of reduction of order must be applied after having found one solution .
In case 3, the solution is
For .
This form of the solution is derived by setting and using Euler's formula
Second order – solution through change of variables
We operate the variable substitution defined by
Differentiating gives
Substituting the differential equation becomes
This equation in is solved via its characteristic polynomial
Now let and denote the two roots of this polynomial. We analyze the case where there are distinct roots and the case where there is a repeated root:
If the roots are distinct, the general solution is where the exponentials may be complex.
If the roots are equal, the general solution is
In both cases, the solution may be found by setting .
Hence, in the first case, and in the second case,
Example
Given
we substitute the simple solution :
For to be a solution, either , which gives the trivial solution, or the coefficient of is zero. Solving the quadratic equation, we get . The general solution is therefore
Difference equation analogue
There is a difference equation analogue to the Cauchy–Euler equation. For a fixed , define the sequence as
Applying the difference operator to , we find that
If we do this times, we find that
where the superscript denotes applying the difference operator times. Comparing this to the fact that the -th derivative of equals
suggests that we can solve the N-th order difference equation
in a similar manner to t
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https://en.wikipedia.org/wiki/Cut%20rule
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In mathematical logic, the cut rule is an inference rule of sequent calculus. It is a generalisation of the classical modus ponens inference rule. Its meaning is that, if a formula A appears as a conclusion in one proof and a hypothesis in another, then another proof in which the formula A does not appear can be deduced. In the particular case of the modus ponens, for example occurrences of man are eliminated of Every man is mortal, Socrates is a man to deduce Socrates is mortal.
Formal notation
Formal notation in sequent calculus notation :
cut
Elimination
The cut rule is the subject of an important theorem, the cut elimination theorem. It states that any judgement that possesses a proof in the sequent calculus that makes use of the cut rule also possesses a cut-free proof, that is, a proof that does not make use of the cut rule.
Rules of inference
Logical calculi
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https://en.wikipedia.org/wiki/Sixth%20government%20of%20Jordi%20Pujol
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The colors indicate the political party affiliation of each member:
So the statistics of the Government composition are:
Cabinets of Catalonia
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https://en.wikipedia.org/wiki/129%20%28number%29
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129 (one hundred [and] twenty-nine) is the natural number following 128 and preceding 130.
In mathematics
129 is the sum of the first ten prime numbers. It is the smallest number that can be expressed as a sum of three squares in four different ways: , , , and .
129 is the product of only two primes, 3 and 43, making 129 a semiprime. Since 3 and 43 are both Gaussian primes, this means that 129 is a Blum integer.
129 is a repdigit in base 6 (333).
129 is a happy number.
129 is a centered octahedral number.
In the military
Raytheon AGM-129 ACM (Advanced Cruise Missile) was a low observable, sub-sonic, jet-powered, air-launched cruise missile used by the United States Air Force
Soviet submarine K-129 (1960) was a Soviet Pacific Fleet nuclear submarine that sank in 1968
was a United States Navy Mission Buenaventura-class fleet oilers during World War II
was a Crosley-class high speed transport of the United States Navy
was the lead ship of her class of destroyer escort in the United States Navy
was a United States Navy Haskell-class attack transport during World War II
was a United States Navy Crater-class cargo ship during World War II
was a United States Navy Auk-class minesweeper for removing naval mines laid in the water
Agusta A129 Mangusta is an attack helicopter originally designed and produced by Italian company Agusta
The 129th Rescue Wing (129 RQW) is a unit of the California Air National Guard
In transportation
LZ 129 Hindenburg was a German zeppelin which went up in flames while landing on May 6, 1937
London Buses route 129 is a Transport for London contracted bus route in London
STS-129 was a Space Shuttle mission to the International Space Station, flown in November 2009 by the shuttle Atlantis''.
In other fields
129 is also:
The year AD 129 or 129 BC
129 AH is a year in the Islamic calendar that corresponds to 746–747 CE
129 Antigone is a main belt asteroid
The atomic number of unbiennium, an element yet to be discovered
A film format: 129 film
Sonnet 129 by William Shakespeare
See also
List of highways numbered 129
United Nations Security Council Resolution 129
References
Integers
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https://en.wikipedia.org/wiki/Confidence%20region
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In statistics, a confidence region is a multi-dimensional generalization of a confidence interval. It is a set of points in an n-dimensional space, often represented as an ellipsoid around a point which is an estimated solution to a problem, although other shapes can occur.
Interpretation
The confidence region is calculated in such a way that if a set of measurements were repeated many times and a confidence region calculated in the same way on each set of measurements, then a certain percentage of the time (e.g. 95%) the confidence region would include the point representing the "true" values of the set of variables being estimated. However, unless certain assumptions about prior probabilities are made, it does not mean, when one confidence region has been calculated, that there is a 95% probability that the "true" values lie inside the region, since we do not assume any particular probability distribution of the "true" values and we may or may not have other information about where they are likely to lie.
The case of independent, identically normally-distributed errors
Suppose we have found a solution to the following overdetermined problem:
where Y is an n-dimensional column vector containing observed values of the dependent variable, X is an n-by-p matrix of observed values of independent variables (which can represent a physical model) which is assumed to be known exactly, is a column vector containing the p parameters which are to be estimated, and is an n-dimensional column vector of errors which are assumed to be independently distributed with normal distributions with zero mean and each having the same unknown variance .
A joint 100(1 − α) % confidence region for the elements of is represented by the set of values of the vector b which satisfy the following inequality:
where the variable b represents any point in the confidence region, p is the number of parameters, i.e. number of elements of the vector is the vector of estimated parameters, and s2 is the reduced chi-squared, an unbiased estimate of equal to
Further, F is the quantile function of the F-distribution, with p and degrees of freedom, is the statistical significance level, and the symbol means the transpose of .
The expression can be rewritten as:
where is the least-squares scaled covariance matrix of .
The above inequality defines an ellipsoidal region in the p-dimensional Cartesian parameter space Rp. The centre of the ellipsoid is at the estimate . According to Press et al., it is easier to plot the ellipsoid after doing singular value decomposition. The lengths of the axes of the ellipsoid are proportional to the reciprocals of the values on the diagonals of the diagonal matrix, and the directions of these axes are given by the rows of the 3rd matrix of the decomposition.
Weighted and generalised least squares
Now consider the more general case where some distinct elements of have known nonzero covariance (in other words, the errors in the obs
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https://en.wikipedia.org/wiki/Glossary%20of%20mathematical%20jargon
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The language of mathematics has a vast vocabulary of specialist and technical terms. It also has a certain amount of jargon: commonly used phrases which are part of the culture of mathematics, rather than of the subject. Jargon often appears in lectures, and sometimes in print, as informal shorthand for rigorous arguments or precise ideas. Much of this is common English, but with a specific non-obvious meaning when used in a mathematical sense.
Some phrases, like "in general", appear below in more than one section.
Philosophy of mathematics
abstract nonsenseA tongue-in-cheek reference to category theory, using which one can employ arguments that establish a (possibly concrete) result without reference to any specifics of the present problem. For that reason, it's also known as general abstract nonsense or generalized abstract nonsense.
canonicalA reference to a standard or choice-free presentation of some mathematical object (e.g., canonical map, canonical form, or canonical ordering). The same term can also be used more informally to refer to something "standard" or "classic". For example, one might say that Euclid's proof is the "canonical proof" of the infinitude of primes.
deepA result is called "deep" if its proof requires concepts and methods that are advanced beyond the concepts needed to formulate the result. For example, the prime number theorem — originally proved using techniques of complex analysis — was once thought to be a deep result until elementary proofs were found. On the other hand, the fact that π is irrational is usually known to be a deep result, because it requires a considerable development of real analysis before the proof can be established — even though the claim itself can be stated in terms of simple number theory and geometry.
elegantAn aesthetic term referring to the ability of an idea to provide insight into mathematics, whether by unifying disparate fields, introducing a new perspective on a single field, or by providing a technique of proof which is either particularly simple, or which captures the intuition or imagination as to why the result it proves is true. In some occasions, the term "beautiful" can also be used to the same effect, though Gian-Carlo Rota distinguished between elegance of presentation and beauty of concept, saying that for example, some topics could be written about elegantly although the mathematical content is not beautiful, and some theorems or proofs are beautiful but may be written about inelegantly.
elementaryA proof or a result is called "elementary" if it only involves basic concepts and methods in the field, and is to be contrasted with deep results which require more development within or outside the field. The concept of "elementary proof" is used specifically in number theory, where it usually refers to a proof that does not resort to methods from complex analysis.
folklore A result is called "folklore" if it is non-obvious, non-published, yet somehow generally known
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https://en.wikipedia.org/wiki/National%20Center%20for%20Health%20Statistics
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The National Center for Health Statistics (NCHS) is a U.S. government agency that provides statistical information to guide actions and policies to improve the public health of the American people. It is a unit of the Centers for Disease Control and Prevention (CDC) and a principal agency of the U.S. Federal Statistical System. It is headquartered at University Town Center in Hyattsville, Maryland, just outside Washington, D.C.
History
The Marine Hospital Service, predecessor of the Public Health Service (PHS), began collecting data on communicable diseases and performing surveillance of the incidence and distribution of diseases due to an 1878 act of Congress. In 1893, another law provided for weekly collection of data from state and municipal authorities.
The Division of Sanitary Reports and Statistics was established in 1899 as part of the initial establishment of internal divisions within the Marine Hospital Service. Separately, the Division of Public Health Methods was formed in 1937 within the National Institute of Health. In 1943, these two divisions were merged, retaining the name Division of Public Health Methods but being transferred into the Office of the Surgeon General.
In 1946, the Division of Public Health Methods absorbed the Vital Statistics Division, which dated from 1903, from the Bureau of the Census in the Department of Commerce. The merged division was renamed the National Office of Vital Statistics. It was then transferred into the PHS Bureau of State Services in 1949.
In 1960, the National Office of Vital Statistics and the National Health Survey merged to form the National Center for Health Statistics. The National Health Survey had been created within PHS in 1956 through the National Health Survey Act (); it was the successor to a seminal national health survey performed by the Works Progress Administration during 1935–1936, which had multiple supplemental studies carried out in the intervening decades.
During the PHS reorganizations of 1966-1973, the National Center for Health Statistics was part of the Health Services and Mental Health Administration (HSMHA), and afterwards was part of the Health Resources Administration. Since 1987, it has been part of the Centers for Disease Control and Prevention (CDC).
Data collection programs
NCHS collects data with surveys, from other agencies and U.S. states, from administrative sources, and from partnerships with private health partners. NCHS collects data from birth and death records, medical records, interview surveys, and through direct physical examinations and laboratory testing. These diverse sources give perspectives to help understand the U.S. population's health, health outcomes, and influences on health.
There are four major data collection programs at NCHS:
National Vital Statistics System
The National Vital Statistics System (NVSS) collects official vital statistics data based on the collection and registration of birth and death events at the stat
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https://en.wikipedia.org/wiki/131%20%28number%29
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131 (one hundred [and] thirty-one) is the natural number following 130 and preceding 132.
In mathematics
131 is a Sophie Germain prime, an irregular prime, the second 3-digit palindromic prime, and also a permutable prime with 113 and 311. It can be expressed as the sum of three consecutive primes, 131 = 41 + 43 + 47. 131 is an Eisenstein prime with no imaginary part and real part of the form . Because the next odd number, 133, is a semiprime, 131 is a Chen prime. 131 is an Ulam number.
131 is a full reptend prime in base 10 (and also in base 2). The decimal expansion of 1/131 repeats the digits 007633587786259541984732824427480916030534351145038167938931 297709923664122137404580152671755725190839694656488549618320 6106870229 indefinitely.
In the military
Convair C-131 Samaritan was an American military transport produced from 1954 to 1956
Strike Fighter Squadron (VFA-131) is a United States Navy F/A-18C Hornet fighter squadron stationed at Naval Air Station Oceana
Tiger 131 is a German Tiger I heavy tank captured in Tunisia by the British 48th Royal Tank Regiment during World War II
was a Mission Buenaventura-class fleet oiler during World War II
was a is a United States Navy ship during World War II
was a United States Navy
was a United States Navy General G. O. Squier-class transport ship during World War II
was a United States Navy during World War II
was a United States Navy during World War II
was a ship of the United States Navy during World War II
ZIL-131 is a 3.5-ton 6x6 army truck
In transportation
London Buses route 131 is a Transport for London contracted bus route in London
The Fiat 131 Mirafiori small/medium family car produced from 1974 to 1984
STS-131 is a NASA Contingency Logistic Flight (CLF) of the Space Shuttle Atlantis which launched in April 2010
In other fields
131 is also:
The year AD 131 or 131 BC
131 AH is a year in the Islamic calendar that corresponds to 748 – 749 CE.
131 Vala is an inner main belt asteroid
Iodine-131, or radioiodine, is a radioisotope of iodine for medical and pharmaceutical use
ACP-131 is the controlling publication for listing of Q codes and Z codes, as published by NATO Allied countries
Sonnet 131 by William Shakespeare
131 is the medical emergency telephone number in Chile
United States Citizenship and Immigration Services Form I-131 to apply for a travel document, reentry permit, refugee travel document or advance parole
131 is the ID3v1 tag equivalent to Indie music
See also
List of highways numbered 131
United Nations Security Council Resolution 131
References
Integers
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https://en.wikipedia.org/wiki/Bertrand%27s%20ballot%20theorem
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In combinatorics, Bertrand's ballot problem is the question: "In an election where candidate A receives p votes and candidate B receives q votes with p > q, what is the probability that A will be strictly ahead of B throughout the count?" The answer is
The result was first published by W. A. Whitworth in 1878, but is named after Joseph Louis François Bertrand who rediscovered it in 1887.
In Bertrand's original paper, he sketches a proof based on a general formula for the number of favourable sequences using a recursion relation. He remarks that it seems probable that such a simple result could be proved by a more direct method. Such a proof was given by Désiré André, based on the observation that the unfavourable sequences can be divided into two equally probable cases, one of which (the case where B receives the first vote) is easily computed; he proves the equality by an explicit bijection. A variation of his method is popularly known as André's reflection method, although André did not use any reflections.
Bertrand's ballot theorem is related to the cycle lemma. They give similar formulas, but the cycle lemma considers circular shifts of a given ballot counting order rather than all permutations.
Example
Suppose there are 5 voters, of whom 3 vote for candidate A and 2 vote for candidate B (so p = 3 and q = 2). There are ten equally likely orders in which the votes could be counted:
AAABB
AABAB
ABAAB
BAAAB
AABBA
ABABA
BAABA
ABBAA
BABAA
BBAAA
For the order AABAB, the tally of the votes as the election progresses is:
For each column the tally for A is always larger than the tally for B, so A is always strictly ahead of B. For the order AABBA the tally of the votes as the election progresses is:
For this order, B is tied with A after the fourth vote, so A is not always strictly ahead of B.
Of the 10 possible orders, A is always ahead of B only for AAABB and AABAB. So the probability that A will always be strictly ahead is
and this is indeed equal to as the theorem predicts.
Equivalent problems
Favourable orders
Rather than computing the probability that a random vote counting order has the desired property, one can instead compute the number of favourable counting orders, then divide by the total number of ways in which the votes could have been counted. (This is the method used by Bertrand.) The total number of ways is the binomial coefficient ; Bertrand's proof shows that the number of favourable orders in which to count the votes is (though he does not give this number explicitly). And indeed after division this gives .
Random walks
Another equivalent problem is to calculate the number of random walks on the integers that consist of n steps of unit length, beginning at the origin and ending at the point m, that never become negative. As n and m have the same parity and , this number is
When and is even, this gives the Catalan number . Thus the probability that the a random walk is never negative and returns to origin at ti
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https://en.wikipedia.org/wiki/Michel%20Lo%C3%A8ve
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Michel Loève (January 22, 1907 – February 17, 1979) was a French-American probabilist and mathematical statistician, of Jewish origin. He is known in mathematical statistics and probability theory for the Karhunen–Loève theorem and Karhunen–Loève transform.
Michel Loève was born in Jaffa (then part of the Ottoman Empire) in 1907, to a Jewish family. He passed most of his childhood years in Egypt and received his primary and secondary education there in French schools. Later, after achieving the grades of B.L. in 1931 and A.B. in 1936, he studied mathematics at the Université de Paris under Paul Lévy, and received his Doctorat ès Sciences (Mathématiques) in 1941. In 1936 was employed as actuaire of the University of Lyon.
Because of his Jewish origin, he was arrested during the German occupation of France and sent to Drancy internment camp. One of his books is dedicated "To Line and To the students and teachers of the School in the Camp de Drancy". Having survived the Holocaust, after the liberation he became between 1944 and 1946 chief of research at the Institut Henri Poincaré at Paris University, then until 1948 worked at the University of London.
After one term as a visiting professor at Columbia University he accepted the position of professor of mathematics at Berkeley, in 1955 adding the title professor of statistics.
He is the author of one of the earliest books on measure-theoretic probability theory and one of the best known textbooks. He is memorialized via the Loève Prize created by his widow Line.
See also
Kari Karhunen
Harold Hotelling
References
External links
University of California in Memoriam
Photographs
Photograph from Portraits of Statisticians
1907 births
1979 deaths
French statisticians
Probability theorists
University of Paris alumni
Academics of the University of London
Columbia University staff
University of California, Berkeley College of Letters and Science faculty
20th-century French mathematicians
Egyptian emigrants to France
French emigrants to the United States
Emigrants from the Ottoman Empire to Egypt
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https://en.wikipedia.org/wiki/Nonlinear%20regression
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In statistics, nonlinear regression is a form of regression analysis in which observational data are modeled by a function which is a nonlinear combination of the model parameters and depends on one or more independent variables. The data are fitted by a method of successive approximations.
