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Wikipedia:Spiric section#0
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In geometry, a spiric section, sometimes called a spiric of Perseus, is a quartic plane curve defined by equations of the form ( x 2 + y 2 ) 2 = d x 2 + e y 2 + f . {\displaystyle (x^{2}+y^{2})^{2}=dx^{2}+ey^{2}+f.\,} Equivalently, spiric sections can be defined as bicircular quartic curves that are symmetric with respect to the x and y-axes. Spiric sections are included in the family of toric sections and include the family of hippopedes and the family of Cassini ovals. The name is from σπειρα meaning torus in ancient Greek. A spiric section is sometimes defined as the curve of intersection of a torus and a plane parallel to its rotational symmetry axis. However, this definition does not include all of the curves given by the previous definition unless imaginary planes are allowed. Spiric sections were first described by the ancient Greek geometer Perseus in roughly 150 BC, and are assumed to be the first toric sections to be described. The name spiric is due to the ancient notation spira of a torus., == Equations == Start with the usual equation for the torus: ( x 2 + y 2 + z 2 + b 2 − a 2 ) 2 = 4 b 2 ( x 2 + y 2 ) . {\displaystyle (x^{2}+y^{2}+z^{2}+b^{2}-a^{2})^{2}=4b^{2}(x^{2}+y^{2}).\,} Interchanging y and z so that the axis of revolution is now on the xy-plane, and setting z=c to find the curve of intersection gives ( x 2 + y 2 − a 2 + b 2 + c 2 ) 2 = 4 b 2 ( x 2 + c 2 ) . {\displaystyle (x^{2}+y^{2}-a^{2}+b^{2}+c^{2})^{2}=4b^{2}(x^{2}+c^{2}).\,} In this formula, the torus is formed by rotating a circle of radius a with its center following another circle of radius b (not necessarily larger than a, self-intersection is permitted). The parameter c is the distance from the intersecting plane to the axis of revolution. There are no spiric sections with c > b + a, since there is no intersection; the plane is too far away from the torus to intersect it. Expanding the equation gives the form seen in the definition ( x 2 + y 2 ) 2 = d x 2 + e y 2 + f {\displaystyle (x^{2}+y^{2})^{2}=dx^{2}+ey^{2}+f\,} where d = 2 ( a 2 + b 2 − c 2 ) , e = 2 ( a 2 − b 2 − c 2 ) , f = − ( a 4 + b 4 + c 4 − 2 a 2 b 2 − 2 a 2 c 2 − 2 b 2 c 2 ) . {\displaystyle d=2(a^{2}+b^{2}-c^{2}),\ e=2(a^{2}-b^{2}-c^{2}),\ f=-(a^{4}+b^{4}+c^{4}-2a^{2}b^{2}-2a^{2}c^{2}-2b^{2}c^{2}).\,} In polar coordinates this becomes ( r 2 − a 2 + b 2 + c 2 ) 2 = 4 b 2 ( r 2 cos 2 θ + c 2 ) {\displaystyle (r^{2}-a^{2}+b^{2}+c^{2})^{2}=4b^{2}(r^{2}\cos ^{2}\theta +c^{2})\,} or r 4 = r 2 ( d cos 2 θ + e sin 2 θ ) + f . {\displaystyle r^{4}=r^{2}(d\cos ^{2}\theta +e\sin ^{2}\theta )+f.} == Spiric sections on a spindle torus == Spiric sections on a spindle torus, whose planes intersect the spindle (inner part), consist of an outer and an inner curve (s. picture). == Spiric sections as isoptics == Isoptics of ellipses and hyperbolas are spiric sections. (S. also weblink The Mathematics Enthusiast.) == Examples of spiric sections == Examples include the hippopede and the Cassini oval and their relatives, such as the lemniscate of Bernoulli. The Cassini oval has the remarkable property that the product of distances to two foci are constant. For comparison, the sum is constant in ellipses, the difference is constant in hyperbolae and the ratio is constant in circles. == References == Weisstein, Eric W. "Spiric Section". MathWorld. MacTutor history 2Dcurves.com description MacTutor biography of Perseus The Mathematics Enthusiast Number 9, article 4 Specific
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Wikipedia:Spiridon Popescu#0
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Spiridon Popescu (August 13, 1864 – May 8, 1933) was a Romanian prose writer. Born in Rogojeni, Galați County, his parents were the peasant Constantin Dumitrașcu al Popei and his wife Safta (née Tofan). He attended seminary in Galați and at Socola Monastery in Iași, earning his high school degree at age 26. He studied physics and mathematics at the University of Iași and took courses at the higher normal school, earning a mathematics degree at age 31. He taught mathematics in Bârlad, Vaslui, Tulcea, Galați and, from 1904, in the national capital, Bucharest. Within the Education Ministry, Popescu was responsible for normal primary education nationwide. He was elected both to the Assembly of Deputies and to the Senate. Popescu made his literary debut in the Iași-based Arhiva in 1890. Encouraged by his brother-in-law Constantin Stere, he wrote for Evenimentul literar, and for Albina. As an active participant in the policy of village education promoted by Spiru Haret, he also contributed to Viața Românească, where he published the Poporanist novellas Moș Gheorghe la expoziție (1907) and Rătăcirea din Stoborăni (1909). His first book was Considerațiuni psihologice din viața poporului român (1893), followed by Din povestirile unui vânător de lupi (1905), Moș Gheorghe la expoziție (1912) and the short story collection Zori de iulie (1912). == Notes ==
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Wikipedia:Spiru Haret#0
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Spiru C. Haret (Romanian pronunciation: [ˈspiru haˈret]; 15 February 1851 – 17 December 1912) was a Romanian mathematician, astronomer, and politician. He made a fundamental contribution to the n-body problem in celestial mechanics by proving that using a third degree approximation for the disturbing forces implies instability of the major axes of the orbits, and by introducing the concept of secular perturbations in relation to this. As a politician, during his three terms as Minister of Education, Haret ran deep reforms, building the modern Romanian education system. He was made a full member of the Romanian Academy in 1892. He also founded the Bucharest Astronomical Observatory, appointing Nicolae Coculescu as its first director. The crater Haret on the Moon is named after him. == Life == Haret was born in Iași, Moldavia, to Constantin and Smaranda Haret, who were of Armenian origin. His baptismal record listed his name as Spiridon Haret. He started his studies in Dorohoi Iași, and in 1862 moved to Saint Sava High School in Bucharest. He showed an early talent for mathematics, publishing two textbooks (one in algebra and one in trigonometry) when he was still a high school student. In 1869 he entered the University of Bucharest, where he studied physics and mathematics. In 1870, while a student in his second term, he became teacher of mathematics at the Nifon Seminary in Bucharest, but quit the following year in order to continue his studies. In 1874, at age 23, he graduated with a degree in physics and mathematics. After graduation, Haret won a scholarship competition organized by Titu Maiorescu and went to Paris in order to study mathematics at the Sorbonne. There he earned a mathematics diploma in 1875 and a physics diploma in 1876. Two years later (on 18 January 1878), he earned his Ph.D. by defending his thesis, Sur l’invariabilité des grandes axes des orbites planétaires (On the invariability of the major axis of planetary orbits), in front of examiners led by Victor Puiseux, his Ph.D. advisor. In this work he proved a result fundamental for the n-body problem in astronomy, the thesis being published in volume 18 of Annales de l'Observatoire de Paris. Haret was the first Romanian to obtain a Ph.D. degree in Paris. After his return to Romania in 1878, Haret largely abandoned scientific research and dedicated the rest of his life to improving Romanian education, which was heavily underdeveloped at the time, both as professor and as politician. He was appointed professor of rational mechanics at the Faculty of Science of the University of Bucharest. The next year (1879), Haret became a correspondent member of the Romanian Academy, receiving full membership in 1892. He kept the professorship at the Faculty of Science until his retirement in 1910, when he was followed as professor of mechanics by Dimitrie Pompeiu. From 1882 he was also a professor of analytical geometry at the School of Bridges and Roads in Bucharest. After retirement, Haret occasionally lectured at the informal People's University. Haret was the Minister of Public Education in three liberal governments, between 1897 and 1899, 1901–1904, and 1907–1910. As Minister of Education he ran a complete reform, basically building the modern Romanian education system. The folk song "Cântă cucu-n Bucovina" ("Sings the Cuckoo in Bukovina") was composed in 1904 by Constantin Mandicevschi at Haret's request for commemorating the 400th anniversary of the death of Prince of Moldavia Stephen the Great. In January 1883, he married in Buzău a local, Ana Popescu, 15 years his junior. The two had a son, Ion, who died at age 1, and later adopted a child, Mihai. Haret died in Bucharest in 1912 of cancer, and was buried in the city's Bellu Cemetery; Ana Haret died in 1941, aged 74. == Scientific activity == Haret's major scientific contribution was made in 1878, in his Ph.D. thesis Sur l’invariabilité des grandes axes des orbites planétaires. At the time it was known that planets disturb each other's orbits, thus deviating from the elliptic motion described by Johannes Kepler’s First Law. Pierre Laplace (in 1773) and Joseph Louis Lagrange (in 1776) had already studied the problem, both of them showing that the major axes of the orbits are stable, by using a first degree approximation of the perturbing forces. In 1808 Siméon Denis Poisson had proved that the stability also holds when using second degree approximations. In his thesis, Haret proved by using third degree approximations that the axes are not stable as previously believed, but instead feature a time variability, which he called secular perturbations. This result implies that planetary motion is not absolutely stable. Henri Poincaré considered this result a great surprise and continued Haret’s research, which eventually led him to the creation of chaos theory. Haret established the instability of the model of the n-body problem assuming frequencies to be incommensurable; Poincaré also took into account commensurabilities, and using generalized Fourier series (which generate quasi-periodic solutions), he proved the divergence of these series (which means instability), thus confirming Haret’s result. Félix Tisserand recommended the extension of Haret's method to other astronomic problems and, much later, in 1955, Jean Meffroy restarted Haret’s research using new techniques. Soon after his return to Romania, Haret abandoned research, focusing for the rest of his life on teaching and, as Minister of Education, on the reform of the education system. He only published an article on the secular acceleration of the Moon in 1880 and one on Jupiter’s Great Red Spot (1912). In 1910 he published Social mechanics, which used mathematics to explain social behaviour (somehow anticipating the fictional "psychohistory" branch of mathematics developed by Hari Seldon, the fictional character of Isaac Asimov's Foundation, published 40 years later). == References == Ion Bulei (1990), Atunci când veacul se năștea... lumea româneasca 1900–1908 [When the century was born... the Romanian world 1900–1908] (in Romanian), Editura Eminescu, pp. 82–96 == External links == Constantin Schifirneț, "Spiru Haret, reformatorul societății românești" [Spiru Haret, the reformer of Romanian society], Studiu Introductiv la Operele Lui Spiru Haret, Vol. I, Editura Comunicare.ro, 2009, Pp. 13-42 (in Romanian) Constantin Schifirneț (2014), "Spiru Haret, Education and School Legislation Reform" (PDF), Revista română de sociologie, XXV (3–4): 311–326 Sorin-Avram Vîrtop (2019), "Beyond mythology and tradition of an educational reform or about the realism of Spiru Haret's educational reform (1851–1912)" (PDF), Analele Universității "Constantin Brâncuși" din Târgu Jiu, 2 (2) "Spiru Haret". bsclupan.asm.md (in Romanian). Andrei Lupan Central Scientific Library. Retrieved 21 February 2022. "Liga Spiru Haret". Archived from the original on 11 April 2009.
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Wikipedia:Split exact sequence#0
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The term split exact sequence is used in two different ways by different people. Some people mean a short exact sequence that right-splits (thus corresponding to a semidirect product) and some people mean a short exact sequence that left-splits (which implies it right-splits, and corresponds to a direct product). This article takes the latter approach, but both are in common use. When reading a book or paper, it is important to note precisely which of the two meanings is in use. In mathematics, a split exact sequence is a short exact sequence in which the middle term is built out of the two outer terms in the simplest possible way. == Equivalent characterizations == A short exact sequence of abelian groups or of modules over a fixed ring, or more generally of objects in an abelian category 0 → A → a B → b C → 0 {\displaystyle 0\to A\mathrel {\stackrel {a}{\to }} B\mathrel {\stackrel {b}{\to }} C\to 0} is called split exact if it is isomorphic to the exact sequence where the middle term is the direct sum of the outer ones: 0 → A → i A ⊕ C → p C → 0 {\displaystyle 0\to A\mathrel {\stackrel {i}{\to }} A\oplus C\mathrel {\stackrel {p}{\to }} C\to 0} The requirement that the sequence is isomorphic means that there is an isomorphism f : B → A ⊕ C {\displaystyle f:B\to A\oplus C} such that the composite f ∘ a {\displaystyle f\circ a} is the natural inclusion i : A → A ⊕ C {\displaystyle i:A\to A\oplus C} and such that the composite p ∘ f {\displaystyle p\circ f} equals b. This can be summarized by a commutative diagram as: The splitting lemma provides further equivalent characterizations of split exact sequences. == Examples == A trivial example of a split short exact sequence is 0 → M 1 → q M 1 ⊕ M 2 → p M 2 → 0 {\displaystyle 0\to M_{1}\mathrel {\stackrel {q}{\to }} M_{1}\oplus M_{2}\mathrel {\stackrel {p}{\to }} M_{2}\to 0} where M 1 , M 2 {\displaystyle M_{1},M_{2}} are R-modules, q {\displaystyle q} is the canonical injection and p {\displaystyle p} is the canonical projection. Any short exact sequence of vector spaces is split exact. This is a rephrasing of the fact that any set of linearly independent vectors in a vector space can be extended to a basis. The exact sequence 0 → Z → 2 Z → Z / 2 Z → 0 {\displaystyle 0\to \mathbf {Z} \mathrel {\stackrel {2}{\to }} \mathbf {Z} \to \mathbf {Z} /2\mathbf {Z} \to 0} (where the first map is multiplication by 2) is not split exact. == Related notions == Pure exact sequences can be characterized as the filtered colimits of split exact sequences. == References == == Sources == Fuchs, László (2015), Abelian Groups, Springer Monographs in Mathematics, Springer, ISBN 9783319194226 Sharp, R. Y., Rodney (2001), Steps in Commutative Algebra, 2nd ed., London Mathematical Society Student Texts, Cambridge University Press, ISBN 0521646235
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Wikipedia:Split-complex number#0
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In algebra, a split-complex number (or hyperbolic number, also perplex number, double number) is based on a hyperbolic unit j satisfying j 2 = 1 {\displaystyle j^{2}=1} , where j ≠ ± 1 {\displaystyle j\neq \pm 1} . A split-complex number has two real number components x and y, and is written z = x + y j . {\displaystyle z=x+yj.} The conjugate of z is z ∗ = x − y j . {\displaystyle z^{*}=x-yj.} Since j 2 = 1 , {\displaystyle j^{2}=1,} the product of a number z with its conjugate is N ( z ) := z z ∗ = x 2 − y 2 , {\displaystyle N(z):=zz^{*}=x^{2}-y^{2},} an isotropic quadratic form. The collection D of all split-complex numbers z = x + y j {\displaystyle z=x+yj} for x , y ∈ R {\displaystyle x,y\in \mathbb {R} } forms an algebra over the field of real numbers. Two split-complex numbers w and z have a product wz that satisfies N ( w z ) = N ( w ) N ( z ) . {\displaystyle N(wz)=N(w)N(z).} This composition of N over the algebra product makes (D, +, ×, *) a composition algebra. A similar algebra based on R 2 {\displaystyle \mathbb {R} ^{2}} and component-wise operations of addition and multiplication, ( R 2 , + , × , x y ) , {\displaystyle (\mathbb {R} ^{2},+,\times ,xy),} where xy is the quadratic form on R 2 , {\displaystyle \mathbb {R} ^{2},} also forms a quadratic space. The ring isomorphism D → R 2 x + y j ↦ ( x − y , x + y ) {\displaystyle {\begin{aligned}D&\to \mathbb {R} ^{2}\\x+yj&\mapsto (x-y,x+y)\end{aligned}}} is an isometry of quadratic spaces. Split-complex numbers have many other names; see § Synonyms below. See the article Motor variable for functions of a split-complex number. == Definition == A split-complex number is an ordered pair of real numbers, written in the form z = x + j y {\displaystyle z=x+jy} where x and y are real numbers and the hyperbolic unit j satisfies j 2 = + 1 {\displaystyle j^{2}=+1} In the field of complex numbers the imaginary unit i satisfies i 2 = − 1. {\displaystyle i^{2}=-1.} The change of sign distinguishes the split-complex numbers from the ordinary complex ones. The hyperbolic unit j is not a real number but an independent quantity. The collection of all such z is called the split-complex plane. Addition and multiplication of split-complex numbers are defined by ( x + j y ) + ( u + j v ) = ( x + u ) + j ( y + v ) ( x + j y ) ( u + j v ) = ( x u + y v ) + j ( x v + y u ) . {\displaystyle {\begin{aligned}(x+jy)+(u+jv)&=(x+u)+j(y+v)\\(x+jy)(u+jv)&=(xu+yv)+j(xv+yu).\end{aligned}}} This multiplication is commutative, associative and distributes over addition. === Conjugate, modulus, and bilinear form === Just as for complex numbers, one can define the notion of a split-complex conjugate. If z = x + j y , {\displaystyle z=x+jy~,} then the conjugate of z is defined as z ∗ = x − j y . {\displaystyle z^{*}=x-jy~.} The conjugate is an involution which satisfies similar properties to the complex conjugate. Namely, ( z + w ) ∗ = z ∗ + w ∗ ( z w ) ∗ = z ∗ w ∗ ( z ∗ ) ∗ = z . {\displaystyle {\begin{aligned}(z+w)^{*}&=z^{*}+w^{*}\\(zw)^{*}&=z^{*}w^{*}\\\left(z^{*}\right)^{*}&=z.\end{aligned}}} The squared modulus of a split-complex number z = x + j y {\displaystyle z=x+jy} is given by the isotropic quadratic form ‖ z ‖ 2 = z z ∗ = z ∗ z = x 2 − y 2 . {\displaystyle \lVert z\rVert ^{2}=zz^{*}=z^{*}z=x^{2}-y^{2}~.} It has the composition algebra property: ‖ z w ‖ = ‖ z ‖ ‖ w ‖ . {\displaystyle \lVert zw\rVert =\lVert z\rVert \lVert w\rVert ~.} However, this quadratic form is not positive-definite but rather has signature (1, −1), so the modulus is not a norm. The associated bilinear form is given by ⟨ z , w ⟩ = R e ( z w ∗ ) = R e ( z ∗ w ) = x u − y v , {\displaystyle \langle z,w\rangle =\operatorname {\mathrm {Re} } \left(zw^{*}\right)=\operatorname {\mathrm {Re} } \left(z^{*}w\right)=xu-yv~,} where z = x + j y {\displaystyle z=x+jy} and w = u + j v . {\displaystyle w=u+jv.} Here, the real part is defined by R e ( z ) = 1 2 ( z + z ∗ ) = x {\displaystyle \operatorname {\mathrm {Re} } (z)={\tfrac {1}{2}}(z+z^{*})=x} . Another expression for the squared modulus is then ‖ z ‖ 2 = ⟨ z , z ⟩ . {\displaystyle \lVert z\rVert ^{2}=\langle z,z\rangle ~.} Since it is not positive-definite, this bilinear form is not an inner product; nevertheless the bilinear form is frequently referred to as an indefinite inner product. A similar abuse of language refers to the modulus as a norm. A split-complex number is invertible if and only if its modulus is nonzero ( ‖ z ‖ ≠ 0 {\displaystyle \lVert z\rVert \neq 0} ), thus numbers of the form x ± j x have no inverse. The multiplicative inverse of an invertible element is given by z − 1 = z ∗ ‖ z ‖ 2 . {\displaystyle z^{-1}={\frac {z^{*}}{{\lVert z\rVert }^{2}}}~.} Split-complex numbers which are not invertible are called null vectors. These are all of the form (a ± j a) for some real number a. === The diagonal basis === There are two nontrivial idempotent elements given by e = 1 2 ( 1 − j ) {\displaystyle e={\tfrac {1}{2}}(1-j)} and e ∗ = 1 2 ( 1 + j ) . {\displaystyle e^{*}={\tfrac {1}{2}}(1+j).} Idempotency means that e e = e {\displaystyle ee=e} and e ∗ e ∗ = e ∗ . {\displaystyle e^{*}e^{*}=e^{*}.} Both of these elements are null: ‖ e ‖ = ‖ e ∗ ‖ = e ∗ e = 0 . {\displaystyle \lVert e\rVert =\lVert e^{*}\rVert =e^{*}e=0~.} It is often convenient to use e and e∗ as an alternate basis for the split-complex plane. This basis is called the diagonal basis or null basis. The split-complex number z can be written in the null basis as z = x + j y = ( x − y ) e + ( x + y ) e ∗ . {\displaystyle z=x+jy=(x-y)e+(x+y)e^{*}~.} If we denote the number z = a e + b e ∗ {\displaystyle z=ae+be^{*}} for real numbers a and b by (a, b), then split-complex multiplication is given by ( a 1 , b 1 ) ( a 2 , b 2 ) = ( a 1 a 2 , b 1 b 2 ) . {\displaystyle \left(a_{1},b_{1}\right)\left(a_{2},b_{2}\right)=\left(a_{1}a_{2},b_{1}b_{2}\right)~.} The split-complex conjugate in the diagonal basis is given by ( a , b ) ∗ = ( b , a ) {\displaystyle (a,b)^{*}=(b,a)} and the squared modulus by ‖ ( a , b ) ‖ 2 = a b . {\displaystyle \lVert (a,b)\rVert ^{2}=ab.} === Isomorphism === On the basis {e, e*} it becomes clear that the split-complex numbers are ring-isomorphic to the direct sum R ⊕ R {\displaystyle \mathbb {R} \oplus \mathbb {R} } with addition and multiplication defined pairwise. The diagonal basis for the split-complex number plane can be invoked by using an ordered pair (x, y) for z = x + j y {\displaystyle z=x+jy} and making the mapping ( u , v ) = ( x , y ) ( 1 1 1 − 1 ) = ( x , y ) S . {\displaystyle (u,v)=(x,y){\begin{pmatrix}1&1\\1&-1\end{pmatrix}}=(x,y)S~.} Now the quadratic form is u v = ( x + y ) ( x − y ) = x 2 − y 2 . {\displaystyle uv=(x+y)(x-y)=x^{2}-y^{2}~.} Furthermore, ( cosh a , sinh a ) ( 1 1 1 − 1 ) = ( e a , e − a ) {\displaystyle (\cosh a,\sinh a){\begin{pmatrix}1&1\\1&-1\end{pmatrix}}=\left(e^{a},e^{-a}\right)} so the two parametrized hyperbolas are brought into correspondence with S. The action of hyperbolic versor e b j {\displaystyle e^{bj}\!} then corresponds under this linear transformation to a squeeze mapping σ : ( u , v ) ↦ ( r u , v r ) , r = e b . {\displaystyle \sigma :(u,v)\mapsto \left(ru,{\frac {v}{r}}\right),\quad r=e^{b}~.} Though lying in the same isomorphism class in the category of rings, the split-complex plane and the direct sum of two real lines differ in their layout in the Cartesian plane. The isomorphism, as a planar mapping, consists of a counter-clockwise rotation by 45° and a dilation by √2. The dilation in particular has sometimes caused confusion in connection with areas of a hyperbolic sector. Indeed, hyperbolic angle corresponds to area of a sector in the R ⊕ R {\displaystyle \mathbb {R} \oplus \mathbb {R} } plane with its "unit circle" given by { ( a , b ) ∈ R ⊕ R : a b = 1 } . {\displaystyle \{(a,b)\in \mathbb {R} \oplus \mathbb {R} :ab=1\}.} The contracted unit hyperbola { cosh a + j sinh a : a ∈ R } {\displaystyle \{\cosh a+j\sinh a:a\in \mathbb {R} \}} of the split-complex plane has only half the area in the span of a corresponding hyperbolic sector. Such confusion may be perpetuated when the geometry of the split-complex plane is not distinguished from that of R ⊕ R {\displaystyle \mathbb {R} \oplus \mathbb {R} } . == Geometry == A two-dimensional real vector space with the Minkowski inner product is called (1 + 1)-dimensional Minkowski space, often denoted R 1 , 1 . {\displaystyle \mathbb {R} ^{1,1}.} Just as much of the geometry of the Euclidean plane R 2 {\displaystyle \mathbb {R} ^{2}} can be described with complex numbers, the geometry of the Minkowski plane R 1 , 1 {\displaystyle \mathbb {R} ^{1,1}} can be described with split-complex numbers. The set of points { z : ‖ z ‖ 2 = a 2 } {\displaystyle \left\{z:\lVert z\rVert ^{2}=a^{2}\right\}} is a hyperbola for every nonzero a in R . {\displaystyle \mathbb {R} .} The hyperbola consists of a right and left branch passing through (a, 0) and (−a, 0). The case a = 1 is called the unit hyperbola. The conjugate hyperbola is given by { z : ‖ z ‖ 2 = − a 2 } {\displaystyle \left\{z:\lVert z\rVert ^{2}=-a^{2}\right\}} with an upper and lower branch passing through (0, a) and (0, −a). The hyperbola and conjugate hyperbola are separated by two diagonal asymptotes which form the set of null elements: { z : ‖ z ‖ = 0 } . {\displaystyle \left\{z:\lVert z\rVert =0\right\}.} These two lines (sometimes called the null cone) are perpendicular in R 2 {\displaystyle \mathbb {R} ^{2}} and have slopes ±1. Split-complex numbers z and w are said to be hyperbolic-orthogonal if ⟨z, w⟩ = 0. While analogous to ordinary orthogonality, particularly as it is known with ordinary complex number arithmetic, this condition is more subtle. It forms the basis for the simultaneous hyperplane concept in spacetime. The analogue of Euler's formula for the split-complex numbers is exp ( j θ ) = cosh ( θ ) + j sinh ( θ ) . {\displaystyle \exp(j\theta )=\cosh(\theta )+j\sinh(\theta ).} This formula can be derived from a power series expansion using the fact that cosh has only even powers while that for sinh has odd powers. For all real values of the hyperbolic angle θ the split-complex number λ = exp(jθ) has norm 1 and lies on the right branch of the unit hyperbola. Numbers such as λ have been called hyperbolic versors. Since λ has modulus 1, multiplying any split-complex number z by λ preserves the modulus of z and represents a hyperbolic rotation (also called a Lorentz boost or a squeeze mapping). Multiplying by λ preserves the geometric structure, taking hyperbolas to themselves and the null cone to itself. The set of all transformations of the split-complex plane which preserve the modulus (or equivalently, the inner product) forms a group called the generalized orthogonal group O(1, 1). This group consists of the hyperbolic rotations, which form a subgroup denoted SO+(1, 1), combined with four discrete reflections given by z ↦ ± z {\displaystyle z\mapsto \pm z} and z ↦ ± z ∗ . {\displaystyle z\mapsto \pm z^{*}.} The exponential map exp : ( R , + ) → S O + ( 1 , 1 ) {\displaystyle \exp \colon (\mathbb {R} ,+)\to \mathrm {SO} ^{+}(1,1)} sending θ to rotation by exp(jθ) is a group isomorphism since the usual exponential formula applies: e j ( θ + ϕ ) = e j θ e j ϕ . {\displaystyle e^{j(\theta +\phi )}=e^{j\theta }e^{j\phi }.