General
In nonlinear regression, a statistical model of the form,
relates a vector of independent variables, , and its associated observed dependent variables, . The function is nonlinear in the components of the vector of parameters , but otherwise arbitrary. For example, the Michaelis–Menten model for enzyme kinetics has two parameters and one independent variable, related by by:
This function is nonlinear because it cannot be expressed as a linear combination of the two s.
Systematic error may be present in the independent variables but its treatment is outside the scope of regression analysis. If the independent variables are not error-free, this is an errors-in-variables model, also outside this scope.
Other examples of nonlinear functions include exponential functions, logarithmic functions, trigonometric functions, power functions, Gaussian function, and Lorentz distributions. Some functions, such as the exponential or logarithmic functions, can be transformed so that they are linear. When so transformed, standard linear regression can be performed but must be applied with caution. See Linearization§Transformation, below, for more details.
In general, there is no closed-form expression for the best-fitting parameters, as there is in linear regression. Usually numerical optimization algorithms are applied to determine the best-fitting parameters. Again in contrast to linear regression, there may be many local minima of the function to be optimized and even the global minimum may produce a biased estimate. In practice, estimated values of the parameters are used, in conjunction with the optimization algorithm, to attempt to find the global minimum of a sum of squares.
For details concerning nonlinear data modeling see least squares and non-linear least squares.
Regression statistics
The assumption underlying this procedure is that the model can be approximated by a linear function, namely a first-order Taylor series:
where . It follows from this that the least squares estimators are given by
compare generalized least squares with covariance matrix proportional to the unit matrix. The nonlinear regression statistics are computed and used as in linear regression statistics, but using J in place of X in the formulas.
When the function itself is not known analytically, but needs to be linearly approximated from , or more, known values (where is the number of estimators), the best estimator is obtained directly from the Linear Template Fit as (see also linear least squares).
The linear approximation introduces bias into the statistics. Therefore, more caution than usual is required in interpreting statistics derived from a nonlinear model.
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https://en.wikipedia.org/wiki/Multinomial%20distribution
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In probability theory, the multinomial distribution is a generalization of the binomial distribution. For example, it models the probability of counts for each side of a k-sided die rolled n times. For n independent trials each of which leads to a success for exactly one of k categories, with each category having a given fixed success probability, the multinomial distribution gives the probability of any particular combination of numbers of successes for the various categories.
When k is 2 and n is 1, the multinomial distribution is the Bernoulli distribution. When k is 2 and n is bigger than 1, it is the binomial distribution. When k is bigger than 2 and n is 1, it is the categorical distribution. The term "multinoulli" is sometimes used for the categorical distribution to emphasize this four-way relationship (so n determines the suffix, and k the prefix).
The Bernoulli distribution models the outcome of a single Bernoulli trial. In other words, it models whether flipping a (possibly biased) coin one time will result in either a success (obtaining a head) or failure (obtaining a tail). The binomial distribution generalizes this to the number of heads from performing n independent flips (Bernoulli trials) of the same coin. The multinomial distribution models the outcome of n experiments, where the outcome of each trial has a categorical distribution, such as rolling a k-sided die n times.
Let k be a fixed finite number. Mathematically, we have k possible mutually exclusive outcomes, with corresponding probabilities p1, ..., pk, and n independent trials. Since the k outcomes are mutually exclusive and one must occur we have pi ≥ 0 for i = 1, ..., k and . Then if the random variables Xi indicate the number of times outcome number i is observed over the n trials, the vector X = (X1, ..., Xk) follows a multinomial distribution with parameters n and p, where p = (p1, ..., pk). While the trials are independent, their outcomes Xi are dependent because they must be summed to n.
Definitions
Probability mass function
Suppose one does an experiment of extracting n balls of k different colors from a bag, replacing the extracted balls after each draw. Balls of the same color are equivalent. Denote the variable which is the number of extracted balls of color i (i = 1, ..., k) as Xi, and denote as pi the probability that a given extraction will be in color i. The probability mass function of this multinomial distribution is:
for non-negative integers x1, ..., xk.
The probability mass function can be expressed using the gamma function as:
This form shows its resemblance to the Dirichlet distribution, which is its conjugate prior.
Example
Suppose that in a three-way election for a large country, candidate A received 20% of the votes, candidate B received 30% of the votes, and candidate C received 50% of the votes. If six voters are selected randomly, what is the probability that there will be exactly one supporter for candidate A, two supporters f
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https://en.wikipedia.org/wiki/133%20%28number%29
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133 (one hundred [and] thirty-three) is the natural number following 132 and preceding 134.
In mathematics
133 is an n whose divisors (excluding n itself) added up divide φ(n). It is an octagonal number and a happy number.
133 is a Harshad number, because it is divisible by the sum of its digits.
133 is a repdigit in base 11 (111) and base 18 (77), whilst in base 20 it is a cyclic number formed from the reciprocal of the number three.
133 is a semiprime: a product of two prime numbers, namely 7 and 19. Since those prime factors are Gaussian primes, this means that 133 is a Blum integer.
133 is the number of compositions of 13 into distinct parts.
In the military
Douglas C-133 Cargomaster was a United States cargo aircraft built between 1956 and 1961
is a heavy landing craft which launched in 1972
was a United States Navy Mission Buenaventura-class fleet oilers during World War II
was a United States Navy during World War II
was a United States Navy during World War II
was a United States Navy General G. O. Squier-class transport ship during World War II
was a United States Navy during World War I
was a United States Navy during World War II
was a United States Navy during World War II
was a United States Navy S-class submarine during World War II
was a United States Navy during World War II
was a United States Navy heavy cruiser during the Korean War
Frontstalag 133 was a temporary German prisoner of war camp during World War II located near Rennes, northern France
Caproni Ca.133 was a three-engine transport/bomber aircraft used by the Italian Regia Aeronautica from the Second Italo-Abyssinian War until World War II
Naval Mobile Construction Battalion 133 is an active duty Seabee battalion originally commissioned during World War II as the 133rd Naval Construction Battalion(NCB)
In transportation
London Buses route 133 is a Transport for London contracted bus route in London
RATB route 133 is bus route run by RATB in Bucharest, Romania
The Fiat 133, also called SEAT 133, was a small rear-engine car developed in Spain between 1974 and 1979
STS-133 is a Space Shuttle Endeavour contingency mission which was Discoverys 39th and final mission. The mission launched on 24 February 2011, and landed on 9 March 2011.
The Bücker Bü 133 Jungmeister was a single-engine, single-seat biplane trainer of the Luftwaffe in the 1930s
In other fields133''' is also:
The nickname of Craig Jones
The year AD 133 or 133 BC
133 AH is a year in the Islamic calendar that corresponds to 750 – 751 CE
133P/Elst-Pizarro is a body with characteristics of both asteroids and comets, a prototype of main-belt comets
133 Cyrene is an S-type main belt asteroid
The atomic number of an element temporarily called untritrium.
Xenon-133 is an isotope of xenon. It is a radionuclide that is inhaled to assess pulmonary function, and to image the lungs
133 is the police emergency telephone number in Chile
"133" is the name of a David G
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https://en.wikipedia.org/wiki/Projection-valued%20measure
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In mathematics, particularly in functional analysis, a projection-valued measure (PVM) is a function defined on certain subsets of a fixed set and whose values are self-adjoint projections on a fixed Hilbert space. Projection-valued measures are formally similar to real-valued measures, except that their values are self-adjoint projections rather than real numbers. As in the case of ordinary measures, it is possible to integrate complex-valued functions with respect to a PVM; the result of such an integration is a linear operator on the given Hilbert space.
Projection-valued measures are used to express results in spectral theory, such as the important spectral theorem for self-adjoint operators, in which case the PVM is sometimes referred to as the spectral measure. The Borel functional calculus for self-adjoint operators is constructed using integrals with respect to PVMs. In quantum mechanics, PVMs are the mathematical description of projective measurements. They are generalized by positive operator valued measures (POVMs) in the same sense that a mixed state or density matrix generalizes the notion of a pure state.
Formal definition
A projection-valued measure on a measurable space
, where is a σ-algebra of subsets of , is a mapping from to the set of self-adjoint projections on a Hilbert space (i.e. the orthogonal projections) such that
(where is the identity operator of ) and for every , the following function
is a complex measure on (that is, a complex-valued countably additive function).
We denote this measure by
.
Note that is a real-valued measure, and a probability measure when has length one.
If is a projection-valued measure and
then the images , are orthogonal to each other. From this follows that in general,
and they commute.
Example. Suppose is a measure space. Let, for every measurable subset in ,
be the operator of multiplication by the indicator function on L2(X). Then is a projection-valued measure. For example, if , , and there is then the associated complex measure which takes a measurable function and gives the integral
Extensions of projection-valued measures, integrals and the spectral theorem
If is a projection-valued measure on a measurable space (X, M), then the map
extends to a linear map on the vector space of step functions on X. In fact, it is easy to check that this map is a ring homomorphism. This map extends in a canonical way to all bounded complex-valued measurable functions on X, and we have the following.
Theorem. For any bounded M-measurable function f on X, there exists a unique bounded linear operator
such that for all where denotes the complex measure from the definition of .
The map
is a homomorphism of rings.
An integral notation is often used for , as in
The theorem is also correct for unbounded measurable functions f, but then will be an unbounded linear operator on the Hilbert space H.
The spectral theorem says that every se
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https://en.wikipedia.org/wiki/Nerve%20%28disambiguation%29
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A nerve is a part of the peripheral nervous system.
Nerve or Nerves may also refer to:
Mathematics
Nerve of a covering, a construction in mathematical topology
Nerve (category theory), a construction in category theory
Film and television
Nerves (film), a 1919 film by the Austrian director and novelist Robert Reinert
"Nerve" (Farscape), a 2000 episode of Farscape
Nerve (2013 film), a 2013 Australian psychological thriller film
Nerve (2016 film), a 2016 American drama thriller film
Books
Nerve (magazine), a Liverpool-based arts and social issues magazine
Nerve (Francis novel), a 1964 novel by Dick Francis
Nerve (Ryan novel), a 2012 young adult thriller by Jeanne Ryan
Computing
Nerve Software, a video game developer
Nerve (website), a website and magazine
Music
Artists
The Nerves, an American power pop band
Nerve, an American band founded by Jojo Mayer
Nerve, an industrial rock band that Junkie XL was a member of
Songs
"nerve", by Bis from Brand-new idol Society, 2011
"Nerve", by Blindside from Blindside, 1997
"Nerve", by Charlotte Church from Two, 2013
"Nerve", by Don Broco from Automatic, 2015
"Nerve", by Half Moon Run from Dark Eyes, 2012
"Nerve", by Soilwork from Stabbing the Drama, 2005
"Nerve", by The Story So Far from The Story So Far, 2015
"Nerves", by Bauhaus from In the Flat Field, 1980
"Nerves", by Maths Class, 2008
"Nerves", by Silkworm from Firewater, 1996
"The Nerve", by George Strait from Carrying Your Love with Me, 1997
"The Nerve", by Kaiser Chiefs from Education, Education, Education & War, 2014
Other uses
Nerve (botany), another word for the vein of a leaf
See also
Nerv (disambiguation)
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https://en.wikipedia.org/wiki/Recursive%20definition
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In mathematics and computer science, a recursive definition, or inductive definition, is used to define the elements in a set in terms of other elements in the set (Aczel 1977:740ff). Some examples of recursively-definable objects include factorials, natural numbers, Fibonacci numbers, and the Cantor ternary set.
A recursive definition of a function defines values of the function for some inputs in terms of the values of the same function for other (usually smaller) inputs. For example, the factorial function is defined by the rules
This definition is valid for each natural number , because the recursion eventually reaches the base case of 0. The definition may also be thought of as giving a procedure for computing the value of the function , starting from and proceeding onwards with etc.
The recursion theorem states that such a definition indeed defines a function that is unique. The proof uses mathematical induction.
An inductive definition of a set describes the elements in a set in terms of other elements in the set. For example, one definition of the set of natural numbers is:
1 is in
If an element n is in then is in
is the intersection of all sets satisfying (1) and (2).
There are many sets that satisfy (1) and (2) – for example, the set satisfies the definition. However, condition (3) specifies the set of natural numbers by removing the sets with extraneous members. Note that this definition assumes that is contained in a larger set (such as the set of real numbers) — in which the operation + is defined.
Properties of recursively defined functions and sets can often be proved by an induction principle that follows the recursive definition. For example, the definition of the natural numbers presented here directly implies the principle of mathematical induction for natural numbers: if a property holds of the natural number 0 (or 1), and the property holds of whenever it holds of , then the property holds of all natural numbers (Aczel 1977:742).
Form of recursive definitions
Most recursive definitions have two foundations: a base case (basis) and an inductive clause.
The difference between a circular definition and a recursive definition is that a recursive definition must always have base cases, cases that satisfy the definition without being defined in terms of the definition itself, and that all other instances in the inductive clauses must be "smaller" in some sense (i.e., closer to those base cases that terminate the recursion) — a rule also known as "recur only with a simpler case".
In contrast, a circular definition may have no base case, and even may define the value of a function in terms of that value itself — rather than on other values of the function. Such a situation would lead to an infinite regress.
That recursive definitions are valid – meaning that a recursive definition identifies a unique function – is a theorem of set theory known as the recursion theorem, the proof of which is non-trivial. Where th
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https://en.wikipedia.org/wiki/Complete%20variety
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In mathematics, in particular in algebraic geometry, a complete algebraic variety is an algebraic variety , such that for any variety the projection morphism
is a closed map (i.e. maps closed sets onto closed sets). This can be seen as an analogue of compactness in algebraic geometry: a topological space is compact if and only if the above projection map is closed with respect to topological products.
The image of a complete variety is closed and is a complete variety. A closed subvariety of a complete variety is complete.
A complex variety is complete if and only if it is compact as a complex-analytic variety.
The most common example of a complete variety is a projective variety, but there do exist complete non-projective varieties in dimensions 2 and higher. While any complete nonsingular surface is projective, there exist nonsingular complete varieties in dimension 3 and higher which are not projective. The first examples of non-projective complete varieties were given by Masayoshi Nagata and Heisuke Hironaka. An affine space of positive dimension is not complete.
The morphism taking a complete variety to a point is a proper morphism, in the sense of scheme theory. An intuitive justification of "complete", in the sense of "no missing points", can be given on the basis of the valuative criterion of properness, which goes back to Claude Chevalley.
See also
Chow's lemma
Theorem of the cube
Fano variety
Notes
References
Sources
Section II.4 of
Chapter 7 of
Section I.9 of
Algebraic varieties
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https://en.wikipedia.org/wiki/University%20of%20Bia%C5%82ystok
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The University of Bialystok is the largest university in the north-eastern region of Poland, educating in various fields of study, including humanities, social and natural sciences and mathematics. It has nine faculties, including a foreign one in Vilnius. Four faculties have been awarded the highest scientific category “A”. The University of Bialystok has the right to confer doctoral degrees in ten fields, as well as postdoctoral degrees in law, economics, chemistry, biology, history and physics.
Over 13,000 students are being educated in 31 fields of study, including doctoral studies and postgraduate studies. The university employs nearly 800 academics, almost 200 professors among them.
Every year the university carries out approximately 60 research projects, financed from domestic and foreign funds; it also benefits from the structural funds. Among the university's many accomplishments are its participation in 6th and 7th Framework Programme for Research, Technological Development and Demonstration, Horizon 2020, Comenius and Aspera as well as the DAPHNE III programme.
History
University of Białystok was opened on June 19, 1997. The university was established as a result of a transformation of the Branch of the University of Warsaw in Białystok after 29 years of its existence.
The university has a branch in Vilnius, Lithuania.
Foundation
The University of Bialystok Foundation, Universitas Bialostocensis () - independent, non-profit, non-governmental organization located in Białystok, Poland.
Foundation was chartered on April 22, 2004, by founders from the academic circles of the University of Bialystok. Foundation is governed by an independent Board of Directors. Foundation aims organizational, material and financial support for the academic excellence and future development of the University of Bialystok. It runs such activities as: lectures, seminars, conferences, courses and workshops, business and legal consulting, participation in EU funding programs, providing financial support for academic projects and student scholarships, support students’ organizations at the University of Bialystok
Rectors
Adam Jamróz (1997–2002)
Marek Gębczyński (2002–2005)
Jerzy Nikitorowicz (2005–2012)
Leonard Etel (2012-2016)
Robert CIborowski (since 2016)
Staff
Professors: 162
Habilitated doctors: 7
Senior lecturers: 291
Teachers (total): 348
Total staff: 808
Number of students: 15 034
International cooperation
International cooperation is also carried out based on about 200 bilateral agreements with institutions from the EU within the framework of the Erasmus programme.
As the first university in the country the University of Bialystok launched a foreign branch in Vilnius, Lithuania; the Faculty of Economics and Informatics has been created there, which educates people of mostly Polish origin, but it is also increasingly popular among the Lithuanian youth.
In 2013 the University of Bialystok initiated the creation of an international consortium o
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https://en.wikipedia.org/wiki/Muirhead%27s%20Inequality
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In mathematics, Muirhead's inequality, named after Robert Franklin Muirhead, also known as the "bunching" method, generalizes the inequality of arithmetic and geometric means.
Preliminary definitions
a-mean
For any real vector
define the "a-mean" [a] of positive real numbers x1, ..., xn by
where the sum extends over all permutations σ of { 1, ..., n }.
When the elements of a are nonnegative integers, the a-mean can be equivalently defined via the monomial symmetric polynomial as
where ℓ is the number of distinct elements in a, and k1, ..., kℓ are their multiplicities.
Notice that the a-mean as defined above only has the usual properties of a mean (e.g., if the mean of equal numbers is equal to them) if . In the general case, one can consider instead , which is called a Muirhead mean.
Examples
For a = (1, 0, ..., 0), the a-mean is just the ordinary arithmetic mean of x1, ..., xn.
For a = (1/n, ..., 1/n), the a-mean is the geometric mean of x1, ..., xn.
For a = (x, 1 − x), the a-mean is the Heinz mean.
The Muirhead mean for a = (−1, 0, ..., 0) is the harmonic mean.
Doubly stochastic matrices
An n × n matrix P is doubly stochastic precisely if both P and its transpose PT are stochastic matrices. A stochastic matrix is a square matrix of nonnegative real entries in which the sum of the entries in each column is 1. Thus, a doubly stochastic matrix is a square matrix of nonnegative real entries in which the sum of the entries in each row and the sum of the entries in each column is 1.
Statement
Muirhead's inequality states that [a] ≤ [b] for all x such that xi > 0 for every i ∈ { 1, ..., n } if and only if there is some doubly stochastic matrix P for which a = Pb.
Furthermore, in that case we have [a] = [b] if and only if a = b or all xi are equal.
The latter condition can be expressed in several equivalent ways; one of them is given below.
The proof makes use of the fact that every doubly stochastic matrix is a weighted average of permutation matrices (Birkhoff-von Neumann theorem).
Another equivalent condition
Because of the symmetry of the sum, no generality is lost by sorting the exponents into decreasing order:
Then the existence of a doubly stochastic matrix P such that a = Pb is equivalent to the following system of inequalities:
(The last one is an equality; the others are weak inequalities.)
The sequence is said to majorize the sequence .
Symmetric sum notation
It is convenient to use a special notation for the sums. A success in reducing an inequality in this form means that the only condition for testing it is to verify whether one exponent sequence () majorizes the other one.
This notation requires developing every permutation, developing an expression made of n! monomials, for instance:
Examples
Arithmetic-geometric mean inequality
Let
and
We have
Then
[aA] ≥ [aG],
which is
yielding the inequality.
Other examples
We seek to prove that x2 + y2 ≥ 2xy by using bunching (Muirhead's inequality).
We t
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https://en.wikipedia.org/wiki/Algebraic%20normal%20form
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In Boolean algebra, the algebraic normal form (ANF), ring sum normal form (RSNF or RNF), Zhegalkin normal form, or Reed–Muller expansion is a way of writing propositional logic formulas in one of three subforms:
The entire formula is purely true or false:
One or more variables are combined into a term by AND (), then one or more terms are combined by XOR () together into ANF. Negations are not permitted:
The previous subform with a purely true term:
Formulas written in ANF are also known as Zhegalkin polynomials and Positive Polarity (or Parity) Reed–Muller expressions (PPRM).
Common uses
ANF is a canonical form, which means that two logically equivalent formulas will convert to the same ANF, easily showing whether two formulas are equivalent for automated theorem proving. Unlike other normal forms, it can be represented as a simple list of lists of variable names—conjunctive and disjunctive normal forms also require recording whether each variable is negated or not. Negation normal form is unsuitable for determining equivalence, since on negation normal forms, equivalence does not imply equality: a ∨ ¬a is not reduced to the same thing as 1, even though they are logically equivalent.
Putting a formula into ANF also makes it easy to identify linear functions (used, for example, in linear-feedback shift registers): a linear function is one that is a sum of single literals. Properties of nonlinear-feedback shift registers can also be deduced from certain properties of the feedback function in ANF.
Performing operations within algebraic normal form
There are straightforward ways to perform the standard boolean operations on ANF inputs in order to get ANF results.
XOR (logical exclusive disjunction) is performed directly:
() ⊕ ()
⊕
1 ⊕ 1 ⊕ x ⊕ x ⊕ y
y
NOT (logical negation) is XORing 1:
1 ⊕ 1 ⊕ x ⊕ y
x ⊕ y
AND (logical conjunction) is distributed algebraically
( ⊕ )
⊕
(1 ⊕ x ⊕ y) ⊕ (x ⊕ x ⊕ xy)
1 ⊕ x ⊕ x ⊕ x ⊕ y ⊕ xy
1 ⊕ x ⊕ y ⊕ xy
OR (logical disjunction) uses either 1 ⊕ (1 ⊕ a)(1 ⊕ b) (easier when both operands have purely true terms) or a ⊕ b ⊕ ab (easier otherwise):
() + ()
1 ⊕ (1 ⊕ )(1 ⊕ )
1 ⊕ x(x ⊕ y)
1 ⊕ x ⊕ xy
Converting to algebraic normal form
Each variable in a formula is already in pure ANF, so one only needs to perform the formula's boolean operations as shown above to get the entire formula into ANF. For example:
x + (y ⋅ ¬z)
x + (y(1 ⊕ z))
x + (y ⊕ yz)
x ⊕ (y ⊕ yz) ⊕ x(y ⊕ yz)
x ⊕ y ⊕ xy ⊕ yz ⊕ xyz
Formal representation
ANF is sometimes described in an equivalent way:
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where fully describes .
Recursively deriving multiargument Boolean functions
There are only four functions with one argument:
To represent a function with multiple arguments one can use the following equality:
, where
Indeed,
if then and so
if then and so
Since both and have fewer arguments than it follows that using this process recur
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https://en.wikipedia.org/wiki/Apollonian%20gasket
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In mathematics, an Apollonian gasket or Apollonian net is a fractal generated by starting with a triple of circles, each tangent to the other two, and successively filling in more circles, each tangent to another three. It is named after Greek mathematician Apollonius of Perga.
Construction
The construction of the Apollonian gasket starts with three circles , , and (black in the figure), that are each tangent to the other two, but that do not have a single point of triple tangency. These circles may be of different sizes to each other, and it is allowed for two to be inside the third, or for all three to be outside each other. As Apollonius discovered, there exist two more circles and (red) that are tangent to all three of the original circles – these are called Apollonian circles. These five circles are separated from each other by six curved triangular regions, each bounded by the arcs from three pairwise-tangent circles. The construction continues by adding six more circles, one in each of these six curved triangles, tangent to its three sides. These in turn create 18 more curved triangles, and the construction continues by again filling these with tangent circles, ad infinitum.
Continued stage by stage in this way, the construction adds new circles at stage , giving a total of circles after stages. In the limit, this set of circles is an Apollonian gasket. In it, each pair of tangent circles has an infinite Pappus chain of circles tangent to both circles in the pair.
The size of each new circle is determined by Descartes' theorem, which states that, for any four mutually tangent circles, the radii of the circles obeys the equation
This equation may have a solution with a negative radius; this means that one of the circles (the one with negative radius) surrounds the other three.
One or two of the initial circles of this construction, or the circles resulting from this construction, can degenerate to a straight line, which can be thought of as a circle with infinite radius. When there are two lines, they must be parallel, and are considered to be tangent at a point at infinity. When the gasket includes two lines on the -axis and one unit above it, and a circle of unit diameter tangent to both lines centered on the -axis, then the circles that are tangent to the -axis are the Ford circles, important in number theory.
The Apollonian gasket has a Hausdorff dimension of about 1.3057. Because it has a well-defined fractional dimension, even though it is not precisely self-similar, it can be thought of as a fractal.
Symmetries
The Möbius transformations of the plane preserve the shapes and tangencies of circles, and therefore preserve the structure of an Apollonian gasket.
Any two triples of mutually tangent circles in an Apollonian gasket may be mapped into each other by a Möbius transformation, and any two Apollonian gaskets may be mapped into each other by a Möbius transformation. In particular, for any two tangent circles in any Ap
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https://en.wikipedia.org/wiki/Wolfgang%20Gr%C3%B6bner
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Wolfgang Gröbner (11 February 1899 – 20 August 1980) was an Austrian mathematician. His name is best known for the Gröbner basis, used for computations in algebraic geometry. However, the theory of Gröbner bases for polynomial rings was developed by his student Bruno Buchberger in 1965, who named them for Gröbner. Gröbner is also known for the Alekseev-Gröbner formula, which was actually proven by him.
Early life
Gröbner was born in Gossensaß, which at that time was in part of the County of Tyrol of the Austro-Hungarian Empire and is now part of Italy.
Gröbner first studied engineering at the University of Technology in Graz, Austria, but switched in 1929 to mathematics.
Career
He wrote his dissertation Ein Beitrag zum Problem der Minimalbasen in 1932 at the University of Vienna; his advisor was Phillip Furtwängler. After his promotion, he did further studies at the University of Göttingen under Emmy Noether, in what is now known as commutative algebra.
Awards
Wilhelm Exner Medal, 1969.
References
1899 births
1980 deaths
20th-century Austrian mathematicians
University of Vienna alumni
Academic staff of the University of Innsbruck
People from Brenner
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https://en.wikipedia.org/wiki/Quasidihedral%20group
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In mathematics, the quasi-dihedral groups, also called semi-dihedral groups, are certain non-abelian groups of order a power of 2. For every positive integer n greater than or equal to 4, there are exactly four isomorphism classes of non-abelian groups of order 2n which have a cyclic subgroup of index 2. Two are well known, the generalized quaternion group and the dihedral group. One of the remaining two groups is often considered particularly important, since it is an example of a 2-group of maximal nilpotency class. In Bertram Huppert's text Endliche Gruppen, this group is called a "Quasidiedergruppe". In Daniel Gorenstein's text, Finite Groups, this group is called the "semidihedral group". Dummit and Foote refer to it as the "quasidihedral group"; we adopt that name in this article. All give the same presentation for this group:
.
The other non-abelian 2-group with cyclic subgroup of index 2 is not given a special name in either text, but referred to as just G or Mm(2). When this group has order 16, Dummit and Foote refer to this group as the "modular group of order 16", as its lattice of subgroups is modular. In this article this group will be called the modular maximal-cyclic group of order . Its presentation is:
.
Both these two groups and the dihedral group are semidirect products of a cyclic group <r> of order 2n−1 with a cyclic group <s> of order 2. Such a non-abelian semidirect product is uniquely determined by an element of order 2 in the group of units of the ring and there are precisely three such elements, , , and , corresponding to the dihedral group, the quasidihedral, and the modular maximal-cyclic group.
The generalized quaternion group, the dihedral group, and the quasidihedral group of order 2n all have nilpotency class n − 1, and are the only isomorphism classes of groups of order 2n with nilpotency class n − 1. The groups of order pn and nilpotency class n − 1 were the beginning of the classification of all p-groups via coclass. The modular maximal-cyclic group of order 2n always has nilpotency class 2. This makes the modular maximal-cyclic group less interesting, since most groups of order pn for large n have nilpotency class 2 and have proven difficult to understand directly.
The generalized quaternion, the dihedral, and the quasidihedral group are the only 2-groups whose derived subgroup has index 4. The Alperin–Brauer–Gorenstein theorem classifies the simple groups, and to a degree the finite groups, with quasidihedral Sylow 2-subgroups.
Examples
The Sylow 2-subgroups of the following groups are quasidihedral:
PSL3(Fq) for q ≡ 3 mod 4,
PSU3(Fq) for q ≡ 1 mod 4,
the Mathieu group M11,
GL2(Fq) for q ≡ 3 mod 4.
References
Finite groups
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https://en.wikipedia.org/wiki/Laver%20table
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In mathematics, Laver tables (named after Richard Laver, who discovered them towards the end of the 1980s in connection with his works on set theory) are tables of numbers that have certain properties of algebraic and combinatorial interest. They occur in the study of racks and quandles.
Definition
For any nonnegative integer n, the n-th Laver table is the 2n × 2n table whose entry in the cell at row p and column q (1 ≤ p,q ≤ 2n) is defined as
where is the unique binary operation that satisfies the following two equations for all p, q in {1,...,2n}:
and
Note: Equation () uses the notation to mean the unique member of {1,...,2n} congruent to x modulo 2n.
Equation () is known as the (left) self-distributive law, and a set endowed with any binary operation satisfying this law is called a shelf. Thus, the n-th Laver table is just the multiplication table for the unique shelf ({1,...,2n}, ) that satisfies Equation ().
Examples: Following are the first five Laver tables, i.e. the multiplication tables for the shelves ({1,...,2n}, ), n = 0, 1, 2, 3, 4:
There is no known closed-form expression to calculate the entries of a Laver table directly, but Patrick Dehornoy provides a simple algorithm for filling out Laver tables.
Properties
For all p, q in {1,...,2n}: .
For all p in {1,...,2n}: is periodic with period πn(p) equal to a power of two.
For all p in {1,...,2n}: is strictly increasing from to .
For all p,q:
Are the first-row periods unbounded?
Looking at just the first row in the n-th Laver table, for n = 0, 1, 2, ..., the entries in each first row are seen to be periodic with a period that's always a power of two, as mentioned in Property 2 above. The first few periods are 1, 1, 2, 4, 4, 8, 8, 8, 8, 16, 16, ... . This sequence is nondecreasing, and in 1995 Richard Laver proved, under the assumption that there exists a rank-into-rank (a large cardinal property), that it actually increases without bound. (It is not known whether this is also provable in ZFC without the additional large-cardinal axiom.) In any case, it grows extremely slowly; Randall Dougherty showed that 32 cannot appear in this sequence (if it ever does) until n > A(9, A(8, A(8, 254))), where A denotes the Ackermann–Péter function.
References
Further reading
.
.
Shelves and the infinite: https://johncarlosbaez.wordpress.com/2016/05/06/shelves-and-the-infinite/
Mathematical logic
Combinatorics
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https://en.wikipedia.org/wiki/Szemer%C3%A9di%E2%80%93Trotter%20theorem
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The Szemerédi–Trotter theorem is a mathematical result in the field of Discrete geometry. It asserts that given points and lines in the Euclidean plane, the number of incidences (i.e., the number of point-line pairs, such that the point lies on the line) is
This bound cannot be improved, except in terms of the implicit constants.
As for the implicit constants, it was shown by János Pach, Radoš Radoičić, Gábor Tardos, and Géza Tóth that the upper bound holds. Since then better constants are known due to better crossing lemma constants; the current best is 2.44. On the other hand, Pach and Tóth showed that the statement does not hold true if one replaces the coefficient 2.5 with 0.42.
An equivalent formulation of the theorem is the following. Given points and an integer , the number of lines which pass through at least of the points is
The original proof of Endre Szemerédi and William T. Trotter was somewhat complicated, using a combinatorial technique known as cell decomposition. Later, László Székely discovered a much simpler proof using the crossing number inequality for graphs. (See below.)
The Szemerédi–Trotter theorem has a number of consequences, including Beck's theorem in incidence geometry and the Erdős-Szemerédi sum-product problem in additive combinatorics.
Proof of the first formulation
We may discard the lines which contain two or fewer of the points, as they can contribute at most incidences to the total number. Thus we may assume that every line contains at least three of the points.
If a line contains points, then it will contain line segments which connect two consecutive points along the line. Because after discarding the two-point lines, it follows that , so the number of these line segments on each line is at least half the number of incidences on that line. Summing over all of the lines, the number of these line segments is again at least half the total number of incidences. Thus if denotes the number of such line segments, it will suffice to show that
Now consider the graph formed by using the points as vertices, and the line segments as edges. Since each line segment lies on one of lines, and any two lines intersect in at most one point, the crossing number of this graph is at most the number of points where two lines intersect, which is at most . The crossing number inequality implies that either , or that . In either case , giving the desired bound
Proof of the second formulation
Since every pair of points can be connected by at most one line, there can be at most lines which can connect at or more points, since . This bound will prove the theorem when is small (e.g. if for some absolute constant ). Thus, we need only consider the case when is large, say .
Suppose that there are m lines that each contain at least points. These lines generate at least incidences, and so by the first formulation of the Szemerédi–Trotter theorem, we have
and so at least one of the statements , or is true
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https://en.wikipedia.org/wiki/Logarithmic%20form
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In algebraic geometry and the theory of complex manifolds, a logarithmic differential form is a differential form with poles of a certain kind. The concept was introduced by Pierre Deligne. In short, logarithmic differentials have the mildest possible singularities needed in order to give information about an open submanifold (the complement of the divisor of poles). (This idea is made precise by several versions of de Rham's theorem discussed below.)
Let X be a complex manifold, D ⊂ X a reduced divisor (a sum of distinct codimension-1 complex subspaces), and ω a holomorphic p-form on X−D. If both ω and dω have a pole of order at most 1 along D, then ω is said to have a logarithmic pole along D. ω is also known as a logarithmic p-form. The p-forms with log poles along D form a subsheaf of the meromorphic p-forms on X, denoted
The name comes from the fact that in complex analysis, ; here is a typical example of a 1-form on the complex numbers C with a logarithmic pole at the origin. Differential forms such as make sense in a purely algebraic context, where there is no analog of the logarithm function.
Logarithmic de Rham complex
Let X be a complex manifold and D a reduced divisor on X. By definition of and the fact that the exterior derivative d satisfies d2 = 0, one has
for every open subset U of X. Thus the logarithmic differentials form a complex of sheaves , known as the logarithmic de Rham complex associated to the divisor D. This is a subcomplex of the direct image , where is the inclusion and is the complex of sheaves of holomorphic forms on X−D.
Of special interest is the case where D has normal crossings: that is, D is locally a sum of codimension-1 complex submanifolds that intersect transversely. In this case, the sheaf of logarithmic differential forms is the subalgebra of generated by the holomorphic differential forms together with the 1-forms for holomorphic functions that are nonzero outside D. Note that
Concretely, if D is a divisor with normal crossings on a complex manifold X, then each point x has an open neighborhood U on which there are holomorphic coordinate functions such that x is the origin and D is defined by the equation for some . On the open set U, sections of are given by
This describes the holomorphic vector bundle on . Then, for any , the vector bundle is the kth exterior power,
The logarithmic tangent bundle means the dual vector bundle to . Explicitly, a section of is a holomorphic vector field on X that is tangent to D at all smooth points of D.