} If a split-complex number z does not lie on one of the diagonals, then z has a polar decomposition. == Algebraic properties == In abstract algebra terms, the split-complex numbers can be described as the quotient of the polynomial ring R [ x ] {\displaystyle \mathbb {R} [x]} by the ideal generated by the polynomial x 2 − 1 , {\displaystyle x^{2}-1,} R [ x ] / ( x 2 − 1 ) . {\displaystyle \mathbb {R} [x]/(x^{2}-1).} The image of x in the quotient is the "imaginary" unit j. With this description, it is clear that the split-complex numbers form a commutative algebra over the real numbers. The algebra is not a field since the null elements are not invertible. All of the nonzero null elements are zero divisors. Since addition and multiplication are continuous operations with respect to the usual topology of the plane, the split-complex numbers form a topological ring. The algebra of split-complex numbers forms a composition algebra since ‖ z w ‖ = ‖ z ‖ ‖ w ‖ {\displaystyle \lVert zw\rVert =\lVert z\rVert \lVert w\rVert ~} for any numbers z and w. From the definition it is apparent that the ring of split-complex numbers is isomorphic to the group ring R [ C 2 ] {\displaystyle \mathbb {R} [C_{2}]} of the cyclic group C2 over the real numbers R . {\displaystyle \mathbb {R} .} Elements of the identity component in the group of units in D have four square roots.: say p = exp ( q ) , q ∈ D . then ± exp ( q 2 ) {\displaystyle p=\exp(q),\ \ q\in D.{\text{then}}\pm \exp({\frac {q}{2}})} are square roots of p. Further, ± j exp ( q 2 ) {\displaystyle \pm j\exp({\frac {q}{2}})} are also square roots of p. The idempotents 1 ± j 2 {\displaystyle {\frac {1\pm j}{2}}} are their own square roots, and the square root of s 1 ± j 2 , s > 0 , is s 1 ± j 2 {\displaystyle s{\frac {1\pm j}{2}},\ \ s>0,\ {\text{is}}\ {\sqrt {s}}{\frac {1\pm j}{2}}} == Matrix representations == One can easily represent split-complex numbers by matrices. The split-complex number z = x + j y {\displaystyle z=x+jy} can be represented by the matrix z ↦ ( x y y x ) . {\displaystyle z\mapsto {\begin{pmatrix}x&y\\y&x\end{pmatrix}}.} Addition and multiplication of split-complex numbers are then given by matrix addition and multiplication. The squared modulus of z is given by the determinant of the corresponding matrix. In fact there are many representations of the split-complex plane in the four-dimensional ring of 2x2 real matrices. The real multiples of the identity matrix form a real line in the matrix ring M(2,R). Any hyperbolic unit m provides a basis element with which to extend the real line to the split-complex plane. The matrices m = ( a c b − a ) {\displaystyle m={\begin{pmatrix}a&c\\b&-a\end{pmatrix}}} which square to the identity matrix satisfy a 2 + b c = 1. {\displaystyle a^{2}+bc=1.} For example, when a = 0, then (b,c) is a point on the standard hyperbola. More generally, there is a hypersurface in M(2,R) of hyperbolic units, any one of which serves in a basis to represent the split-complex numbers as a subring of M(2,R). The number z = x + j y {\displaystyle z=x+jy} can be represented by the matrix x I + y m . {\displaystyle x\ I+y\ m.} == History == The use of split-complex numbers dates back to 1848 when James Cockle revealed his tessarines. William Kingdon Clifford used split-complex numbers to represent sums of spins. Clifford introduced the use of split-complex numbers as coefficients in a quaternion algebra now called split-biquaternions. He called its elements "motors", a term in parallel with the "rotor" action of an ordinary complex number taken from the circle group. Extending the analogy, functions of a motor variable contrast to functions of an ordinary complex variable. Since the late twentieth century, the split-complex multiplication has commonly been seen as a Lorentz boost of a spacetime plane. In that model, the number z = x + y j represents an event in a spatio-temporal plane, where x is measured in seconds and y in light-seconds. The future corresponds to the quadrant of events {z : |y| < x}, which has the split-complex polar decomposition z = ρ e a j {\displaystyle z=\rho e^{aj}\!} . The model says that z can be reached from the origin by entering a frame of reference of rapidity a and waiting ρ nanoseconds. The split-complex equation e a j e b j = e ( a + b ) j {\displaystyle e^{aj}\ e^{bj}=e^{(a+b)j}} expressing products on the unit hyperbola illustrates the additivity of rapidities for collinear velocities. Simultaneity of events depends on rapidity a; { z = σ j e a j : σ ∈ R } {\displaystyle \{z=\sigma je^{aj}:\sigma \in \mathbb {R} \}} is the line of events simultaneous with the origin in the frame of reference with rapidity a. Two events z and w are hyperbolic-orthogonal when z ∗ w + z w ∗ = 0. {\displaystyle z^{*}w+zw^{*}=0.} Canonical events exp(aj) and j exp(aj) are hyperbolic orthogonal and lie on the axes of a frame of reference in which the events simultaneous with the origin are proportional to j exp(aj). In 1933 Max Zorn was using the split-octonions and noted the composition algebra property. He realized that the Cayley–Dickson construction, used to generate division algebras, could be modified (with a factor gamma, γ) to construct other composition algebras including the split-octonions. His innovation was perpetuated by Adrian Albert, Richard D. Schafer, and others. The gamma factor, with R as base field, builds split-complex numbers as a composition algebra. Reviewing Albert for Mathematical Reviews, N. H. McCoy wrote that there was an "introduction of some new algebras of order 2e over F generalizing Cayley–Dickson algebras." Taking F = R and e = 1 corresponds to the algebra of this article. In 1935 J.C. Vignaux and A. Durañona y Vedia developed the split-complex geometric algebra and function theory in four articles in Contribución a las Ciencias Físicas y Matemáticas, National University of La Plata, República Argentina (in Spanish). These expository and pedagogical essays presented the subject for broad appreciation. In 1941 E.F. Allen used the split-complex geometric arithmetic to establish the nine-point hyperbola of a triangle inscribed in zz∗ = 1. In 1956 Mieczyslaw Warmus published "Calculus of Approximations" in Bulletin de l’Académie polonaise des sciences (see link in References). He developed two algebraic systems, each of which he called "approximate numbers", the second of which forms a real algebra. D. H. Lehmer reviewed the article in Mathematical Reviews and observed that this second system was isomorphic to the "hyperbolic complex" numbers, the subject of this article. In 1961 Warmus continued his exposition, referring to the components of an approximate number as midpoint and radius of the interval denoted. == Synonyms == Different authors have used a great variety of names for the split-complex numbers. Some of these include: (real) tessarines, James Cockle (1848) (algebraic) motors, W.K. Clifford (1882) hyperbolic complex numbers, J.C. Vignaux (1935), G. Cree (1949) bireal numbers, U. Bencivenga (1946) real hyperbolic numbers, N. Smith (1949) approximate numbers, Warmus (1956), for use in interval analysis double numbers, I.M. Yaglom (1968), Kantor and Solodovnikov (1989), Hazewinkel (1990), Rooney (2014) hyperbolic numbers, W. Miller & R. Boehning (1968), G. Sobczyk (1995) anormal-complex numbers, W. Benz (1973) perplex numbers, P. Fjelstad (1986) and Poodiack & LeClair (2009) countercomplex or hyperbolic, Carmody (1988) Lorentz numbers, F.R. Harvey (1990) semi-complex numbers, F. Antonuccio (1994) paracomplex numbers, Cruceanu, Fortuny & Gadea (1996) split-complex numbers, B. Rosenfeld (1997) spacetime numbers, N. Borota (2000) Study numbers, P. Lounesto (2001) twocomplex numbers, S. Olariu (2002) split binarions, K. McCrimmon (2004) == See also == Minkowski space Split-quaternion Hypercomplex number == References == == Further reading == Bencivenga, Uldrico (1946) "Sulla rappresentazione geometrica delle algebre doppie dotate di modulo", Atti della Reale Accademia delle Scienze e Belle-Lettere di Napoli, Ser (3) v.2 No7. MR0021123. Walter Benz (1973) Vorlesungen uber Geometrie der Algebren, Springer N. A. Borota, E. Flores, and T. J. Osler (2000) "Spacetime numbers the easy way", Mathematics and Computer Education 34: 159–168. N. A. Borota and T. J. Osler (2002) "Functions of a spacetime variable", Mathematics and Computer Education 36: 231–239. K. Carmody, (1988) "Circular and hyperbolic quaternions, octonions, and sedenions", Appl. Math. Comput. 28:47–72. K. Carmody, (1997) "Circular and hyperbolic quaternions, octonions, and sedenions – further results", Appl. Math. Comput. 84:27–48. William Kingdon Clifford (1882) Mathematical Works, A. W. Tucker editor, page 392, "Further Notes on Biquaternions" V.Cruceanu, P. Fortuny & P.M. Gadea (1996) A Survey on Paracomplex Geometry, Rocky Mountain Journal of Mathematics 26(1): 83–115, link from Project Euclid. De Boer, R. (1987) "An also known as list for perplex numbers", American Journal of Physics 55(4):296. Anthony A. Harkin & Joseph B. Harkin (2004) Geometry of Generalized Complex Numbers, Mathematics Magazine 77(2):118–29. F. Reese Harvey. Spinors and calibrations. Academic Press, San Diego. 1990. ISBN 0-12-329650-1. Contains a description of normed algebras in indefinite signature, including the Lorentz numbers. Hazewinkle, M. (1994) "Double and dual numbers", Encyclopaedia of Mathematics, Soviet/AMS/Kluwer, Dordrect. Kevin McCrimmon (2004) A Taste of Jordan Algebras, pp 66, 157, Universitext, Springer ISBN 0-387-95447-3 MR2014924 C. Musès, "Applied hypernumbers: Computational concepts", Appl. Math. Comput. 3 (1977) 211–226. C. Musès, "Hypernumbers II—Further concepts and computational applications", Appl. Math. Comput. 4 (1978) 45–66. Olariu, Silviu (2002) Complex Numbers in N Dimensions, Chapter 1: Hyperbolic Complex Numbers in Two Dimensions, pages 1–16, North-Holland Mathematics Studies #190, Elsevier ISBN 0-444-51123-7. Poodiack, Robert D. & Kevin J. LeClair (2009) "Fundamental theorems of algebra for the perplexes", The College Mathematics Journal 40(5):322–35. Isaak Yaglom (1968) Complex Numbers in Geometry, translated by E. Primrose from 1963 Russian original, Academic Press, pp. 18–20. J. Rooney (2014). "Generalised Complex Numbers in Mechanics". In Marco Ceccarelli and Victor A. Glazunov (ed.). Advances on Theory and Practice of Robots and Manipulators: Proceedings of Romansy 2014 XX CISM-IFToMM Symposium on Theory and Practice of Robots and Manipulators. Mechanisms and Machine Science. Vol. 22. Springer. pp. 55–62. doi:10.1007/978-3-319-07058-2_7. ISBN 978-3-319-07058-2.
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Wikipedia:Splitting lemma (functions)#0
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In mathematics, especially in singularity theory, the splitting lemma is a useful result due to René Thom which provides a way of simplifying the local expression of a function usually applied in a neighbourhood of a degenerate critical point. == Formal statement == Let f : ( R n , 0 ) → ( R , 0 ) {\displaystyle f:(\mathbb {R} ^{n},0)\to (\mathbb {R} ,0)} be a smooth function germ, with a critical point at 0 (so ( ∂ f / ∂ x i ) ( 0 ) = 0 {\displaystyle (\partial f/\partial x_{i})(0)=0} for i = 1 , … , n {\displaystyle i=1,\dots ,n} ). Let V be a subspace of R n {\displaystyle \mathbb {R} ^{n}} such that the restriction f |V is non-degenerate, and write B for the Hessian matrix of this restriction. Let W be any complementary subspace to V. Then there is a change of coordinates Φ ( x , y ) {\displaystyle \Phi (x,y)} of the form Φ ( x , y ) = ( ϕ ( x , y ) , y ) {\displaystyle \Phi (x,y)=(\phi (x,y),y)} with x ∈ V , y ∈ W {\displaystyle x\in V,y\in W} , and a smooth function h on W such that f ∘ Φ ( x , y ) = 1 2 x T B x + h ( y ) . {\displaystyle f\circ \Phi (x,y)={\frac {1}{2}}x^{T}Bx+h(y).} This result is often referred to as the parametrized Morse lemma, which can be seen by viewing y as the parameter. It is the gradient version of the implicit function theorem. == Extensions == There are extensions to infinite dimensions, to complex analytic functions, to functions invariant under the action of a compact group, ... == References == Poston, Tim; Stewart, Ian (1979), Catastrophe Theory and Its Applications, Pitman, ISBN 978-0-273-08429-7. Brocker, Th (1975), Differentiable Germs and Catastrophes, Cambridge University Press, ISBN 978-0-521-20681-5.
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Wikipedia:Spread of a matrix#0
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In mathematics, and more specifically matrix theory, the spread of a matrix is the largest distance in the complex plane between any two eigenvalues of the matrix. == Definition == Let A {\displaystyle A} be a square matrix with eigenvalues λ 1 , … , λ n {\displaystyle \lambda _{1},\ldots ,\lambda _{n}} . That is, these values λ i {\displaystyle \lambda _{i}} are the complex numbers such that there exists a vector v i {\displaystyle v_{i}} on which A {\displaystyle A} acts by scalar multiplication: A v i = λ i v i . {\displaystyle Av_{i}=\lambda _{i}v_{i}.} Then the spread of A {\displaystyle A} is the non-negative number s ( A ) = max { | λ i − λ j | : i , j = 1 , … n } . {\displaystyle s(A)=\max\{|\lambda _{i}-\lambda _{j}|:i,j=1,\ldots n\}.} == Examples == For the zero matrix and the identity matrix, the spread is zero. The zero matrix has only zero as its eigenvalues, and the identity matrix has only one as its eigenvalues. In both cases, all eigenvalues are equal, so no two eigenvalues can be at nonzero distance from each other. For a projection, the only eigenvalues are zero and one. A projection matrix therefore has a spread that is either 0 {\displaystyle 0} (if all eigenvalues are equal) or 1 {\displaystyle 1} (if there are two different eigenvalues). All eigenvalues of a unitary matrix A {\displaystyle A} lie on the unit circle. Therefore, in this case, the spread is at most equal to the diameter of the circle, the number 2. The spread of a matrix depends only on the spectrum of the matrix (its multiset of eigenvalues). If a second matrix B {\displaystyle B} of the same size is invertible, then B A B − 1 {\displaystyle BAB^{-1}} has the same spectrum as A {\displaystyle A} . Therefore, it also has the same spread as A {\displaystyle A} . == See also == Field of values == References == Marvin Marcus and Henryk Minc, A survey of matrix theory and matrix inequalities, Dover Publications, 1992, ISBN 0-486-67102-X. Chap.III.4.
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Wikipedia:Square (algebra)#0
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In mathematics, a square is the result of multiplying a number by itself. The verb "to square" is used to denote this operation. Squaring is the same as raising to the power 2, and is denoted by a superscript 2; for instance, the square of 3 may be written as 32, which is the number 9. In some cases when superscripts are not available, as for instance in programming languages or plain text files, the notations x^2 (caret) or x**2 may be used in place of x2. The adjective which corresponds to squaring is quadratic. The square of an integer may also be called a square number or a perfect square. In algebra, the operation of squaring is often generalized to polynomials, other expressions, or values in systems of mathematical values other than the numbers. For instance, the square of the linear polynomial x + 1 is the quadratic polynomial (x + 1)2 = x2 + 2x + 1. One of the important properties of squaring, for numbers as well as in many other mathematical systems, is that (for all numbers x), the square of x is the same as the square of its additive inverse −x. That is, the square function satisfies the identity x2 = (−x)2. This can also be expressed by saying that the square function is an even function. == In real numbers == The squaring operation defines a real function called the square function or the squaring function. Its domain is the whole real line, and its image is the set of nonnegative real numbers. The square function preserves the order of positive numbers: larger numbers have larger squares. In other words, the square is a monotonic function on the interval [0, +∞). On the negative numbers, numbers with greater absolute value have greater squares, so the square is a monotonically decreasing function on (−∞,0]. Hence, zero is the (global) minimum of the square function. The square x2 of a number x is less than x (that is x2 < x) if and only if 0 < x < 1, that is, if x belongs to the open interval (0,1). This implies that the square of an integer is never less than the original number x. Every positive real number is the square of exactly two numbers, one of which is strictly positive and the other of which is strictly negative. Zero is the square of only one number, itself. For this reason, it is possible to define the square root function, which associates with a non-negative real number the non-negative number whose square is the original number. No square root can be taken of a negative number within the system of real numbers, because squares of all real numbers are non-negative. The lack of real square roots for the negative numbers can be used to expand the real number system to the complex numbers, by postulating the imaginary unit i, which is one of the square roots of −1. The property "every non-negative real number is a square" has been generalized to the notion of a real closed field, which is an ordered field such that every non-negative element is a square and every polynomial of odd degree has a root. The real closed fields cannot be distinguished from the field of real numbers by their algebraic properties: every property of the real numbers, which may be expressed in first-order logic (that is expressed by a formula in which the variables that are quantified by ∀ or ∃ represent elements, not sets), is true for every real closed field, and conversely every property of the first-order logic, which is true for a specific real closed field is also true for the real numbers. == In geometry == There are several major uses of the square function in geometry. The name of the square function shows its importance in the definition of the area: it comes from the fact that the area of a square with sides of length l is equal to l2. The area depends quadratically on the size: the area of a shape n times larger is n2 times greater. This holds for areas in three dimensions as well as in the plane: for instance, the surface area of a sphere is proportional to the square of its radius, a fact that is manifested physically by the inverse-square law describing how the strength of physical forces such as gravity varies according to distance. The square function is related to distance through the Pythagorean theorem and its generalization, the parallelogram law. Euclidean distance is not a smooth function: the three-dimensional graph of distance from a fixed point forms a cone, with a non-smooth point at the tip of the cone. However, the square of the distance (denoted d2 or r2), which has a paraboloid as its graph, is a smooth and analytic function. The dot product of a Euclidean vector with itself is equal to the square of its length: v⋅v = v2. This is further generalised to quadratic forms in linear spaces via the inner product. The inertia tensor in mechanics is an example of a quadratic form. It demonstrates a quadratic relation of the moment of inertia to the size (length). There are infinitely many Pythagorean triples, sets of three positive integers such that the sum of the squares of the first two equals the square of the third. Each of these triples gives the integer sides of a right triangle. == In abstract algebra and number theory == The square function is defined in any field or ring. An element in the image of this function is called a square, and the inverse images of a square are called square roots. The notion of squaring is particularly important in the finite fields Z/pZ formed by the numbers modulo an odd prime number p. A non-zero element of this field is called a quadratic residue if it is a square in Z/pZ, and otherwise, it is called a quadratic non-residue. Zero, while a square, is not considered to be a quadratic residue. Every finite field of this type has exactly (p − 1)/2 quadratic residues and exactly (p − 1)/2 quadratic non-residues. The quadratic residues form a group under multiplication. The properties of quadratic residues are widely used in number theory. More generally, in rings, the square function may have different properties that are sometimes used to classify rings. Zero may be the square of some non-zero elements. A commutative ring such that the square of a non zero element is never zero is called a reduced ring. More generally, in a commutative ring, a radical ideal is an ideal I such that x 2 ∈ I {\displaystyle x^{2}\in I} implies x ∈ I {\displaystyle x\in I} . Both notions are important in algebraic geometry, because of Hilbert's Nullstellensatz. An element of a ring that is equal to its own square is called an idempotent. In any ring, 0 and 1 are idempotents. There are no other idempotents in fields and more generally in integral domains. However, the ring of the integers modulo n has 2k idempotents, where k is the number of distinct prime factors of n. A commutative ring in which every element is equal to its square (every element is idempotent) is called a Boolean ring; an example from computer science is the ring whose elements are binary numbers, with bitwise AND as the multiplication operation and bitwise XOR as the addition operation. In a totally ordered ring, x2 ≥ 0 for any x. Moreover, x2 = 0 if and only if x = 0. In a supercommutative algebra where 2 is invertible, the square of any odd element equals zero. If A is a commutative semigroup, then one has ∀ x , y ∈ A ( x y ) 2 = x y x y = x x y y = x 2 y 2 . {\displaystyle \forall x,y\in A\quad (xy)^{2}=xyxy=xxyy=x^{2}y^{2}.} In the language of quadratic forms, this equality says that the square function is a "form permitting composition". In fact, the square function is the foundation upon which other quadratic forms are constructed which also permit composition. The procedure was introduced by L. E. Dickson to produce the octonions out of quaternions by doubling. The doubling method was formalized by A. A. Albert who started with the real number field R {\displaystyle \mathbb {R} } and the square function, doubling it to obtain the complex number field with quadratic form x2 + y2, and then doubling again to obtain quaternions. The doubling procedure is called the Cayley–Dickson construction, and has been generalized to form algebras of dimension 2n over a field F with involution. The square function z2 is the "norm" of the composition algebra C {\displaystyle \mathbb {C} } , where the identity function forms a trivial involution to begin the Cayley–Dickson constructions leading to bicomplex, biquaternion, and bioctonion composition algebras. == In complex numbers == On complex numbers, the square function z → z 2 {\displaystyle z\to z^{2}} is a twofold cover in the sense that each non-zero complex number has exactly two square roots. The square of the absolute value of a complex number is called its absolute square, squared modulus, or squared magnitude. It is the product of the complex number with its complex conjugate, and equals the sum of the squares of the real and imaginary parts of the complex number. The absolute square of a complex number is always a nonnegative real number, that is zero if and only if the complex number is zero. It is easier to compute than the absolute value (no square root), and is a smooth real-valued function. Because of these two properties, the absolute square is often preferred to the absolute value for explicit computations and when methods of mathematical analysis are involved (for example optimization or integration). For complex vectors, the dot product can be defined involving the conjugate transpose, leading to the squared norm. == Other uses == Squares are ubiquitous in algebra, more generally, in almost every branch of mathematics, and also in physics where many units are defined using squares and inverse squares: see below. Least squares is the standard method used with overdetermined systems. Squaring is used in statistics and probability theory in determining the standard deviation of a set of values, or a random variable. The deviation of each value xi from the mean x ¯ {\displaystyle {\overline {x}}} of the set is defined as the difference x i − x ¯ {\displaystyle x_{i}-{\overline {x}}} . These deviations are squared, then a mean is taken of the new set of numbers (each of which is positive). This mean is the variance, and its square root is the standard deviation. == See also == Cube (algebra) Euclidean distance Exponentiation by squaring Hilbert's seventeenth problem, for the representation of positive polynomials as a sum of squares of rational functions Metric tensor Polynomial ring Polynomial SOS, the representation of a non-negative polynomial as the sum of squares of polynomials Quadratic equation Square-free polynomial Sums of squares (disambiguation page with various relevant links) === Related identities === Algebraic (need a commutative ring) Brahmagupta–Fibonacci identity, related to complex numbers in the sense discussed above Degen's eight-square identity, related to octonions in the same way Difference of two squares Euler's four-square identity, related to quaternions in the same way Lagrange's identity Other Parseval's identity Pythagorean trigonometric identity === Related physical quantities === acceleration, length per square time coupling constant (has square charge in the denominator, and may be expressed with square distance in the numerator) cross section (physics), an area-dimensioned quantity kinetic energy (quadratic dependence on velocity) specific energy, a (square velocity)-dimensioned quantity == Footnotes == == Further reading == Marshall, Murray Positive polynomials and sums of squares. Mathematical Surveys and Monographs, 146. American Mathematical Society, Providence, RI, 2008. xii+187 pp. ISBN 978-0-8218-4402-1, ISBN 0-8218-4402-4 Rajwade, A. R. (1993). Squares. London Mathematical Society Lecture Note Series. Vol. 171. Cambridge University Press. ISBN 0-521-42668-5. Zbl 0785.11022.
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Wikipedia:Square class#0
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In mathematics, specifically abstract algebra, a square class of a field F {\displaystyle F} is an element of the square class group, the quotient group F × / F × 2 {\displaystyle F^{\times }/F^{\times 2}} of the multiplicative group of nonzero elements in the field modulo the square elements of the field. Each square class is a subset of the nonzero elements (a coset of the multiplicative group) consisting of the elements of the form xy2 where x is some particular fixed element and y ranges over all nonzero field elements. For instance, if F = R {\displaystyle F=\mathbb {R} } , the field of real numbers, then F × {\displaystyle F^{\times }} is just the group of all nonzero real numbers (with the multiplication operation) and F × 2 {\displaystyle F^{\times 2}} is the subgroup of positive numbers (as every positive number has a real square root). The quotient of these two groups is a group with two elements, corresponding to two cosets: the set of positive numbers and the set of negative numbers. Thus, the real numbers have two square classes, the positive numbers and the negative numbers. Square classes are frequently studied in relation to the theory of quadratic forms. The reason is that if V {\displaystyle V} is an F {\displaystyle F} -vector space and q : V → F {\displaystyle q:V\to F} is a quadratic form and v {\displaystyle v} is an element of V {\displaystyle V} such that q ( v ) = a ∈ F × {\displaystyle q(v)=a\in F^{\times }} , then for all u ∈ F × {\displaystyle u\in F^{\times }} , q ( u v ) = a u 2 {\displaystyle q(uv)=au^{2}} and thus it is sometimes more convenient to talk about the square classes which the quadratic form represents. Every element of the square class group is an involution. It follows that, if the number of square classes of a field is finite, it must be a power of two. == References ==
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Wikipedia:Square-free polynomial#0
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In mathematics, a square-free polynomial is a univariate polynomial (over a field or an integral domain) that has no multiple root in an algebraically closed field containing its coefficients. In characteristic 0, or over a finite field, a univariate polynomial is square free if and only if it does not have as a divisor any square of a non-constant polynomial. In applications in physics and engineering, a square-free polynomial is commonly called a polynomial with no repeated roots. The product rule implies that, if p2 divides f, then p divides the formal derivative f ′ of f. The converse is also true and hence, f {\displaystyle f} is square-free if and only if 1 {\displaystyle 1} is a greatest common divisor of the polynomial and its derivative. A square-free decomposition or square-free factorization of a polynomial is a factorization into powers of square-free polynomials f = a 1 a 2 2 a 3 3 ⋯ a n n = ∏ k = 1 n a k k {\displaystyle f=a_{1}a_{2}^{2}a_{3}^{3}\cdots a_{n}^{n}=\prod _{k=1}^{n}a_{k}^{k}\,} where those of the ak that are non-constant are pairwise coprime square-free polynomials (here, two polynomials are said coprime is their greatest common divisor is a constant; in other words that is the coprimality over the field of fractions of the coefficients that is considered). Every non-zero polynomial admits a square-free factorization, which is unique up to the multiplication and division of the factors by non-zero constants. The square-free factorization is much easier to compute than the complete factorization into irreducible factors, and is thus often preferred when the complete factorization is not really needed, as for the partial fraction decomposition and the symbolic integration of rational fractions. Square-free factorization is the first step of the polynomial factorization algorithms that are implemented in computer algebra systems. Therefore, the algorithm of square-free factorization is basic in computer algebra. Over a field of characteristic 0, the quotient of f {\displaystyle f} by its greatest common divisor (GCD) with its derivative is the product of the a i {\displaystyle a_{i}} in the above square-free decomposition. Over a perfect field of non-zero characteristic p, this quotient is the product of the a i {\displaystyle a_{i}} such that i is not a multiple of p. Further GCD computations and exact divisions allow computing the square-free factorization (see square-free factorization over a finite field). In characteristic zero, a better algorithm is known, Yun's algorithm, which is described below. Its computational complexity is, at most, twice that of the GCD computation of the input polynomial and its derivative. More precisely, if T n {\displaystyle T_{n}} is the time needed to compute the GCD of two polynomials of degree n {\displaystyle n} and the quotient of these polynomials by the GCD, then 2 T n {\displaystyle 2T_{n}} is an upper bound for the time needed to compute the complete square free decomposition. There are also known algorithms for square-free decomposition of multivariate polynomials, that proceed generally by considering a multivariate polynomial as a univariate polynomial with polynomial coefficients, and applying recursively a univariate algorithm. == Yun's algorithm == This section describes Yun's algorithm for the square-free decomposition of univariate polynomials over a field of characteristic 0. It proceeds by a succession of GCD computations and exact divisions. The input is thus a non-zero polynomial f, and the first step of the algorithm consists of computing the GCD a0 of f and its formal derivative f'. If f = a 1 a 2 2 a 3 3 ⋯ a k k {\displaystyle f=a_{1}a_{2}^{2}a_{3}^{3}\cdots a_{k}^{k}} is the desired factorization, we have thus a 0 = a 2 1 a 3 2 ⋯ a k k − 1 , {\displaystyle a_{0}=a_{2}^{1}a_{3}^{2}\cdots a_{k}^{k-1},} f / a 0 = a 1 a 2 a 3 ⋯ a k {\displaystyle f/a_{0}=a_{1}a_{2}a_{3}\cdots a_{k}} and f ′ / a 0 = ∑ i = 1 k i a i ′ a 1 ⋯ a i − 1 a i + 1 ⋯ a k . {\displaystyle f'/a_{0}=\sum _{i=1}^{k}ia_{i}'a_{1}\cdots a_{i-1}a_{i+1}\cdots a_{k}.} If we set b 1 = f / a 0 {\displaystyle b_{1}=f/a_{0}} , c 1 = f ′ / a 0 {\displaystyle c_{1}=f'/a_{0}} and d 1 = c 1 − b 1 ′ {\displaystyle d_{1}=c_{1}-b_{1}'} , we get that gcd ( b 1 , d 1 ) = a 1 , {\displaystyle \gcd(b_{1},d_{1})=a_{1},} b 2 = b 1 / a 1 = a 2 a 3 ⋯ a n , {\displaystyle b_{2}=b_{1}/a_{1}=a_{2}a_{3}\cdots a_{n},} and c 2 = d 1 / a 1 = ∑ i = 2 k ( i − 1 ) a i ′ a 2 ⋯ a i − 1 a i + 1 ⋯ a k . {\displaystyle c_{2}=d_{1}/a_{1}=\sum _{i=2}^{k}(i-1)a_{i}'a_{2}\cdots a_{i-1}a_{i+1}\cdots a_{k}.} Iterating this process until b k + 1 = 1 {\displaystyle b_{k+1}=1} we find all the a i . {\displaystyle a_{i}.} This is formalized into an algorithm as follows: a 0 := gcd ( f , f ′ ) ; b 1 := f / a 0 ; c 1 := f ′ / a 0 ; d 1 := c 1 − b 1 ′ ; i := 1 ; {\displaystyle a_{0}:=\gcd(f,f');\quad b_{1}:=f/a_{0};\quad c_{1}:=f'/a_{0};\quad d_{1}:=c_{1}-b_{1}';\quad i:=1;} repeat a i := gcd ( b i , d i ) ; b i + 1 := b i / a i ; c i + 1 := d i / a i ; i := i + 1 ; d i := c i − b i ′ ; {\displaystyle a_{i}:=\gcd(b_{i},d_{i});\quad b_{i+1}:=b_{i}/a_{i};\quad c_{i+1}:=d_{i}/a_{i};\quad i:=i+1;\quad d_{i}:=c_{i}-b_{i}';} until b i = 1 ; {\displaystyle b_{i}=1;} Output a 1 , … , a i − 1 . {\displaystyle a_{1},\ldots ,a_{i-1}.} The degree of c i {\displaystyle c_{i}} and d i {\displaystyle d_{i}} is one less than the degree of b i . {\displaystyle b_{i}.} As f {\displaystyle f} is the product of the b i , {\displaystyle b_{i},} the sum of the degrees of the b i {\displaystyle b_{i}} is the degree of f . {\displaystyle f.} As the complexity of GCD computations and divisions increase more than linearly with the degree, it follows that the total running time of the "repeat" loop is less than the running time of the first line of the algorithm, and that the total running time of Yun's algorithm is upper bounded by twice the time needed to compute the GCD of f {\displaystyle f} and f ′ {\displaystyle f'} and the quotient of f {\displaystyle f} and f ′ {\displaystyle f'} by their GCD. == Square root == In general, a polynomial has no polynomial square root. More precisely, most polynomials cannot be written as the square of another polynomial. A polynomial has a square root if and only if all exponents of the square-free decomposition are even. In this case, a square root is obtained by dividing these exponents by 2. Thus the problem of deciding if a polynomial has a square root, and of computing it if it exists, is a special case of square-free factorization. == References ==
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Wikipedia:Square-integrable function#0
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In mathematics, a square-integrable function, also called a quadratically integrable function or L 2 {\displaystyle L^{2}} function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite. Thus, square-integrability on the real line ( − ∞ , + ∞ ) {\displaystyle (-\infty ,+\infty )} is defined as follows. One may also speak of quadratic integrability over bounded intervals such as [ a , b ] {\displaystyle [a,b]} for a ≤ b {\displaystyle a\leq b} . An equivalent definition is to say that the square of the function itself (rather than of its absolute value) is Lebesgue integrable. For this to be true, the integrals of the positive and negative portions of the real part must both be finite, as well as those for the imaginary part. The vector space of (equivalence classes of) square integrable functions (with respect to Lebesgue measure) forms the L p {\displaystyle L^{p}} space with p = 2. {\displaystyle p=2.} Among the L p {\displaystyle L^{p}} spaces, the class of square integrable functions is unique in being compatible with an inner product, which allows notions like angle and orthogonality to be defined. Along with this inner product, the square integrable functions form a Hilbert space, since all of the L p {\displaystyle L^{p}} spaces are complete under their respective p {\displaystyle p} -norms. Often the term is used not to refer to a specific function, but to equivalence classes of functions that are equal almost everywhere. == Properties == The square integrable functions (in the sense mentioned in which a "function" actually means an equivalence class of functions that are equal almost everywhere) form an inner product space with inner product given by ⟨ f , g ⟩ = ∫ A f ( x ) g ( x ) ¯ d x , {\displaystyle \langle f,g\rangle =\int _{A}f(x){\overline {g(x)}}\,\mathrm {d} x,} where f {\displaystyle f} and g {\displaystyle g} are square integrable functions, f ( x ) ¯ {\displaystyle {\overline {f(x)}}} is the complex conjugate of f ( x ) , {\displaystyle f(x),} A {\displaystyle A} is the set over which one integrates—in the first definition (given in the introduction above), A {\displaystyle A} is ( − ∞ , + ∞ ) {\displaystyle (-\infty ,+\infty )} , in the second, A {\displaystyle A} is [ a , b ] {\displaystyle [a,b]} . Since | a | 2 = a ⋅ a ¯ {\displaystyle |a|^{2}=a\cdot {\overline {a}}} , square integrability is the same as saying ⟨ f , f ⟩ < ∞ . {\displaystyle \langle f,f\rangle <\infty .\,} It can be shown that square integrable functions form a complete metric space under the metric induced by the inner product defined above. A complete metric space is also called a Cauchy space, because sequences in such metric spaces converge if and only if they are Cauchy. A space that is complete under the metric induced by a norm is a Banach space. Therefore, the space of square integrable functions is a Banach space, under the metric induced by the norm, which in turn is induced by the inner product. As we have the additional property of the inner product, this is specifically a Hilbert space, because the space is complete under the metric induced by the inner product. This inner product space is conventionally denoted by ( L 2 , ⟨ ⋅ , ⋅ ⟩ 2 ) {\displaystyle \left(L_{2},\langle \cdot ,\cdot \rangle _{2}\right)} and many times abbreviated as L 2 . {\displaystyle L_{2}.} Note that L 2 {\displaystyle L_{2}} denotes the set of square integrable functions, but no selection of metric, norm or inner product are specified by this notation. The set, together with the specific inner product ⟨ ⋅ , ⋅ ⟩ 2 {\displaystyle \langle \cdot ,\cdot \rangle _{2}} specify the inner product space. The space of square integrable functions is the L p {\displaystyle L^{p}} space in which p = 2. {\displaystyle p=2.} == Examples == The function 1 x n , {\displaystyle {\tfrac {1}{x^{n}}},} defined on ( 0 , 1 ) , {\displaystyle (0,1),} is in L 2 {\displaystyle L^{2}} for n < 1 2 {\displaystyle n<{\tfrac {1}{2}}} but not for n = 1 2 . {\displaystyle n={\tfrac {1}{2}}.} The function 1 x , {\displaystyle {\tfrac {1}{x}},} defined on [ 1 , ∞ ) , {\displaystyle [1,\infty ),} is square-integrable. Bounded functions, defined on [ 0 , 1 ] , {\displaystyle [0,1],} are square-integrable. These functions are also in L p , {\displaystyle L^{p},} for any value of p . {\displaystyle p.} === Non-examples === The function 1 x , {\displaystyle {\tfrac {1}{x}},} defined on [ 0 , 1 ] , {\displaystyle [0,1],} where the value at 0 {\displaystyle 0} is arbitrary. Furthermore, this function is not in L p {\displaystyle L^{p}} for any value of p {\displaystyle p} in [ 1 , ∞ ) . {\displaystyle [1,\infty ).} == See also == Inner product space L p {\displaystyle L^{p}} space – Function spaces generalizing finite-dimensional p norm spaces == References ==
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Wikipedia:Squaring the circle#0
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Squaring the circle is a problem in geometry first proposed in Greek mathematics. It is the challenge of constructing a square with the area of a given circle by using only a finite number of steps with a compass and straightedge. The difficulty of the problem raised the question of whether specified axioms of Euclidean geometry concerning the existence of lines and circles implied the existence of such a square. In 1882, the task was proven to be impossible, as a consequence of the Lindemann–Weierstrass theorem, which proves that pi ( π {\displaystyle \pi } ) is a transcendental number. That is, π {\displaystyle \pi } is not the root of any polynomial with rational coefficients. It had been known for decades that the construction would be impossible if π {\displaystyle \pi } were transcendental, but that fact was not proven until 1882. Approximate constructions with any given non-perfect accuracy exist, and many such constructions have been found. Despite the proof that it is impossible, attempts to square the circle have been common in pseudomathematics (i.e. the work of mathematical cranks). The expression "squaring the circle" is sometimes used as a metaphor for trying to do the impossible. The term quadrature of the circle is sometimes used as a synonym for squaring the circle. It may also refer to approximate or numerical methods for finding the area of a circle. In general, quadrature or squaring may also be applied to other plane figures. == History == Methods to calculate the approximate area of a given circle, which can be thought of as a precursor problem to squaring the circle, were known already in many ancient cultures. These methods can be summarized by stating the approximation to π that they produce. In around 2000 BCE, the Babylonian mathematicians used the approximation π ≈ 25 8 = 3.125 {\displaystyle \pi \approx {\tfrac {25}{8}}=3.125} , and at approximately the same time the ancient Egyptian mathematicians used π ≈ 256 81 ≈ 3.16 {\displaystyle \pi \approx {\tfrac {256}{81}}\approx 3.16} . Over 1000 years later, the Old Testament Books of Kings used the simpler approximation π ≈ 3 {\displaystyle \pi \approx 3} . Ancient Indian mathematics, as recorded in the Shatapatha Brahmana and Shulba Sutras, used several different approximations to π {\displaystyle \pi } . Archimedes proved a formula for the area of a circle, according to which 3 10 71 ≈ 3.141 < π < 3 1 7 ≈ 3.143 {\displaystyle 3\,{\tfrac {10}{71}}\approx 3.141<\pi <3\,{\tfrac {1}{7}}\approx 3.143} . In Chinese mathematics, in the third century CE, Liu Hui found even more accurate approximations using a method similar to that of Archimedes, and in the fifth century Zu Chongzhi found π ≈ 355 / 113 ≈ 3.141593 {\displaystyle \pi \approx 355/113\approx 3.141593} , an approximation known as Milü. The problem of constructing a square whose area is exactly that of a circle, rather than an approximation to it, comes from Greek mathematics. Greek mathematicians found compass and straightedge constructions to convert any polygon into a square of equivalent area. They used this construction to compare areas of polygons geometrically, rather than by the numerical computation of area that would be more typical in modern mathematics. As Proclus wrote many centuries later, this motivated the search for methods that would allow comparisons with non-polygonal shapes: The first known Greek to study the problem was Anaxagoras, who worked on it while in prison. Hippocrates of Chios attacked the problem by finding a shape bounded by circular arcs, the lune of Hippocrates, that could be squared. Antiphon the Sophist believed that inscribing regular polygons within a circle and doubling the number of sides would eventually fill up the area of the circle (this is the method of exhaustion). Since any polygon can be squared, he argued, the circle can be squared. In contrast, Eudemus argued that magnitudes can be divided up without limit, so the area of the circle would never be used up. Contemporaneously with Antiphon, Bryson of Heraclea argued that, since larger and smaller circles both exist, there must be a circle of equal area; this principle can be seen as a form of the modern intermediate value theorem. The more general goal of carrying out all geometric constructions using only a compass and straightedge has often been attributed to Oenopides, but the evidence for this is circumstantial. The problem of finding the area under an arbitrary curve, now known as integration in calculus, or quadrature in numerical analysis, was known as squaring before the invention of calculus. Since the techniques of calculus were unknown, it was generally presumed that a squaring should be done via geometric constructions, that is, by compass and straightedge. For example, Newton wrote to Oldenburg in 1676 "I believe M. Leibnitz will not dislike the theorem towards the beginning of my letter pag. 4 for squaring curve lines geometrically". In modern mathematics the terms have diverged in meaning, with quadrature generally used when methods from calculus are allowed, while squaring the curve retains the idea of using only restricted geometric methods. A 1647 attempt at squaring the circle, Opus geometricum quadraturae circuli et sectionum coni decem libris comprehensum by Grégoire de Saint-Vincent, was heavily criticized by Vincent Léotaud. Nevertheless, de Saint-Vincent succeeded in his quadrature of the hyperbola, and in doing so was one of the earliest to develop the natural logarithm. James Gregory, following de Saint-Vincent, attempted another proof of the impossibility of squaring the circle in Vera Circuli et Hyperbolae Quadratura (The True Squaring of the Circle and of the Hyperbola) in 1667. Although his proof was faulty, it was the first paper to attempt to solve the problem using algebraic properties of π {\displaystyle \pi } . Johann Heinrich Lambert proved in 1761 that π {\displaystyle \pi } is an irrational number. It was not until 1882 that Ferdinand von Lindemann succeeded in proving more strongly that π is a transcendental number, and by doing so also proved the impossibility of squaring the circle with compass and straightedge. After Lindemann's impossibility proof, the problem was considered to be settled by professional mathematicians, and its subsequent mathematical history is dominated by pseudomathematical attempts at circle-squaring constructions, largely by amateurs, and by the debunking of these efforts. As well, several later mathematicians including Srinivasa Ramanujan developed compass and straightedge constructions that approximate the problem accurately in few steps. Two other classical problems of antiquity, famed for their impossibility, were doubling the cube and trisecting the angle. Like squaring the circle, these cannot be solved by compass and straightedge. However, they have a different character than squaring the circle, in that their solution involves the root of a cubic equation, rather than being transcendental. Therefore, more powerful methods than compass and straightedge constructions, such as neusis construction or mathematical paper folding, can be used to construct solutions to these problems. == Impossibility == The solution of the problem of squaring the circle by compass and straightedge requires the construction of the number π {\displaystyle {\sqrt {\pi }}} , the length of the side of a square whose area equals that of a unit circle. If π {\displaystyle {\sqrt {\pi }}} were a constructible number, it would follow from standard compass and straightedge constructions that π {\displaystyle \pi } would also be constructible. In 1837, Pierre Wantzel showed that lengths that could be constructed with compass and straightedge had to be solutions of certain polynomial equations with rational coefficients. Thus, constructible lengths must be algebraic numbers. If the circle could be squared using only compass and straightedge, then π {\displaystyle \pi } would have to be an algebraic number. It was not until 1882 that Ferdinand von Lindemann proved the transcendence of π {\displaystyle \pi } and so showed the impossibility of this construction. Lindemann's idea was to combine the proof of transcendence of Euler's number e {\displaystyle e} , shown by Charles Hermite in 1873, with Euler's identity e i π = − 1. {\displaystyle e^{i\pi }=-1.} This identity immediately shows that π {\displaystyle \pi } is an irrational number, because a rational power of a transcendental number remains transcendental. Lindemann was able to extend this argument, through the Lindemann–Weierstrass theorem on linear independence of algebraic powers of e {\displaystyle e} , to show that π {\displaystyle \pi } is transcendental and therefore that squaring the circle is impossible. Bending the rules by introducing a supplemental tool, allowing an infinite number of compass-and-straightedge operations or by performing the operations in certain non-Euclidean geometries makes squaring the circle possible in some sense. For example, Dinostratus' theorem uses the quadratrix of Hippias to square the circle, meaning that if this curve is somehow already given, then a square and circle of equal areas can be constructed from it. The Archimedean spiral can be used for another similar construction. Although the circle cannot be squared in Euclidean space, it sometimes can be in hyperbolic geometry under suitable interpretations of the terms. The hyperbolic plane does not contain squares (quadrilaterals with four right angles and four equal sides), but instead it contains regular quadrilaterals, shapes with four equal sides and four equal angles sharper than right angles. There exist in the hyperbolic plane (countably) infinitely many pairs of constructible circles and constructible regular quadrilaterals of equal area, which, however, are constructed simultaneously. There is no method for starting with an arbitrary regular quadrilateral and constructing the circle of equal area. Symmetrically, there is no method for starting with an arbitrary circle and constructing a regular quadrilateral of equal area, and for sufficiently large circles no such quadrilateral exists. == Approximate constructions == Although squaring the circle exactly with compass and straightedge is impossible, approximations to squaring the circle can be given by constructing lengths close to π {\displaystyle \pi } . It takes only elementary geometry to convert any given rational approximation of π {\displaystyle \pi } into a corresponding compass and straightedge construction, but such constructions tend to be very long-winded in comparison to the accuracy they achieve. After the exact problem was proven unsolvable, some mathematicians applied their ingenuity to finding approximations to squaring the circle that are particularly simple among other imaginable constructions that give similar precision. === Construction by Kochański === One of many early historical approximate compass-and-straightedge constructions is from a 1685 paper by Polish Jesuit Adam Adamandy Kochański, producing an approximation diverging from π {\displaystyle \pi } in the 5th decimal place. Although much more precise numerical approximations to π {\displaystyle \pi } were already known, Kochański's construction has the advantage of being quite simple. In the left diagram | P 3 P 9 | = | P 1 P 2 | 40 3 − 2 3 ≈ 3.141 5 33 338 ⋅ | P 1 P 2 | ≈ π r . {\displaystyle |P_{3}P_{9}|=|P_{1}P_{2}|{\sqrt {{\frac {40}{3}}-2{\sqrt {3}}}}\approx 3.141\,5{\color {red}33\,338}\cdot |P_{1}P_{2}|\approx \pi r.} In the same work, Kochański also derived a sequence of increasingly accurate rational approximations for π {\displaystyle \pi } . === Constructions using 355/113 === Jacob de Gelder published in 1849 a construction based on the approximation π ≈ 355 113 = 3.141 592 920 … {\displaystyle \pi \approx {\frac {355}{113}}=3.141\;592{\color {red}\;920\;\ldots }} This value is accurate to six decimal places and has been known in China since the 5th century as Milü, and in Europe since the 17th century. Gelder did not construct the side of the square; it was enough for him to find the value A H ¯ = 4 2 7 2 + 8 2 . {\displaystyle {\overline {AH}}={\frac {4^{2}}{7^{2}+8^{2}}}.} The illustration shows de Gelder's construction. In 1914, Indian mathematician Srinivasa Ramanujan gave another geometric construction for the same approximation. === Constructions using the golden ratio === An approximate construction by E. W. Hobson in 1913 is accurate to three decimal places. Hobson's construction corresponds to an approximate value of 6 5 ⋅ ( 1 + φ ) = 3.141 640 … , {\displaystyle {\frac {6}{5}}\cdot \left(1+\varphi \right)=3.141\;{\color {red}640\;\ldots },} where φ {\displaystyle \varphi } is the golden ratio, φ = ( 1 + 5 ) / 2 {\displaystyle \varphi =(1+{\sqrt {5}})/2} . The same approximate value appears in a 1991 construction by Robert Dixon. In 2022 Frédéric Beatrix presented a geometrographic construction in 13 steps. === Second construction by Ramanujan === In 1914, Ramanujan gave a construction which was equivalent to taking the approximate value for π {\displaystyle \pi } to be ( 9 2 + 19 2 22 ) 1 4 = 2143 22 4 = 3.141 592 65 2 582 … {\displaystyle \left(9^{2}+{\frac {19^{2}}{22}}\right)^{\frac {1}{4}}={\sqrt[{4}]{\frac {2143}{22}}}=3.141\;592\;65{\color {red}2\;582\;\ldots }} giving eight decimal places of π {\displaystyle \pi } . He describes the construction of line segment OS as follows. == Incorrect constructions == In his old age, the English philosopher Thomas Hobbes convinced himself that he had succeeded in squaring the circle, a claim refuted by John Wallis as part of the Hobbes–Wallis controversy. During the 18th and 19th century, the false notions that the problem of squaring the circle was somehow related to the longitude problem, and that a large reward would be given for a solution, became prevalent among would-be circle squarers. In 1851, John Parker published a book Quadrature of the Circle in which he claimed to have squared the circle. His method actually produced an approximation of π {\displaystyle \pi } accurate to six digits. The Victorian-age mathematician, logician, and writer Charles Lutwidge Dodgson, better known by his pseudonym Lewis Carroll, also expressed interest in debunking illogical circle-squaring theories. In one of his diary entries for 1855, Dodgson listed books he hoped to write, including one called "Plain Facts for Circle-Squarers". In the introduction to "A New Theory of Parallels", Dodgson recounted an attempt to demonstrate logical errors to a couple of circle-squarers, stating: A ridiculing of circle squaring appears in Augustus De Morgan's book A Budget of Paradoxes, published posthumously by his widow in 1872. Having originally published the work as a series of articles in The Athenæum, he was revising it for publication at the time of his death. Circle squaring declined in popularity after the nineteenth century, and it is believed that De Morgan's work helped bring this about. Even after it had been proved impossible, in 1894, amateur mathematician Edwin J. Goodwin claimed that he had developed a method to square the circle. The technique he developed did not accurately square the circle, and provided an incorrect area of the circle which essentially redefined π {\displaystyle \pi } as equal to 3.2. Goodwin then proposed the Indiana pi bill in the Indiana state legislature allowing the state to use his method in education without paying royalties to him. The bill passed with no objections in the state house, but the bill was tabled and never voted on in the Senate, amid increasing ridicule from the press. The mathematical crank Carl Theodore Heisel also claimed to have squared the circle in his 1934 book, "Behold! : the grand problem no longer unsolved: the circle squared beyond refutation." Paul Halmos referred to the book as a "classic crank book." == In literature == The problem of squaring the circle has been mentioned over a wide range of literary eras, with a variety of metaphorical meanings. Its literary use dates back at least to 414 BC, when the play The Birds by Aristophanes was first performed. In it, the character Meton of Athens mentions squaring the circle, possibly to indicate the paradoxical nature of his utopian city. Dante's Paradise, canto XXXIII, lines 133–135, contain the verse: For Dante, squaring the circle represents a task beyond human comprehension, which he compares to his own inability to comprehend Paradise. Dante's image also calls to mind a passage from Vitruvius, famously illustrated later in Leonardo da Vinci's Vitruvian Man, of a man simultaneously inscribed in a circle and a square. Dante uses the circle as a symbol for God, and may have mentioned this combination of shapes in reference to the simultaneous divine and human nature of Jesus. Earlier, in canto XIII, Dante calls out Greek circle-squarer Bryson as having sought knowledge instead of wisdom. Several works of 17th-century poet Margaret Cavendish elaborate on the circle-squaring problem and its metaphorical meanings, including a contrast between unity of truth and factionalism, and the impossibility of rationalizing "fancy and female nature". By 1742, when Alexander Pope published the fourth book of his Dunciad, attempts at circle-squaring had come to be seen as "wild and fruitless": Similarly, the Gilbert and Sullivan comic opera Princess Ida features a song which satirically lists the impossible goals of the women's university run by the title character, such as finding perpetual motion. One of these goals is "And the circle – they will square it/Some fine day." The sestina, a poetic form first used in the 12th century by Arnaut Daniel, has been said to metaphorically square the circle in its use of a square number of lines (six stanzas of six lines each) with a circular scheme of six repeated words. Spanos (1978) writes that this form invokes a symbolic meaning in which the circle stands for heaven and the square stands for the earth. A similar metaphor was used in "Squaring the Circle", a 1908 short story by O. Henry, about a long-running family feud. In the title of this story, the circle represents the natural world, while the square represents the city, the world of man. In later works, circle-squarers such as Leopold Bloom in James Joyce's novel Ulysses and Lawyer Paravant in Thomas Mann's The Magic Mountain are seen as sadly deluded or as unworldly dreamers, unaware of its mathematical impossibility and making grandiose plans for a result they will never attain. == See also == Mrs. Miniver's problem – Problem on areas of intersecting circles Round square copula – Philosophical treatment of oxymoronsPages displaying short descriptions of redirect targets Squircle – Shape between a square and a circle Tarski's circle-squaring problem – Problem of cutting and reassembling a disk into a square == References == == Further reading and external links == Bogomolny, Alexander. "Squaring the Circle". cut-the-knot. Grime, James (25 March 2013). "Squaring the Circle". Numberphile. Brady Haran – via YouTube. Harper, Suzanne; Driskell, Shannon (August 2010). "An Investigation of Historical Geometric Constructions". Convergence. Mathematical Association of America. O'Connor, J J; Robertson, E F (April 1999). "Squaring the circle". MacTutor History of Mathematics archive. Otero, Daniel E. (July 2010). "The Quadrature of the Circle and Hippocrates' Lunes". Convergence. Mathematical Association of America. Polster, Burkard (29 June 2019). "2000 years unsolved: Why is doubling cubes and squaring circles impossible?". Mathologer – via YouTube.