Logarithmic differentials and singular cohomology
Let X be a complex manifold and D a divisor with normal crossings on X. Deligne proved a holomorphic analog of de Rham's theorem in terms of logarithmic differentials. Namely,
where the left side denotes the cohomology of X with coefficients in a complex of sheaves, sometimes called hypercohomology. This follows from the natural inclusion of complexes of sheaves
being a quasi-isomorphism.
Logar
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https://en.wikipedia.org/wiki/Euler%20system
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In mathematics, an Euler system is a collection of compatible elements of Galois cohomology groups indexed by fields. They were introduced by in his work on Heegner points on modular elliptic curves, which was motivated by his earlier paper and the work of . Euler systems are named after Leonhard Euler because the factors relating different elements of an Euler system resemble the Euler factors of an Euler product.
Euler systems can be used to construct annihilators of ideal class groups or Selmer groups, thus giving bounds on their orders, which in turn has led to deep theorems such as the finiteness of some Tate-Shafarevich groups. This led to Karl Rubin's new proof of the main conjecture of Iwasawa theory, considered simpler than the original proof due to Barry Mazur and Andrew Wiles.
Definition
Although there are several definitions of special sorts of Euler system, there seems to be no published definition of an Euler system that covers all known cases. But it is possible to say roughly what an Euler system is, as follows:
An Euler system is given by collection of elements cF. These elements are often indexed by certain number fields F containing some fixed number field K, or by something closely related such as square-free integers. The elements cF are typically elements of some Galois cohomology group such as H1(F, T) where T is a p-adic representation of the absolute Galois group of K.
The most important condition is that the elements cF and cG for two different fields F ⊆ G are related by a simple formula, such as
Here the "Euler factor" P(τ|B;x) is defined to be the element det(1-τx|B) considered as an element of O[x], which when x happens to act on B is not the same as det(1-τx|B) considered as an element of O.
There may be other conditions that the cF have to satisfy, such as congruence conditions.
Kazuya Kato refers to the elements in an Euler system as "arithmetic incarnations of zeta" and describes the property of being an Euler system as "an arithmetic reflection of the fact that these incarnations are related to special values of Euler products".
Examples
Cyclotomic units
For every square-free positive integer n pick an n-th root ζn of 1, with ζmn = ζmζn for m,n coprime. Then the cyclotomic Euler system is the set of numbers
αn = 1 − ζn. These satisfy the relations
modulo all primes above l
where l is a prime not dividing n and Fl is a Frobenius automorphism with Fl(ζn) = ζ.
Kolyvagin used this Euler system to give an elementary proof of the Gras conjecture.
Gauss sums
Elliptic units
Heegner points
Kolyvagin constructed an Euler system from the Heegner points of an elliptic curve, and used this to show that in some cases the Tate-Shafarevich group is finite.
Kato's Euler system
Kato's Euler system consists of certain elements occurring in the algebraic K-theory of modular curves. These elements—named Beilinson elements after Alexander Beilinson who introduced them in —were used by Kazuya Kato in to prove on
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https://en.wikipedia.org/wiki/Sylvester%E2%80%93Gallai%20theorem
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The Sylvester–Gallai theorem in geometry states that every finite set of points in the Euclidean plane has a line that passes through exactly two of the points or a line that passes through all of them. It is named after James Joseph Sylvester, who posed it as a problem in 1893, and Tibor Gallai, who published one of the first proofs of this theorem in 1944.
A line that contains exactly two of a set of points is known as an ordinary line. Another way of stating the theorem is that every finite set of points that is not collinear has an ordinary line. According to a strengthening of the theorem, every finite point set (not all on one line) has at least a linear number of ordinary lines. An algorithm can find an ordinary line in a set of points in time .
History
The Sylvester–Gallai theorem was posed as a problem by . suggests that Sylvester may have been motivated by a related phenomenon in algebraic geometry, in which the inflection points of a cubic curve in the complex projective plane form a configuration of nine points and twelve lines (the Hesse configuration) in which each line determined by two of the points contains a third point. The Sylvester–Gallai theorem implies that it is impossible for all nine of these points to have real coordinates.
claimed to have a short proof of the Sylvester–Gallai theorem, but it was already noted to be incomplete at the time of publication. proved the theorem (and actually a slightly stronger result) in an equivalent formulation, its projective dual. Unaware of Melchior's proof, again stated the conjecture, which was subsequently proved by Tibor Gallai, and soon afterwards by other authors.
In a 1951 review, Erdős called the result "Gallai's theorem", but it was already called the Sylvester–Gallai theorem in a 1954 review by Leonard Blumenthal. It is one of many mathematical topics named after Sylvester.
Equivalent versions
The question of the existence of an ordinary line can also be posed for points in the real projective plane RP2 instead of the Euclidean plane. The projective plane can be formed from the Euclidean plane by adding extra points "at infinity" where lines that are parallel in the Euclidean plane intersect each other, and by adding a single line "at infinity" containing all the added points. However, the additional points of the projective plane cannot help create non-Euclidean finite point sets with no ordinary line, as any finite point set in the projective plane can be transformed into a Euclidean point set with the same combinatorial pattern of point-line incidences. Therefore, any pattern of finitely many intersecting points and lines that exists in one of these two types of plane also exists in the other. Nevertheless, the projective viewpoint allows certain configurations to be described more easily. In particular, it allows the use of projective duality, in which the roles of points and lines in statements of projective geometry can be exchanged for each other. Under pr
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https://en.wikipedia.org/wiki/Statistical%20learning%20theory
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Statistical learning theory is a framework for machine learning drawing from the fields of statistics and functional analysis. Statistical learning theory deals with the statistical inference problem of finding a predictive function based on data. Statistical learning theory has led to successful applications in fields such as computer vision, speech recognition, and bioinformatics.
Introduction
The goals of learning are understanding and prediction. Learning falls into many categories, including supervised learning, unsupervised learning, online learning, and reinforcement learning. From the perspective of statistical learning theory, supervised learning is best understood. Supervised learning involves learning from a training set of data. Every point in the training is an input–output pair, where the input maps to an output. The learning problem consists of inferring the function that maps between the input and the output, such that the learned function can be used to predict the output from future input.
Depending on the type of output, supervised learning problems are either problems of regression or problems of classification. If the output takes a continuous range of values, it is a regression problem. Using Ohm's law as an example, a regression could be performed with voltage as input and current as an output. The regression would find the functional relationship between voltage and current to be , such that
Classification problems are those for which the output will be an element from a discrete set of labels. Classification is very common for machine learning applications. In facial recognition, for instance, a picture of a person's face would be the input, and the output label would be that person's name. The input would be represented by a large multidimensional vector whose elements represent pixels in the picture.
After learning a function based on the training set data, that function is validated on a test set of data, data that did not appear in the training set.
Formal description
Take to be the vector space of all possible inputs, and to be the vector space of all possible outputs. Statistical learning theory takes the perspective that there is some unknown probability distribution over the product space , i.e. there exists some unknown . The training set is made up of samples from this probability distribution, and is notated
Every is an input vector from the training data, and is the output that corresponds to it.
In this formalism, the inference problem consists of finding a function such that . Let be a space of functions called the hypothesis space. The hypothesis space is the space of functions the algorithm will search through. Let be the loss function, a metric for the difference between the predicted value and the actual value . The expected risk is defined to be
The target function, the best possible function that can be chosen, is given by the that satisfies
Because the probability distribution is
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https://en.wikipedia.org/wiki/G%26T
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G&T can mean:
Gin and tonic
Geometry & Topology — a peer-refereed, international mathematics research journal.
Geometry and trigonometry
the Gifted And Talented
a Gifted And Talented program
Generation & Transmission cooperative (wholesale energy provider)
Gramophone & Typewriter Ltd
G&T Crampton, an Irish construction company
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https://en.wikipedia.org/wiki/Stein%20manifold
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In mathematics, in the theory of several complex variables and complex manifolds, a Stein manifold is a complex submanifold of the vector space of n complex dimensions. They were introduced by and named after . A Stein space is similar to a Stein manifold but is allowed to have singularities. Stein spaces are the analogues of affine varieties or affine schemes in algebraic geometry.
Definition
Suppose is a complex manifold of complex dimension and let denote the ring of holomorphic functions on We call a Stein manifold if the following conditions hold:
is holomorphically convex, i.e. for every compact subset , the so-called holomorphically convex hull,
is also a compact subset of .
is holomorphically separable, i.e. if are two points in , then there exists such that
Non-compact Riemann surfaces are Stein manifolds
Let X be a connected, non-compact Riemann surface. A deep theorem of Heinrich Behnke and Stein (1948) asserts that X is a Stein manifold.
Another result, attributed to Hans Grauert and Helmut Röhrl (1956), states moreover that every holomorphic vector bundle on X is trivial. In particular, every line bundle is trivial, so . The exponential sheaf sequence leads to the following exact sequence:
Now Cartan's theorem B shows that , therefore .
This is related to the solution of the second Cousin problem.
Properties and examples of Stein manifolds
The standard complex space is a Stein manifold.
Every domain of holomorphy in is a Stein manifold.
It can be shown quite easily that every closed complex submanifold of a Stein manifold is a Stein manifold, too.
The embedding theorem for Stein manifolds states the following: Every Stein manifold of complex dimension can be embedded into by a biholomorphic proper map.
These facts imply that a Stein manifold is a closed complex submanifold of complex space, whose complex structure is that of the ambient space (because the embedding is biholomorphic).
Every Stein manifold of (complex) dimension n has the homotopy type of an n-dimensional CW-complex.
In one complex dimension the Stein condition can be simplified: a connected Riemann surface is a Stein manifold if and only if it is not compact. This can be proved using a version of the Runge theorem for Riemann surfaces, due to Behnke and Stein.
Every Stein manifold is holomorphically spreadable, i.e. for every point , there are holomorphic functions defined on all of which form a local coordinate system when restricted to some open neighborhood of .
Being a Stein manifold is equivalent to being a (complex) strongly pseudoconvex manifold. The latter means that it has a strongly pseudoconvex (or plurisubharmonic) exhaustive function, i.e. a smooth real function on (which can be assumed to be a Morse function) with , such that the subsets are compact in for every real number . This is a solution to the so-called Levi problem, named after Eugenio Levi (1911). The function invites a generalization of S
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https://en.wikipedia.org/wiki/KSEG%20%28software%29
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KSEG is a free (GPL) interactive geometry software for exploring Euclidean geometry. It was created by Ilya Baran.
It runs on Unix-based platforms. It also compiles and runs on Mac OS X and should run on anything that Qt supports. Additionally, it was also ported to Microsoft Windows.
KSEG is a tool designed to let you easily visualize dynamic properties of compass and straightedge construction and to make geometric exploration as fast and easy as possible. KSEG was inspired by the Geometer's Sketchpad, but it goes beyond the functionality that Sketchpad provides.
KSEG can be used in the classroom, for personal exploration of geometry, or for making high-quality figures for LaTeX.
See also
Kig – KDE 4 / Qt 4 geometry software
References
External links
contains source and windows executable.
KSEG for Mac OS X.
Standalone binary for Mac OS X which also includes Qt.
kseg – freeBSD port.
Free interactive geometry software
Software that uses Qt
Free educational software
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https://en.wikipedia.org/wiki/Sheaf%20cohomology
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In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking, sheaf cohomology describes the obstructions to solving a geometric problem globally when it can be solved locally. The central work for the study of sheaf cohomology is Grothendieck's 1957 Tôhoku paper.
Sheaves, sheaf cohomology, and spectral sequences were introduced by Jean Leray at the prisoner-of-war camp Oflag XVII-A in Austria. From 1940 to 1945, Leray and other prisoners organized a "université en captivité" in the camp.
Leray's definitions were simplified and clarified in the 1950s. It became clear that sheaf cohomology was not only a new approach to cohomology in algebraic topology, but also a powerful method in complex analytic geometry and algebraic geometry. These subjects often involve constructing global functions with specified local properties, and sheaf cohomology is ideally suited to such problems. Many earlier results such as the Riemann–Roch theorem and the Hodge theorem have been generalized or understood better using sheaf cohomology.
Definition
The category of sheaves of abelian groups on a topological space X is an abelian category, and so it makes sense to ask when a morphism f: B → C of sheaves is injective (a monomorphism) or surjective (an epimorphism). One answer is that f is injective (respectively surjective) if and only if the associated homomorphism on stalks Bx → Cx is injective (respectively surjective) for every point x in X. It follows that f is injective if and only if the homomorphism B(U) → C(U) of sections over U is injective for every open set U in X. Surjectivity is more subtle, however: the morphism f is surjective if and only if for every open set U in X, every section s of C over U, and every point x in U, there is an open neighborhood V of x in U such that s restricted to V is the image of some section of B over V. (In words: every section of C lifts locally to sections of B.)
As a result, the question arises: given a surjection B → C of sheaves and a section s of C over X, when is s the image of a section of B over X? This is a model for all kinds of local-vs.-global questions in geometry. Sheaf cohomology gives a satisfactory general answer. Namely, let A be the kernel of the surjection B → C, giving a short exact sequence
of sheaves on X. Then there is a long exact sequence of abelian groups, called sheaf cohomology groups:
where H0(X,A) is the group A(X) of global sections of A on X. For example, if the group H1(X,A) is zero, then this exact sequence implies that every global section of C lifts to a global section of B. More broadly, the exact sequence makes knowledge of higher cohomology groups a fundamental tool in aiming to understand sections of sheaves.
Grothendieck's definition of sheaf cohomology, now standard, uses the language of homological algebra. The essential point is to fix a topological space X and think of cohomolog
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https://en.wikipedia.org/wiki/Japanese%20mathematics
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denotes a distinct kind of mathematics which was developed in Japan during the Edo period (1603–1867). The term wasan, from wa ("Japanese") and san ("calculation"), was coined in the 1870s and employed to distinguish native Japanese mathematical theory from Western mathematics (洋算 yōsan).
In the history of mathematics, the development of wasan falls outside the Western realm. At the beginning of the Meiji period (1868–1912), Japan and its people opened themselves to the West. Japanese scholars adopted Western mathematical technique, and this led to a decline of interest in the ideas used in wasan.
History
The Japanese mathematical schema evolved during a period when Japan's people were isolated from European influences, but instead borrowed from ancient mathematical texts written in China, including those from the Yuan dynasty and earlier. The Japanese mathematicians Yoshida Shichibei Kōyū, Imamura Chishō, and Takahara Kisshu are among the earliest known Japanese mathematicians. They came to be known to their contemporaries as "the Three Arithmeticians".
Yoshida was the author of the oldest extant Japanese mathematical text, the 1627 work called Jinkōki. The work dealt with the subject of soroban arithmetic, including square and cube root operations. Yoshida's book significantly inspired a new generation of mathematicians, and redefined the Japanese perception of educational enlightenment, which was defined in the Seventeen Article Constitution as "the product of earnest meditation".
Seki Takakazu founded enri (円理: circle principles), a mathematical system with the same purpose as calculus at a similar time to calculus's development in Europe. However Seki's investigations did not proceed from the same foundations as those used in Newton's studies in Europe.
Mathematicians like Takebe Katahiro played and important role in developing Enri (" circle principle"), a crude analog to the Western calculus. He obtained power series expansion of in 1722, 15 years earlier than Euler. He used Richardson extrapolation in 1695, about 200 years earlier than Richardson. He also computed 41 digits of π, based on polygon approximation and Richardson extrapolation.
Select mathematicians
The following list encompasses mathematicians whose work was derived from wasan.
Yoshida Mitsuyoshi (1598–1672)
Seki Takakazu (1642–1708)
Takebe Kenkō (1664–1739)
Matsunaga Ryohitsu (fl. 1718-1749)
Kurushima Kinai (d. 1757)
Arima Raido (1714–1783)
Fujita Sadasuke (1734-1807)
Ajima Naonobu (1739–1783)
Aida Yasuaki (1747–1817)
Sakabe Kōhan (1759–1824)
Fujita Kagen (1765–1821)
Hasegawa Ken (c. 1783-1838)
Wada Nei (1787–1840)
Shiraishi Chochu (1796–1862)
Koide Shuke (1797–1865)
Omura Isshu (1824–1871)
See also
Japanese theorem for cyclic polygons
Japanese theorem for cyclic quadrilaterals
Sangaku, the custom of presenting mathematical problems, carved in wood tablets, to the public in Shinto shrines
Soroban, a Japanese abacus
:Category:Japanese mathem
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https://en.wikipedia.org/wiki/Completely%20multiplicative%20function
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In number theory, functions of positive integers which respect products are important and are called completely multiplicative functions or totally multiplicative functions. A weaker condition is also important, respecting only products of coprime numbers, and such functions are called multiplicative functions. Outside of number theory, the term "multiplicative function" is often taken to be synonymous with "completely multiplicative function" as defined in this article.
Definition
A completely multiplicative function (or totally multiplicative function) is an arithmetic function (that is, a function whose domain is the natural numbers), such that f(1) = 1 and f(ab) = f(a)f(b) holds for all positive integers a and b.
Without the requirement that f(1) = 1, one could still have f(1) = 0, but then f(a) = 0 for all positive integers a, so this is not a very strong restriction.
The definition above can be rephrased using the language of algebra: A completely multiplicative function is a homomorphism from the monoid (that is, the positive integers under multiplication) to some other monoid.
Examples
The easiest example of a completely multiplicative function is a monomial with leading coefficient 1: For any particular positive integer n, define f(a) = an. Then f(bc) = (bc)n = bncn = f(b)f(c), and f(1) = 1n = 1.