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Wikipedia:Squeeze mapping#0
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In linear algebra, a squeeze mapping, also called a squeeze transformation, is a type of linear map that preserves Euclidean area of regions in the Cartesian plane, but is not a rotation or shear mapping. For a fixed positive real number a, the mapping ( x , y ) ↦ ( a x , y / a ) {\displaystyle (x,y)\mapsto (ax,y/a)} is the squeeze mapping with parameter a. Since { ( u , v ) : u v = c o n s t a n t } {\displaystyle \{(u,v)\,:\,uv=\mathrm {constant} \}} is a hyperbola, if u = ax and v = y/a, then uv = xy and the points of the image of the squeeze mapping are on the same hyperbola as (x,y) is. For this reason it is natural to think of the squeeze mapping as a hyperbolic rotation, as did Émile Borel in 1914, by analogy with circular rotations, which preserve circles. == Logarithm and hyperbolic angle == The squeeze mapping sets the stage for development of the concept of logarithms. The problem of finding the area bounded by a hyperbola (such as xy = 1) is one of quadrature. The solution, found by Grégoire de Saint-Vincent and Alphonse Antonio de Sarasa in 1647, required the natural logarithm function, a new concept. Some insight into logarithms comes through hyperbolic sectors that are permuted by squeeze mappings while preserving their area. The area of a hyperbolic sector is taken as a measure of a hyperbolic angle associated with the sector. The hyperbolic angle concept is quite independent of the ordinary circular angle, but shares a property of invariance with it: whereas circular angle is invariant under rotation, hyperbolic angle is invariant under squeeze mapping. Both circular and hyperbolic angle generate invariant measures but with respect to different transformation groups. The hyperbolic functions, which take hyperbolic angle as argument, perform the role that circular functions play with the circular angle argument. == Group theory == In 1688, long before abstract group theory, the squeeze mapping was described by Euclid Speidell in the terms of the day: "From a Square and an infinite company of Oblongs on a Superficies, each Equal to that square, how a curve is begotten which shall have the same properties or affections of any Hyperbola inscribed within a Right Angled Cone." If r and s are positive real numbers, the composition of their squeeze mappings is the squeeze mapping of their product. Therefore, the collection of squeeze mappings forms a one-parameter group isomorphic to the multiplicative group of positive real numbers. An additive view of this group arises from consideration of hyperbolic sectors and their hyperbolic angles. From the point of view of the classical groups, the group of squeeze mappings is SO+(1,1), the identity component of the indefinite orthogonal group of 2×2 real matrices preserving the quadratic form u2 − v2. This is equivalent to preserving the form xy via the change of basis x = u + v , y = u − v , {\displaystyle x=u+v,\quad y=u-v\,,} and corresponds geometrically to preserving hyperbolae. The perspective of the group of squeeze mappings as hyperbolic rotation is analogous to interpreting the group SO(2) (the connected component of the definite orthogonal group) preserving quadratic form x2 + y2 as being circular rotations. Note that the "SO+" notation corresponds to the fact that the reflections u ↦ − u , v ↦ − v {\displaystyle u\mapsto -u,\quad v\mapsto -v} are not allowed, though they preserve the form (in terms of x and y these are x ↦ y, y ↦ x and x ↦ −x, y ↦ −y); the additional "+" in the hyperbolic case (as compared with the circular case) is necessary to specify the identity component because the group O(1,1) has 4 connected components, while the group O(2) has 2 components: SO(1,1) has 2 components, while SO(2) only has 1. The fact that the squeeze transforms preserve area and orientation corresponds to the inclusion of subgroups SO ⊂ SL – in this case SO(1,1) ⊂ SL(2) – of the subgroup of hyperbolic rotations in the special linear group of transforms preserving area and orientation (a volume form). In the language of Möbius transformations, the squeeze transformations are the hyperbolic elements in the classification of elements. A geometric transformation is called conformal when it preserves angles. Hyperbolic angle is defined using area under y = 1/x. Since squeeze mappings preserve areas of transformed regions such as hyperbolic sectors, the angle measure of sectors is preserved. Thus squeeze mappings are conformal in the sense of preserving hyperbolic angle. == Applications == Here some applications are summarized with historic references. === Relativistic spacetime === Spacetime geometry is conventionally developed as follows: Select (0,0) for a "here and now" in a spacetime. Light radiant left and right through this central event tracks two lines in the spacetime, lines that can be used to give coordinates to events away from (0,0). Trajectories of lesser velocity track closer to the original timeline (0,t). Any such velocity can be viewed as a zero velocity under a squeeze mapping called a Lorentz boost. This insight follows from a study of split-complex number multiplications and the diagonal basis which corresponds to the pair of light lines. Formally, a squeeze preserves the hyperbolic metric expressed in the form xy; in a different coordinate system. This application in the theory of relativity was noted in 1912 by Wilson and Lewis, by Werner Greub, and by Louis Kauffman. Furthermore, the squeeze mapping form of Lorentz transformations was used by Gustav Herglotz (1909/10) while discussing Born rigidity, and was popularized by Wolfgang Rindler in his textbook on relativity, who used it in his demonstration of their characteristic property. The term squeeze transformation was used in this context in an article connecting the Lorentz group with Jones calculus in optics. === Corner flow === In fluid dynamics one of the fundamental motions of an incompressible flow involves bifurcation of a flow running up against an immovable wall. Representing the wall by the axis y = 0 and taking the parameter r = exp(t) where t is time, then the squeeze mapping with parameter r applied to an initial fluid state produces a flow with bifurcation left and right of the axis x = 0. The same model gives fluid convergence when time is run backward. Indeed, the area of any hyperbolic sector is invariant under squeezing. For another approach to a flow with hyperbolic streamlines, see Potential flow § Power laws with n = 2. In 1989 Ottino described the "linear isochoric two-dimensional flow" as v 1 = G x 2 v 2 = K G x 1 {\displaystyle v_{1}=Gx_{2}\quad v_{2}=KGx_{1}} where K lies in the interval [−1, 1]. The streamlines follow the curves x 2 2 − K x 1 2 = c o n s t a n t {\displaystyle x_{2}^{2}-Kx_{1}^{2}=\mathrm {constant} } so negative K corresponds to an ellipse and positive K to a hyperbola, with the rectangular case of the squeeze mapping corresponding to K = 1. Stocker and Hosoi described their approach to corner flow as follows: we suggest an alternative formulation to account for the corner-like geometry, based on the use of hyperbolic coordinates, which allows substantial analytical progress towards determination of the flow in a Plateau border and attached liquid threads. We consider a region of flow forming an angle of π/2 and delimited on the left and bottom by symmetry planes. Stocker and Hosoi then recall Moffatt's consideration of "flow in a corner between rigid boundaries, induced by an arbitrary disturbance at a large distance." According to Stocker and Hosoi, For a free fluid in a square corner, Moffatt's (antisymmetric) stream function ... [indicates] that hyperbolic coordinates are indeed the natural choice to describe these flows. === Bridge to transcendentals === The area-preserving property of squeeze mapping has an application in setting the foundation of the transcendental functions natural logarithm and its inverse the exponential function: Definition: Sector(a,b) is the hyperbolic sector obtained with central rays to (a, 1/a) and (b, 1/b). Lemma: If bc = ad, then there is a squeeze mapping that moves the sector(a,b) to sector(c,d). Proof: Take parameter r = c/a so that (u,v) = (rx, y/r) takes (a, 1/a) to (c, 1/c) and (b, 1/b) to (d, 1/d). Theorem (Gregoire de Saint-Vincent 1647) If bc = ad, then the quadrature of the hyperbola xy = 1 against the asymptote has equal areas between a and b compared to between c and d. Proof: An argument adding and subtracting triangles of area 1⁄2, one triangle being {(0,0), (0,1), (1,1)}, shows the hyperbolic sector area is equal to the area along the asymptote. The theorem then follows from the lemma. Theorem (Alphonse Antonio de Sarasa 1649) As area measured against the asymptote increases in arithmetic progression, the projections upon the asymptote increase in geometric sequence. Thus the areas form logarithms of the asymptote index. For instance, for a standard position angle which runs from (1, 1) to (x, 1/x), one may ask "When is the hyperbolic angle equal to one?" The answer is the transcendental number x = e. A squeeze with r = e moves the unit angle to one between (e, 1/e) and (ee, 1/ee) which subtends a sector also of area one. The geometric progression e, e2, e3, ..., en, ... corresponds to the asymptotic index achieved with each sum of areas 1,2,3, ..., n,... which is a proto-typical arithmetic progression A + nd where A = 0 and d = 1 . === Lie transform === Following Pierre Ossian Bonnet's (1867) investigations on surfaces of constant curvatures, Sophus Lie (1879) found a way to derive new pseudospherical surfaces from a known one. Such surfaces satisfy the Sine-Gordon equation: d 2 Θ d s d σ = K sin Θ , {\displaystyle {\frac {d^{2}\Theta }{ds\ d\sigma }}=K\sin \Theta ,} where ( s , σ ) {\displaystyle (s,\sigma )} are asymptotic coordinates of two principal tangent curves and Θ {\displaystyle \Theta } their respective angle. Lie showed that if Θ = f ( s , σ ) {\displaystyle \Theta =f(s,\sigma )} is a solution to the Sine-Gordon equation, then the following squeeze mapping (now known as Lie transform) indicates other solutions of that equation: Θ = f ( m s , σ m ) . {\displaystyle \Theta =f\left(ms,\ {\frac {\sigma }{m}}\right).} Lie (1883) noticed its relation to two other transformations of pseudospherical surfaces: The Bäcklund transform (introduced by Albert Victor Bäcklund in 1883) can be seen as the combination of a Lie transform with a Bianchi transform (introduced by Luigi Bianchi in 1879.) Such transformations of pseudospherical surfaces were discussed in detail in the lectures on differential geometry by Gaston Darboux (1894), Luigi Bianchi (1894), or Luther Pfahler Eisenhart (1909). It is known that the Lie transforms (or squeeze mappings) correspond to Lorentz boosts in terms of light-cone coordinates, as pointed out by Terng and Uhlenbeck (2000): Sophus Lie observed that the SGE [Sinus-Gordon equation] is invariant under Lorentz transformations. In asymptotic coordinates, which correspond to light cone coordinates, a Lorentz transformation is ( x , t ) ↦ ( 1 λ x , λ t ) {\displaystyle (x,t)\mapsto \left({\tfrac {1}{\lambda }}x,\lambda t\right)} . This can be represented as follows: − c 2 t 2 + x 2 = − c 2 t ′ 2 + x ′ 2 c t ′ = c t γ − x β γ = c t cosh η − x sinh η x ′ = − c t β γ + x γ = − c t sinh η + x cosh η u = c t + x , v = c t − x , k = 1 + β 1 − β = e η u ′ = u k , v ′ = k v u ′ v ′ = u v {\displaystyle {\begin{matrix}-c^{2}t^{2}+x^{2}=-c^{2}t^{\prime 2}+x^{\prime 2}\\\hline {\begin{aligned}ct'&=ct\gamma -x\beta \gamma &&=ct\cosh \eta -x\sinh \eta \\x'&=-ct\beta \gamma +x\gamma &&=-ct\sinh \eta +x\cosh \eta \end{aligned}}\\\hline u=ct+x,\ v=ct-x,\ k={\sqrt {\tfrac {1+\beta }{1-\beta }}}=e^{\eta }\\u'={\frac {u}{k}},\ v'=kv\\\hline u'v'=uv\end{matrix}}} where k corresponds to the Doppler factor in Bondi k-calculus, η is the rapidity. == See also == Indefinite orthogonal group Isochoric process == References == HSM Coxeter & SL Greitzer (1967) Geometry Revisited, Chapter 4 Transformations, A genealogy of transformation. P. S. Modenov and A. S. Parkhomenko (1965) Geometric Transformations, volume one. See pages 104 to 106. Walter, Scott (1999). "The non-Euclidean style of Minkowskian relativity" (PDF). In J. Gray (ed.). The Symbolic Universe: Geometry and Physics. Oxford University Press. pp. 91–127.(see page 9 of e-link) Learning materials related to Reciprocal Eigenvalues at Wikiversity
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Wikipedia:Squeeze theorem#0
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In calculus, the squeeze theorem (also known as the sandwich theorem, among other names) is a theorem regarding the limit of a function that is bounded between two other functions. The squeeze theorem is used in calculus and mathematical analysis, typically to confirm the limit of a function via comparison with two other functions whose limits are known. It was first used geometrically by the mathematicians Archimedes and Eudoxus in an effort to compute π, and was formulated in modern terms by Carl Friedrich Gauss. == Statement == The squeeze theorem is formally stated as follows. The functions g and h are said to be lower and upper bounds (respectively) of f. Here, a is not required to lie in the interior of I. Indeed, if a is an endpoint of I, then the above limits are left- or right-hand limits. A similar statement holds for infinite intervals: for example, if I = (0, ∞), then the conclusion holds, taking the limits as x → ∞. This theorem is also valid for sequences. Let (an), (cn) be two sequences converging to ℓ, and (bn) a sequence. If ∀ n ≥ N , N ∈ N {\displaystyle \forall n\geq N,N\in \mathbb {N} } we have an ≤ bn ≤ cn, then (bn) also converges to ℓ. === Proof === According to the above hypotheses we have, taking the limit inferior and superior: L = lim x → a g ( x ) ≤ lim inf x → a f ( x ) ≤ lim sup x → a f ( x ) ≤ lim x → a h ( x ) = L , {\displaystyle L=\lim _{x\to a}g(x)\leq \liminf _{x\to a}f(x)\leq \limsup _{x\to a}f(x)\leq \lim _{x\to a}h(x)=L,} so all the inequalities are indeed equalities, and the thesis immediately follows. A direct proof, using the (ε, δ)-definition of limit, would be to prove that for all real ε > 0 there exists a real δ > 0 such that for all x with | x − a | < δ , {\displaystyle |x-a|<\delta ,} we have | f ( x ) − L | < ε . {\displaystyle |f(x)-L|<\varepsilon .} Symbolically, ∀ ε > 0 , ∃ δ > 0 : ∀ x , ( | x − a | < δ ⇒ | f ( x ) − L | < ε ) . {\displaystyle \forall \varepsilon >0,\exists \delta >0:\forall x,(|x-a|<\delta \ \Rightarrow |f(x)-L|<\varepsilon ).} As lim x → a g ( x ) = L {\displaystyle \lim _{x\to a}g(x)=L} means that and lim x → a h ( x ) = L {\displaystyle \lim _{x\to a}h(x)=L} means that then we have g ( x ) ≤ f ( x ) ≤ h ( x ) {\displaystyle g(x)\leq f(x)\leq h(x)} g ( x ) − L ≤ f ( x ) − L ≤ h ( x ) − L {\displaystyle g(x)-L\leq f(x)-L\leq h(x)-L} We can choose δ := min { δ 1 , δ 2 } {\displaystyle \delta :=\min \left\{\delta _{1},\delta _{2}\right\}} . Then, if | x − a | < δ {\displaystyle |x-a|<\delta } , combining (1) and (2), we have − ε < g ( x ) − L ≤ f ( x ) − L ≤ h ( x ) − L < ε , {\displaystyle -\varepsilon <g(x)-L\leq f(x)-L\leq h(x)-L\ <\varepsilon ,} − ε < f ( x ) − L < ε , {\displaystyle -\varepsilon <f(x)-L<\varepsilon ,} which completes the proof. Q.E.D The proof for sequences is very similar, using the ε {\displaystyle \varepsilon } -definition of the limit of a sequence. == Examples == === First example === The limit lim x → 0 x 2 sin ( 1 x ) {\displaystyle \lim _{x\to 0}x^{2}\sin \left({\tfrac {1}{x}}\right)} cannot be determined through the limit law lim x → a ( f ( x ) ⋅ g ( x ) ) = lim x → a f ( x ) ⋅ lim x → a g ( x ) , {\displaystyle \lim _{x\to a}(f(x)\cdot g(x))=\lim _{x\to a}f(x)\cdot \lim _{x\to a}g(x),} because lim x → 0 sin ( 1 x ) {\displaystyle \lim _{x\to 0}\sin \left({\tfrac {1}{x}}\right)} does not exist. However, by the definition of the sine function, − 1 ≤ sin ( 1 x ) ≤ 1. {\displaystyle -1\leq \sin \left({\tfrac {1}{x}}\right)\leq 1.} It follows that − x 2 ≤ x 2 sin ( 1 x ) ≤ x 2 {\displaystyle -x^{2}\leq x^{2}\sin \left({\tfrac {1}{x}}\right)\leq x^{2}} Since lim x → 0 − x 2 = lim x → 0 x 2 = 0 {\displaystyle \lim _{x\to 0}-x^{2}=\lim _{x\to 0}x^{2}=0} , by the squeeze theorem, lim x → 0 x 2 sin ( 1 x ) {\displaystyle \lim _{x\to 0}x^{2}\sin \left({\tfrac {1}{x}}\right)} must also be 0. === Second example === Probably the best-known examples of finding a limit by squeezing are the proofs of the equalities lim x → 0 sin x x = 1 , lim x → 0 1 − cos x x = 0. {\displaystyle {\begin{aligned}&\lim _{x\to 0}{\frac {\sin x}{x}}=1,\\[10pt]&\lim _{x\to 0}{\frac {1-\cos x}{x}}=0.\end{aligned}}} The first limit follows by means of the squeeze theorem from the fact that cos x ≤ sin x x ≤ 1 {\displaystyle \cos x\leq {\frac {\sin x}{x}}\leq 1} for x close enough to 0. The correctness of which for positive x can be seen by simple geometric reasoning (see drawing) that can be extended to negative x as well. The second limit follows from the squeeze theorem and the fact that 0 ≤ 1 − cos x x ≤ x {\displaystyle 0\leq {\frac {1-\cos x}{x}}\leq x} for x close enough to 0. This can be derived by replacing sin x in the earlier fact by 1 − cos 2 x {\textstyle {\sqrt {1-\cos ^{2}x}}} and squaring the resulting inequality. These two limits are used in proofs of the fact that the derivative of the sine function is the cosine function. That fact is relied on in other proofs of derivatives of trigonometric functions. === Third example === It is possible to show that d d θ tan θ = sec 2 θ {\displaystyle {\frac {d}{d\theta }}\tan \theta =\sec ^{2}\theta } by squeezing, as follows. In the illustration at right, the area of the smaller of the two shaded sectors of the circle is sec 2 θ Δ θ 2 , {\displaystyle {\frac {\sec ^{2}\theta \,\Delta \theta }{2}},} since the radius is sec θ and the arc on the unit circle has length Δθ. Similarly, the area of the larger of the two shaded sectors is sec 2 ( θ + Δ θ ) Δ θ 2 . {\displaystyle {\frac {\sec ^{2}(\theta +\Delta \theta )\,\Delta \theta }{2}}.} What is squeezed between them is the triangle whose base is the vertical segment whose endpoints are the two dots. The length of the base of the triangle is tan(θ + Δθ) − tan θ, and the height is 1. The area of the triangle is therefore tan ( θ + Δ θ ) − tan θ 2 . {\displaystyle {\frac {\tan(\theta +\Delta \theta )-\tan \theta }{2}}.} From the inequalities sec 2 θ Δ θ 2 ≤ tan ( θ + Δ θ ) − tan θ 2 ≤ sec 2 ( θ + Δ θ ) Δ θ 2 {\displaystyle {\frac {\sec ^{2}\theta \,\Delta \theta }{2}}\leq {\frac {\tan(\theta +\Delta \theta )-\tan \theta }{2}}\leq {\frac {\sec ^{2}(\theta +\Delta \theta )\,\Delta \theta }{2}}} we deduce that sec 2 θ ≤ tan ( θ + Δ θ ) − tan θ Δ θ ≤ sec 2 ( θ + Δ θ ) , {\displaystyle \sec ^{2}\theta \leq {\frac {\tan(\theta +\Delta \theta )-\tan \theta }{\Delta \theta }}\leq \sec ^{2}(\theta +\Delta \theta ),} provided Δθ > 0, and the inequalities are reversed if Δθ < 0. Since the first and third expressions approach sec2θ as Δθ → 0, and the middle expression approaches d d θ tan θ , {\displaystyle {\tfrac {d}{d\theta }}\tan \theta ,} the desired result follows. === Fourth example === The squeeze theorem can still be used in multivariable calculus but the lower (and upper functions) must be below (and above) the target function not just along a path but around the entire neighborhood of the point of interest and it only works if the function really does have a limit there. It can, therefore, be used to prove that a function has a limit at a point, but it can never be used to prove that a function does not have a limit at a point. lim ( x , y ) → ( 0 , 0 ) x 2 y x 2 + y 2 {\displaystyle \lim _{(x,y)\to (0,0)}{\frac {x^{2}y}{x^{2}+y^{2}}}} cannot be found by taking any number of limits along paths that pass through the point, but since 0 ≤ x 2 x 2 + y 2 ≤ 1 − | y | ≤ y ≤ | y | ⟹ − | y | ≤ x 2 y x 2 + y 2 ≤ | y | lim ( x , y ) → ( 0 , 0 ) − | y | = 0 lim ( x , y ) → ( 0 , 0 ) | y | = 0 ⟹ 0 ≤ lim ( x , y ) → ( 0 , 0 ) x 2 y x 2 + y 2 ≤ 0 {\displaystyle {\begin{array}{rccccc}&0&\leq &\displaystyle {\frac {x^{2}}{x^{2}+y^{2}}}&\leq &1\\[4pt]-|y|\leq y\leq |y|\implies &-|y|&\leq &\displaystyle {\frac {x^{2}y}{x^{2}+y^{2}}}&\leq &|y|\\[4pt]{{\displaystyle \lim _{(x,y)\to (0,0)}-|y|=0} \atop {\displaystyle \lim _{(x,y)\to (0,0)}\ \ \ |y|=0}}\implies &0&\leq &\displaystyle \lim _{(x,y)\to (0,0)}{\frac {x^{2}y}{x^{2}+y^{2}}}&\leq &0\end{array}}} therefore, by the squeeze theorem, lim ( x , y ) → ( 0 , 0 ) x 2 y x 2 + y 2 = 0. {\displaystyle \lim _{(x,y)\to (0,0)}{\frac {x^{2}y}{x^{2}+y^{2}}}=0.} == References == === Notes === === References === == External links == Weisstein, Eric W. "Squeezing Theorem". MathWorld. Squeeze Theorem by Bruce Atwood (Beloit College) after work by, Selwyn Hollis (Armstrong Atlantic State University), the Wolfram Demonstrations Project. Squeeze Theorem on ProofWiki.
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Wikipedia:Srđan Ognjanović#0
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Srđan Ognjanović (Serbian Cyrillic: Срђан Огњановић, English alternatives: Srdjan Ognjanovic, and Srdan Ognjanovic) is a Serbian mathematician. He was a principal of Mathematical Grammar School in Belgrade. == Career == He received his degrees in the field of Mathematical Sciences from the Faculty of Mathematics and Natural Sciences, University of Belgrade. Prior to that, Ognjanović was a student of Mathematical Gymnasium Belgrade, from which he graduated in 1972, in A-division. Ognjanović started his professional career as a teacher of mathematics at Mathematical Gymnasium Belgrade (Serbian: "Matematička Gimnazija") while still a student of mathematics at Faculty of Mathematics and Natural Sciences, University of Belgrade, continued his career after graduation also in Mathematical Gymnasium Belgrade, and devoted his career to teaching mathematics in the same school, now being a professor in Mathematical Gymnasium Belgrade for more than 30 years. Students of professor Ognjanović won numerous prizes at International Science Olympiads in Mathematics, Physics, Informatics, Astronomy, Astrophysics, and Earth Sciences, also at other prestigious competitions around the world, and, accordingly, won many full scholarships at top-ranked universities. Mr Ognjanović is the author of numerous books and collections of problems for elementary and secondary schools, as well as special collections of assignments for preparation for mathematics competitions and mathematics workbooks used as a preparation for admission to faculties. == Awards and legacy == Ognjanović was listed among "300 most powerful people in Serbia" in a list published annually by "Blic" daily newspaper (14 February 2011), member of Axel Springer AG. The criteria were easiness in achieving goals, public awareness, financial and political influence, personal integrity and authority, respectiveness of the institution the person represents, and personal charisma. Among latest awards for his published works Mr Ognjanović received (in 2010): Grand Prize at 16th International Book Fair, in Novi Sad, from a Business Chamber of Vojvodina, and "Stojan Novaković" Prize for the best textbook and set of textbooks published by Zavod - Serbian State Company of Textbooks. == References == == External links == Mathematical Gymnasium Belgrade homepage Principal of Mathematical Gymnasium Belgrade homepage
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Wikipedia:Stahl's theorem#0
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In matrix analysis Stahl's theorem is a theorem proved in 2011 by Herbert Stahl concerning Laplace transforms for special matrix functions. It originated in 1975 as the Bessis-Moussa-Villani (BMV) conjecture by Daniel Bessis, Pierre Moussa, and Marcel Villani. In 2004 Elliott H. Lieb and Robert Seiringer gave two important reformulations of the BMV conjecture. In 2015, Alexandre Eremenko gave a simplified proof of Stahl's theorem. In 2023, Otte Heinävaara proved a structure theorem for Hermitian matrices introducing tracial joint spectral measures that implies Stahl's theorem as a corollary. == Statement of the theorem == Let tr {\displaystyle \operatorname {tr} } denote the trace of a matrix. If A {\displaystyle A} and B {\displaystyle B} are n × n {\displaystyle n\times n} Hermitian matrices and B {\displaystyle B} is positive semidefinite, define f ( t ) = tr ( exp ( A − t B ) ) {\displaystyle \mathbf {f} (t)=\operatorname {tr} (\exp(A-tB))} , for all real t ≥ 0 {\displaystyle t\geq 0} . Then f {\displaystyle \mathbf {f} } can be represented as the Laplace transform of a non-negative Borel measure μ {\displaystyle \mu } on [ 0 , ∞ ) {\displaystyle [0,\infty )} . In other words, for all real t ≥ 0 {\displaystyle t\geq 0} , f {\displaystyle \mathbf {f} } (t) = ∫ [ 0 , ∞ ) e − t s d μ ( s ) {\displaystyle \int _{[0,\infty )}e^{-ts}\,d\mu (s)} , for some non-negative measure μ {\displaystyle \mu } depending upon A {\displaystyle A} and B {\displaystyle B} . == References ==
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Wikipedia:Stan Wagon#0
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Stanley Wagon is a Canadian-American mathematician, a professor emeritus of mathematics at Macalester College in Minnesota. He is the author of multiple books on number theory, geometry, and computational mathematics, and is also known for his snow sculpture. == Biography == Wagon was born in Montreal, to Sam and Diana (Idlovitch) Wagon. His sister Lila (Wagon) Hope-Simpson died in 2021. Wagon did his undergraduate studies at McGill University in Montreal, graduating in 1971. He earned his Ph.D. in 1975 from Dartmouth College, under the supervision of James Earl Baumgartner. He married mathematician Joan Hutchinson, and the two of them shared a single faculty position at Smith College and again at Macalester, where they moved in 1990. == Books == The Banach–Tarski Paradox (Cambridge University Press, 1985) Old and New Unsolved Problems in Plane Geometry and Number Theory (with Victor Klee, Mathematical Association of America, 1991) Mathematica® in Action: Problem Solving Through Visualization and Computation (W.H. Freeman, 1991; 2nd ed., Springer, 1999; 3rd ed., Springer, 2010) Animating Calculus (with E. Packel, TELOS, 1996) Which Way Did the Bicycle Go? (with J. D. E. Konhauser and D. Velleman, Mathematical Association of America, 1996) VisualDSolve: Visualizing Differential Equations with Mathematica (with Dan Schwalbe, TELOS, 1997; 2nd ed., with Schwalbe and Antonin Slavik, Wolfram Research, 2009). A Course in Computational Number Theory (with David Bressoud, Springer, 2000) The Mathematical Explorer (Wolfram Research, Inc., 2001) The SIAM 100-Digit Challenge: A Study in High-Accuracy Numerical Computing (with Laurie, Bornemann, and Waldvogel, SIAM, 2004) == Other activities == Wagon is also known for riding a bicycle with square wheels, for his mathematical snow sculptures, and for having given the name to the 420 Arch, a natural stone arch in southern Utah. == Awards and honors == Wagon won the Lester R. Ford Award of the Mathematical Association of America for his 1988 paper, "Fourteen Proofs of a Result about Tiling a Rectangle". Wagon and his co-authors Ellen Gethner and Brian Wick won the Chauvenet Prize for mathematical exposition in 2002 for their 1998 paper, "A Stroll through the Gaussian Primes". == References == == External links == Official website
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Wikipedia:Stan van Hoesel#0
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Constantinus P. M. (Stan the man) van Hoesel (born 1961) is a Dutch mathematician, and Professor of Operations Research at the Maastricht University, and head of its Quantitative Economics Group, known for his work on mathematical optimization. == Life and work == Born in Tilburg, Stan obtained his Msc in mathematics at the Eindhoven University of Technology in 1986, and in 1991 his PhD at the Erasmus University Rotterdam under Alexander H. G. Rinnooy Kan and Antoon Kolen with the thesis, entitled "Models and Algorithms for Single-Item Lot Sizing Problems." In 1987 Van Hoesel started his academic career at the Erasmus University Rotterdam as assistant professor, and continued his research in mathematical modelling of problems and solutions for production planning. After his graduation he moved to the Eindhoven University of Technology, and later to the Maastricht University, where in 2001 he was appointed professor of operations research. His PhD students were at the Eindhoven University of Technology Cleola van Eijl (graduated in 1996); and at the Maastricht University Arie Koster (graduated in 1999), Reinder Lok (2007), and Bert Marchal (2012). Van Hoesel's research interests are in the field of "optimisation problems in the business world. He uses techniques from mathematics and computer sciences to discover the most efficient solution for various planning problems. Topics related to his research are telecommunications and traffic, where he looks at the migration of various types of networks and the sequence in which customers are served in order to process all orders as quickly as possible." == Selected publications == Stan Van Hoesel. Models and Algorithms for Single-Item Lot Sizing Problems, PhD thesis Erasmus University Rotterdam. Articles, a selection: Wagelmans, Albert, Stan Van Hoesel, and Antoon Kolen. "Economic lot sizing: an O (n log n) algorithm that runs in linear time in the Wagner-Whitin case." Operations Research 40.1-Supplement - 1 (1992): pp. 145–156. Zwaneveld, P. J., Kroon, L. G., Romeijn, H. E., Salomon, M., Dauzere-Péres, S., Van Hoesel, S. P., & Ambergen, H. W. (1996). "Routing trains through railway stations: Model formulation and algorithms." Transportation science, 30(3), 181–194.' CA Koster, Arie M., Hans L. Bodlaender, and Stan PM Van Hoesel. "Treewidth: computational experiments." Electronic Notes in Discrete Mathematics 8 (2001): 54–57. Zwaneveld, Peter J., Leo G. Kroon, and Stan PM Van Hoesel. "Routing trains through a railway station based on a node packing model." European Journal of Operational Research 128.1 (2001): 14–33. Aardal, K. I., Van Hoesel, S. P., Koster, A. M., Mannino, C., & Sassano, A. (2007). "Models and solution techniques for frequency assignment problems." Annals of Operations Research, 153(1), 79-129. == References == == External links == Hoesel, Stan van, Maastricht University
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Wikipedia:Standard basis#0
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In mathematics, the standard basis (also called natural basis or canonical basis) of a coordinate vector space (such as R n {\displaystyle \mathbb {R} ^{n}} or C n {\displaystyle \mathbb {C} ^{n}} ) is the set of vectors, each of whose components are all zero, except one that equals 1. For example, in the case of the Euclidean plane R 2 {\displaystyle \mathbb {R} ^{2}} formed by the pairs (x, y) of real numbers, the standard basis is formed by the vectors e x = ( 1 , 0 ) , e y = ( 0 , 1 ) . {\displaystyle \mathbf {e} _{x}=(1,0),\quad \mathbf {e} _{y}=(0,1).} Similarly, the standard basis for the three-dimensional space R 3 {\displaystyle \mathbb {R} ^{3}} is formed by vectors e x = ( 1 , 0 , 0 ) , e y = ( 0 , 1 , 0 ) , e z = ( 0 , 0 , 1 ) . {\displaystyle \mathbf {e} _{x}=(1,0,0),\quad \mathbf {e} _{y}=(0,1,0),\quad \mathbf {e} _{z}=(0,0,1).} Here the vector ex points in the x direction, the vector ey points in the y direction, and the vector ez points in the z direction. There are several common notations for standard-basis vectors, including {ex, ey, ez}, {e1, e2, e3}, {i, j, k}, and {x, y, z}. These vectors are sometimes written with a hat to emphasize their status as unit vectors (standard unit vectors). These vectors are a basis in the sense that any other vector can be expressed uniquely as a linear combination of these. For example, every vector v in three-dimensional space can be written uniquely as v x e x + v y e y + v z e z , {\displaystyle v_{x}\,\mathbf {e} _{x}+v_{y}\,\mathbf {e} _{y}+v_{z}\,\mathbf {e} _{z},} the scalars v x {\displaystyle v_{x}} , v y {\displaystyle v_{y}} , v z {\displaystyle v_{z}} being the scalar components of the vector v. In the n-dimensional Euclidean space R n {\displaystyle \mathbb {R} ^{n}} , the standard basis consists of n distinct vectors { e i : 1 ≤ i ≤ n } , {\displaystyle \{\mathbf {e} _{i}:1\leq i\leq n\},} where ei denotes the vector with a 1 in the ith coordinate and 0's elsewhere. Standard bases can be defined for other vector spaces, whose definition involves coefficients, such as polynomials and matrices. In both cases, the standard basis consists of the elements of the space such that all coefficients but one are 0 and the non-zero one is 1. For polynomials, the standard basis thus consists of the monomials and is commonly called monomial basis. For matrices M m × n {\displaystyle {\mathcal {M}}_{m\times n}} , the standard basis consists of the m×n-matrices with exactly one non-zero entry, which is 1. For example, the standard basis for 2×2 matrices is formed by the 4 matrices e 11 = ( 1 0 0 0 ) , e 12 = ( 0 1 0 0 ) , e 21 = ( 0 0 1 0 ) , e 22 = ( 0 0 0 1 ) . {\displaystyle \mathbf {e} _{11}={\begin{pmatrix}1&0\\0&0\end{pmatrix}},\quad \mathbf {e} _{12}={\begin{pmatrix}0&1\\0&0\end{pmatrix}},\quad \mathbf {e} _{21}={\begin{pmatrix}0&0\\1&0\end{pmatrix}},\quad \mathbf {e} _{22}={\begin{pmatrix}0&0\\0&1\end{pmatrix}}.} == Properties == By definition, the standard basis is a sequence of orthogonal unit vectors. In other words, it is an ordered and orthonormal basis. However, an ordered orthonormal basis is not necessarily a standard basis. For instance the two vectors representing a 30° rotation of the 2D standard basis described above, i.e. , v 1 = ( 3 2 , 1 2 ) {\displaystyle v_{1}=\left({{\sqrt {3}} \over 2},{1 \over 2}\right)\,} v 2 = ( 1 2 , − 3 2 ) {\displaystyle v_{2}=\left({1 \over 2},{-{\sqrt {3}} \over 2}\right)\,} are also orthogonal unit vectors, but they are not aligned with the axes of the Cartesian coordinate system, so the basis with these vectors does not meet the definition of standard basis. == Generalizations == There is a standard basis also for the ring of polynomials in n indeterminates over a field, namely the monomials. All of the preceding are special cases of the indexed family ( e i ) i ∈ I = ( ( δ i j ) j ∈ I ) i ∈ I {\displaystyle {(e_{i})}_{i\in I}=((\delta _{ij})_{j\in I})_{i\in I}} where I {\displaystyle I} is any set and δ i j {\displaystyle \delta _{ij}} is the Kronecker delta, equal to zero whenever i ≠ j and equal to 1 if i = j. This family is the canonical basis of the R-module (free module) R ( I ) {\displaystyle R^{(I)}} of all families f = ( f i ) {\displaystyle f=(f_{i})} from I into a ring R, which are zero except for a finite number of indices, if we interpret 1 as 1R, the unit in R. == Other usages == The existence of other 'standard' bases has become a topic of interest in algebraic geometry, beginning with work of Hodge from 1943 on Grassmannians. It is now a part of representation theory called standard monomial theory. The idea of standard basis in the universal enveloping algebra of a Lie algebra is established by the Poincaré–Birkhoff–Witt theorem. Gröbner bases are also sometimes called standard bases. In physics, the standard basis vectors for a given Euclidean space are sometimes referred to as the versors of the axes of the corresponding Cartesian coordinate system. == See also == Canonical units Examples of vector spaces § Generalized coordinate space == Citations == == References ==