The Liouville function is a non-trivial example of a completely multiplicative function as are Dirichlet characters, the Jacobi symbol and the Legendre symbol.
Properties
A completely multiplicative function is completely determined by its values at the prime numbers, a consequence of the fundamental theorem of arithmetic. Thus, if n is a product of powers of distinct primes, say n = pa qb ..., then f(n) = f(p)a f(q)b ...
While the Dirichlet convolution of two multiplicative functions is multiplicative, the Dirichlet convolution of two completely multiplicative functions need not be completely multiplicative.
There are a variety of statements about a function which are equivalent to it being completely multiplicative. For example, if a function f is multiplicative then it is completely multiplicative if and only if its Dirichlet inverse is where is the Möbius function.
Completely multiplicative functions also satisfy a distributive law. If f is completely multiplicative then
where * represents the Dirichlet product and represents pointwise multiplication. One consequence of this is that for any completely multiplicative function f one has
which can be deduced from the above by putting both , where is the constant function.
Here is the divisor function.
Proof of distributive property
Dirichlet series
The L-function of completely (or totally) multiplicative Dirichlet series satisfies
which means that the sum all over the natural numbers is equal to the product all over the prime numbers.
See also
Arithmetic function
Dirichlet L-function
Dirichlet series
Multiplicative function
References
Multiplicative functions
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https://en.wikipedia.org/wiki/Fundamental%20theorem%20of%20curves
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In differential geometry, the fundamental theorem of space curves states that every regular curve in three-dimensional space, with non-zero curvature, has its shape (and size or scale) completely determined by its curvature and torsion.
Use
A curve can be described, and thereby defined, by a pair of scalar fields: curvature and torsion , both of which depend on some parameter which parametrizes the curve but which can ideally be the arc length of the curve. From just the curvature and torsion, the vector fields for the tangent, normal, and binormal vectors can be derived using the Frenet–Serret formulas. Then, integration of the tangent field (done numerically, if not analytically) yields the curve.
Congruence
If a pair of curves are in different positions but have the same curvature and torsion, then they are congruent to each other.
See also
Differential geometry of curves
Gaussian curvature
References
Further reading
Theorems about curves
Theorems in differential geometry
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https://en.wikipedia.org/wiki/L%C3%A9vy%20C%20curve
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In mathematics, the Lévy C curve is a self-similar fractal curve that was first described and whose differentiability properties were analysed by Ernesto Cesàro in 1906 and Georg Faber in 1910, but now bears the name of French mathematician Paul Lévy, who was the first to describe its self-similarity properties as well as to provide a geometrical construction showing it as a representative curve in the same class as the Koch curve. It is a special case of a period-doubling curve, a de Rham curve.
L-system construction
If using a Lindenmayer system then the construction of the C curve starts with a straight line. An isosceles triangle with angles of 45°, 90° and 45° is built using this line as its hypotenuse. The original line is then replaced by the other two sides of this triangle.
At the second stage, the two new lines each form the base for another right-angled isosceles triangle, and are replaced by the other two sides of their respective triangle. So, after two stages, the curve takes the appearance of three sides of a rectangle with the same length as the original line, but only half as wide.
At each subsequent stage, each straight line segment in the curve is replaced by the other two sides of a right-angled isosceles triangle built on it. After n stages the curve consists of 2n line segments, each of which is smaller than the original line by a factor of 2n/2.
This L-system can be described as follows:
where "" means "draw forward", "+" means "turn clockwise 45°", and "−" means "turn anticlockwise 45°".
The fractal curve that is the limit of this "infinite" process is the Lévy C curve. It takes its name from its resemblance to a highly ornamented version of the letter "C". The curve resembles the finer details of the Pythagoras tree.
The Hausdorff dimension of the C curve equals 2 (it contains open sets), whereas the boundary has dimension about 1.9340 .
Variations
The standard C curve is built using 45° isosceles triangles. Variations of the C curve can be constructed by using isosceles triangles with angles other than 45°. As long as the angle is less than 60°, the new lines introduced at each stage are each shorter than the lines that they replace, so the construction process tends towards a limit curve. Angles less than 45° produce a fractal that is less tightly "curled".
IFS construction
If using an iterated function system (IFS, or the chaos game IFS-method actually), then the construction of the C curve is a bit easier. It will need a set of two "rules" which are: Two points in a plane (the translators), each associated with a scale factor of 1/. The first rule is a rotation of 45° and the second −45°. This set will iterate a point [x, y] from randomly choosing any of the two rules and use the parameters associated with the rule to scale/rotate and translate the point using a 2D-transform function.
Put into formulae:
from the initial set of points .
Sample Implementation of Levy C Curve
// Java Sample Implementation
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https://en.wikipedia.org/wiki/Einstein%20tensor
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In differential geometry, the Einstein tensor (named after Albert Einstein; also known as the trace-reversed Ricci tensor) is used to express the curvature of a pseudo-Riemannian manifold. In general relativity, it occurs in the Einstein field equations for gravitation that describe spacetime curvature in a manner that is consistent with conservation of energy and momentum.
Definition
The Einstein tensor is a tensor of order 2 defined over pseudo-Riemannian manifolds. In index-free notation it is defined as
where is the Ricci tensor, is the metric tensor and is the scalar curvature, which is computed as the trace of the Ricci Tensor by . In component form, the previous equation reads as
The Einstein tensor is symmetric
and, like the on shell stress–energy tensor, has zero divergence:
Explicit form
The Ricci tensor depends only on the metric tensor, so the Einstein tensor can be defined directly with just the metric tensor. However, this expression is complex and rarely quoted in textbooks. The complexity of this expression can be shown using the formula for the Ricci tensor in terms of Christoffel symbols:
where is the Kronecker tensor and the Christoffel symbol is defined as
and terms of the form represent its partial derivative in the μ-direction, i.e.:
Before cancellations, this formula results in individual terms. Cancellations bring this number down somewhat.
In the special case of a locally inertial reference frame near a point, the first derivatives of the metric tensor vanish and the component form of the Einstein tensor is considerably simplified:
where square brackets conventionally denote antisymmetrization over bracketed indices, i.e.
Trace
The trace of the Einstein tensor can be computed by contracting the equation in the definition with the metric tensor . In dimensions (of arbitrary signature):
Therefore, in the special case of dimensions, . That is, the trace of the Einstein tensor is the negative of the Ricci tensor's trace. Thus, another name for the Einstein tensor is the trace-reversed Ricci tensor. This case is especially relevant in the theory of general relativity.
Use in general relativity
The Einstein tensor allows the Einstein field equations to be written in the concise form:
where is the cosmological constant and is the Einstein gravitational constant.
From the explicit form of the Einstein tensor, the Einstein tensor is a nonlinear function of the metric tensor, but is linear in the second partial derivatives of the metric. As a symmetric order-2 tensor, the Einstein tensor has 10 independent components in a 4-dimensional space. It follows that the Einstein field equations are a set of 10 quasilinear second-order partial differential equations for the metric tensor.
The contracted Bianchi identities can also be easily expressed with the aid of the Einstein tensor:
The (contracted) Bianchi identities automatically ensure the covariant conservation of the stress–energy tensor in c
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https://en.wikipedia.org/wiki/Super-Poincar%C3%A9%20algebra
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In theoretical physics, a super-Poincaré algebra is an extension of the Poincaré algebra to incorporate supersymmetry, a relation between bosons and fermions. They are examples of supersymmetry algebras (without central charges or internal symmetries), and are Lie superalgebras. Thus a super-Poincaré algebra is a Z2-graded vector space with a graded Lie bracket such that the even part is a Lie algebra containing the Poincaré algebra, and the odd part is built from spinors on which there is an anticommutation relation with values in the even part.
Informal sketch
The Poincaré algebra describes the isometries of Minkowski spacetime. From the representation theory of the Lorentz group, it is known that the Lorentz group admits two inequivalent complex spinor representations, dubbed and . Taking their tensor product, one obtains ; such decompositions of tensor products of representations into direct sums is given by the Littlewood–Richardson rule.
Normally, one treats such a decomposition as relating to specific particles: so, for example, the pion, which is a chiral vector particle, is composed of a quark-anti-quark pair. However, one could also identify with Minkowski spacetime itself. This leads to a natural question: if Minkowski space-time belongs to the adjoint representation, then can Poincaré symmetry be extended to the fundamental representation? Well, it can: this is exactly the super-Poincaré algebra. There is a corresponding experimental question: if we live in the adjoint representation, then where is the fundamental representation hiding? This is the program of supersymmetry, which has not been found experimentally.
History
The super-Poincaré algebra was first proposed in the context of the Haag–Łopuszański–Sohnius theorem, as a means of avoiding the conclusions of the Coleman–Mandula theorem. That is, the Coleman–Mandula theorem is a no-go theorem that states that the Poincaré algebra cannot be extended with additional symmetries that might describe the internal symmetries of the observed physical particle spectrum. However, the Coleman–Mandula theorem assumed that the algebra extension would be by means of a commutator; this assumption, and thus the theorem, can be avoided by considering the anti-commutator, that is, by employing anti-commuting Grassmann numbers. The proposal was to consider a supersymmetry algebra, defined as the semidirect product of a central extension of the super-Poincaré algebra by a compact Lie algebra of internal symmetries.
Definition
The simplest supersymmetric extension of the Poincaré algebra contains two Weyl spinors with the following anti-commutation relation:
and all other anti-commutation relations between the Qs and Ps vanish. The operators are known as supercharges. In the above expression are the generators of translation and are the Pauli matrices. The index runs over the values A dot is used over the index to remind that this index transforms according to the inequivalent conjugate
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https://en.wikipedia.org/wiki/Semigroup%20action
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In algebra and theoretical computer science, an action or act of a semigroup on a set is a rule which associates to each element of the semigroup a transformation of the set in such a way that the product of two elements of the semigroup (using the semigroup operation) is associated with the composite of the two corresponding transformations. The terminology conveys the idea that the elements of the semigroup are acting as transformations of the set. From an algebraic perspective, a semigroup action is a generalization of the notion of a group action in group theory. From the computer science point of view, semigroup actions are closely related to automata: the set models the state of the automaton and the action models transformations of that state in response to inputs.
An important special case is a monoid action or act, in which the semigroup is a monoid and the identity element of the monoid acts as the identity transformation of a set. From a category theoretic point of view, a monoid is a category with one object, and an act is a functor from that category to the category of sets. This immediately provides a generalization to monoid acts on objects in categories other than the category of sets.
Another important special case is a transformation semigroup. This is a semigroup of transformations of a set, and hence it has a tautological action on that set. This concept is linked to the more general notion of a semigroup by an analogue of Cayley's theorem.
(A note on terminology: the terminology used in this area varies, sometimes significantly, from one author to another. See the article for details.)
Formal definitions
Let S be a semigroup. Then a (left) semigroup action (or act) of S is a set X together with an operation which is compatible with the semigroup operation ∗ as follows:
for all s, t in S and x in X, .
This is the analogue in semigroup theory of a (left) group action, and is equivalent to a semigroup homomorphism into the set of functions on X. Right semigroup actions are defined in a similar way using an operation satisfying .
If M is a monoid, then a (left) monoid action (or act) of M is a (left) semigroup action of M with the additional property that
for all x in X: e • x = x
where e is the identity element of M. This correspondingly gives a monoid homomorphism. Right monoid actions are defined in a similar way. A monoid M with an action on a set is also called an operator monoid.
A semigroup action of S on X can be made into monoid act by adjoining an identity to the semigroup and requiring that it acts as the identity transformation on X.
Terminology and notation
If S is a semigroup or monoid, then a set X on which S acts as above (on the left, say) is also known as a (left) S-act, S-set, S-action, S-operand, or left act over S. Some authors do not distinguish between semigroup and monoid actions, by regarding the identity axiom () as empty when there is no identity element, or by using the term unitary S-act
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https://en.wikipedia.org/wiki/Orthogonal%20complement
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In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace W of a vector space V equipped with a bilinear form B is the set W⊥ of all vectors in V that are orthogonal to every vector in W. Informally, it is called the perp, short for perpendicular complement. It is a subspace of V.
Example
Let be the vector space equipped with the usual dot product (thus making it an inner product space), and let with
then its orthogonal complement can also be defined as being
The fact that every column vector in is orthogonal to every column vector in can be checked by direct computation. The fact that the spans of these vectors are orthogonal then follows by bilinearity of the dot product. Finally, the fact that these spaces are orthogonal complements follows from the dimension relationships given below.
General bilinear forms
Let be a vector space over a field equipped with a bilinear form We define to be left-orthogonal to , and to be right-orthogonal to when For a subset of define the left orthogonal complement to be
There is a corresponding definition of right orthogonal complement. For a reflexive bilinear form, where implies for all and in the left and right complements coincide. This will be the case if is a symmetric or an alternating form.
The definition extends to a bilinear form on a free module over a commutative ring, and to a sesquilinear form extended to include any free module over a commutative ring with conjugation.
Properties
An orthogonal complement is a subspace of ;
If then ;
The radical of is a subspace of every orthogonal complement;
;
If is non-degenerate and is finite-dimensional, then
If are subspaces of a finite-dimensional space and then
Inner product spaces
This section considers orthogonal complements in an inner product space
Two vectors and are called if which happens if and only if for all scalars
If is any subset of an inner product space then its is the vector subspace
which is always a closed subset of that satisfies and also and
If is a vector subspace of an inner product space then
If is a closed vector subspace of a Hilbert space then
where is called the of into and and it indicates that is a complemented subspace of with complement
Properties
The orthogonal complement is always closed in the metric topology. In finite-dimensional spaces, that is merely an instance of the fact that all subspaces of a vector space are closed. In infinite-dimensional Hilbert spaces, some subspaces are not closed, but all orthogonal complements are closed. If is a vector subspace of an inner product space the orthogonal complement of the orthogonal complement of is the closure of that is,
Some other useful properties that always hold are the following. Let be a Hilbert space and let and be its linear subspaces. Then:
;
if then ;
;
;
if is a closed linear subspace of then ;
if is a
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https://en.wikipedia.org/wiki/Simple%20polygon
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In geometry, a simple polygon is a polygon that does not intersect itself and has no holes. That is, it is a piecewise-linear Jordan curve consisting of finitely many line segments. These polygons include as special cases the convex polygons, star-shaped polygons, and monotone polygons.
The sum of external angles of a simple polygon is . Every simple polygon with sides can be triangulated by of its diagonals, and by the art gallery theorem its interior is visible from some of its vertices.
Simple polygons are commonly seen as the input to computational geometry problems, including point in polygon testing, area computation, the convex hull of a simple polygon, triangulation, and Euclidean shortest paths.
Other constructions in geometry related to simple polygons include Schwarz–Christoffel mapping, used to find conformal maps involving simple polygons, polygonalization of point sets, constructive solid geometry formulas for polygons, and visibility graphs of polygons.
Definitions
A simple polygon is a closed curve in the Euclidean plane consisting of straight line segments, meeting end-to-end to form a polygonal chain. Other than the shared endpoints of consecutive line segments in this chain, no two of the line segments may intersect each other. The qualifier simple is sometimes omitted, with the word polygon assumed to mean a simple polygon.
The line segments that form a polygon are called its edges or sides. An endpoint of a segment is called a vertex (plural: vertices) or a corner. Edges and vertices are more formal, but may be ambiguous in contexts that also involve the edges and vertices of a graph; the more colloquial terms sides and corners can be used to avoid this ambiguity. Exactly two edges meet at each vertex, and the number of edges always equals the number of vertices. Some sources allow two line segments to form a straight angle (180°), while others disallow this, instead requiring collinear segments of a closed polygonal chain to be merged into a single longer side. Two vertices are neighbors if they are the two endpoints of one of the sides of the polygon.
Simple polygons are sometimes called Jordan polygons, because they are Jordan curves; the Jordan curve theorem can be used to prove that such a polygon divides the plane into two regions. Indeed, Camille Jordan's original proof of this theorem took the special case of simple polygons (stated without proof) as its starting point. The region inside the polygon (its interior) forms a bounded set topologically equivalent to an open disk by the Jordan–Schönflies theorem, with a finite but nonzero area. The polygon itself is topologically equivalent to a circle, and the region outside (the exterior) is an unbounded connected open set, with infinite area. Although the formal definition of a simple polygon is typically as a system of line segments, it is also possible (and common in informal usage) to define a simple polygon as a closed set in the plane, the union of these l
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https://en.wikipedia.org/wiki/Sun%20Zhihong
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Sun Zhihong (, born October 16, 1965) is a Chinese mathematician, working primarily on number theory, combinatorics, and graph theory.
Sun and his twin brother Sun Zhiwei proved a theorem about what are now known as the Wall–Sun–Sun primes that guided the search for counterexamples to Fermat's Last Theorem.
External links
Zhi-Hong Sun's homepage
1965 births
Living people
Mathematicians from Jiangsu
20th-century Chinese mathematicians
21st-century Chinese mathematicians
Number theorists
Academic staff of Huaiyin Normal University
Scientists from Huai'an
Educators from Huai'an
Chinese twins
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https://en.wikipedia.org/wiki/Sun%20Zhiwei
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Sun Zhiwei (, born October 16, 1965) is a Chinese mathematician, working primarily in number theory, combinatorics, and group theory. He is a professor at Nanjing University.
Biography
Sun Zhiwei was born in Huai'an, Jiangsu. Sun and his twin brother Sun Zhihong proved a theorem about what are now known as the Wall–Sun–Sun primes.
Sun proved Sun's curious identity in 2002. In 2003, he presented a unified approach to three topics of Paul Erdős in combinatorial number theory: covering systems, restricted sumsets, and zero-sum problems or EGZ Theorem.
With Stephen Redmond, he posed the Redmond–Sun conjecture in 2006.
In 2013, he published a paper containing many conjectures on primes, one of which states that for any positive integer there are consecutive primes not exceeding such that , where denotes the -th prime.
He is the Editor-in-Chief of the Journal of Combinatorics and Number Theory.
Notes
External links
Zhi-Wei Sun's homepage
1965 births
20th-century Chinese mathematicians
21st-century Chinese mathematicians
Mathematicians from Jiangsu
Combinatorialists
Living people
Academic staff of Nanjing University
Number theorists
Scientists from Huai'an
Squares in number theory
Educators from Huai'an
Chinese twins
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https://en.wikipedia.org/wiki/Simplicial%20homology
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In algebraic topology, simplicial homology is the sequence of homology groups of a simplicial complex. It formalizes the idea of the number of holes of a given dimension in the complex. This generalizes the number of connected components (the case of dimension 0).
Simplicial homology arose as a way to study topological spaces whose building blocks are n-simplices, the n-dimensional analogs of triangles. This includes a point (0-simplex), a line segment (1-simplex), a triangle (2-simplex) and a tetrahedron (3-simplex). By definition, such a space is homeomorphic to a simplicial complex (more precisely, the geometric realization of an abstract simplicial complex). Such a homeomorphism is referred to as a triangulation of the given space. Many topological spaces of interest can be triangulated, including every smooth manifold (Cairns and Whitehead).
Simplicial homology is defined by a simple recipe for any abstract simplicial complex. It is a remarkable fact that simplicial homology only depends on the associated topological space. As a result, it gives a computable way to distinguish one space from another.
Definitions
Orientations
A key concept in defining simplicial homology is the notion of an orientation of a simplex. By definition, an orientation of a k-simplex is given by an ordering of the vertices, written as (), with the rule that two orderings define the same orientation if and only if they differ by an even permutation. Thus every simplex has exactly two orientations, and switching the order of two vertices changes an orientation to the opposite orientation. For example, choosing an orientation of a 1-simplex amounts to choosing one of the two possible directions, and choosing an orientation of a 2-simplex amounts to choosing what "counterclockwise" should mean.
Chains
Let be a simplicial complex. A simplicial -chain is a finite formal sum
where each is an integer and is an oriented -simplex. In this definition, we declare that each oriented simplex is equal to the negative of the simplex with the opposite orientation. For example,
The group of -chains on is written . This is a free abelian group which has a basis in one-to-one correspondence with the set of -simplices in . To define a basis explicitly, one has to choose an orientation of each simplex. One standard way to do this is to choose an ordering of all the vertices and give each simplex the orientation corresponding to the induced ordering of its vertices.
Boundaries and cycles
Let be an oriented -simplex, viewed as a basis element of . The boundary operator
is the homomorphism defined by:
where the oriented simplex
is the face of , obtained by deleting its vertex.
In , elements of the subgroup
are referred to as cycles, and the subgroup
is said to consist of boundaries.
Boundaries of boundaries
Because , where is the second face removed, . In geometric terms, this says that the boundary of anything has no boundary. Equivalently, the abelian groups
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https://en.wikipedia.org/wiki/Basic%20Linear%20Algebra%20Subprograms
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Basic Linear Algebra Subprograms (BLAS) is a specification that prescribes a set of low-level routines for performing common linear algebra operations such as vector addition, scalar multiplication, dot products, linear combinations, and matrix multiplication. They are the de facto standard low-level routines for linear algebra libraries; the routines have bindings for both C ("CBLAS interface") and Fortran ("BLAS interface"). Although the BLAS specification is general, BLAS implementations are often optimized for speed on a particular machine, so using them can bring substantial performance benefits. BLAS implementations will take advantage of special floating point hardware such as vector registers or SIMD instructions.
It originated as a Fortran library in 1979 and its interface was standardized by the BLAS Technical (BLAST) Forum, whose latest BLAS report can be found on the netlib website. This Fortran library is known as the reference implementation (sometimes confusingly referred to as the BLAS library) and is not optimized for speed but is in the public domain.
Most computing libraries that offer linear algebra routines conform to common BLAS user interface command structures, thus queries to those libraries (and the associated results) are often portable between BLAS library branches, such as cuBLAS (nvidia GPU, GPGPU), rocBLAS (amd GPU, GPGP), and OpenBLAS. This interoperability is then the basis of functioning homogenous code implementations between heterzygous cascades of computing architectures (such as those found in some advanced clustering implementations). Examples of CPU-based BLAS library branches include: OpenBLAS, BLIS (BLAS-like Library Instantiation Software), Arm Performance Libraries, ATLAS, and Intel Math Kernel Library (iMKL). AMD maintains a fork of BLIS that is optimized for the AMD platform, although it is unclear whether integrated ombudsmen resources are present in that particular software-hardware implementation. ATLAS is a portable library that automatically optimizes itself for an arbitrary architecture. iMKL is a freeware and proprietary vendor library optimized for x86 and x86-64 with a performance emphasis on Intel processors. OpenBLAS is an open-source library that is hand-optimized for many of the popular architectures. The LINPACK benchmarks rely heavily on the BLAS routine gemm for its performance measurements.
Many numerical software applications use BLAS-compatible libraries to do linear algebra computations, including LAPACK, LINPACK, Armadillo, GNU Octave, Mathematica, MATLAB, NumPy, R, Julia and Lisp-Stat.
Background
With the advent of numerical programming, sophisticated subroutine libraries became useful. These libraries would contain subroutines for common high-level mathematical operations such as root finding, matrix inversion, and solving systems of equations. The language of choice was FORTRAN. The most prominent numerical programming library was IBM's Scientific Subroutine Package (SSP).
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https://en.wikipedia.org/wiki/Residual%20strength
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Residual strength is the load or force (usually mechanical) that a damaged object or material can still carry without failing. Material toughness, fracture size and geometry as well as its orientation all contribute to residual strength.
References
Materials science
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https://en.wikipedia.org/wiki/GEMM
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GEMM may refer to:
General matrix multiply gemm, one of the Basic Linear Algebra Subprograms
Genetically engineered mouse model
Gilt-edged market maker
Global Electronic Music Marketplace, a former online music market
CFU-GEMM, granulocyte-erythrocyte-monocyte-megakaryocyte colony forming unit
See also
Gem (disambiguation)
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https://en.wikipedia.org/wiki/Associated%20Legendre%20polynomials
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In mathematics, the associated Legendre polynomials are the canonical solutions of the general Legendre equation
or equivalently
where the indices ℓ and m (which are integers) are referred to as the degree and order of the associated Legendre polynomial respectively. This equation has nonzero solutions that are nonsingular on only if ℓ and m are integers with 0 ≤ m ≤ ℓ, or with trivially equivalent negative values. When in addition m is even, the function is a polynomial. When m is zero and ℓ integer, these functions are identical to the Legendre polynomials. In general, when ℓ and m are integers, the regular solutions are sometimes called "associated Legendre polynomials", even though they are not polynomials when m is odd. The fully general class of functions with arbitrary real or complex values of ℓ and m are Legendre functions. In that case the parameters are usually labelled with Greek letters.
The Legendre ordinary differential equation is frequently encountered in physics and other technical fields. In particular, it occurs when solving Laplace's equation (and related partial differential equations) in spherical coordinates. Associated Legendre polynomials play a vital role in the definition of spherical harmonics.
Definition for non-negative integer parameters and
These functions are denoted , where the superscript indicates the order and not a power of P. Their most straightforward definition is in terms
of derivatives of ordinary Legendre polynomials (m ≥ 0)
The factor in this formula is known as the Condon–Shortley phase. Some authors omit it. That the functions described by this equation satisfy the general Legendre differential equation with the indicated values of the parameters ℓ and m follows by differentiating m times the Legendre equation for :
Moreover, since by Rodrigues' formula,
the P can be expressed in the form
This equation allows extension of the range of m to: . The definitions of , resulting from this expression by substitution of , are proportional. Indeed, equate the coefficients of equal powers on the left and right hand side of
then it follows that the proportionality constant is
so that
Alternative notations
The following alternative notations are also used in literature:
Closed Form
The Associated Legendre Polynomial can also be written as:
with simple monomials and the generalized form of the binomial coefficient.
Orthogonality
The associated Legendre polynomials are not mutually orthogonal in general. For example, is not orthogonal to . However, some subsets are orthogonal. Assuming 0 ≤ m ≤ ℓ, they satisfy the orthogonality condition for fixed m:
Where is the Kronecker delta.
Also, they satisfy the orthogonality condition for fixed :
Negative and/or negative
The differential equation is clearly invariant under a change in sign of m.
The functions for negative m were shown above to be proportional to those of positive m:
(This followed from the Rodrigues' formula definition. T
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https://en.wikipedia.org/wiki/Mark%20Kac
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Mark Kac ( ; Polish: Marek Kac; August 3, 1914 – October 26, 1984) was a Polish American mathematician. His main interest was probability theory. His question, "Can one hear the shape of a drum?" set off research into spectral theory, the idea of understanding the extent to which the spectrum allows one to read back the geometry. (In the end, the answer was "no", in general.)
Biography
He was born to a Polish-Jewish family; their town, Kremenets (Polish: "Krzemieniec"), changed hands from the Russian Empire (by then Soviet Ukraine) to Poland after the Peace of Riga, when Kac was a child.
Kac completed his Ph.D. in mathematics at the Polish University of Lwów in 1937 under the direction of Hugo Steinhaus. While there, he was a member of the Lwów School of Mathematics. After receiving his degree, he began to look for a position abroad, and in 1938 was granted a scholarship from the Parnas Foundation, which enabled him to go work in the United States. He arrived in New York City in November 1938.
With the onset of World War II in Europe, Kac was able to stay in America, while his parents and brother, who had remained in Kremenets, were murdered by the Germans in mass executions in August 1942.
From 1939 to 1961, Kac taught at Cornell University, first as an instructor, then from 1943 as an assistant professor and from 1947 as a full professor. While there, he became a naturalized US citizen in 1943. From 1943 to 1945, he also worked in the MIT Radiation Laboratory, together with George Uhlenbeck. During the 1951–1952 academic year, Kac was on sabbatical at the Institute for Advanced Study. In 1952, Kac, with Theodore H. Berlin, introduced the spherical model of a ferromagnet (a variant of the Ising model) and, with J. C. Ward, found an exact solution of the Ising model using a combinatorial method. In 1961, Kac left Cornell and went to The Rockefeller University in New York City. In the early 1960s, he worked with George Uhlenbeck and P. C. Hemmer on the mathematics of a van der Waals gas. After twenty years at Rockefeller, he moved to the University of Southern California where he spent the rest of his career.
Work
In his 1966 article titled "Can one hear the shape of the drum", Kac asked whether the geometric shape of a drum is uniquely defined by its sound. The answer was negative, meaning two different resonators can have identical set of eigenfrequencies.
In 1956, he introduced a simplified mathematical model known as the Kac ring, which features the emergence of macroscopic irreversibility from completely time-symmetric microscopic laws. Using the model as an analogy to molecular motion, he provided an explanation for Loschmidt's paradox.
Reminiscences
His definition of a profound truth. "A truth is a statement whose negation is false. A profound truth is a truth whose negation is also a profound truth." (Also attributed to Niels Bohr)
He preferred to work on results that were robust, meaning that they were true under many different as
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https://en.wikipedia.org/wiki/Categorical%20logic
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Categorical logic is the branch of mathematics in which tools and concepts from category theory are applied to the study of mathematical logic. It is also notable for its connections to theoretical computer science.
In broad terms, categorical logic represents both syntax and semantics by a category, and an interpretation by a functor. The categorical framework provides a rich conceptual background for logical and type-theoretic constructions. The subject has been recognisable in these terms since around 1970.
Overview
There are three important themes in the categorical approach to logic:
Categorical semantics Categorical logic introduces the notion of structure valued in a category C with the classical model theoretic notion of a structure appearing in the particular case where C is the category of sets and functions. This notion has proven useful when the set-theoretic notion of a model lacks generality and/or is inconvenient. R.A.G. Seely's modeling of various impredicative theories, such as System F, is an example of the usefulness of categorical semantics.
It was found that the connectives of pre-categorical logic were more clearly understood using the concept of adjoint functor, and that the quantifiers were also best understood using adjoint functors.
Internal languages This can be seen as a formalization and generalization of proof by diagram chasing. One defines a suitable internal language naming relevant constituents of a category, and then applies categorical semantics to turn assertions in a logic over the internal language into corresponding categorical statements. This has been most successful in the theory of toposes, where the internal language of a topos together with the semantics of intuitionistic higher-order logic in a topos enables one to reason about the objects and morphisms of a topos "as if they were sets and functions". This has been successful in dealing with toposes that have "sets" with properties incompatible with classical logic. A prime example is Dana Scott's model of untyped lambda calculus in terms of objects that retract onto their own function space. Another is the Moggi–Hyland model of system F by an internal full subcategory of the effective topos of Martin Hyland.
Term-model constructions In many cases, the categorical semantics of a logic provide a basis for establishing a correspondence between theories in the logic and instances of an appropriate kind of category. A classic example is the correspondence between theories of βη-equational logic over simply typed lambda calculus and Cartesian closed categories. Categories arising from theories via term-model constructions can usually be characterized up to equivalence by a suitable universal property. This has enabled proofs of meta-theoretical properties of some logics by means of an appropriate categorical algebra. For instance, Freyd gave a proof of the disjunction and existence properties of intuitionistic logic this way.
See also
History of
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https://en.wikipedia.org/wiki/Ranked%20list%20of%20French%20regions
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The following are ranked lists of French regions.
Population figures are from the 2016 census, with the exception of Mayotte, whose statistics are as of 2017.
Region boundaries are as of 2018.
By population
These figures are from the census in 2016. Statistics for Mayotte are from 2017.
By area
The total area of France is 632,734 km², of which 543,940 km² (86.0%) is in Europe (Metropolitan France).
By density
In 2016, the official population of France had a density of 104.8 people per square kilometre, including the overseas regions, and 118.5 people per square kilometre excluding them.
See also
List of French regions and overseas collectivities by GDP
References
France
List ranked
Regions, ranked
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https://en.wikipedia.org/wiki/Bose%20gas
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An ideal Bose gas is a quantum-mechanical phase of matter, analogous to a classical ideal gas. It is composed of bosons, which have an integer value of spin, and abide by Bose–Einstein statistics. The statistical mechanics of bosons were developed by Satyendra Nath Bose for a photon gas, and extended to massive particles by Albert Einstein who realized that an ideal gas of bosons would form a condensate at a low enough temperature, unlike a classical ideal gas. This condensate is known as a Bose–Einstein condensate.
Introduction and examples
Bosons are quantum mechanical particles that follow Bose–Einstein statistics, or equivalently, that possess integer spin. These particles can be classified as elementary: these are the Higgs boson, the photon, the gluon, the W/Z and the hypothetical graviton; or composite like the atom of hydrogen, the atom of 16O, the nucleus of deuterium, mesons etc. Additionally, some quasiparticles in more complex systems can also be considered bosons like the plasmons (quanta of charge density waves).
The first model that treated a gas with several bosons, was the photon gas, a gas of photons, developed by Bose. This model leads to a better understanding of Planck's law and the black-body radiation. The photon gas can be easily expanded to any kind of ensemble of massless non-interacting bosons. The phonon gas, also known as Debye model, is an example where the normal modes of vibration of the crystal lattice of a metal, can be treated as effective massless bosons. Peter Debye used the phonon gas model to explain the behaviour of heat capacity of metals at low temperature.
An interesting example of a Bose gas is an ensemble of helium-4 atoms. When a system of 4He atoms is cooled down to temperature near absolute zero, many quantum mechanical effects are present. Below 2.17 kelvins, the ensemble starts to behave as a superfluid, a fluid with almost zero viscosity. The Bose gas is the most simple quantitative model that explains this phase transition. Mainly when a gas of bosons is cooled down, it forms a Bose–Einstein condensate, a state where a large number of bosons occupy the lowest energy, the ground state, and quantum effects are macroscopically visible like wave interference.
The theory of Bose-Einstein condensates and Bose gases can also explain some features of superconductivity where charge carriers couple in pairs (Cooper pairs) and behave like bosons. As a result, superconductors behave like having no electrical resistivity at low temperatures.
The equivalent model for half-integer particles (like electrons or helium-3 atoms), that follow Fermi–Dirac statistics, is called the Fermi gas (an ensemble of non-interacting fermions). At low enough particle number density and high temperature, both the Fermi gas and the Bose gas behave like a classical ideal gas.
Macroscopic limit
The thermodynamics of an ideal Bose gas is best calculated using the grand canonical ensemble. The grand potential for a Bose gas i
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https://en.wikipedia.org/wiki/Superreal%20number
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In abstract algebra, the superreal numbers are a class of extensions of the real numbers, introduced by H. Garth Dales and W. Hugh Woodin as a generalization of the hyperreal numbers and primarily of interest in non-standard analysis, model theory, and the study of Banach algebras. The field of superreals is itself a subfield of the surreal numbers.
Dales and Woodin's superreals are distinct from the super-real numbers of David O. Tall, which are lexicographically ordered fractions of formal power series over the reals.
Formal definition
Suppose X is a Tychonoff space and C(X) is the algebra of continuous real-valued functions on X. Suppose P is a prime ideal in C(X). Then the factor algebra A = C(X)/P is by definition an integral domain that is a real algebra and that can be seen to be totally ordered. The field of fractions F of A is a superreal field if F strictly contains the real numbers , so that F is not order isomorphic to .
If the prime ideal P is a maximal ideal, then F is a field of hyperreal numbers (Robinson's hyperreals being a very special case).