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Wikipedia:Standard flag#0
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In heraldry and vexillology, a heraldic flag is a flag containing coats of arms, heraldic badges, or other devices used for personal identification. Heraldic flags include banners, standards, pennons and their variants, gonfalons, guidons, and pinsels. Specifications governing heraldic flags vary from country to country, and have varied over time. == Types == === Pennon === The pennon is a small elongated flag, either pointed or swallow-tailed (when swallow-tailed it may be described as a banderole). It was charged with the heraldic badge or some other armorial ensign of the owner, and displayed on his own lance, as a personal ensign. The pennoncelle was a modification of the pennon. In contemporary Scots usage, the pennon is 120 cm (four feet) in length. It tapers either to a point or to a rounded end as the owner chooses. It is assigned by the Lord Lyon King of Arms to any armiger who wishes to apply for it. === Banner === The banner of arms (also simply called banner) is square or oblong and larger than the pennon, bearing the entire coat of arms of the owner, composed precisely as upon a shield but in a square or rectangular shape. In the olden time, when a Knight had distinguished himself by conspicuous gallantry, it was the custom to mark his meritorious conduct by prompt advancement on the very field of battle. In such a case, the point or points of the good Knight's Pennon were rent off, and thus the ... small Flag was reduced to the square form of the Banner, by which thenceforth he was to be distinguished The banners of members of Orders of Chivalry are typically displayed in the Order's chapel. Banners of Knights of the Order of the Thistle are hung in the 1911 chapel of the Order in St Giles High Kirk in Edinburgh. Banners of Knights of the Order of the Garter are displayed in St George's Chapel at Windsor Castle. From Victorian times Garter banners have been approximately 1.5 m × 1.5 m (5 ft × 5 ft). Banners became available to all English armigers as a result of a report by Garter to the Earl Marshal dated 29 January 1906. The report stated that the size of a banner for Esquires and Gentlemen should be considered in the future. Until that date, they were available to all noblemen and knights banneret. In 2011, Garter Woodcock said that the banner for an Esquire or Gentleman should be the same size as a Marquess's and those of a lower rank down to Knight, that is, 90 cm × 90 cm (3 ft × 3 ft). In Scotland, the size of personal banners, excluding any fringes, are specified by the Lord Lyon. === Heraldic standard === The heraldic standard appeared around the middle of the fourteenth century, and it was in general use by personages of high rank during the two following centuries. The standard appears to have been adopted for the special purpose of displaying badges. "The badge was worn on his livery by a servant as retainer, and consequently the standard by which he mustered in camp was of the livery colours, and bore the badge, with both of which the retainer was familiar." Heraldic flags that are used by individuals, like a monarch or president, as a means of identification are often called 'standards' (e.g. royal standard). These flags, usually banners, are not standards in a strict heraldic sense but have come to be known as such. The heraldic standard is not rectangular – it tapers, usually from 120 to 60 cm (4 to 2 ft), and the fly edge is rounded (lanceolate). In England, any armiger who has been granted a badge is entitled to fly a standard. The medieval English standard was larger than the other flags, and its size varied with the owner's rank. The Cross of Saint George usually appeared next to the staff, and the rest of the field was generally divided per fess (horizontally) into two colours, in most cases the livery colours of the owner. "With some principal figure or device occupying a prominent position, various badges are displayed over the whole field, a motto, which is placed bend-wise, having divided the standard into compartments. The edges are fringed throughout, and the extremity is sometimes swallow-tailed, and sometimes rounded." The Royal standards of England were used by the kings of England as a headquarters symbol for their armies. Modern usage of the heraldic standard includes the flag of the Master Gunner, St James's Park and the flag of the Port of London Authority (used by the chairman and the Vice Chairman). The Oriflamme was the battle standard of the King of France during the Middle Ages. In Scotland, a standard requires a separate grant by the Lord Lyon. Such a grant is made only if certain conditions are met. The length of the standard depends upon one's noble rank. === Banderole === A Banderole (Fr. for a "little banner"), has both a literal descriptive meaning for its use by knights and ships, and is also heraldic device for representing bishops. === Gonfalone === A gonfalone or gonfalon is a vertically hung banner emblazoned with a coat of arms. Gonfalons have wide use in civic, religious, and academic heraldry. The term originated in Florence, Italy, where communities, or neighborhoods, traditionally displayed gonfaloni in public ceremonies. === Guidon === The Scots guidon is similar in shape to the standard and pennon. At 1.98 metres (6 feet 6 inches) long, it is smaller than the standard and twice the size of the pennon. Guidons are assigned by the Lord Lyon to those individuals who qualify for a grant of supporters to their Arms and to other individuals who have a following such as individuals who occupy a position of leadership or a long-term official position commanding the loyalty of more than a handful of people. The Guidon tapers to a round, unsplit end at the fly. A guidon can also refer to a cavalry troop's banner, such as that which survived the Custer massacre. === Pinsel === The Scottish pinsel is triangular in shape, 76 cm (2+1⁄2 ft) high at the hoist and 140 cm (4+1⁄2 ft) in width tapering to a point. This is the flag denoting a person to whom a Clan Chief has delegated authority for a particular occasion, such as a clan gathering when the Chief is absent. This flag is allotted only to Chiefs or very special Chieftain-Barons for practical use, and only upon the specific authority of the Lord Lyon King of Arms. == See also == Banners of the members of the Garter Royal standard of Cambodia Royal standards of Canada Flag of the governor general of Canada Flags of the lieutenant governors of Canada Royal Standard of Norway Royal Standard of Spain Royal Standard of the United Kingdom Royal standards of England Royal Banner of England Royal Banner of Scotland Oriflamme Personal Command Sign of the Swedish Monarch Royal Standard of Thailand King's Flag for Australia Flag of the governor-general of Australia Flags of the governors of the Australian states Queen's Personal New Zealand Flag Flag of the governor-general of New Zealand Japanese heraldic banners: Fūrinkazan Hata-jirushi Nobori Sashimono Uma-jirushi == Notes == == References == Berry, W. (1830). Encyclopaedia heraldica or complete dictionary of heraldry. Vol. 1. London: Sherwood, Gilbert & Piper. Boutell, Charles (1914). Fox-Davies, A. C. (ed.). The Handbook to English Heraldry (11th ed.). London: Reeves & Turner. Burnett, C.J.; Hodgson, L. (2001). Stall Plates of the Most Ancient and Most Noble Order of the Thistle in the Chapel of the Order within St Giles' Cathedral, The High Kirk of Edinburgh. Edinburgh: Heraldry Society of Scotland. ISBN 0-9525258-3-6. Chisholm, Hugh, ed. (1911). "Banderole" . Encyclopædia Britannica. Vol. 3 (11th ed.). Cambridge University Press. p. 312. Johnston, L. (2011). "Emperor Akihito and the heraldic achievements of the Garter". Lipskey, Glenn Edgard (1972). "The Chronicle of Alfonso the Emperor, The Poem of Almeria". Retrieved 15 December 2014. "The Court of the Lord Lyon - Further Guidance on Flags". lyon-court.org. Retrieved 3 March 2019. Woodcock, T (2011). "Garter King of Arms". College of Arms. Personal communication dated 23 December 2011. Attribution This article incorporates text from A. C. Fox-Davies' 1914 edition of Charles Boutell's The Handbook to English Heraldry at Project Gutenberg, which is in the public domain in the United States. == Further reading == Fox-Davies, Arthur Charles (1976) [1904]. The Art of Heraldry (facsimile ed.). Arno Press. Nelson, Phil (1 February 2010). "Banderole". Dictionary of heraldic terms. OED staff (September 2011). "banderol[e] | bandrol | bannerol, n.". Oxford English Dictionary (Second 1989; online version September 2011. ed.). Earlier version first published in New English Dictionary, 1885.
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Wikipedia:Stanislas Ouaro#0
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Stanislas Ouaro (born 19 January 1975) is a Burkinabé politician and mathematician. == Biography == Stanislas Ouaro was born on 19 January 1975. He graduated with a doctor's degree from University of Ouagadougou in 2001 with his thesis titled Etude de problèmes elliptiques-paraboliques nonlinéaires en une dimension d'espace. Before he joined government, he was the president of University of Ouaga II since 2012. On 31 January 2018, he was appointed the Minister of National Education and Literacy, replacing Jean-Martin Coulibaly. On 19 January 2019, he resigned together with other members of Thieba cabinet. On 24 January, he was appointed the Minister of National Education, Literacy and Promotion of National Languages. == Health == During the 2020 coronavirus outbreak, on 21 March, Ouaro contracted the coronavirus. == References == == External links == Stanislas Ouaro on Facebook
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Wikipedia:Stanislav Vydra#0
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Stanislav Vydra (13 November 1741 in Hradec Králové – 2 December 1804 in Prague) was a Bohemian Jesuit priest, writer, and mathematician. == Life == Vydra entered the Jesuit novitiate of Hradec Králové in 1757. After two years in Brno, he studied philosophy and mathematics from 1762 to 1764 at Charles University. His teachers included Joseph Stepling and Jan Tesánek. In 1765, he went as a teacher to Jičín and became Stepling's assistant a year later. He ministered as parish priest in Vilémov from 1771 to 1772. In 1772, Vydra was appointed professor of mathematics in Charles University in Prague. Here he taught until 1773. From 1789 to 1799, he was appointed to the mathematics faculty and served as dean of the Faculty of Arts. He became the rector of the university in 1800. He went blind in 1803 and died one year later. He is buried in Prague at the Olšany Cemetery in Prague. == Teachings == Stanislav Vydra taught elementary mathematics, a compulsory subject for the students at the philosophical faculty since 1752. He published “Elementa calcvli differentialis et integralis” in 1783, which became a well-known calculus textbook in Prague. After his death, his pupil and successor Josef Ladislav Jandera published his book Pocátkowé Arytmetyky, which was the first text book of elementary mathematics in Bohemia. == Selected works == Historia Matheseos in Bohemia et Moravia cultae, 1778 Elementa Calcvli Differentialis, et Integralis, 1783 Počátkowé Arytmetyky, 1806 == References == Between elementary mathematics and national wiedergeburt – 274 sides, Broschur George Schuppener, Karel Macek: Stanislav Vydra (1741–1804), Leipzig University publishing house (2004)
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Wikipedia:Stanisław Grzepski#0
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Stanisław Grzepski (1524–1570) was a Polish humanist and mathematician. == Sources == Linke, Waldemar (2019). "'The Sarmatian In Languages Trained'. Staniskaw Grzepski (1524-1570) As A Researcher Of The Hebrew Bible And The Septuagint". Studia Theologica Varsaviensia – via Academia.edu. Linke, Waldemar (2023). "A Year in Stanisław Grzepski's (1524–1570) "De multiplici siclo et talento hebraico"". Verbum Vitae – via Academia.edu. Bibliografia polska (in Polish)
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Wikipedia:Stanisław Jaśkowski#0
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Stanisław Jaśkowski (Polish pronunciation: [staˈɲsvaf jaɕˈkɔfskʲi]; 22 April 1906, in Warsaw – 16 November 1965, in Warsaw) was a Polish logician who made important contributions to proof theory and formal semantics. He was a student of Jan Łukasiewicz and a member of the Lwów–Warsaw School of Logic. He is regarded as one of the founders of natural deduction, which he discovered independently of Gerhard Gentzen in the 1930s. He is also known for his research into paraconsistent logic. Upon his death, his name was added to the Genius Wall of Fame. He was the President (rector) of the Nicolaus Copernicus University in Toruń. == Life and career == He was born in 1906 in Warsaw to father Feliks Jaśkowski and mother Kazimiera (nee Dzierzbicka). In 1924, he graduated from high school in Zakopane and enrolled at the University of Warsaw to study mathematics. He was taught mathematical logic under Jan Łukasiewicz and participated in the Polish Mathematicians' Congresses in Lviv (1927) and Vilnius (1931). After the outbreak of World War II, he participated in the September Campaign as a volunteer. In 1942, he was briefly imprisoned by the Germans. In 1945, he continued his scientific career at the University of Toruń where he defended his habilitation and assumed the post of the head of the Faculty of Mathematical Logic. Since 1950, he collaborated with the State Institute of Mathematics of the Polish Academy of Sciences (PAN). Between 1959–1962, he served as the Rector of the University. He was among the founders and served as the first President of the Polish Mathematical Society's branch in Toruń. Jaśkowski is considered to be one of the founders of natural deduction, which he discovered independently of Gerhard Gentzen in the 1930s. Gentzen's approach initially became more popular with logicians because it could be used to prove the cut-elimination theorem. However, Jaśkowski's is closer to the way that proofs are done in practice. He was also one of the first to propose a formal calculus of inconsistency-tolerant (or paraconsistent) logic. Furthermore, Jaśkowski was a pioneer in the investigation of both intuitionistic logic and free logic. He died in 1965 in Warsaw and was buried at the Powązki Cemetery. == Works == On the Rules of Suppositions in Formal Logic Studia Logica 1, 1934 pp. 5–32 (reprinted in: Storrs McCall (ed.), Polish Logic 1920-1939, Oxford University Press, 1967 pp. 232–258 Investigations into the System of Intuitionist Logic 1936 (translated in: Storrs McCall (ed.), Polish Logic 1920-1939, Oxford University Press, 1967 pp. 259–263 A propositional Calculus for Inconsistent Deductive Systems 1948 (reprinted in: Studia Logica, 24 1969, pp 143–157 and in: Logic and Logical Philosophy 7, 1999 pp. 35–56) On the Discussive Conjunction in the Propositional Calculus for Inconsistent Deductive Systems 1949 (reprinted in: Logic and Logical Philosophy 7, 1999 pp. 57–59) On Formulas in which no Individual Variable occurs more than Twice, Journal of Symbolic Logic, 31, 1966, pp. 1–6) in Polish O symetrii w zdobnictwie i przyrodzie - matematyczna teoria ornamentów (English title: On Symmetry in Art and Nature), PWS, Warszawa, 1952 (book 168 pages) Matematyczna teoria ornamentów (English title: Mathematical Theory of Ornaments), PWN, Warszawa, 1957 (book 100 pages) == See also == List of Polish mathematicians Timeline of Polish science and technology == References == == Sources == Jerzy Perzanowski (1999). "Fifty Years of Parainconsistent Logics" (PDF). Logic and Logical Philosophy. 7: 21–24. Archived from the original (PDF) on 2006-04-04. Woleński, Jan (2003). "Lvov-Warsaw School". The Stanford Encyclopedia of Philosophy (Summer 2003 Edition). Retrieved 2006-03-11. Jerzy Kotas, August Pieczkowski. Scientific works of Stanisław Jaśkowski, Studia Logica 21, 1967, 7-15 == External links == Polish Logic of the Postwar Period
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Wikipedia:Stanisław Krajewski#0
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Stanisław Krajewski (born 1950) is a Polish philosopher, mathematician and writer, activist of the Jewish minority in Poland. == Biography == He is professor of philosophy at the University of Warsaw, author, leader of the Jewish community in Poland and co-chairman of the Polish Council of Christians and Jews. Born in Warsaw in 1950, he studied at the Faculty of Mathematics, University of Warsaw, obtained Ph.D. in mathematics at the Institute of Mathematics of the Polish Academy of Sciences, and later Habilitation degree in philosophy at the Faculty of Philosophy and Sociology of the University of Warsaw. In 2012 he was awarded the title of “professor of humanities” by the President of Poland. Krajewski taught at the Bialystok branch of the University of Warsaw (1975 - 1981), later at the Institute of Mathematics, Polish Academy of Sciences. Since 1997 he has taught at the Institute of Philosophy of the University of Warsaw and chaired the Institute’s Scientific Council since 2012. Krajewski was involved in dissident activities during the communist period, was member of “Solidarity” from the beginning in 1980 till 1990, the underground period included. Immediately after the fall of communism in 1989, Krajewski was among the founders of the Polish-Israeli Friendship Society as well as the Polish Council of Christians and Jews. He has been the Jewish co-chairman of the Council since its inception in 1989. - He served on the board of the Union of Jewish Religious Communities in Poland (1997-2006) - was the Polish consultant to the American Jewish Committee (1992-2009) - was member of the International Council of the Auschwitz Camp Museum and Memorial (from its beginning in 1990 until 2006) - He has also been involved in devising the post-World War II section of the core exhibition in the Warsaw Museum of the History of Polish Jews, POLIN, opened in 2014. Krajewski is author of publications in the field of logic and philosophy of mathematics as well as numerous books and articles on Judaism, Jewish experience and Christian-Jewish dialogue. A recipient (jointly with his wife) of the Lifetime Achievement Award of the Taube Foundation for Jewish Life & Culture and American Jewish Committee, presented during the 23rd Jewish Culture Festival in Kraków. Married to Monika Krajewska., they have two sons. == Books == 2024: Small Numbers, Big Presence: Jews in Poland after World War II (Peter Lang, Berlin, ISBN 978-3-631-90084-0) 2022: Wybranie. O rozumieniu kilku terminów języka świętego (Austeria, Kraków, ISBN 978-83-7866-526-7) 2019: Żydzi w Polsce – i w Tatrach też (Austeria, Kraków, ISBN 978-83-7866-242-6) 2018: Was ich dem interreligiösen Dialog und dem Christentum verdanke, 5-77, Co zawdzięczam dialogowi międzyreligijnemu i chrześcijaństwu, 79-141(Fundacja Judaica, Kraków, ISBN 978-83-936339-4-4). 2017: Co zawdzięczam dialogowi międzyreligijnemu i chrześcijaństwu, 1-67, What I Owe to Interreligious Dialogue and Christianity, 71-127 (Fundacja Judaica, Kraków, ISBN 978-83-936339-2-0). 2014: Żydzi i... (Austeria, ISBN 978-83-7866-029-3) 2014 Czy fizyka i matematyka to nauki humanistyczne? (Do Physics and Mathematics Belong to the Humanities? with Michał Heller, in Polish, Copernicus Center Press, ISBN 978-83-7886-078-5) 2011: Czy matematyka jest nauką humanistyczną? (Does Mathematics Belong to the Humanities?, in Polish, Copernicus Center Press, ISBN 978-83-62259-13-7) 2010: Nasza żydowskość (Our Jewishness, in Polish, Austeria, ISBN 978-83-61978-44-2) 2007: Tajemnica Izraela a tajemnica Kościoła (The Mystery of Israel and the Mystery of the Church, in Polish, Biblioteka "Więzi", ISBN 978-83-60356-39-5) 2005: Poland and the Jews: reflections of a Polish Polish Jew (in English, Austeria, ISBN 83-89129-22-1) 2004: 54 komentarze do Tory dla nawet najmniej religijnych spośród nas (54 Commentaries on the Torah for Even the Least Religious Among Us, in Polish, Austeria, ISBN 83-89129-02-7) 2003: Twierdzenie Gödla i jego interpretacje filozoficzne: od mechanicyzmu do postmodernizmu (Goedel’s Theorem and Its Philosophical Interpretations: from Mechanism to Post-Modernism, in Polish, Wyd. Instytutu Filozofii i Socjologii PAN, ISBN 83-7388-017-8) 1997: Żydzi, judaizm, Polska (Jews, Judaism, Poland, in Polish, Vocatio, ISBN 83-7146-073-2) Co-editor of Theological Discourse and Logic, Logica universalis 13(4)/2019, eds. Marcin Trepczyński and Stanisław Krajewski. Journal of Applied Logics — IfCoLog Journal of Logics and their Applications, Special Issue “Concept of God”, Guest Editors: Stanisław Krajewski and Ricardo Silvestre, vol 6 (6)/2019. Poznanie i religia: epistemołogia religioznowo opyta w ruskoj i jewrejskoj fiłosofskoj mysli XX wieka (Russian), eds. Janusz Dobieszewski, Stanisław Krajewski, Jakub Mach (Wyd. UW WFiS, Warszawa) Epistemologia doświadczenia religijnego w XX-wiecznej filozofii rosyjskiej i żydowskiej, red. Janusz Dobieszewski, Stanisław Krajewski, Jakub Mach (Kraków: Universitas; ISBN 97883-242-3425-7) Studies in Logic, Grammar and Rhetoric 44 (57), 2016, Theology in Mathematics?(ed. by Stanisław Krajewski and Kazimierz Trzęsicki); ISBN 978-83-7431-480-0 Studies in Logic, Grammar and Rhetoric 27 (40), Papers on Logic and Rationality: Festschrift in Honour of Andrzej Grzegorczyk (2012, in English, Bialystok: Univ. of Bialystok); Abraham Joshua Heschel: Philosophy, Theology and Interreligious Dialogue, ed. by S. Krajewski and A. Lipszyc (2009, in English, Wiesbaden: Harrassowitz Verlag). Common Rejoicing in the Torah (2008, in Polish), Topics in Logic, Philosophy and Foundations of Mathematics and Computer Science. In Recognition of Professor Andrzej Grzegorczyk (2007, in English, Amsterdam: IOS), == References == == External links == http://spispracownikow.uw.edu.pl/index2.php?szukaj=startp http://www.woolf.cam.ac.uk/uploads/krajeski.pdf https://uw.academia.edu/StanislawKrajewski/Papers
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Wikipedia:Stanisław Leśniewski#0
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Stanisław Leśniewski (Polish: [lɛɕˈɲɛfskʲi]; 30 March 1886 – 13 May 1939) was a Polish mathematician, philosopher and logician. A professor of mathematics at the University of Warsaw, he was a leading representative of the Lwów–Warsaw School of Logic and is known for coining and introducing the concept of mereology as part of a comprehensive framework for logic and mathematics. == Life == Leśniewski was born on 28 March 1886 at Serpukhov, near Moscow, to father Izydor, an engineer working on the construction of the Trans-Siberian Railway, and mother Helena (née Palczewska). Leśniewski went to a high school in Irkutsk. Later he attended lectures by Hans Cornelius at the Ludwig Maximilian University of Munich and lectures by Wacław Sierpiński at Lviv University. Leśniewski belonged to the first generation of the Lwów–Warsaw School of logic founded by Kazimierz Twardowski. Together with Alfred Tarski and Jan Łukasiewicz, he formed a trio which made the University of Warsaw, during the interbellum, perhaps the most important research center in the world for formal logic. Despite that, Leśniewski's growing anti-semitism later caused the deterioration of the relationship with Tarski. His main contribution was the construction of three nested formal systems, to which he gave the Greek-derived names of protothetic, ontology, and mereology. ("Calculus of names" is sometimes used instead of ontology, a term widely employed in metaphysics in a very different sense.) A good textbook presentation of these systems is that by Simons (1987), who compares and contrasts them with the variants of mereology, more popular nowadays, descending from the calculus of individuals of Leonard and Goodman. Simons clarifies something that is very difficult to determine by reading Leśniewski and his students, namely that Polish mereology is a first-order theory equivalent to what is now called classical extensional mereology (modulo choice of language). While he did publish a fair body of work (Leśniewski, 1992, is his collected works in English translation), some of it in German, the leading language for mathematics of his day, his writings had limited impact because of their enigmatic style and highly idiosyncratic notation. Leśniewski was also a radical nominalist: he rejected axiomatic set theory at a time when that theory was in full flower. He pointed to Russell's paradox and the like in support of his rejection, and devised his three formal systems as a concrete alternative to set theory. Even though Alfred Tarski was his sole doctoral pupil, Leśniewski nevertheless strongly influenced an entire generation of Polish logicians and mathematicians via his teaching at the University of Warsaw. It is mainly thanks to the writings of his students (e.g., Srzednicki and Rickey 1984) that Leśniewski's thought is known. During the Polish–Soviet War of 1919-21, Leśniewski served the cause of Poland's independence by breaking Soviet Russian ciphers for the Polish General Staff's Cipher Bureau. Leśniewski died suddenly of cancer, shortly before the German invasion of Poland, which resulted in the destruction of his Nachlass. He was buried at Warsaw's Powązki Cemetery. == Works == 1988. Lecture Notes in Logic. Kluwer. Table of Contents. 1992. Collected Works. 2 vols. Kluwer. Table of Contents. 1929, "Über Funktionen, deren Felder Gruppen mit Rücksicht auf diese Funktionen sind", Fundamenta Mathematicae 13: 319-32. 1929, "Grundzüge eines neuen Systems der Grundlagen der Mathematik", Fundamenta Mathematicae 14: 1-81. 1929, "Über Funktionen, deren Felder Abelsche Gruppen in bezug auf diese Funktionen sind", Fundamenta Mathematicae 14: 242-51. == See also == History of philosophy in Poland List of Poles == References == == Bibliography == Ivor Grattan-Guinness, 2000. In Search of Mathematical Roots. Princeton: Princeton University Press. Luschei, Eugene, 1962. The Logical Systems of Lesniewski. Amsterdam: North-Holland. Miéville, Denis, 1984. "Un Développement des Systèmes Logiques de Stanislas Lesniewski", Peter Lang, European University Studies. Simons, Peter, 1987. Parts: A Study in Ontology. New York: Oxford University Press. Srzednicki, J. T. J., and Rickey, V. F., (eds.), 1984. Lesniewski's Systems: Ontology and Mereology. Dordrecht: Kluwer. Surma, Stanislaw J. (editor) (1977/8) "On Leśniewski's Systems, Proceedings of XXII Conference on History of Logic", Studia Logica 36(4): 247–426 MR0476370 Urbaniak, Rafal, 2013. Leśniewski's Systems of Logic and Foundations of Mathematics, Dordrecht: Springer. Wolenski, Jan, 1989. Logic and Philosophy in the Lwow-Warsaw School. Dordrecht: Kluwer. == External links == "Lesniewski: Logic". Internet Encyclopedia of Philosophy. Simons, Peter. "Lesniewski". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy. Woleński, Jan. "Lvov-Warsaw school". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy. Betti, Arianna, 2001, "Sempiternal Truth: The Bolzano-Twardowski-Lesniewski connection." Polish Philosophy: Stanislaw Lesniewski by Francesco Coniglione and Arianna Betti. Raul Corazzon's Theory and History of Ontology web page: Lesniewski. Selected bibliography of and about Lesniewski. Includes the English translations and selected bibliography of the secondary literature. O'Connor, John J.; Robertson, Edmund F., "Stanisław Leśniewski", MacTutor History of Mathematics Archive, University of St Andrews
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Wikipedia:Stanisław Radziszowski#0
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Stanisław P. Radziszowski (born June 7, 1953) is a Polish-American mathematician and computer scientist, best known for his work in Ramsey theory. Radziszowski was born in Gdańsk, Poland, and received his PhD from the Institute of Informatics of the University of Warsaw in 1980. His thesis topic was "Logic and Complexity of Synchronous Parallel Computations". From 1976 to 1980 he worked as a visiting professor in various universities in Mexico City. In 1984, he moved to the United States, where he took up a position in the Department of Computer Science at the Rochester Institute of Technology. Radziszowski has published many papers in graph theory, Ramsey theory, block designs, number theory and computational complexity. In a 1995 paper with Brendan McKay he determined the Ramsey number R(4,5)=25. His survey of Ramsey numbers, last updated in March 2017, is a standard reference on the subject and published at the Electronic Journal of Combinatorics. == References == == External links == Radziszowski's survey of small Ramsey numbers Home Page Sound file of Radziszowski speaking his own name (au format)
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Wikipedia:Stanisław Saks#0
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Stanisław Saks (30 December 1897 – 23 November 1942) was a Polish mathematician and university tutor, a member of the Lwów School of Mathematics, known primarily for his membership in the Scottish Café circle, an extensive monograph on the theory of integrals, his works on measure theory and the Vitali–Hahn–Saks theorem. == Life and work == Stanisław Saks was born on 30 December 1897 in Kalisz, Congress Poland, to an assimilated Polish-Jewish family. In 1915 he graduated from a local gymnasium and joined the newly recreated Warsaw University. In 1922 he received a doctorate of his alma mater with a prestigious distinction maxima cum laude. Soon afterwards he also passed his habilitation and received the Rockefeller fellowship, which allowed him to travel to the United States. Around that time he started publishing articles in various mathematical journals, mostly the Fundamenta Mathematicae, but also in the Transactions of the American Mathematical Society. He participated in the Silesian Uprisings and was awarded the Cross of the Valorous and the Medal of Independence for his bravery. Following the end of the uprising he returned to Warsaw and resumed his academic career. For most of it he studied the theories of functions and functionals in particular. In 1930 he published his most notable book, the Zarys teorii całki (Sketch on the Theory of the Integral), which later got expanded and translated into several languages, including English (Theory of the Integral), French (Théorie de l'Intégrale) and Russian (Teoriya Integrala). Despite his successes, Saks was never awarded the title of professor and remained an ordinary tutor, initially at his alma mater and the Warsaw University of Technology, and later at the Lwów University and Wilno University. He was also an active socialist and a journalist at the Robotnik weekly (1919–1926) and later a collaborator of the Association of Socialist Youth. Saks wrote a mathematics book with Antoni Zygmund, Analytic Functions, in 1933. It was translated into English in 1952 by E. J. Scott. In the preface to the English edition, Zygmund writes: Stanislaw Saks was a man of moral as well as physical courage, of rare intelligence and wit. To his colleagues and pupils he was an inspiration not only as a mathematician but as a human being. In the period between the two world wars he exerted great influence upon a whole generation of Polish mathematicians in Warsaw and Lwów. In November 1942, at the age of 45, Saks died in a Warsaw prison, victim of a policy of extermination. After the outbreak of World War II and the occupation of Poland by Germany, Saks joined the Polish underground. Arrested in November 1942, he was executed on 23 November 1942 by the German Gestapo in Warsaw. == Publications == Saks, Stanisław (1937). Theory of the Integral. Monografie Matematyczne. Vol. 7 (2nd ed.). Warsaw-Lwów: G.E. Stechert & Co. pp. VI+347. JFM 63.0183.05. Zbl 0017.30004.. English translation by Laurence Chisholm Young, with two additional notes by Stefan Banach. Saks, Stanisław; Zygmund, Antoni (1965). Analytic functions. Monografie Matematyczne. Vol. 28 (Second ed.). Warsaw: Państwowe Wydawnietwo Naukowe. MR 0180658. == See also == Lwów School of Mathematics == Notes == == References == O'Connor, John J.; Robertson, Edmund F., "Stanisław Saks", MacTutor History of Mathematics Archive, University of St Andrews Zygmund, Antoni (1987), "Stanislaw Saks, 1897–1942", The Mathematical Intelligencer, 9, Springer New York: 36–41, doi:10.1007/BF03023571, ISSN 0343-6993, S2CID 119349092
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Wikipedia:Stanisław Solski#0
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Stanisław Solski (Kalisz, September 21, 1622 – Kraków, 9 August, 1701) was a Polish Jesuit mathematician and architect. He published several works in Polish and Latin. == Life == There aren't information on early life and origin. Solski joined the Jesuit Order in 1638, before he studied in a school in Kalisz. He studied philosophy in Kalisz and then theology in Poznań. From 1652 to 1653 he was a teacher of poetry in Krosno and from 1653 to 1654 he taught poetry and rhetoric in Kamieniec Podolski. In 1670 he left the mansion and move to Cracow where he occupied primarily architectural work, because he was the architect of the bishop Jan Malachowski. He designed and supervised the reconstruction and construction of churches and monasteries, including the church of St. Barbara. == Works == Machina motum perpetuum exhibens, 1661 Machina exhibendo motui perpetuo artificiali idonea, 1663 Geometra Polski: Księga 1, Księga 2, Księga 3, 1683-1686 Architekt Polski, 1688 Solski, Stanislaw (1688). Praxis nova et expeditissima mensurandi geometrice (in Latin). Cracoviae: Franciszek Cezary. == External links == Grzebień, Ludwik. "Stanisław Solski" (in Polish). Internetowy Polski Słownik Biograficzny. Retrieved July 5, 2017.