References
Bibliography
Field (mathematics)
Real closed field
Infinity
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https://en.wikipedia.org/wiki/Real%20closed%20field
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In mathematics, a real closed field is a field F that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers, the field of real algebraic numbers, and the field of hyperreal numbers.
Definitions
A real closed field is a field F in which any of the following equivalent conditions is true:
F is elementarily equivalent to the real numbers. In other words, it has the same first-order properties as the reals: any sentence in the first-order language of fields is true in F if and only if it is true in the reals.
There is a total order on F making it an ordered field such that, in this ordering, every positive element of F has a square root in F and any polynomial of odd degree with coefficients in F has at least one root in F.
F is a formally real field such that every polynomial of odd degree with coefficients in F has at least one root in F, and for every element a of F there is b in F such that a = b2 or a = −b2.
F is not algebraically closed, but its algebraic closure is a finite extension.
F is not algebraically closed but the field extension is algebraically closed.
There is an ordering on F that does not extend to an ordering on any proper algebraic extension of F.
F is a formally real field such that no proper algebraic extension of F is formally real. (In other words, the field is maximal in an algebraic closure with respect to the property of being formally real.)
There is an ordering on F making it an ordered field such that, in this ordering, the intermediate value theorem holds for all polynomials over F with degree ≥ 0.
F is a weakly o-minimal ordered field.
If F is an ordered field, the Artin–Schreier theorem states that F has an algebraic extension, called the real closure K of F, such that K is a real closed field whose ordering is an extension of the given ordering on F, and is unique up to a unique isomorphism of fields identical on F (note that every ring homomorphism between real closed fields automatically is order preserving, because x ≤ y if and only if ∃z : y = x + z2). For example, the real closure of the ordered field of rational numbers is the field of real algebraic numbers. The theorem is named for Emil Artin and Otto Schreier, who proved it in 1926.
If (F, P) is an ordered field, and E is a Galois extension of F, then by Zorn's lemma there is a maximal ordered field extension (M, Q) with M a subfield of E containing F and the order on M extending P. This M, together with its ordering Q, is called the relative real closure of (F, P) in E. We call (F, P) real closed relative to E if M is just F. When E is the algebraic closure of F the relative real closure of F in E is actually the real closure of F described earlier.
If F is a field (no ordering compatible with the field operations is assumed, nor is it assumed that F is orderable) then F still has a real closure, which may not be a field anymore, but just a
real closed ring. For example, the real closu
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https://en.wikipedia.org/wiki/Characteristic%20%28algebra%29
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In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest positive number of copies of the ring's multiplicative identity () that will sum to the additive identity (). If no such number exists, the ring is said to have characteristic zero.
That is, is the smallest positive number such that:
if such a number exists, and otherwise.
Motivation
The special definition of the characteristic zero is motivated by the equivalent definitions characterized in the next section, where the characteristic zero is not required to be considered separately.
The characteristic may also be taken to be the exponent of the ring's additive group, that is, the smallest positive integer such that:
for every element of the ring (again, if exists; otherwise zero). This definition applies in the more general class of a rngs (see ); for (unital) rings the two definitions are equivalent due to their distributive law.
Equivalent characterizations
The characteristic is the natural number such that is the kernel of the unique ring homomorphism from to .
The characteristic is the natural number such that contains a subring isomorphic to the factor ring , which is the image of the above homomorphism.
When the non-negative integers are partially ordered by divisibility, then is the smallest and is the largest. Then the characteristic of a ring is the smallest value of for which . If nothing "smaller" (in this ordering) than will suffice, then the characteristic is . This is the appropriate partial ordering because of such facts as that is the least common multiple of and , and that no ring homomorphism exists unless divides .
The characteristic of a ring is precisely if the statement for all implies that is a multiple of .
Case of rings
If and are rings and there exists a ring homomorphism , then the characteristic of divides the characteristic of . This can sometimes be used to exclude the possibility of certain ring homomorphisms. The only ring with characteristic is the zero ring, which has only a single element . If a nontrivial ring does not have any nontrivial zero divisors, then its characteristic is either or prime. In particular, this applies to all fields, to all integral domains, and to all division rings. Any ring of characteristic is infinite.
The ring of integers modulo has characteristic . If is a subring of , then and have the same characteristic. For example, if is prime and is an irreducible polynomial with coefficients in the field with elements, then the quotient ring is a field of characteristic . Another example: The field of complex numbers contains , so the characteristic of is .
A -algebra is equivalently a ring whose characteristic divides . This is because for every ring there is a ring homomorphism , and this map factors through if and only if the characteristic of divides . In this case for any in the ring, then adding to itself times gives .
If a
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https://en.wikipedia.org/wiki/Salem%20number
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In mathematics, a Salem number is a real algebraic integer whose conjugate roots all have absolute value no greater than 1, and at least one of which has absolute value exactly 1. Salem numbers are of interest in Diophantine approximation and harmonic analysis. They are named after Raphaël Salem.
Properties
Because it has a root of absolute value 1, the minimal polynomial for a Salem number must be reciprocal. This implies that is also a root, and that all other roots have absolute value exactly one. As a consequence α must be a unit in the ring of algebraic integers, being of norm 1.
Every Salem number is a Perron number (a real algebraic number greater than one all of whose conjugates have smaller absolute value).
Relation with Pisot–Vijayaraghavan numbers
The smallest known Salem number is the largest real root of Lehmer's polynomial (named after Derrick Henry Lehmer)
which is about : it is conjectured that it is indeed the smallest Salem number, and the smallest possible Mahler measure of an irreducible non-cyclotomic polynomial.
Lehmer's polynomial is a factor of the shorter 12th-degree polynomial,
all twelve roots of which satisfy the relation
Salem numbers can be constructed from Pisot–Vijayaraghavan numbers. To recall, the smallest of the latter is the unique real root of the cubic polynomial,
known as the plastic number and approximately equal to 1.324718. This can be used to generate a family of Salem numbers including the smallest one found so far. The general approach is to take the minimal polynomial of a Pisot–Vijayaraghavan number and its reciprocal polynomial, , and solve the equation,
for integral above a bound. Subtracting one side from the other, factoring, and disregarding trivial factors will then yield the minimal polynomial of certain Salem numbers. For example, using the negative case of the above,
then for , this factors as,
where the decic is Lehmer's polynomial. Using higher will yield a family with a root approaching the plastic number. This can be better understood by taking th roots of both sides,
so as goes higher, will approach the solution of . If the positive case is used, then approaches the plastic number from the opposite direction. Using the minimal polynomial of the next smallest Pisot–Vijayaraghavan number gives,
which for factors as,
a decic not generated in the previous and has the root which is the 5th smallest known Salem number. As , this family in turn tends towards the larger real root of .
References
Chap. 3.
Algebraic numbers
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https://en.wikipedia.org/wiki/Formally%20real%20field
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In mathematics, in particular in field theory and real algebra, a formally real field is a field that can be equipped with a (not necessarily unique) ordering that makes it an ordered field.
Alternative definitions
The definition given above is not a first-order definition, as it requires quantifiers over sets. However, the following criteria can be coded as (infinitely many) first-order sentences in the language of fields and are equivalent to the above definition.
A formally real field F is a field that also satisfies one of the following equivalent properties:
−1 is not a sum of squares in F. In other words, the Stufe of F is infinite. (In particular, such a field must have characteristic 0, since in a field of characteristic p the element −1 is a sum of 1s.) This can be expressed in first-order logic by , , etc., with one sentence for each number of variables.
There exists an element of F that is not a sum of squares in F, and the characteristic of F is not 2.
If any sum of squares of elements of F equals zero, then each of those elements must be zero.
It is easy to see that these three properties are equivalent. It is also easy to see that a field that admits an ordering must satisfy these three properties.
A proof that if F satisfies these three properties, then F admits an ordering uses the notion of prepositive cones and positive cones. Suppose −1 is not a sum of squares; then a Zorn's Lemma argument shows that the prepositive cone of sums of squares can be extended to a positive cone . One uses this positive cone to define an ordering: if and only if belongs to P.
Real closed fields
A formally real field with no formally real proper algebraic extension is a real closed field. If K is formally real and Ω is an algebraically closed field containing K, then there is a real closed subfield of Ω containing K. A real closed field can be ordered in a unique way, and the non-negative elements are exactly the squares.
Notes
References
Field (mathematics)
Ordered groups
pl:Ciało (formalnie) rzeczywiste
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https://en.wikipedia.org/wiki/Hilbert%E2%80%93Speiser%20theorem
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In mathematics, the Hilbert–Speiser theorem is a result on cyclotomic fields, characterising those with a normal integral basis. More generally, it applies to any finite abelian extension of , which by the Kronecker–Weber theorem are isomorphic to subfields of cyclotomic fields.
Hilbert–Speiser Theorem. A finite abelian extension has a normal integral basis if and only if it is tamely ramified over .
This is the condition that it should be a subfield of where is a squarefree odd number. This result was introduced by in his Zahlbericht and by .
In cases where the theorem states that a normal integral basis does exist, such a basis may be constructed by means of Gaussian periods. For example if we take a prime number , has a normal integral basis consisting of all the -th roots of unity other than . For a field contained in it, the field trace can be used to construct such a basis in also (see the article on Gaussian periods). Then in the case of squarefree and odd, is a compositum of subfields of this type for the primes dividing (this follows from a simple argument on ramification). This decomposition can be used to treat any of its subfields.
proved a converse to the Hilbert–Speiser theorem:
Each finite tamely ramified abelian extension of a fixed number field has a relative normal integral basis if and only if .
There is an elliptic analogue of the theorem proven by .
It is now called the Srivastav-Taylor theorem .
References
Cyclotomic fields
Theorems in algebraic number theory
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https://en.wikipedia.org/wiki/L%C3%A9vy%27s%20constant
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In mathematics Lévy's constant (sometimes known as the Khinchin–Lévy constant) occurs in an expression for the asymptotic behaviour of the denominators of the convergents of continued fractions.
In 1935, the Soviet mathematician Aleksandr Khinchin showed that the denominators qn of the convergents of the continued fraction expansions of almost all real numbers satisfy
Soon afterward, in 1936, the French mathematician Paul Lévy found the explicit expression for the constant, namely
The term "Lévy's constant" is sometimes used to refer to (the logarithm of the above expression), which is approximately equal to 1.1865691104… The value derives from the asymptotic expectation of the logarithm of the ratio of successive denominators, using the Gauss-Kuzmin distribution. In particular, the ratio has the asymptotic density function
for and zero otherwise. This gives Lévy's constant as
.
The base-10 logarithm of Lévy's constant, which is approximately 0.51532041…, is half of the reciprocal of the limit in Lochs' theorem.
See also
Khinchin's constant
References
Further reading
External links
Continued fractions
Mathematical constants
Paul Lévy (mathematician)
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https://en.wikipedia.org/wiki/Multimodal%20distribution
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In statistics, a multimodal distribution is a probability distribution with more than one mode. These appear as distinct peaks (local maxima) in the probability density function, as shown in Figures 1 and 2. Categorical, continuous, and discrete data can all form multimodal distributions. Among univariate analyses, multimodal distributions are commonly bimodal.
Terminology
When the two modes are unequal the larger mode is known as the major mode and the other as the minor mode. The least frequent value between the modes is known as the antimode. The difference between the major and minor modes is known as the amplitude. In time series the major mode is called the acrophase and the antimode the batiphase.
Galtung's classification
Galtung introduced a classification system (AJUS) for distributions:
A: unimodal distribution – peak in the middle
J: unimodal – peak at either end
U: bimodal – peaks at both ends
S: bimodal or multimodal – multiple peaks
This classification has since been modified slightly:
J: (modified) – peak on right
L: unimodal – peak on left
F: no peak (flat)
Under this classification bimodal distributions are classified as type S or U.
Examples
Bimodal distributions occur both in mathematics and in the natural sciences.
Probability distributions
Important bimodal distributions include the arcsine distribution and the beta distribution (iff both parameters a and b are less than 1). Others include the U-quadratic distribution.
The ratio of two normal distributions is also bimodally distributed. Let
where a and b are constant and x and y are distributed as normal variables with a mean of 0 and a standard deviation of 1. R has a known density that can be expressed as a confluent hypergeometric function.
The distribution of the reciprocal of a t distributed random variable is bimodal when the degrees of freedom are more than one. Similarly the reciprocal of a normally distributed variable is also bimodally distributed.
A t statistic generated from data set drawn from a Cauchy distribution is bimodal.
Occurrences in nature
Examples of variables with bimodal distributions include the time between eruptions of certain geysers, the color of galaxies, the size of worker weaver ants, the age of incidence of Hodgkin's lymphoma, the speed of inactivation of the drug isoniazid in US adults, the absolute magnitude of novae, and the circadian activity patterns of those crepuscular animals that are active both in morning and evening twilight. In fishery science multimodal length distributions reflect the different year classes and can thus be used for age distribution- and growth estimates of the fish population. Sediments are usually distributed in a bimodal fashion. When sampling mining galleries crossing either the host rock and the mineralized veins, the distribution of geochemical variables would be bimodal. Bimodal distributions are also seen in traffic analysis, where traffic peaks in during the AM rush hour and then
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https://en.wikipedia.org/wiki/Parabolic%20geometry
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Parabolic geometry may refer to:
Parabolic geometry, former name for Euclidean geometry, a comprehensive and deductive mathematical system
Parabolic geometry (differential geometry): The homogeneous space defined by a semisimple Lie group modulo a parabolic subgroup, or the curved analog of such a space
Cartan parabolic geometry, geometry induced by parabolic Cartan inclusions
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https://en.wikipedia.org/wiki/Cohomology%20ring
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In mathematics, specifically algebraic topology, the cohomology ring of a topological space X is a ring formed from the cohomology groups of X together with the cup product serving as the ring multiplication. Here 'cohomology' is usually understood as singular cohomology, but the ring structure is also present in other theories such as de Rham cohomology. It is also functorial: for a continuous mapping of spaces one obtains a ring homomorphism on cohomology rings, which is contravariant.
Specifically, given a sequence of cohomology groups Hk(X;R) on X with coefficients in a commutative ring R (typically R is Zn, Z, Q, R, or C) one can define the cup product, which takes the form
The cup product gives a multiplication on the direct sum of the cohomology groups
This multiplication turns H•(X;R) into a ring. In fact, it is naturally an N-graded ring with the nonnegative integer k serving as the degree. The cup product respects this grading.
The cohomology ring is graded-commutative in the sense that the cup product commutes up to a sign determined by the grading. Specifically, for pure elements of degree k and ℓ; we have
A numerical invariant derived from the cohomology ring is the cup-length, which means the maximum number of graded elements of degree ≥ 1 that when multiplied give a non-zero result. For example a complex projective space has cup-length equal to its complex dimension.
Examples
where .
where .
where .
where .
where .
where .
By the Künneth formula, the mod 2 cohomology ring of the cartesian product of n copies of is a polynomial ring in n variables with coefficients in .
The reduced cohomology ring of wedge sums is the direct product of their reduced cohomology rings.
The cohomology ring of suspensions vanishes except for the degree 0 part.
See also
Quantum cohomology
References
Homology theory
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https://en.wikipedia.org/wiki/Divide
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Divide may refer to:
Mathematics
Division (mathematics)
Divides, redirects to Divisor
Geography
Drainage divide, a line separating two drainage basins
Great Divide Basin, in Wyoming
Places
Divide, Saskatchewan, Canada
Divide, Colorado, community
Divide, Illinois, an unincorporated community
Divide, Montana, a rural community
Divide, Oregon, an unincorporated community
Divide, West Virginia, an unincorporated community
Divide County, North Dakota
Music
"Divide", a song by All That Remains from The Order of Things
"Divide", a song by Bastille from Doom Days
"Divide", a song by Disturbed from Indestructible
"Divide", a song by Vision of Disorder from Vision of Disorder
÷ (album), a 2017 album by Ed Sheeran
Divides, album by The Virginmarys 2016
The Continental Divide (album)
The Divide, album by Tom Waits and Scott Vestal 2011
See also
Continental divide (disambiguation)
Div (disambiguation)
Divided (disambiguation)
Division (disambiguation)
Division sign (÷)
The Divide (disambiguation)
Vertical line (dividing line)
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https://en.wikipedia.org/wiki/Jean%20Dieudonn%C3%A9
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Jean Alexandre Eugène Dieudonné (; 1 July 1906 – 29 November 1992) was a French mathematician, notable for research in abstract algebra, algebraic geometry, and functional analysis, for close involvement with the Nicolas Bourbaki pseudonymous group and the Éléments de géométrie algébrique project of Alexander Grothendieck, and as a historian of mathematics, particularly in the fields of functional analysis and algebraic topology. His work on the classical groups (the book La Géométrie des groupes classiques was published in 1955), and on formal groups, introducing what now are called Dieudonné modules, had a major effect on those fields.
He was born and brought up in Lille, with a formative stay in England where he was introduced to algebra. In 1924 he was admitted to the École Normale Supérieure, where André Weil was a classmate. He began working in complex analysis. In 1934 he was one of the group of normaliens convened by Weil, which would become 'Bourbaki'.
Education and teaching
He served in the French Army during World War II, and then taught in Clermont-Ferrand until the liberation of France. After holding professorships at the University of São Paulo (1946–47), the University of Nancy (1948–1952) and the University of Michigan (1952–53), he joined the Department of Mathematics at Northwestern University in 1953, before returning to France as a founding member of the Institut des Hautes Études Scientifiques. He moved to the University of Nice to found the Department of Mathematics in 1964, and retired in 1970. He was elected as a member of the Académie des Sciences in 1968.