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Wikipedia:Stanisław Szarek#0
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Stanisław J. Szarek (born November 13, 1953) is a Polish professor of mathematics at both Case Western Reserve University in the USA (since 1983) and Pierre and Marie Curie University in France (since 1996). His research concerns convex geometry and functional analysis. Szarek was born in Lądek-Zdrój, Poland. He earned a master's degree from the University of Warsaw in 1976, and a Ph.D. from the Polish Academy of Sciences in 1979 under the supervision of Aleksander Pełczyński. He continued at the Polish Academy as a research fellow for four years before taking a faculty position at Case, where he is now the Kerr Professor of Mathematics. Szarek won a gold medal in the 1971 International Mathematical Olympiad. He was an invited speaker at the 2006 International Congress of Mathematicians. In 2007 he won the Langevin Prize of the French Academy of Sciences. In 2012 he became one of the inaugural fellows of the American Mathematical Society and in 2017 he was awarded the Sierpiński medal. == References ==
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Wikipedia:Stanisław Trybuła#0
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Stanisław Czesław Trybuła (2 January 1932 – 28 January 2008) was a Polish mathematician and statistician. == Early life and education == Trybuła was a pupil of state high school in Rypin, Poland, and he graduated from The First High School in Toruń in 1950. He studied mathematics in Nicolaus Copernicus University in Toruń and Wrocław University. He defended his master thesis on some problems of the game theory prepared under supervision of Hugo Steinhaus at Wrocław University in 1955. == Academic career == In 1955 Trybuła became a faculty member at Department of Mathematics, Wrocław University of Technology. In 1959 he was distinguished as the candidate of science and in 1960 he defended his PhD on minimax estimation under supervision of Hugo Steinhaus. For many years he collaborated with or was on the staff of Institute of Power Systems (IASE) in Wrocław. He worked out the original method of identification of the complex power systems. Since 1968 he was faculty member of the Institute of Mathematics, Faculty of Fundamental Problems of Technology, Wrocław University of Technology. Trybuła got habilitation at Faculty of Mathematics, Physics and Chemistry, Wrocław University in 1968 based on his seminal works on sequential analysis for stochastic processes. In 1973, he became an associate professor, and in 1988, a full professor of mathematical sciences. == Contributions == Trybuła was the advisor of 14 PhD theses. He published 102 works independently and 38 co-authored ones He took early retirement in 1998 and was writing academic books on statistics and the game theory. He is the co-author of the WJ bidding system in the bridge, known also as Polish Club (see also, following Stayman convention, Trybula transfers, Wesolowski texas, Gawrys fourth suit forcing). == References ==
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Wikipedia:Stanisław Zaremba (mathematician)#0
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Stanisław Zaremba (3 October 1863 – 23 November 1942) was a Polish mathematician and engineer. His research in partial differential equations, applied mathematics and classical analysis, particularly on harmonic functions, gained him a wide recognition. He was one of the mathematicians who contributed to the success of the Polish School of Mathematics through his teaching and organizational skills as well as through his research. Apart from his research works, Zaremba wrote many university textbooks and monographies. He was a professor of the Jagiellonian University (since 1900), member of Academy of Learning (since 1903), co-founder and president of the Polish Mathematical Society (1919), and the first editor of the Annales de la Société Polonaise de Mathématique. He should not be confused with his son Stanisław Krystyn Zaremba, also a mathematician. == Biography == Zaremba was born on 3 October 1863 in Romanówka, present-day Ukraine. The son of an engineer, he was educated at a grammar school in Saint Petersburg and studied at the Institute of Technology of the same city obtaining his diploma in engineering in 1886. The same year he left Saint Petersburg and went to Paris to study mathematics: he received his degree from the Sorbonne in 1889. He stayed in France until 1900, when he joined the faculty at the Jagiellonian University in Kraków. His years in France enabled him to establish a strong bridge between Polish mathematicians and those in France. He died on 23 November 1942 in Kraków, during the German occupation of Poland. == Work == === Research activity === Zaremba's magneficent discoveries found a broad response and deep appreciation in the world. Stanisław Zaremba is the pride of Polish science. At the same time, by his pioneering activity in such an important field as mathematical analysis, he became one of the creators of modern Polish mathematics. == Selected publication == — (1907). Zarys pierwszych zasad teoryi liczb całkowitych (in Polish). Kraków: Akademia Umiejętności. OCLC 69628334. — (1909). Pogląd na historyę rozwoju i stan obecny teoryi równań fizyki w czterech odczytach (in Polish). Warsaw: Druk. J. Sikorskiego. OCLC 69346520. — (1909). Teorya wyznaczników i równań liniowych (in Polish). Kraków: Kółko Matematyczno-Fizyczne Uczniów Uniwersytetu Jagiellońskiego. OCLC 69616537. — (1909), "Sur l'unicité de la solution du problème de Dirichlet", Bulletin International de l'Académie des Sciences de Cracovie. Classe des Sciences Mathématiques et Naturelles, Serie A: Sciences mathématiques (in French): 561–563, JFM 40.0452.02 — (1910), "Sur un problème mixte relatif à l' équation de Laplace", Bulletin International de l'Académie des Sciences de Cracovie. Classe des Sciences Mathématiques et Naturelles, Serie A: Sciences mathématiques (in French): 313–344, JFM 41.0854.12, translated in Russian as Zaremba, S. (1946), Об одной смешанной задаче, относящейся к уравнению Лапласа, Uspekhi Matematicheskikh Nauk (in Russian), 1 (3–4(13–14)): 125–146, MR 0025032, Zbl 0061.23010. — (1912). Arytmetyka teoretyczna (in Polish). Kraków: Akademia Umiejętności. OCLC 69511091. — (1914). Ogólne zasady analizy matematycznej. Cz. 2: Rachunek całkowy (in Polish). Kraków: Kółko Matematyczno-Fizyczne Uczniów Uniwersytetu Jagiellońskiego. — (1915). Wstęp do analizy. Cz. 1: Pojęcie dowodu matematycznego oraz inne wiadomości pomocnicze (in Polish). Warsaw: skł. gł. w Księgarni Gebethnera i Wolffa. OCLC 69586726. — (1918). Wstęp do analizy. Cz. 2: Teorya liczb rzeczywistych (in Polish). Warsaw: skł. gł. w Księgarni Gebethnera i Wolffa. OCLC 69586727. — (1933). Zarys mechaniki teoretycznej. Tom 1: Wiadomości pomocnicze i kinematyka (in Polish). Kraków: Akademia Umiejętności. OCLC 69538136. == See also == Kraków School of Mathematics Mixed boundary condition == Notes == == References == "Zaremba Stanisław". Internetowa encyklopedia PWN (in Polish). Wydawnictwo Naukowe PWN. Retrieved 2007-11-26.. "Zaremba Stanisław". WIEM Encyklopedia (in Polish). Retrieved 2007-11-26.. Koroński, Jan (2016), "Stanisław Zaremba (1863‒1942) and his results in the field of differential equations", Technical Transactions, 2016 (Fundamental Sciences Issue 2-NP (20) 2015), doi:10.4467/2353737XCT.15.217.4422 (inactive 1 November 2024){{citation}}: CS1 maint: DOI inactive as of November 2024 (link). Kuratowski, Kazimierz (1980), A Half Century of Polish Mathematics: Remembrances and Reflections, International Series in Pure and Applied Mathematics, vol. 108, Warsaw / Oxford: Wydawnictwo Naukowe PWN / Pergamon Press, pp. VIII+204, ISBN 83-01-00819-9, MR 0565253, Zbl 0438.01006, ISBN 0-08-023046-6. == External links == O'Connor, John J.; Robertson, Edmund F., "Stanisław Zaremba (mathematician)", MacTutor History of Mathematics Archive, University of St Andrews Stanisław Zaremba at the Mathematics Genealogy Project
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Wikipedia:Stanisław Świerczkowski#0
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Stanisław (Stash) Świerczkowski (16 July 1932 – 30 September 2015) was a Polish mathematician famous for his solutions to two iconic problems posed by Hugo Steinhaus: the three-gap theorem and the non-tetratorus theorem. == Early life and education == Stanisław (Stash) Świerczkowski was born in Toruń, Poland. His parents were divorced during his infancy. When war broke out his father was captured in Soviet-controlled Poland and murdered in the 1940 Katyń Massacre. He belonged to the Polish nobility; Świerczkowski's mother belonged to the upper middle class and would have probably suffered deportation and murder by the Nazis. However she had German connections and was able to gain relatively privileged class 2 Volksliste citizenship. At the end of the war Świerczkowski's mother was forced into hiding near Toruń until she was confident that she could win exoneration from the Soviet-controlled government for her Volksliste status and be rehabilitated as a Polish citizen. Meanwhile, Świerczkowski lived in a rented room in Toruń and attended school there. Świerczkowski won a university place to study astronomy at the University of Wrocław but switched to mathematics to avoid the drudgery of astronomical calculations. He discovered a natural ability through his friendship with Jan Mycielski and was able to remain at Wrocław to complete his masters under Jan Mikusiński. He graduated with a PhD in 1960, his dissertation including the now-famous three-gap theorem, which he proved in 1956 in answer to a question of Hugo Steinhaus. == Noted mathematical results == The three-gap theorem says: take arbitrarily finitely many integer multiples of an irrational number between zero and one and plot them as points around a circle of unit circumference; then at most three different distances will occur between consecutive points. This answered a question of Hugo Steinhaus. The theorem belongs to the field of Diophantine approximation since the smallest of the three distances observed may be used to give a rational approximation to the chosen irrational number. It has been extended and generalised in many ways. The non-tetratorus theorem, published by Świerczkowski in 1958, states that it is impossible to construct a closed chain (torus) of regular tetrahedra, placed face to face. Again this answered a question of Hugo Steinhaus. The result is attractive and counter-intuitive, since the tetrahedron is unique among the Platonic solids in having this property. Recent work by Michael Elgersma and Stan Wagon has sparked new interest in this result by showing that one can create chains of tetrahedra that are arbitrarily close to being closed. In 1964, in a joint work with Jan Mycielski, he established one of the early results on the axiom of determinacy (AD), namely that AD implies that all sets of real numbers are Lebesgue measurable. Świerczkowski's last mathematical work was on proving Gödel's incompleteness theorems using hereditarily finite sets instead of encoding of finite sequences of natural numbers. It is these proofs that were the basis for the production, in 2015, of mechanised proofs of Gödel's two famous theorems. == Career == Świerczkowski had a very migratory career. He was allowed abroad from Poland to study at Dundee University, where his work with Alexander Murray MacBeath would later attract the attention of André Weil. He then took up a research fellowship at Glasgow University before being obliged to return to Poland. When the Polish Academy of Sciences granted him a passport to attend a conference in Stuttgart he used this as an opportunity to leave Poland for good in 1961, first resuming his fellowship in Glasgow before taking a job in the recently created University of Sussex. In 1963 he visited André Weil at the Institute for Advanced Study and thereafter, between 1964 and 1973, held posts at the University of Washington, the Australian National University and Queen's University in Canada. In 1973 he left mathematics, moved to the Netherlands and built a yacht in which he sailed around the world for ten years. The period 1986 to 1997 was again spent teaching mathematics, at Sultan Qaboos University. His last post was at the University of Colorado at Boulder (1998–2001). Thereafter he retired to Tasmania. == References == == External links == Stanisław Świerczkowski at the Mathematics Genealogy Project O'Connor, John J.; Robertson, Edmund F., "Stanisław Świerczkowski", MacTutor History of Mathematics Archive, University of St Andrews
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Wikipedia:Stanko Bilinski#0
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Stanko Bilinski (22 April 1909 in Našice – 6 April 1998 in Zagreb) was a Croatian mathematician and academician. He was a professor at the University of Zagreb and a fellow of the Croatian Academy of Sciences and Arts. In 1960, he discovered a rhombic dodecahedron of the second kind, the Bilinski dodecahedron. Like the standard rhombic dodecahedron, this convex polyhedron has 12 congruent rhombus sides, but they are differently shaped and arranged. Bilinski's discovery corrected a 75-year-old omission in Evgraf Fedorov's classification of convex polyhedra with congruent rhombic faces. == References == == Further reading == "In memoriam: Stanko Bilinski (22.4.1909.–6.4.1998.)" (PDF). Glasnik Matematički (in Croatian). 33 (2). Croatian Mathematical Society: 323–333. December 1998. "Stanko Bilinski (1909. – 2009.)" (PDF). miš – matematika i škola (in Croatian).
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Wikipedia:Stanley symmetric function#0
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In mathematics and especially in algebraic combinatorics, the Stanley symmetric functions are a family of symmetric functions introduced by Richard Stanley (1984) in his study of the symmetric group of permutations. Formally, the Stanley symmetric function Fw(x1, x2, ...) indexed by a permutation w is defined as a sum of certain fundamental quasisymmetric functions. Each summand corresponds to a reduced decomposition of w, that is, to a way of writing w as a product of a minimal possible number of adjacent transpositions. They were introduced in the course of Stanley's enumeration of the reduced decompositions of permutations, and in particular his proof that the permutation w0 = n(n − 1)...21 (written here in one-line notation) has exactly ( n 2 ) ! 1 n − 1 ⋅ 3 n − 2 ⋅ 5 n − 3 ⋯ ( 2 n − 3 ) 1 {\displaystyle {\frac {{\binom {n}{2}}!}{1^{n-1}\cdot 3^{n-2}\cdot 5^{n-3}\cdots (2n-3)^{1}}}} reduced decompositions. (Here ( n 2 ) {\displaystyle {\binom {n}{2}}} denotes the binomial coefficient n(n − 1)/2 and ! denotes the factorial.) == Properties == The Stanley symmetric function Fw is homogeneous with degree equal to the number of inversions of w. Unlike other nice families of symmetric functions, the Stanley symmetric functions have many linear dependencies and so do not form a basis of the ring of symmetric functions. When a Stanley symmetric function is expanded in the basis of Schur functions, the coefficients are all non-negative integers. The Stanley symmetric functions have the property that they are the stable limit of Schubert polynomials F w ( x ) = lim n → ∞ S 1 n × w ( x ) {\displaystyle F_{w}(x)=\lim _{n\to \infty }{\mathfrak {S}}_{1^{n}\times w}(x)} where we treat both sides as formal power series, and take the limit coefficientwise. == References == Stanley, Richard P. (1984), "On the number of reduced decompositions of elements of Coxeter groups" (PDF), European Journal of Combinatorics, 5 (4): 359–372, doi:10.1016/s0195-6698(84)80039-6, ISSN 0195-6698, MR 0782057
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Wikipedia:Star domain#0
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In geometry, a set S {\displaystyle S} in the Euclidean space R n {\displaystyle \mathbb {R} ^{n}} is called a star domain (or star-convex set, star-shaped set or radially convex set) if there exists an s 0 ∈ S {\displaystyle s_{0}\in S} such that for all s ∈ S , {\displaystyle s\in S,} the line segment from s 0 {\displaystyle s_{0}} to s {\displaystyle s} lies in S . {\displaystyle S.} This definition is immediately generalizable to any real, or complex, vector space. Intuitively, if one thinks of S {\displaystyle S} as a region surrounded by a wall, S {\displaystyle S} is a star domain if one can find a vantage point s 0 {\displaystyle s_{0}} in S {\displaystyle S} from which any point s {\displaystyle s} in S {\displaystyle S} is within line-of-sight. A similar, but distinct, concept is that of a radial set. == Definition == Given two points x {\displaystyle x} and y {\displaystyle y} in a vector space X {\displaystyle X} (such as Euclidean space R n {\displaystyle \mathbb {R} ^{n}} ), the convex hull of { x , y } {\displaystyle \{x,y\}} is called the closed interval with endpoints x {\displaystyle x} and y {\displaystyle y} and it is denoted by [ x , y ] := { t y + ( 1 − t ) x : 0 ≤ t ≤ 1 } = x + ( y − x ) [ 0 , 1 ] , {\displaystyle \left[x,y\right]~:=~\left\{ty+(1-t)x:0\leq t\leq 1\right\}~=~x+(y-x)[0,1],} where z [ 0 , 1 ] := { z t : 0 ≤ t ≤ 1 } {\displaystyle z[0,1]:=\{zt:0\leq t\leq 1\}} for every vector z . {\displaystyle z.} A subset S {\displaystyle S} of a vector space X {\displaystyle X} is said to be star-shaped at s 0 ∈ S {\displaystyle s_{0}\in S} if for every s ∈ S , {\displaystyle s\in S,} the closed interval [ s 0 , s ] ⊆ S . {\displaystyle \left[s_{0},s\right]\subseteq S.} A set S {\displaystyle S} is star shaped and is called a star domain if there exists some point s 0 ∈ S {\displaystyle s_{0}\in S} such that S {\displaystyle S} is star-shaped at s 0 . {\displaystyle s_{0}.} A set that is star-shaped at the origin is sometimes called a star set. Such sets are closely related to Minkowski functionals. == Examples == Any line or plane in R n {\displaystyle \mathbb {R} ^{n}} is a star domain. A line or a plane with a single point removed is not a star domain. If A {\displaystyle A} is a set in R n , {\displaystyle \mathbb {R} ^{n},} the set B = { t a : a ∈ A , t ∈ [ 0 , 1 ] } {\displaystyle B=\{ta:a\in A,t\in [0,1]\}} obtained by connecting all points in A {\displaystyle A} to the origin is a star domain. A cross-shaped figure is a star domain but is not convex. A star-shaped polygon is a star domain whose boundary is a sequence of connected line segments. == Properties == Convexity: any non-empty convex set is a star domain. A set is convex if and only if it is a star domain with respect to each point in that set. Closure and interior: The closure of a star domain is a star domain, but the interior of a star domain is not necessarily a star domain. Contraction: Every star domain is a contractible set, via a straight-line homotopy. In particular, any star domain is a simply connected set. Shrinking: Every star domain, and only a star domain, can be "shrunken into itself"; that is, for every dilation ratio r < 1 , {\displaystyle r<1,} the star domain can be dilated by a ratio r {\displaystyle r} such that the dilated star domain is contained in the original star domain. Union and intersection: The union or intersection of two star domains is not necessarily a star domain. Balance: Given W ⊆ X , {\displaystyle W\subseteq X,} the set ⋂ | u | = 1 u W {\displaystyle \bigcap _{|u|=1}uW} (where u {\displaystyle u} ranges over all unit length scalars) is a balanced set whenever W {\displaystyle W} is a star shaped at the origin (meaning that 0 ∈ W {\displaystyle 0\in W} and r w ∈ W {\displaystyle rw\in W} for all 0 ≤ r ≤ 1 {\displaystyle 0\leq r\leq 1} and w ∈ W {\displaystyle w\in W} ). Diffeomorphism: A non-empty open star domain S {\displaystyle S} in R n {\displaystyle \mathbb {R} ^{n}} is diffeomorphic to R n . {\displaystyle \mathbb {R} ^{n}.} Binary operators: If A {\displaystyle A} and B {\displaystyle B} are star domains, then so is the Cartesian product A × B {\displaystyle A\times B} , and the sum A + B {\displaystyle A+B} . Linear transformations: If A {\displaystyle A} is a star domain, then so is every linear transformation of A {\displaystyle A} . == See also == Absolutely convex set – Convex and balanced set Absorbing set – Set that can be "inflated" to reach any point Art gallery problem – Mathematical problem Balanced set – Construct in functional analysis Bounded set (topological vector space) – Generalization of boundedness Convex set – In geometry, set whose intersection with every line is a single line segment Minkowski functional – Function made from a set Radial set – Topological set Star polygon – Regular non-convex polygon Symmetric set – Property of group subsets (mathematics) Star-shaped preferences == References == Ian Stewart, David Tall, Complex Analysis. Cambridge University Press, 1983, ISBN 0-521-28763-4, MR0698076 C.R. Smith, A characterization of star-shaped sets, American Mathematical Monthly, Vol. 75, No. 4 (April 1968). p. 386, MR0227724, JSTOR 2313423 Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. Schechter, Eric (1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. ISBN 978-0-12-622760-4. OCLC 175294365. == External links == Humphreys, Alexis. "Star convex". MathWorld.