Career
Dieudonné drafted much of the Bourbaki series of texts, the many volumes of the EGA algebraic geometry series, and nine volumes of his own Éléments d'Analyse. The first volume of the Traité is a French version of the book Foundations of Modern Analysis (1960), which had become a graduate textbook on functional analysis.
He also wrote individual monographs on Infinitesimal Calculus, Linear Algebra and Elementary Geometry, invariant theory, commutative algebra, algebraic geometry, and formal groups.
With Laurent Schwartz he supervised the early research of Alexander Grothendieck. Later from 1959 to 1964 he was at the Institut des Hautes Études Scientifiques alongside Grothendieck, and collaborating on the expository work needed to support the project of refounding algebraic geometry on the new basis of schemes.
Selected works
9 volumes of Éléments d'analyse (1960-1982), éd. Gauthier-Villars
; Eng. trans:
(a reprint of )
; Eng. trans:
References
External links
A talk on the history of Algebraic Geometry given by Jean Dieudonné at the Department of Mathematics of the University of Wisconsin-Milwaukee in 1972 has been recently restored and is available here
Dieudonné appears in the Horizon BBC documentary A Mathematical Mystery Tour
1906 births
1992 deaths
20th-century French mathematicians
Algebraic geometers
French historians of mathematics
Éc
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https://en.wikipedia.org/wiki/Total%20derivative
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In mathematics, the total derivative of a function at a point is the best linear approximation near this point of the function with respect to its arguments. Unlike partial derivatives, the total derivative approximates the function with respect to all of its arguments, not just a single one. In many situations, this is the same as considering all partial derivatives simultaneously. The term "total derivative" is primarily used when is a function of several variables, because when is a function of a single variable, the total derivative is the same as the ordinary derivative of the function.
The total derivative as a linear map
Let be an open subset. Then a function is said to be (totally) differentiable at a point if there exists a linear transformation such that
The linear map is called the (total) derivative or (total) differential of at . Other notations for the total derivative include and . A function is (totally) differentiable if its total derivative exists at every point in its domain.
Conceptually, the definition of the total derivative expresses the idea that is the best linear approximation to at the point . This can be made precise by quantifying the error in the linear approximation determined by . To do so, write
where equals the error in the approximation. To say that the derivative of at is is equivalent to the statement
where is little-o notation and indicates that is much smaller than as . The total derivative is the unique linear transformation for which the error term is this small, and this is the sense in which it is the best linear approximation to .
The function is differentiable if and only if each of its components is differentiable, so when studying total derivatives, it is often possible to work one coordinate at a time in the codomain. However, the same is not true of the coordinates in the domain. It is true that if is differentiable at , then each partial derivative exists at . The converse does not hold: it can happen that all of the partial derivatives of at exist, but is not differentiable at . This means that the function is very "rough" at , to such an extreme that its behavior cannot be adequately described by its behavior in the coordinate directions. When is not so rough, this cannot happen. More precisely, if all the partial derivatives of at exist and are continuous in a neighborhood of , then is differentiable at . When this happens, then in addition, the total derivative of is the linear transformation corresponding to the Jacobian matrix of partial derivatives at that point.
The total derivative as a differential form
When the function under consideration is real-valued, the total derivative can be recast using differential forms. For example, suppose that is a differentiable function of variables . The total derivative of at may be written in terms of its Jacobian matrix, which in this instance is a row matrix:
The linear approximation pro
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https://en.wikipedia.org/wiki/Fr%C3%A4nkel
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Fränkel (or Fraenkel) is a surname. Notable people with the surname include:
Abraham Fraenkel (1891–1965), German-Israeli mathematician, known for Zermelo–Fraenkel set theory
Albert Fränkel (1848–1916), German physician
Aviezri Fraenkel (born 1929), Israeli mathematician
Baruch Fränkel-Teomim (1760–1828), rabbi, Talmudist
David ben Naphtali Fränkel (c. 1704–1762), German rabbi
Eduard Fraenkel (1888–1965), German-English classical scholar
Ernst Fränkel (physician) (1844–1921), German gynaecologist
Ernst Fraenkel (linguist) (1881–1957), German linguist
Ernst Fraenkel (political scientist) (1898–1975), German political scientist
Hermann Fränkel (1888–1977), German-American classicist
Heinz Fraenkel-Conrat (1910–1999), German biochemist
Iwan Fränkel (born 1941), Surinamese footballer
Jonas Fränkel (1773–1846), German banker and philanthropist
Knut Frænkel (1870–1897), Swedish engineer and arctic explorer
Leó Frankel (Léo Fränkel) (1844–1896), Hungarian communist revolutionary
Naftali Frenkel, 16-year-old killed in the 2014 kidnapping and murder of Israeli teenagers
Purrel Fränkel (born 1976), Surinamese footballer
Ray Fränkel (born 1982), Dutch Surinamese footballer
Samuel Fränkel (1801–1881), German textile manufacturer
Sándor Ferenczi (1873–1933), Hungarian psychoanalyst, born Alexander Fränkel
See also
Frankel
Frankl
Frenkel
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https://en.wikipedia.org/wiki/Hyperbolic%20quaternion
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In abstract algebra, the algebra of hyperbolic quaternions is a nonassociative algebra over the real numbers with elements of the form
where the squares of i, j, and k are +1 and distinct elements of {i, j, k} multiply with the anti-commutative property.
The four-dimensional algebra of hyperbolic quaternions incorporates some of the features of the older and larger algebra of biquaternions. They both contain subalgebras isomorphic to the split-complex number plane. Furthermore, just as the quaternion algebra H can be viewed as a union of complex planes, so the hyperbolic quaternion algebra is a union of split-complex number planes sharing the same real line.
It was Alexander Macfarlane who promoted this concept in the 1890s as his Algebra of Physics, first through the American Association for the Advancement of Science in 1891, then through his 1894 book of five Papers in Space Analysis, and in a series of lectures at Lehigh University in 1900.
Algebraic structure
Like the quaternions, the set of hyperbolic quaternions form a vector space over the real numbers of dimension 4. A linear combination
is a hyperbolic quaternion when and are real numbers and the basis set has these products:
Using the distributive property, these relations can be used to multiply any two hyperbolic quaternions.
Unlike the ordinary quaternions, the hyperbolic quaternions are not associative. For example, , while . In fact, this example shows that the hyperbolic quaternions are not even an alternative algebra.
The first three relations show that products of the (non-real) basis elements are anti-commutative. Although this basis set does not form a group, the set
forms a quasigroup. One also notes that any subplane of the set M of hyperbolic quaternions that contains the real axis forms a plane of split-complex numbers. If
is the conjugate of , then the product
is the quadratic form used in spacetime theory. In fact, for events p and q, the bilinear form
arises as the negative of the real part of the hyperbolic quaternion product pq*, and is used in Minkowski space.
Note that the set of units U = {q : qq* ≠ 0 } is not closed under multiplication. See the references (external link) for details.
Discussion
The hyperbolic quaternions form a nonassociative ring; the failure of associativity in this algebra curtails the facility of this algebra in transformation theory. Nevertheless,
this algebra put a focus on analytical kinematics by suggesting a mathematical model:
When one selects a unit vector r in the hyperbolic quaternions, then r 2 = +1. The plane with hyperbolic quaternion multiplication is a commutative and associative subalgebra isomorphic to the split-complex number plane.
The hyperbolic versor transforms Dr by
Since the direction r in space is arbitrary, this hyperbolic quaternion multiplication can express any Lorentz boost using the parameter a called rapidity. However, the hyperbolic quaternion algebra is deficient for representing th
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https://en.wikipedia.org/wiki/Median%20%28disambiguation%29
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Median may refer to:
Mathematics and statistics
Median (statistics), in statistics, a number that separates the lowest- and highest-value halves
Median (geometry), in geometry, a line joining a vertex of a triangle to the midpoint of the opposite side
Median (graph theory), a vertex m(a,b,c) that belongs to shortest paths between each pair of a, b, and c
Median algebra, an algebraic triple product generalising the algebraic properties of the majority function
Median graph, undirected graph in which every three vertices a, b, and c have a unique median
Geometric median, a point minimizing the sum of distances to a given set of points
People
Median (rapper), a rapper from the U.S. city of Raleigh, North Carolina
Science and technology
Median (biology), an anatomical term of location, meaning at or towards the central plane of a bilaterally symmetrical organism or structure
Median filter, a nonlinear digital filtering technique used to reduce noise in images
Median nerve, a nerve in humans and other animals located in the upper limb, one of the five main nerves originating from the brachial plexus
Other
Median language, the extinct Northwestern Iranian language of the Medes people
Median Empire or Median Kingdom, an ancient Iranian empire predating the First Persian Empire
Median consonant, a consonant sound that is produced when air flows across the center of the mouth over the tongue
Median strip, the portion of a divided roadway used to separate opposing traffic; the British equivalent is central reservation
Median triangle, an archaeological term referring to the area bounded by Hamadān, Malāyer and Kangāvar in Iran
See also
Medes, an ancient Iranian people who originated in Media, in the northwest of present-day Iran
Mediant, in music, the note halfway between the tonic and the dominant
Mediant (mathematics), a fraction created from the sums of the numerators and denominators of two other fractions
Medial (disambiguation)
Medium (disambiguation)
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https://en.wikipedia.org/wiki/Median%20%28geometry%29
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In geometry, a median of a triangle is a line segment joining a vertex to the midpoint of the opposite side, thus bisecting that side. Every triangle has exactly three medians, one from each vertex, and they all intersect each other at the triangle's centroid. In the case of isosceles and equilateral triangles, a median bisects any angle at a vertex whose two adjacent sides are equal in length.
The concept of a median extends to tetrahedra.
Relation to center of mass
Each median of a triangle passes through the triangle's centroid, which is the center of mass of an infinitely thin object of uniform density coinciding with the triangle. Thus the object would balance on the intersection point of the medians. The centroid is twice as close along any median to the side that the median intersects as it is to the vertex it emanates from.
Equal-area division
Each median divides the area of the triangle in half; hence the name, and hence a triangular object of uniform density would balance on any median. (Any other lines which divide the area of the triangle into two equal parts do not pass through the centroid.) The three medians divide the triangle into six smaller triangles of equal area.
Proof of equal-area property
Consider a triangle ABC. Let D be the midpoint of , E be the midpoint of , F be the midpoint of , and O be the centroid (most commonly denoted G).
By definition, . Thus and , where represents the area of triangle ; these hold because in each case the two triangles have bases of equal length and share a common altitude from the (extended) base, and a triangle's area equals one-half its base times its height.
We have:
Thus, and
Since , therefore, .
Using the same method, one can show that .
Three congruent triangles
In 2014 Lee Sallows discovered the following theorem:
The medians of any triangle dissect it into six equal area smaller triangles as in the figure above where three adjacent pairs of triangles meet at the midpoints D, E and F. If the two triangles in each such pair are rotated about their common midpoint until they meet so as to share a common side, then the three new triangles formed by the union of each pair are congruent.
Formulas involving the medians' lengths
The lengths of the medians can be obtained from Apollonius' theorem as:
where and are the sides of the triangle with respective medians and from their midpoints.
These formulas imply the relationships:
Other properties
Let ABC be a triangle, let G be its centroid, and let D, E, and F be the midpoints of BC, CA, and AB, respectively. For any point P in the plane of ABC then
The centroid divides each median into parts in the ratio 2:1, with the centroid being twice as close to the midpoint of a side as it is to the opposite vertex.
For any triangle with sides and medians
The medians from sides of lengths and are perpendicular if and only if
The medians of a right triangle with hypotenuse satisfy
Any triangle's area T can be expresse
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https://en.wikipedia.org/wiki/Grothendieck%20universe
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In mathematics, a Grothendieck universe is a set U with the following properties:
If x is an element of U and if y is an element of x, then y is also an element of U. (U is a transitive set.)
If x and y are both elements of U, then is an element of U.
If x is an element of U, then P(x), the power set of x, is also an element of U.
If is a family of elements of U, and if is an element of U, then the union is an element of U.
A Grothendieck universe is meant to provide a set in which all of mathematics can be performed. (In fact, uncountable Grothendieck universes provide models of set theory with the natural ∈-relation, natural powerset operation etc.). Elements of a Grothendieck universe are sometimes called small sets. The idea of universes is due to Alexander Grothendieck, who used them as a way of avoiding proper classes in algebraic geometry.
The existence of a nontrivial Grothendieck universe goes beyond the usual axioms of Zermelo–Fraenkel set theory; in particular it would imply the existence of strongly inaccessible cardinals.
Tarski–Grothendieck set theory is an axiomatic treatment of set theory, used in some automatic proof systems, in which every set belongs to a Grothendieck universe.
The concept of a Grothendieck universe can also be defined in a topos.
Properties
As an example, we will prove an easy proposition.
Proposition. If and , then .
Proof. because . because , so .
It is similarly easy to prove that any Grothendieck universe U contains:
All singletons of each of its elements,
All products of all families of elements of U indexed by an element of U,
All disjoint unions of all families of elements of U indexed by an element of U,
All intersections of all families of elements of U indexed by an element of U,
All functions between any two elements of U, and
All subsets of U whose cardinal is an element of U.
In particular, it follows from the last axiom that if U is non-empty, it must contain all of its finite subsets and a subset of each finite cardinality. One can also prove immediately from the definitions that the intersection of any class of universes is a universe.
Grothendieck universes and inaccessible cardinals
There are two simple examples of Grothendieck universes:
The empty set, and
The set of all hereditarily finite sets .
Other examples are more difficult to construct. Loosely speaking, this is because Grothendieck universes are equivalent to strongly inaccessible cardinals. More formally, the following two axioms are equivalent:
(U) For each set x, there exists a Grothendieck universe U such that x ∈ U.
(C) For each cardinal κ, there is a strongly inaccessible cardinal λ that is strictly larger than κ.
To prove this fact, we introduce the function c(U). Define:
where by |x| we mean the cardinality of x. Then for any universe U, c(U) is either zero or strongly inaccessible. Assuming it is non-zero, it is a strong limit cardinal because the power set of any element of U is an
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https://en.wikipedia.org/wiki/Calculus%20of%20negligence
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In the United States, the calculus of negligence, also known as the Hand rule, Hand formula, or BPL formula, is a term coined by Judge Learned Hand which describes a process for determining whether a legal duty of care has been breached (see negligence). The original description of the calculus was in United States v. Carroll Towing Co., in which an improperly secured barge had drifted away from a pier and caused damage to several other boats.
Articulation of the rule
Hand stated:
This relationship has been formalized by the law and economics school as such: an act is in breach of the duty of care if:
where B is the cost (burden) of taking precautions, and P is the probability of loss (L). L is the gravity of loss. The product of P x L must be a greater amount than B to create a duty of due care for the defendant.
Rationale
The calculus of negligence is based on the Coase theorem. The tort system acts as if, before the injury or damage, a contract had been made between the parties under the assumption that a rational, cost-minimizing individual will not spend money on taking precautions if those precautions are more expensive than the costs of the harm that they prevent. In other words, rather than spending money on safety, the individual will simply allow harm to occur and pay for the costs of that harm, because that will be more cost-efficient than taking precautions. This represents cases where B is greater than PL.
If the harm could be avoided for less than the cost of the harm (B is less than PL), then the individual should take the precautions, rather than allowing the harm to occur. If precautions were not taken, we find that a legal duty of care has been breached, and we impose liability on the individual to pay for the harm.
This approach, in theory, leads to an optimal allocation of resources; where harm can be cheaply avoided, the legal system requires precautions. Where precautions are prohibitively expensive, it does not. In marginal-cost terms, we require individuals to invest one unit of precautions up until the point that those precautions prevent exactly one unit of harm, and no less.
Mathematical rationale
The Hand rule attempts to formalize the intuitive notion that when the expected loss exceeds the cost of taking precautions, the duty of care has been breached:To assess the expected loss, statistical methods, such as regression analysis, may be used. A common metric for quantifying losses in the case of work accidents is the present value of lost future earnings and medical costs associated with the accident. In the case when the probability of loss is assumed to be a single number , and is the loss from the event occurring, the familiar form of the Hand rule is recovered. More generally, for continuous outcomes the Hand rule takes form:where is the domain for losses and is the probability density function of losses. Assuming that losses are positive, common choices for loss distributions include the ga
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https://en.wikipedia.org/wiki/LIS
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LIS or LiS may refer to:
Computing
LIS (programming language)
Lis (linear algebra library), library of iterative solvers for linear systems
Laboratory information system, databases oriented towards medical laboratories
Land information system, land mapping and cadastre GIS used by local governments
Language-independent specification, a programming language specification
Legume Information System, online resources and exploratory tools for legume researchers and breeders
Linear Integrated Systems, American manufacturer of semiconductors
Local information systems, collect, store, and disseminate information about small geographic areas
Location information server, provides location information
Longest increasing subsequence, algorithm to find the longest increasing subsequence in an array of numbers
Science
Laser Isotope Separation, a means of producing enriched uranium from uranium ore
Lateral internal sphincterotomy, an operation for the treatment of chronic anal fissure
Lightning Imaging Sensor, an instrument on the TRMM satellite and on the International Space Station
Liquid-impregnated surface
Locked-in syndrome, a type of paralysis
Library and information science
Other
Lis (given name)
Lis (surname)
Lis River, a river in Portugal
Lis coat of arms, of Polish Clan Lis
Lis, Albania, in the Mat municipality of Dibër County
Italian Sign Language (Lingua dei Segni Italiana)
Life Is Strange (series), a series of episodic graphic adventure games.
Life Is Strange (2015 video game)
Lithium-sulphur battery (Li-S)
Locate in Scotland (1981–2001), replaced by Scottish Development International
London Interdisciplinary School, alternative university in England
Lisbon Airport, Portugal, IATA code
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