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Wikipedia:Stationary phase approximation#0
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In mathematics, the stationary phase approximation is a basic principle of asymptotic analysis, applying to functions given by integration against a rapidly-varying complex exponential. This method originates from the 19th century, and is due to George Gabriel Stokes and Lord Kelvin. It is closely related to Laplace's method and the method of steepest descent, but Laplace's contribution precedes the others. == Basics == The main idea of stationary phase methods relies on the cancellation of sinusoids with rapidly varying phase. If many sinusoids have the same phase and they are added together, they will add constructively. If, however, these same sinusoids have phases which change rapidly as the frequency changes, they will add incoherently, varying between constructive and destructive addition at different times. == Formula == Letting Σ {\displaystyle \Sigma } denote the set of critical points of the function f {\displaystyle f} (i.e. points where ∇ f = 0 {\displaystyle \nabla f=0} ), under the assumption that g {\displaystyle g} is either compactly supported or has exponential decay, and that all critical points are nondegenerate (i.e. det ( H e s s ( f ( x 0 ) ) ) ≠ 0 {\displaystyle \det(\mathrm {Hess} (f(x_{0})))\neq 0} for x 0 ∈ Σ {\displaystyle x_{0}\in \Sigma } ) we have the following asymptotic formula, as k → ∞ {\displaystyle k\to \infty } : ∫ R n g ( x ) e i k f ( x ) d x = ∑ x 0 ∈ Σ e i k f ( x 0 ) | det ( H e s s ( f ( x 0 ) ) ) | − 1 / 2 e i π 4 s g n ( H e s s ( f ( x 0 ) ) ) ( 2 π / k ) n / 2 g ( x 0 ) + o ( k − n / 2 ) {\displaystyle \int _{\mathbb {R} ^{n}}g(x)e^{ikf(x)}dx=\sum _{x_{0}\in \Sigma }e^{ikf(x_{0})}|\det({\mathrm {Hess} }(f(x_{0})))|^{-1/2}e^{{\frac {i\pi }{4}}\mathrm {sgn} (\mathrm {Hess} (f(x_{0})))}(2\pi /k)^{n/2}g(x_{0})+o(k^{-n/2})} Here H e s s ( f ) {\displaystyle \mathrm {Hess} (f)} denotes the Hessian of f {\displaystyle f} , and s g n ( H e s s ( f ) ) {\displaystyle \mathrm {sgn} (\mathrm {Hess} (f))} denotes the signature of the Hessian, i.e. the number of positive eigenvalues minus the number of negative eigenvalues. For n = 1 {\displaystyle n=1} , this reduces to: ∫ R g ( x ) e i k f ( x ) d x = ∑ x 0 ∈ Σ g ( x 0 ) e i k f ( x 0 ) + s i g n ( f ″ ( x 0 ) ) i π / 4 ( 2 π k | f ″ ( x 0 ) | ) 1 / 2 + o ( k − 1 / 2 ) {\displaystyle \int _{\mathbb {R} }g(x)e^{ikf(x)}dx=\sum _{x_{0}\in \Sigma }g(x_{0})e^{ikf(x_{0})+\mathrm {sign} (f''(x_{0}))i\pi /4}\left({\frac {2\pi }{k|f''(x_{0})|}}\right)^{1/2}+o(k^{-1/2})} In this case the assumptions on f {\displaystyle f} reduce to all the critical points being non-degenerate. This is just the Wick-rotated version of the formula for the method of steepest descent. == An example == Consider a function f ( x , t ) = 1 2 π ∫ R F ( ω ) e i [ k ( ω ) x − ω t ] d ω {\displaystyle f(x,t)={\frac {1}{2\pi }}\int _{\mathbb {R} }F(\omega )e^{i[k(\omega )x-\omega t]}\,d\omega } . The phase term in this function, ϕ = k ( ω ) x − ω t {\displaystyle \phi =k(\omega )x-\omega t} , is stationary when d d ω ( k ( ω ) x − ω t ) = 0 {\displaystyle {\frac {d}{d\omega }}{\mathopen {}}\left(k(\omega )x-\omega t\right){\mathclose {}}=0} or equivalently, d k ( ω ) d ω | ω = ω 0 = t x {\displaystyle {\frac {dk(\omega )}{d\omega }}{\Big |}_{\omega =\omega _{0}}={\frac {t}{x}}} . Solutions to this equation yield dominant frequencies ω 0 {\displaystyle \omega _{0}} for some x {\displaystyle x} and t {\displaystyle t} . If we expand ϕ {\displaystyle \phi } as a Taylor series about ω 0 {\displaystyle \omega _{0}} and neglect terms of order higher than ( ω − ω 0 ) 2 {\displaystyle (\omega -\omega _{0})^{2}} , we have ϕ = [ k ( ω 0 ) x − ω 0 t ] + 1 2 x k ″ ( ω 0 ) ( ω − ω 0 ) 2 + ⋯ {\displaystyle \phi =\left[k(\omega _{0})x-\omega _{0}t\right]+{\frac {1}{2}}xk''(\omega _{0})(\omega -\omega _{0})^{2}+\cdots } where k ″ {\displaystyle k''} denotes the second derivative of k {\displaystyle k} . When x {\displaystyle x} is relatively large, even a small difference ( ω − ω 0 ) {\displaystyle (\omega -\omega _{0})} will generate rapid oscillations within the integral, leading to cancellation. Therefore we can extend the limits of integration beyond the limit for a Taylor expansion. If we use the formula, ∫ R e 1 2 i c x 2 d x = 2 i π c = 2 π | c | e ± i π 4 {\displaystyle \int _{\mathbb {R} }e^{{\frac {1}{2}}icx^{2}}dx={\sqrt {\frac {2i\pi }{c}}}={\sqrt {\frac {2\pi }{|c|}}}e^{\pm i{\frac {\pi }{4}}}} . f ( x , t ) ≈ 1 2 π e i [ k ( ω 0 ) x − ω 0 t ] | F ( ω 0 ) | ∫ R e 1 2 i x k ″ ( ω 0 ) ( ω − ω 0 ) 2 d ω {\displaystyle f(x,t)\approx {\frac {1}{2\pi }}e^{i\left[k(\omega _{0})x-\omega _{0}t\right]}\left|F(\omega _{0})\right|\int _{\mathbb {R} }e^{{\frac {1}{2}}ixk''(\omega _{0})(\omega -\omega _{0})^{2}}\,d\omega } . This integrates to f ( x , t ) ≈ | F ( ω 0 ) | 2 π 2 π x | k ″ ( ω 0 ) | cos [ k ( ω 0 ) x − ω 0 t ± π 4 ] {\displaystyle f(x,t)\approx {\frac {\left|F(\omega _{0})\right|}{2\pi }}{\sqrt {\frac {2\pi }{x\left|k''(\omega _{0})\right|}}}\cos \left[k(\omega _{0})x-\omega _{0}t\pm {\frac {\pi }{4}}\right]} . == Reduction steps == The first major general statement of the principle involved is that the asymptotic behaviour of I(k) depends only on the critical points of f. If by choice of g the integral is localised to a region of space where f has no critical point, the resulting integral tends to 0 as the frequency of oscillations is taken to infinity. See for example Riemann–Lebesgue lemma. The second statement is that when f is a Morse function, so that the singular points of f are non-degenerate and isolated, then the question can be reduced to the case n = 1. In fact, then, a choice of g can be made to split the integral into cases with just one critical point P in each. At that point, because the Hessian determinant at P is by assumption not 0, the Morse lemma applies. By a change of co-ordinates f may be replaced by ( x 1 2 + x 2 2 + ⋯ + x j 2 ) − ( x j + 1 2 + x j + 2 2 + ⋯ + x n 2 ) {\displaystyle (x_{1}^{2}+x_{2}^{2}+\cdots +x_{j}^{2})-(x_{j+1}^{2}+x_{j+2}^{2}+\cdots +x_{n}^{2})} . The value of j is given by the signature of the Hessian matrix of f at P. As for g, the essential case is that g is a product of bump functions of xi. Assuming now without loss of generality that P is the origin, take a smooth bump function h with value 1 on the interval [−1, 1] and quickly tending to 0 outside it. Take g ( x ) = ∏ i h ( x i ) {\displaystyle g(x)=\prod _{i}h(x_{i})} , then Fubini's theorem reduces I(k) to a product of integrals over the real line like J ( k ) = ∫ h ( x ) e i k f ( x ) d x {\displaystyle J(k)=\int h(x)e^{ikf(x)}\,dx} with f(x) = ±x2. The case with the minus sign is the complex conjugate of the case with the plus sign, so there is essentially one required asymptotic estimate. In this way asymptotics can be found for oscillatory integrals for Morse functions. The degenerate case requires further techniques (see for example Airy function). == One-dimensional case == The essential statement is this one: ∫ − 1 1 e i k x 2 d x = π k e i π / 4 + O ( 1 k ) {\displaystyle \int _{-1}^{1}e^{ikx^{2}}\,dx={\sqrt {\frac {\pi }{k}}}e^{i\pi /4}+{\mathcal {O}}{\mathopen {}}\left({\frac {1}{k}}\right){\mathclose {}}} . In fact by contour integration it can be shown that the main term on the right hand side of the equation is the value of the integral on the left hand side, extended over the range [ − ∞ , ∞ ] {\displaystyle [-\infty ,\infty ]} (for a proof see Fresnel integral). Therefore it is the question of estimating away the integral over, say, [ 1 , ∞ ] {\displaystyle [1,\infty ]} . This is the model for all one-dimensional integrals I ( k ) {\displaystyle I(k)} with f {\displaystyle f} having a single non-degenerate critical point at which f {\displaystyle f} has second derivative > 0 {\displaystyle >0} . In fact the model case has second derivative 2 at 0. In order to scale using k {\displaystyle k} , observe that replacing k {\displaystyle k} by c k {\displaystyle ck} where c {\displaystyle c} is constant is the same as scaling x {\displaystyle x} by c {\displaystyle {\sqrt {c}}} . It follows that for general values of f ″ ( 0 ) > 0 {\displaystyle f''(0)>0} , the factor π / k {\displaystyle {\sqrt {\pi /k}}} becomes 2 π k f ″ ( 0 ) {\displaystyle {\sqrt {\frac {2\pi }{kf''(0)}}}} . For f ″ ( 0 ) < 0 {\displaystyle f''(0)<0} one uses the complex conjugate formula, as mentioned before. == Lower-order terms == As can be seen from the formula, the stationary phase approximation is a first-order approximation of the asymptotic behavior of the integral. The lower-order terms can be understood as a sum of over Feynman diagrams with various weighting factors, for well behaved f {\displaystyle f} . == See also == Common integrals in quantum field theory Laplace's method Method of steepest descent == Notes == == References == Bleistein, N. and Handelsman, R. (1975), Asymptotic Expansions of Integrals, Dover, New York. Victor Guillemin and Shlomo Sternberg (1990), Geometric Asymptotics, (see Chapter 1). Hörmander, L. (1976), Linear Partial Differential Operators, Volume 1, Springer-Verlag, ISBN 978-3-540-00662-6. Aki, Keiiti; & Richards, Paul G. (2002), Quantitative Seismology (2nd ed.), pp 255–256. University Science Books, ISBN 0-935702-96-2 Wong, R. (2001), Asymptotic Approximations of Integrals, Classics in Applied Mathematics, Vol. 34. Corrected reprint of the 1989 original. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA. xviii+543 pages, ISBN 0-89871-497-4. Dieudonné, J. (1980), Calcul Infinitésimal, Hermann, Paris Paris, Richard Bruce (2011), Hadamard Expansions and Hyperasymptotic Evaluation: An Extension of the Method of Steepest Descents, Cambridge University Press, ISBN 978-1-107-00258-6 == External links == "Stationary phase, method of the", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
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Wikipedia:Statutory Professor in the Analysis of Partial Differential Equations#0
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The Statutory Professorship in the Analysis of Partial Differential Equations is a chair at the Mathematical Institute of the University of Oxford, England. Since its inception in 2009, the chair has been held by Professor Gui-Qiang Chen. It is associated with Keble College, Oxford. == Holders of the chair == 2009–0000 Prof. Gui-Qiang George Chen == See also == List of professorships at the University of Oxford == References == == External links == * Professor Gui-Qiang G. Chen's profile at Oxford
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Wikipedia:Stefan Brands#0
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Stefan Brands is the designer of the core cryptographic protocols of Microsoft's U-Prove technology. == Business career == Following his academic research on these protocols during the nineties, they were implemented and marketed under the U-Prove name by Credentica until Microsoft acquired the technology. Prior to Credentica, earlier versions of Brands' protocols were implemented by DigiCash, by Zero-Knowledge Systems, and by two consortiums made up of academic research groups, European banks, and large IT organizations. === Early career and affiliations === Brands has worked at DigiCash, at Zero-Knowledge Systems, and at Microsoft Corp. He has also served as an adjunct professor at McGill University, and as an advisor to Canada's data protection commissioner and to the Electronic Privacy Information Center. == References == == External links == Stefan Brands at the Mathematics Genealogy Project
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Wikipedia:Stefan E. Warschawski#0
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Stefan Emanuel "Steve" Warschawski (April 18, 1904 – May 5, 1989) was a Russian-born American mathematician, a professor and department chair at the University of Minnesota and the founder of the mathematics department at the University of California, San Diego. == Early life and education == Warschawski was born in Lida, now in Belarus; at the time of his birth Lida was part of the Russian Empire. His father was a Russian medical doctor, and his mother was ethnically German; the family spoke German at home. In 1915, his family moved to Königsberg, in Prussia (now Kaliningrad, Russia), the home of his mother's family. Warschawski studied at the University of Königsberg until 1926 and then moved to the University of Göttingen for his doctoral studies under the supervision of Alexander Ostrowski. Ostrowski moved to the University of Basel and Warschawski followed him there to complete his studies. == Career == After receiving his Ph.D., Warschawski took a position at Göttingen in 1930 but, due to the rise of Hitler and his own Jewish ancestry, he soon moved to Utrecht University in Utrecht, Netherlands and then Columbia University in New York City. After a sequence of temporary positions, he found a permanent faculty position at Washington University in St. Louis in 1939. During World War II he moved to Brown University and then the University of Minnesota, where he remained until his 1963 move to San Diego, where he was the founding chair of the mathematics department. Warschawski stepped down as chair in 1967, and retired in 1971, but remained active in research: approximately one third of his research publications were written after his retirement. Over the course of his career, he advised 19 Ph.D. students, all but one at either Minnesota or San Diego. Vernor Vinge is among Warschawski's doctoral students. == Research == Warschawski was known for his research on complex analysis and in particular on conformal maps. He also made contributions to the theory of minimal surfaces and harmonic functions. The Noshiro–Warschawski theorem is named after Warschawski and Noshiro, who discovered it independently; it states that, if f is an analytic function on the open unit disk such that the real part of its first derivative is positive, then f is one-to-one. In 1980, he solved the Visser–Ostrowski problem for derivatives of conformal mappings at the boundary. == Legacy == Warschawski was honored in 1978 by the creation of the Stefan E. Warschawski Assistant Professorship at San Diego. The Stephen E. Warschawski Memorial Scholarship was also given in his name in 1999–2000 to four UCSD undergraduates as a one-time award. His wife, Ilse, died in 2009 and left a US$1 million bequest to UCSD, part of which went towards endowing a professorship in the mathematics department. == References ==
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Wikipedia:Stefan Grigorievich Samko#0
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Stefan Grigorievich Samko (Russian: Стефан Григорьевич Самко; born March 28, 1941) is a mathematician active in the field of functional analysis, function spaces and operator theory. He is a retired professor of Mathematics at Algarve University and Rostov State University. == Career == === Research activity === S. Samko has more than 260 research papers spread throughout the areas of, Harmonic Analysis and Operator Theory in Variable Exponent Function Spaces; Function spaces; Potential type operators; Hypersingualr integrals and the method of approximative inverse operators; Fractional calculus of one and many variables; Integral equations of the first kind (including multi-dimensional ones). === Teaching activity === He was the adviser for 21 PhD students, from Russia and Portugal. The complete list is: == Bibliography == Samko, Stefan; Kilbas, Anatoly A.; Marichev, Oleg I. (1993). Fractional Integrals and Derivatives: Theory and Applications. Gordon & Breach Science Publishers. ISBN 978-2881248641. Samko, Stefan (2002). Hypersingular Integrals and Their Applications. CRC Press. ISBN 9780415272681. Karapetiants, Nikolai; Samko, Stefan (2002). Equations with Involutive Operators and Their Applications. Birkhäuser. Kokilashvili, Vakhtang; Meskhi, Alexander; Rafeiro, Humberto; Samko, Stefan (2016). Integral Operators in Non-Standard Function Spaces: Volume 1: Variable Exponent Lebesgue and Amalgam Spaces. Birkhäuser. ISBN 978-3319210148. Kokilashvili, Vakhtang; Meskhi, Alexander; Rafeiro, Humberto; Samko, Stefan (2016). Integral Operators in Non-Standard Function Spaces. Volume 2: Variable Exponent Hölder, Morrey–Campanato and Grand Spaces. Birkhäuser. ISBN 978-3-319-21018-6. == References == == External links == Stefan Grigorievich Samko at the Mathematics Genealogy Project Stefan Samko's home page
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Wikipedia:Stefan Langerman#0
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Stefan Langerman false Swarzberg is a Belgian computer scientist and mathematician whose research topics include computational geometry, data structures, and recreational mathematics. He is professor and co-head of the algorithms research group at the Université libre de Bruxelles (ULB) with Jean Cardinal. He is a director of research for the Belgian Fonds de la Recherche Scientifique (FRS–FNRS). == Education and career == Langerman left his Belgian secondary school at age 13 and was admitted by examination to the École polytechnique of the Université libre de Bruxelles. He studied civil engineering there for two years before switching his course of study to computer science, and earning a licenciate. After working as a user interface programmer for the Center for Digital Molecular Biophysics in Gembloux, he moved to the US for graduate study at Rutgers University, where he earned a master's degree and then in 2001 a PhD. His doctoral dissertation, Algorithms and Data Structures in Computational Geometry, was supervised by William Steiger. Next, before joining ULB and FNRS, Langerman worked as a postdoctoral researcher at McGill University with computational geometry researchers Luc Devroye and Godfried Toussaint. == Research == Langerman's research is primarily in computational geometry. Known for novel and often playful results such as "Wrapping the Mozartkugel"[WM] which earned him the moniker of a computational chocolatier, Langerman has made a number of scientific advances in fields as diverse as musical similarity,[MMS] polycube unfolding,[CUP] computational archaeology,[WBT] and protein folding. Langerman's work in data structures includes the co-invention of the queap[Q] and the introduction of the notion of retroactive data structures,[RDS] a generalization of the concept of a persistent data structure. He is the author or more than 240 publications, and has led scientific missions with other western scientists to collaborate with colleagues in North Korea. == Family == Langerman is also the founder of Langerman SPRL, a Belgian colored-diamond company based on the collection of Langerman's father Arthur Langerman, a dealer of colored diamonds who is also noted as a collector of anti-semitic materials. He is the co-author with his father of a paper on Morpion solitaire, written jointly with another father-and-son pair, Martin Demaine and Erik Demaine.[MS] Both Stefan Langerman and his father are members of the Board of Trustees of the Arthur Langerman Foundation, a non-profit organization based in Berlin, which makes its founder’s unique collection of visual antisemitica available for research, educational and exhibition purposes. == Selected publications == == References == == External links == Official website Stefan Langerman publications indexed by Google Scholar Arthur Langerman Foundation
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Wikipedia:Stefan Mazurkiewicz#0
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Stefan Mazurkiewicz (25 September 1888 – 19 June 1945) was a Polish mathematician who worked in mathematical analysis, topology, and probability. He was a student of Wacław Sierpiński and a member of the Polish Academy of Learning (PAU). His students included Karol Borsuk, Bronisław Knaster, Kazimierz Kuratowski, Stanisław Saks, and Antoni Zygmund. For a time Mazurkiewicz was a professor at the University of Paris; however, he spent most of his career as a professor at the University of Warsaw. The Hahn-Mazurkiewicz theorem, a basic result on curves prompted by the phenomenon of space-filling curves, is named for Mazurkiewicz and Hans Hahn. His 1935 paper Sur l'existence des continus indécomposables is generally considered the most elegant piece of work in point-set topology. During the Polish–Soviet War (1919–21), Mazurkiewicz as early as 1919 broke the most common Russian cipher for the Polish General Staff's cryptological agency. Thanks to this, orders issued by Soviet commander Mikhail Tukhachevsky's staff were known to Polish Army leaders. This contributed substantially, perhaps decisively, to Polish victory at the critical Battle of Warsaw and possibly to Poland's survival as an independent country. == See also == Biuro Szyfrów List of Polish mathematicians == External links == Stefan Mazurkiewicz at the Mathematics Genealogy Project O'Connor, John J.; Robertson, Edmund F., "Stefan Mazurkiewicz", MacTutor History of Mathematics Archive, University of St Andrews
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Wikipedia:Stefan Nemirovski#0
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Stefan Yuryevich Nemirovski (Russian: Стефан Юрьевич Немировский; born 29 July 1973) is a Russian mathematician. He made notable contributions to topology and complex analysis, and was awarded an EMS Prize in 2000. Nemirovski earned his Ph.D. from Moscow State University in 1998. == References == == External links == EMS Prize Laudatio, Notices AMS Website at the University of Bochum
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Wikipedia:Stefanie Petermichl#0
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Stefanie Petermichl (born 1971) is a German mathematical analyst who works as a professor at the University of Toulouse, in France. Topics of her research include harmonic analysis, several complex variables, stochastic control, and elliptic partial differential equations. == Education and career == Petermichl studied at the Karlsruhe Institute of Technology, and then did her graduate studies at Michigan State University, completing her Ph.D. in 2000 under the supervision of Alexander Volberg. After postdoctoral studies at the Institute for Advanced Study and Brown University, she joined the faculty of the University of Texas at Austin in 2005. She moved to the University of Bordeaux in 2007, and again to Toulouse in 2009. Since 2019, she holds the Humboldt chair at the University of Würzburg. == Recognition == Petermichl won the Salem Prize for 2006 "for her work on several crucial impacts to the theory of vector valued singular operators". She was the first woman to win that prize. In 2012, the French Academy of Sciences gave her their Ernest Déchelle Prize. She became a member of the Institut Universitaire de France in 2013. She is an invited speaker at the 2018 International Congress of Mathematicians, speaking in the section on Analysis and Operator Algebras. In 2016, she was awarded a European Research Council (ERC) grant. == References ==
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Wikipedia:Stefano Bianchini#0
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Stefano Bianchini (born 1970) is an Italian mathematician known for his research on partial differential equations. He won the 2004 EMS Prize for his contributions to the theory of discontinuous solutions of one-dimensional hyperbolic conservation laws. Bianchini earned his PhD from the International School for Advanced Studies in 2000, under supervision of Alberto Bressan. Along with Bressan, he co-authored a paper that led to the solution of the long-standing problem of stability and convergence of vanishing viscosity approximations. == References == == External links == Stefano Bianchini at the Mathematics Genealogy Project
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Wikipedia:Stefano De Marchi#0
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Stefano De Marchi (born 17 December 1962 in Candiana, Padua) is an Italian mathematician who works in numerical analysis and is a professor at the University of Padua. He is managing editor of the open access journal Dolomites Research Notes on Approximation published by the Padua University Press, coordinator of the Constructive Approximation and Applications Research Group, coordinator of the Research Italian network on Approximation, and responsible for the Unione Matematica Italiana Thematic Group on "Approximation Theory and Applications (A.T.A.)". His scientific interests deal mainly with interpolation and approximation of functions and data by polynomials and radial basis functions (RBFs)). == Education and career == Stefano De Marchi studied Bachelor's degree of Mathematics in 1981-1987, Master in Applied Mathematics in 1991 at the University of Padua, and received his doctorate in Computational Mathematics, Consorzio Nord-Orgientale, VI ciclo, University of Padua under Maria Morandi Cecchi and Larry Lee Schumaker supervisions (dissertation: Approssimazione e Interpolazione su "Simplices": Caratterizzazioni, Metodi ed Estensioni) He habilitated in 2017 and became a Full Professor of Numerical Analysis at the Department of Mathematics “Tullio Levi-Civita”, University of Padua in 2022. == Research == Stefano has made contributions to approximation theory such as Weakly Admissible Meshes, Barycentric rational interpolation, Stability issues and greedy algorithms in RBF theory, Rational RBF approximation, Medical image reconstruction, and Fake nodes. He is one of the discoverers of the so called Padua points, which are the only set of quasi-optimal interpolation points explicitly known on the square, for polynomial interpolation of total degree. Their name is due to the University of Padua, where they were originally discovered. He is also author of the books: ′′Funzioni Splines Univariate″, ′′Appunti di Calcolo Numerico″, ′′Meshfree Approximation for Multi-Asset European and American Option Problems″ and the Lecture notes: ′′Four lectures on radial basis functions″ and '′Lectures on multivariate polynomial interpolation″. == References == == External links == Home page at University of Padua Stefano De Marchi - Publications list Stefano De Marchi - Curriculum Vitae Stefano De Marchi publications indexed by Google Scholar Stefano De Marchi - The Mathematics Genealogy Project Padova Neuroscience Center Dolomites Research Notes on Approximation Journal Constructive Approximation and Applications Research Group Research ITalian network on Approximation Unione Matematica Italiana
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Wikipedia:Stephen Bigelow#0
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Stephen John Bigelow is an Australian mathematician and professor of mathematics at the University of California, Santa Barbara. He is known for his proof that braid groups are linear, concurrently with and independently of another proof by Daan Krammer. Bigelow earned bachelor's and master's degrees in 1992 and 1994 from the University of Melbourne. He completed his PhD in 2000 from the University of California, Berkeley under the joint supervision of Robion Kirby and Andrew Casson. He returned to Melbourne for two years as a research fellow before joining the UCSB faculty in 2002. Bigelow was an invited speaker at the International Congress of Mathematicians in 2002, speaking on representations of braid groups. He was a Sloan Research Fellow for 2002–2006. In 2012 he was designated as one of the inaugural fellows of the American Mathematical Society. == References ==
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Wikipedia:Stephen Blyth#0
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Stephen James Blyth is a British mathematician and academic. Since October 2022, he has been Principal of Lady Margaret Hall, Oxford. He had been Professor of the Practice of Statistics at Harvard University since 2012, and was also chief executive officer of the Harvard Management Company between January 2015 and July 2016. == Biography == Blyth matriculated into Christ's College, Cambridge, in 1985 to study the Mathematical Tripos. He graduated with a first class honours Bachelor of Arts (BA) degree, and was the 3rd wrangler in that year. He then moved to the United States, where he studied statistics at Harvard University. He was awarded a Master of Arts (AM) degree in 1989. His doctoral dissertation was titled "Local Regression Coefficients and the Correlation Curve" and his advisor was Kjell Doksum. His Doctor of Philosophy (PhD) degree was awarded in 1992. After graduating with his PhD, he returned to England and was a lecturer in the Department of Mathematics at Imperial College London from 1992 to 1994. He then moved into the financial industry, working at HSBC, Morgan Stanley and Deutsche Bank. In 2006, he returned to Harvard University where he joined the faculty. He taught statistics in the Harvard Faculty of Arts and Sciences, rising to become Professor of the Practice of Statistics in 2012. He also worked with the Harvard Management Company (HMC) which manages Harvard University's endowment and related financial assets. By 2014, he was managing director and head of public markets. In September 2014, he was announced as the next president and chief executive officer of the Harvard Management Company: he took up the post on 1 January 2015. He took medical leave from 23 May 2016, and stepped down as CEO on 27 July 2016. In December 2021, Blyth was announced as the next Principal of Lady Margaret Hall, Oxford. He took up the post on 1 October 2022 and was installed during a service in the College Chapel on 7 October 2022. == Selected works == Blyth, Stephen (2013). Introduction to Quantitative Finance. Oxford: Oxford University Press. ISBN 978-0199666591. == References ==
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Wikipedia:Stephen Childress#0
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William Stephen Childress is an American applied mathematician, author and professor emeritus at the Courant Institute of Mathematical Sciences. He works on classical fluid mechanics, asymptotic methods and singular perturbations, geophysical fluid dynamics, magnetohydrodynamics and dynamo theory, mathematical models in biology, and locomotion in fluids. He is also a co-founder of the Courant Institute of Mathematical Sciences's Applied Mathematics Lab. == Published books == 1977: Mechanics of Swimming and Flying, online ISBN 9780511569593. 1978: Mathematical models in developmental biology with Jerome K. Percus, ISBN 978-1470410803 1987: Topics in Geophysical Fluid Dynamics: Atmospheric Dynamics, Dynamo Theory, and Climate Dynamics, with M. Ghil. Softcover ISBN 978-0-387-96475-1, eBook ISBN 978-1-4612-1052-8. 1995: Stretch, Twist, Fold: The Fast Dynamo with Andrew D. Gilbert, ISBN 978-3662140147, ISBN 3662140144 2009: An Introduction to Theoretical Fluid Mechanics, ISBN 978-0-8218-4888-3. 2012: Natural Locomotion in Fluids and on Surfaces Swimming, Flying, and Sliding. Edited with Anette Hosoi, William W. Schultz, Jane Wang. Hardcover ISBN 978-1-4614-3996-7, Softcover ISBN 978-1-4899-9916-0, eBook ISBN 978-1-4614-3997-4 2018: Construction of Steady-state Hydrodynamic Dynamos. I. Spatially Periodic Fields, ISBN 978-1378904725 == Recognition == 1976 Guggenheim Fellowship for Natural Sciences, US & Canada 2008 Fellow of American Physical Society == References == == External links == William Stephen Childress's home page
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Wikipedia:Stephen Drury (mathematician)#0
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Stephen William Drury is an Anglo-Canadian mathematician and professor of mathematics at McGill University. He specializes in mathematical analysis, harmonic analysis and linear algebra. He received his doctorate from the University of Cambridge in 1970 under the supervision of Nicholas Varopoulos and completed his postdoctoral training at the Faculté des sciences d'Orsay, France. He was recruited to McGill by Professor Carl Herz in 1972. Among other contributions, he solved the Sidon set union problem, worked on restrictions of Fourier and Radon transforms to curves, and generalized von Neumann's inequality. In operator theory, the Drury–Arveson space is named after William Arveson and him. His research now pertains to the interplay between matrix theory and harmonic analysis and their applications to graph theory. == References ==
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Wikipedia:Stephen Gelbart#0
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Stephen Samuel Gelbart (Hebrew: סטיבן סמואל גלברט; born June 12, 1946) is an American-Israeli mathematician who holds the Nicki and J. Ira Harris Professorial Chair in mathematics at the Weizmann Institute of Science in Israel. He was named a fellow of the American Mathematical Society in 2013 "for contributions to the development and dissemination of the Langlands program." == Biography == Gelbart was born in Syracuse, New York. He graduated from Cornell University in 1967, and earned a Ph.D. from Princeton University in 1970, with a dissertation on Fourier analysis supervised by Elias M. Stein. He returned to Cornell as an assistant professor in 1971, was promoted to full professor in 1980, moved to the Weizmann Institute in 1984, and was given his named chair in 1998. He was president of the Israel Mathematical Union from 1994 to 1996. His doctoral students include Erez Lapid. == Selected publications == === Articles === Harmonics on Stiefel manifolds and generalized Hankel transforms. Bull. Amer. Math. Soc. 78 (1972) 451–455. MR0480872 A theory of Stiefel harmonics. Trans. Amer. Math. Soc. 192 (1974) 29–50. MR0425519 An elementary introduction to the Langlands program. Bull. Amer. Math. Soc. 10 (1984) 177–219. MR733692 with Freydoon Shahidi: Boundedness of automorphic L-functions in vertical strips. J. Amer. Math. Soc. 14 (2001) 79–107. MR1800349 with Stephen D. Miller: Riemann's zeta function and beyond. Bull. Amer. Math. Soc. 41 (2004) 59–112. MR2015450 === Books === Automorphic forms on adele groups. Princeton University Press. 1975. ISBN 0691081565. Weil's representation and the spectrum of the metaplectic group. Springer. 1976. with Ilya Piatetski-Shapiro and Stephen Rallis: Explicit constructions of automorphic L-functions. 1987. with Freydoon Shahidi: Analytic properties of automorphic L-functions. Academic Press. 1988. later edition. Elsevier. 2014. ISBN 9781483261034. Lectures on the Arthur-Selberg trace formula. American Mathematical Society. 1996. === as editor === with Joseph Bernstein as editor and 6 contributing authors: Bump, D.; Cogdell, J.W.; de Shalit, E.; Gaitsgory, D.; Kowalski, E.; Kudla, S. S. (20 May 2003). Bernstein, J.; Gelbart, S. (eds.). An Introduction to the Langlands Program. Springer Science & Business Media. ISBN 978-0-8176-3211-3.{{cite book}}: CS1 maint: multiple names: authors list (link) == References == == External links == Home page
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Wikipedia:Stephen Siklos#0
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Stephen Theodore Chesmer Siklos (1950 – 17 August 2019) was a lecturer in the Faculty of Mathematics at the University of Cambridge. He is known for setting up the Sixth Term Examination Papers, used for undergraduate mathematics admissions at several British universities. == Early life == Siklos was born in Epsom, Surrey, England in 1950. His father, Theo Siklos, was an educator and his wife, Ruth Siklos, an almoner. He was educated at Collyer's School before reading the Mathematical Tripos at Pembroke College, University of Cambridge, where he graduated with a masters in mathematics and was awarded the Tyson Medal. == Academic career == In 1973, he began doing research in general relativity under Stephen Hawking, publishing his dissertation titled "Singularities, Invariants and Cosmology" in 1976. From 1980 to 1999 he lectured at Cambridge and was the director of studies at Newnham College. In 1999, he joined Jesus College as a senior tutor, later becoming the college president. == References ==
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Wikipedia:Stephen T. Hedetniemi#0
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Stephen T. Hedetniemi (7 February 1939) is an American mathematician and computer scientist specializing in graph theory. He is professor emeritus of computer science at Clemson University. == Biography == Hedetniemi graduated from the University of Michigan with a bachelor's degree in mathematics in 1960, a master's degree in 1962, and a doctorate in communication sciences in 1966 with Frank Harary. He was in the Computational Logic Group at the University of Michigan and became assistant professor in Computer Science at the University of Iowa in 1967 and associate professor in 1969. From 1972 he was an associate professor at the University of Virginia. In 1972 he spent two months at the Naval Weapons Laboratory in Dahlgren and in 1975/76 he was a visiting professor at the University of Victoria. From 1977 to 1982, he was a professor and head of the Department of Computer Science at the University of Oregon. From 1982 he was a professor at Clemson University. == Selected publications == Homomorphisms of Graphs and Automata, University of Michigan Communications Sciences Program, Technical Report 03105-44-T, 1966 (PhD thesis) MR2615860 Fundamentals of Domination in Graphs, with Teresa W. Haynes & Peter Slater, Marcel Dekker, 1998 Structures of Domination in Graphs, co-edited with Teresa W. Haynes & Michael A. Henning, Springer, 2021 Domination in Graphs: Core Concepts, with Teresa W. Haynes & Michael A. Henning, Springer, 2023 == References == == External links == Clemson University profile
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Wikipedia:Stephen Twinoburyo#0
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Stephen Twinoburyo (8 January 1970 – 1 January 2019) was a Ugandan scientist, mathematician, lecturer, and entrepreneur. He was the CEO of Scimatic Solutions, a South African company which helps students with maths and science tuition. == Early life and education == Twinoburyo was born on 8 January 1970, in Mbarara, Uganda. He was the second of seven children, and his father worked as a town clerk. He attended Ntare School, and was head prefect there in 1989. In 1990, he started studying engineering at Makerere University, and relocated to South Africa. During his time there, he was chairman of Lumumba Hall. He later studied mathematics as a part-time degree at the University of South Africa, completing the course in 2007. == Career == In 1994, Twinoburyo visited Soweto, South Africa, and it inspired him to move to the country in 1997. He lectured at the University of Pretoria, and taught in colleges in Pretoria and Cape Town. In 2008, Twinoburyo decided to found Uganda Professionals Living in South Africa (AUPSA), and worked as their chairman. In 2009, he organised a meeting of Ugandan expatriates in South Africa. The meeting was held in Sandton, South Africa. AUPSA was set up to connect Ugandan expatriates living in South Africa. Twinoburyo also worked for the Ugandan Civil Alliance Network. In 2010, Twinoburyo said that Ugandans were unhappy about the ticket prices for the 2010 FIFA World Cup in South Africa. In 2011, he condemned alleged human rights abuses in Uganda, and asked South African president Jacob Zuma not to attend the inauguration of Ugandan president Yoweri Museveni. In 2014, Twinoburyo set up and became the CEO of Scimatic Solutions, a South African company which helps students with maths and science tuition. He was inspired to set up the company after visiting the California Science Center and National Air and Space Museum in Washington, D.C. The company is based in Hatfield, Pretoria. == Personal life == Twinoburyo and his wife had three children. He died in South Africa on 1 January 2019 of a heart attack, one week before his 49th birthday. His body was repatriated to Uganda. == References ==
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Wikipedia:Stephens' constant#0
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Stephens' constant expresses the density of certain subsets of the prime numbers. Let a {\displaystyle a} and b {\displaystyle b} be two multiplicatively independent integers, that is, a m b n ≠ 1 {\displaystyle a^{m}b^{n}\neq 1} except when both m {\displaystyle m} and n {\displaystyle n} equal zero. Consider the set T ( a , b ) {\displaystyle T(a,b)} of prime numbers p {\displaystyle p} such that p {\displaystyle p} evenly divides a k − b {\displaystyle a^{k}-b} for some power k {\displaystyle k} . Assuming the validity of the generalized Riemann hypothesis, the density of the set T ( a , b ) {\displaystyle T(a,b)} relative to the set of all primes is a rational multiple of C S = ∏ p ( 1 − p p 3 − 1 ) = 0.57595996889294543964316337549249669 … {\displaystyle C_{S}=\prod _{p}\left(1-{\frac {p}{p^{3}-1}}\right)=0.57595996889294543964316337549249669\ldots } (sequence A065478 in the OEIS) Stephens' constant is closely related to the Artin constant C A {\displaystyle C_{A}} that arises in the study of primitive roots. C S = ∏ p ( C A + ( 1 − p 2 p 2 ( p − 1 ) ) ) ( p ( p + 1 + 1 p ) ) {\displaystyle C_{S}=\prod _{p}\left(C_{A}+\left({{1-p^{2}} \over {p^{2}(p-1)}}\right)\right)\left({{p} \over {(p+1+{{1} \over {p}})}}\right)} == See also == Euler product Twin prime constant == References ==
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Wikipedia:Stevan Pilipović#0
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Pilipović is a surname of South Slavic origin, a patronymic of the given name Pilip. Notable people with the surname include: Borislav Pilipović (born 1984), Bosnian-Herzegovinian football player Kristian Pilipovic (born 1994), Croatian-born Austrian handball player Renato Pilipović (born 1977), Croatian football player and coach Stevan Pilipović (born 1950), Serbian mathematician Stojan Pilipović (born 1987), Serbian football player Tamara Pilipović (born 1990), Serbian politician == See also == Pilipovich Filipović
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Wikipedia:Steve Kuzmicich#0
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Stjepan Slavo Raphael Kuzmicich (2 November 1931 – 14 June 2018), was a New Zealand statistician. He served as the New Zealand government statistician from 1984 to 1992. == References ==
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Wikipedia:Steven Hurder#0
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Steven Edmond Hurder is an American mathematician specializing in foliation theory, differential topology, smooth ergodic theory, rigidity of group actions and spectral and index theory of operators. Hurder is a professor emeritus at University of Illinois Chicago. Hurder was named as an inaugural fellow of the American Mathematical Society in 2013. == Education == Hurder received his PhD in 1980 at University of Illinois Urbana-Champaign. His advisor was Franz W. Kamber, and the title of his dissertation was Dual Homotopy Invariants of G-Foliations. == Selected publications == Hurder, Steven (1981). Dual homotopy invariants of G-foliations, Topology, 20(4):365–387. Hurder, Steven; Katok, Anatoly (1990). Differentiability, rigidity and Godbillon-Vey classes for Anosov flows, Inst. Hautes Études Sci. Publ. Math. no. 72, 5–61. Clark, Alex; Hurder, Steven (2013). Homogeneous matchbox manifolds, Trans. Amer. Math. Soc. 365, no. 6, 3151–3191. Hurder, Steven (1992). Rigidity for Anosov actions of higher rank lattices, Ann. of Math. (2) 135, no. 2, 361–410. Hurder, S.; Katok, A (1987). Ergodic theory and Weil measures for foliations, Ann. of Math. (2) 126, no. 2, 221–275. Douglas, Ronald G.; Kaminker, Jerome (1991). Cyclic cocycles, renormalization and eta-invariants, Invent. Math. 103 (1991), no. 1, 101—179. == References == == External links == Steven Hurder's homepage
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Wikipedia:Steven Sperber#0
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Steven Sperber is an American mathematician, academic, and author. He is a Professor at the University of Minnesota. Sperber's research has focused on arithmetic algebraic geometry, p-adic differential equations, and their applications in advanced number theory and mathematical structures. His scholarly contributions include publications in Annals of Mathematics, Inventiones Mathematicae, and Compositio Mathematica, alongside the authorship of P-adic Methods in Number Theory and Algebraic Geometry. == Education == Sperber attended S.J. Tilden High School in Brooklyn and enrolled at Brooklyn College in 1962. He began his graduate studies at Harvard in 1966, then transferred to the University of Pennsylvania in 1967, where he worked with Stephen Shatz. Later, he earned his Ph.D. in 1975 from the University of Pennsylvania, with Bernard Dwork of Princeton University, serving as his doctoral adviser. == Career == After completing his Ph.D., Sperber held a Lecturer position at University of Illinois from 1975 to 1977. Following this, he joined the University of Minnesota, starting as an Assistant Professor from 1977 to 1980. He was subsequently promoted to Associate Professor, a position he held from 1980 to 1983, and has served as a Professor since 1983. == Research == In his doctoral thesis (1975) under Dwork, Sperber developed the p-adic cohomology for the family of multiple (n-variable) Kloosterman sums defined over a finite field of characteristic p. In particular, he related these sums to certain classical confluent hypergeometric differential equations. This relationship generalized the study due to Dwork, linking the one variable Kloosterman sums and it's arithmetic properties to the p-adic Bessel function. The arithmetic information that Sperber's work produced included determining the degree of the associated L-function, proving a functional equation, and determining the precise p-adic size of the roots (or poles, depending on the parity of n) of the L-function. From this work forward, Sperber's work focused on arithmetic properties of exponential sums and the study of p-adic differential equations. In a joint work with Sibuya, he studied power series solutions of an algebraic differential equation having coefficients in a number field. Their result showed that such series solutions have a nontrivial radius of convergence for any non-Archimedean valuation of the number field. Sperber's work with Adolphson has extended over a half-century. Together, they studied the L-functions associated with non-degenerate toric exponential sums over a finite field of characteristic p using p-adic cohomology. In the non-degenerate case, they were able to establish the vanishing of all but middle dimensional p-adic cohomology. From this, they extended the work to the cases of smooth projective hypersurfaces and complete intersections defined over a finite field, obtaining similar results in these cases for the interesting factor of their zeta functions. For exponential sums, they expressed the degree of the L-function (or its reciprocal) given in terms of the volume of the Newton polyhedron of the argument of the exponential sum. They also established that the Newton polygon of the L-function lies over its Hodge polygon, and proved the purity of roots in simplicial cases. In related work, Adolphson and Sperber derived general results even without the hypothesis of non-degeneracy. They obtained estimates for the degree as a rational function and for total degree of the associated L-function for a toric exponential sum, using the p-adic method developed by Bombieri. They also derived (1987) estimates for the divisibility of the exponential sum by powers of p. These results were then extended to estimate the divisibility by powers of p of the number of solutions in a finite field of characteristics p of a finite system of polynomial equations defined over such a field. These results were a refinement and generalization of the Chevalley-Warning theorem. In a series of articles, Adolphson and Sperber developed the p-adic theory of multiplicative character sums and the case of twisted exponential sums involving both multiplicative and additive characters of the underlying finite field of definition. In other works, they studied affine exponential sums and showed that in some cases, the pattern of vanishing p-adic cohomology holds even in the cases where nondegeneracy fails to hold. They generalized Igusa's seminal work, demonstrating that Hasse invariants may be realized in many settings in terms of the reduction mod p of the p-adically bounded solutions of the relevant, related A-hypergeometric systems. Moreover, they considered families of hypersurfaces defined by suitable deformations of a Calabi-Yau (or, more generally, a generalized Calabi-Yau) hypersurface and established a p-adic formula for the unique largest (p-adically) reciprocal root of the zeta function, which has the following form. If H s ¯ {\displaystyle {\overline {s}}} is an ordinary fiber of the family at s ¯ {\displaystyle {\overline {s}}} an element of the finite field and λ is a Teichmüller unit over s ¯ {\displaystyle {\overline {s}}} , then this distinguished reciprocal root is the value at λ of a ratio of p-adic exponential functions having classical significance. In joint work with Doran, Kelly, Salerno, Voight, and Whitcher, Sperber studied an alternative construction of mirror symmetry to the example involving the Dwork family of hypersurfaces. Instead of the Dwork family, they considered various invertible polynomial families suggested by Berglund-Hübsch-Krawitz. Their results showed that in these cases, the interesting factor of the zeta function remains identical, suggesting an underlying arithmetic stability in the mirror correspondence. Haessig and Sperber applied the methods used in the study of toric sums to quite general families of such sums, and particularly to their symmetric power L-functions. Their results include estimates for the degree of the L-function. In the case of generalized Kloosterman sums, they obtained arithmetic estimates and applied some of these method as well to infinite symmetric power L-functions, which carry information for the unit root L-function studied by Dwork and Fu-Wan. Libgober and Sperber considered holomorphic functions from the n-fold complex torus to the complex numbers, defined by a Laurent polynomial. They introduced the zeta function of monodromy at ∞ and showed that, in the non-degenerate case, it closely agreed with the L-function of the exponential sums associated with this function when viewed over a finite field. In these non-degenerate cases, they also established a connection between certain arithmetic invariants that arose in analogous situations across two distinct mathematical contexts. == Bibliography == === Books === P-adic Methods in Number Theory and Algebraic Geometry (1992) ISBN 9780821851456 === Selected articles === Sperber, S. (1980). "Congruence properties of the hyperkloosterman sum". Compositio Mathematica. 42: 3–33. Sperber, Steven (1986). "On the p-adic Theory of Exponential Sums". American Journal of Mathematics. 108 (2): 255–296. doi:10.2307/2374675. JSTOR 2374675. Adolphson, Alan; Sperber, Steven (October 1987). "Newton polyhedra and the degree of the L-function associated to an exponential sum". Inventiones Mathematicae. 88 (3): 555–569. Bibcode:1987InMat..88..555A. doi:10.1007/BF01391831. Adolphson, Alan; Sperber, Steven (1989). "Exponential Sums and Newton Polyhedra: Cohomology and Estimates". Annals of Mathematics. 130 (2): 367–406. doi:10.2307/1971424. JSTOR 1971424. Libgober, A.; Sperber, S. (1995). "On the zeta function of monodromy of a polynomial map". Compositio Mathematica. 99: 287–307. == References ==
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Wikipedia:Stevo Todorčević#0
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Stevo Todorčević (Serbian Cyrillic: Стево Тодорчевић; born February 9, 1955), is a Yugoslavian mathematician specializing in mathematical logic and set theory. He holds a Canada Research Chair in mathematics at the University of Toronto, and a director of research position at the Centre national de la recherche scientifique in Paris. == Early life and education == Todorčević was born in Ubovića Brdo. As a child he moved to Banatsko Novo Selo, and went to school in Pančevo. At Belgrade University, he studied pure mathematics, attending lectures by Đuro Kurepa. He began graduate studies in 1978, and wrote his doctoral thesis in 1979 with Kurepa as his advisor. == Research == Todorčević's work involves mathematical logic, set theory, and their applications to pure mathematics. In Todorčević's 1978 master’s thesis, he constructed a model of MA + ¬wKH in a way to allow him to make the continuum any regular cardinal, and so derived a variety of topological consequences. Here MA is an abbreviation for Martin's axiom and wKH stands for the weak Kurepa Hypothesis. In 1980, Todorčević and Abraham proved the existence of rigid Aronszajn trees and the consistency of MA + the negation of the continuum hypothesis + there exists a first countable S-space. == Awards and honours == Todorčević is the winner of the first prize of the Balkan Mathematical Society for 1980 and 1982, the 2012 CRM-Fields-PIMS prize in mathematical sciences, and the Shoenfield prize of the Association for Symbolic Logic for "outstanding expository writing in the field of logic" in 2013, for his book Introduction to Ramsey Spaces.[IRS] He was selected by the Association for Symbolic Logic as their 2016 Gödel Lecturer. He became a corresponding member of the Serbian Academy of Sciences and Arts as of 1991 and a full member of the Academy in 2009. In 2016 Todorčević became a fellow of the Royal Society of Canada. Todorčević has been described as "the greatest Serbian mathematician" since the time of Mihailo Petrović Alas. == Books == Todorčević is the author of several books in mathematics, including: Partition Problems in Topology. Providence, R.I: American Mathematical Soc. 1989. ISBN 978-0-8218-5091-6. MR 0980949. (with Ilijas Farah) Some Applications of the Method of Forcing. Moscow: Yenisei. 1995. ISBN 978-5-88623-014-7. MR 1486583. Topics in Topology. Lecture Notes in Mathematics. Berlin ; New York: Springer. 1997. ISBN 978-3-540-62611-4. MR 1442262. (with Spiros A. Argyros) Ramsey Methods in Analysis. Basel ; Boston: Springer Science & Business Media. 2005. ISBN 978-3-7643-7264-4. MR 2145246. Walks on Ordinals and Their Characteristics. Basel: Birkhäuser. 2007. ISBN 978-3-7643-8528-6. MR 2355670. OCLC 166357947. Introduction to Ramsey Spaces. Annals of Mathematics Studies. Princeton: Princeton University Press. 2010. ISBN 978-0-691-14542-6. MR 2603812. OCLC 437054050. Notes on Forcing Axioms. New Jersey: World Scientific Publishing Company. 2014. ISBN 978-981-4571-57-9. MR 3184691. == See also == Baumgartner's axiom Kechris–Pestov–Todorčević correspondence Open coloring axiom S and L spaces == References == == Sources == Larson, Jean A. (2012), "Infinite combinatorics", in Gabbay, Dov M.; Kanamori, Akihiro; Woods, John (eds.), Sets and extensions in the twentieth century, Handbook of the History of Logic, vol. 6, Amsterdam: Elsevier/North-Holland, pp. 145–357, doi:10.1016/B978-0-444-51621-3.50003-7, ISBN 978-0-444-51621-3, MR 3409860. RSC Fellowship Citation and Detailed Appraisal: Stevo Todorcevic == External links == CRM Fields PIMS Prize Lecture: Prof. Stevo Todorcevic (photo album) CRM-Fields-PIMS Prize Lecture: Stevo Todorcevic (University of Toronto) Stevo Todorcevic at University of Toronto Stevo Todorcevic at Institut de mathématiques de Jussieu – Paris Rive Gauche Todorčević najcenjeniji (Todorčević most respected)(in Serbian) Dispute over Infinity Divides Mathematicians by Natalie Wolchover, Quanta Magazine, November 26, 2013; contains some comments on choices of axioms for set theory Stevo Todorcevic at Institute for Advanced Study Prof. Todorčević Interview(in Serbian)
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Wikipedia:Stewart Nelson#0
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Stewart Nelson is an American mathematician and programmer from The Bronx who co-founded Systems Concepts. From a young age, Nelson was tinkering with electronics, aided and abetted by his father who was a physicist that had become an engineer. Stewart attended Poughkeepsie High School, graduating in the spring of 1963. From his first few days of High School, Stewart displayed his talents for hacking the international telephone trunk lines, along with an uncanny skill for picking combination locks, although this was always done as innocent entertainment. He simply loved the challenge of seeing how quickly he could accomplish this feat. His quirky sense of humor was always visible, as was his disdain for any rule that got in the way of his gaining knowledge. Stewart was an inspiration to the school's Tech-elec Club, as well as a ringleader in the founding of the school's pirate radio station. Nelson enrolled at MIT in 1963 and quickly became known for hooking up the AI Lab's PDP-1 (and later the PDP-6) to the telephone network, making him one of the first phreakers. Nelson later accomplished other feats like hard-wiring additional instructions into the PDP-1. Nelson was hired by Ed Fredkin's Information International Inc. at the urging of Marvin Minsky to work on PDP-7 programs at the MIT Computer Science and Artificial Intelligence Laboratory. Nelson was known as a brilliant software programmer. He was influential in LISP, the assembly instructions for the Digital Equipment Corporation PDP, and a number of other systems. The group of young hackers was known for working on systems after hours. One night, Nelson and others decided to rewire MIT's PDP-1 as a prank. Later, Margaret Hamilton tried to use the DEC-supplied DECAL assembler on the machine and it crashed repeatedly. == References ==
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Wikipedia:Stieltjes moment problem#0
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In mathematics, the Stieltjes moment problem, named after Thomas Joannes Stieltjes, seeks necessary and sufficient conditions for a sequence (m0, m1, m2, ...) to be of the form m n = ∫ 0 ∞ x n d μ ( x ) {\displaystyle m_{n}=\int _{0}^{\infty }x^{n}\,d\mu (x)} for some measure μ. If such a function μ exists, one asks whether it is unique. The essential difference between this and other well-known moment problems is that this is on a half-line [0, ∞), whereas in the Hausdorff moment problem one considers a bounded interval [0, 1], and in the Hamburger moment problem one considers the whole line (−∞, ∞). == Existence == Let Δ n = [ m 0 m 1 m 2 ⋯ m n m 1 m 2 m 3 ⋯ m n + 1 m 2 m 3 m 4 ⋯ m n + 2 ⋮ ⋮ ⋮ ⋱ ⋮ m n m n + 1 m n + 2 ⋯ m 2 n ] {\displaystyle \Delta _{n}=\left[{\begin{matrix}m_{0}&m_{1}&m_{2}&\cdots &m_{n}\\m_{1}&m_{2}&m_{3}&\cdots &m_{n+1}\\m_{2}&m_{3}&m_{4}&\cdots &m_{n+2}\\\vdots &\vdots &\vdots &\ddots &\vdots \\m_{n}&m_{n+1}&m_{n+2}&\cdots &m_{2n}\end{matrix}}\right]} be a Hankel matrix, and Δ n ( 1 ) = [ m 1 m 2 m 3 ⋯ m n + 1 m 2 m 3 m 4 ⋯ m n + 2 m 3 m 4 m 5 ⋯ m n + 3 ⋮ ⋮ ⋮ ⋱ ⋮ m n + 1 m n + 2 m n + 3 ⋯ m 2 n + 1 ] . {\displaystyle \Delta _{n}^{(1)}=\left[{\begin{matrix}m_{1}&m_{2}&m_{3}&\cdots &m_{n+1}\\m_{2}&m_{3}&m_{4}&\cdots &m_{n+2}\\m_{3}&m_{4}&m_{5}&\cdots &m_{n+3}\\\vdots &\vdots &\vdots &\ddots &\vdots \\m_{n+1}&m_{n+2}&m_{n+3}&\cdots &m_{2n+1}\end{matrix}}\right].} Then { mn : n = 1, 2, 3, ... } is a moment sequence of some measure on [ 0 , ∞ ) {\displaystyle [0,\infty )} with infinite support if and only if for all n, both det ( Δ n ) > 0 a n d det ( Δ n ( 1 ) ) > 0. {\displaystyle \det(\Delta _{n})>0\ \mathrm {and} \ \det \left(\Delta _{n}^{(1)}\right)>0.} { mn : n = 1, 2, 3, ... } is a moment sequence of some measure on [ 0 , ∞ ) {\displaystyle [0,\infty )} with finite support of size m if and only if for all n ≤ m {\displaystyle n\leq m} , both det ( Δ n ) > 0 a n d det ( Δ n ( 1 ) ) > 0 {\displaystyle \det(\Delta _{n})>0\ \mathrm {and} \ \det \left(\Delta _{n}^{(1)}\right)>0} and for all larger n {\displaystyle n} det ( Δ n ) = 0 a n d det ( Δ n ( 1 ) ) = 0. {\displaystyle \det(\Delta _{n})=0\ \mathrm {and} \ \det \left(\Delta _{n}^{(1)}\right)=0.} == Uniqueness == There are several sufficient conditions for uniqueness, for example, Carleman's condition, which states that the solution is unique if ∑ n ≥ 1 m n − 1 / ( 2 n ) = ∞ . {\displaystyle \sum _{n\geq 1}m_{n}^{-1/(2n)}=\infty ~.} == References == Reed, Michael; Simon, Barry (1975), Fourier Analysis, Self-Adjointness, Methods of modern mathematical physics, vol. 2, Academic Press, p. 341 (exercise 25), ISBN 0-12-585002-6
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Wikipedia:Stirling's approximation#0
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In mathematics, Stirling's approximation (or Stirling's formula) is an asymptotic approximation for factorials. It is a good approximation, leading to accurate results even for small values of n {\displaystyle n} . It is named after James Stirling, though a related but less precise result was first stated by Abraham de Moivre. One way of stating the approximation involves the logarithm of the factorial: ln ( n ! ) = n ln n − n + O ( ln n ) , {\displaystyle \ln(n!)=n\ln n-n+O(\ln n),} where the big O notation means that, for all sufficiently large values of n {\displaystyle n} , the difference between ln ( n ! ) {\displaystyle \ln(n!)} and n ln n − n {\displaystyle n\ln n-n} will be at most proportional to the logarithm of n {\displaystyle n} . In computer science applications such as the worst-case lower bound for comparison sorting, it is convenient to instead use the binary logarithm, giving the equivalent form log 2 ( n ! ) = n log 2 n − n log 2 e + O ( log 2 n ) . {\displaystyle \log _{2}(n!)=n\log _{2}n-n\log _{2}e+O(\log _{2}n).} The error term in either base can be expressed more precisely as 1 2 log 2 ( 2 π n ) + O ( 1 n ) {\displaystyle {\tfrac {1}{2}}\log _{2}(2\pi n)+O({\tfrac {1}{n}})} , corresponding to an approximate formula for the factorial itself, n ! ∼ 2 π n ( n e ) n . {\displaystyle n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}.} Here the sign ∼ {\displaystyle \sim } means that the two quantities are asymptotic, that is, their ratio tends to 1 as n {\displaystyle n} tends to infinity. == Derivation == Roughly speaking, the simplest version of Stirling's formula can be quickly obtained by approximating the sum ln ( n ! ) = ∑ j = 1 n ln j {\displaystyle \ln(n!)=\sum _{j=1}^{n}\ln j} with an integral: ∑ j = 1 n ln j ≈ ∫ 1 n ln x d x = n ln n − n + 1. {\displaystyle \sum _{j=1}^{n}\ln j\approx \int _{1}^{n}\ln x\,{\rm {d}}x=n\ln n-n+1.} The full formula, together with precise estimates of its error, can be derived as follows. Instead of approximating n ! {\displaystyle n!} , one considers its natural logarithm, as this is a slowly varying function: ln ( n ! ) = ln 1 + ln 2 + ⋯ + ln n . {\displaystyle \ln(n!)=\ln 1+\ln 2+\cdots +\ln n.} The right-hand side of this equation minus 1 2 ( ln 1 + ln n ) = 1 2 ln n {\displaystyle {\tfrac {1}{2}}(\ln 1+\ln n)={\tfrac {1}{2}}\ln n} is the approximation by the trapezoid rule of the integral ln ( n ! ) − 1 2 ln n ≈ ∫ 1 n ln x d x = n ln n − n + 1 , {\displaystyle \ln(n!)-{\tfrac {1}{2}}\ln n\approx \int _{1}^{n}\ln x\,{\rm {d}}x=n\ln n-n+1,} and the error in this approximation is given by the Euler–Maclaurin formula: ln ( n ! ) − 1 2 ln n = 1 2 ln 1 + ln 2 + ln 3 + ⋯ + ln ( n − 1 ) + 1 2 ln n = n ln n − n + 1 + ∑ k = 2 m ( − 1 ) k B k k ( k − 1 ) ( 1 n k − 1 − 1 ) + R m , n , {\displaystyle {\begin{aligned}\ln(n!)-{\tfrac {1}{2}}\ln n&={\tfrac {1}{2}}\ln 1+\ln 2+\ln 3+\cdots +\ln(n-1)+{\tfrac {1}{2}}\ln n\\&=n\ln n-n+1+\sum _{k=2}^{m}{\frac {(-1)^{k}B_{k}}{k(k-1)}}\left({\frac {1}{n^{k-1}}}-1\right)+R_{m,n},\end{aligned}}} where B k {\displaystyle B_{k}} is a Bernoulli number, and Rm,n is the remainder term in the Euler–Maclaurin formula. Take limits to find that lim n → ∞ ( ln ( n ! ) − n ln n + n − 1 2 ln n ) = 1 − ∑ k = 2 m ( − 1 ) k B k k ( k − 1 ) + lim n → ∞ R m , n . {\displaystyle \lim _{n\to \infty }\left(\ln(n!)-n\ln n+n-{\tfrac {1}{2}}\ln n\right)=1-\sum _{k=2}^{m}{\frac {(-1)^{k}B_{k}}{k(k-1)}}+\lim _{n\to \infty }R_{m,n}.} Denote this limit as y {\displaystyle y} . Because the remainder Rm,n in the Euler–Maclaurin formula satisfies R m , n = lim n → ∞ R m , n + O ( 1 n m ) , {\displaystyle R_{m,n}=\lim _{n\to \infty }R_{m,n}+O\left({\frac {1}{n^{m}}}\right),} where big-O notation is used, combining the equations above yields the approximation formula in its logarithmic form: ln ( n ! ) = n ln ( n e ) + 1 2 ln n + y + ∑ k = 2 m ( − 1 ) k B k k ( k − 1 ) n k − 1 + O ( 1 n m ) . {\displaystyle \ln(n!)=n\ln \left({\frac {n}{e}}\right)+{\tfrac {1}{2}}\ln n+y+\sum _{k=2}^{m}{\frac {(-1)^{k}B_{k}}{k(k-1)n^{k-1}}}+O\left({\frac {1}{n^{m}}}\right).} Taking the exponential of both sides and choosing any positive integer m {\displaystyle m} , one obtains a formula involving an unknown quantity e y {\displaystyle e^{y}} . For m = 1, the formula is n ! = e y n ( n e ) n ( 1 + O ( 1 n ) ) . {\displaystyle n!=e^{y}{\sqrt {n}}\left({\frac {n}{e}}\right)^{n}\left(1+O\left({\frac {1}{n}}\right)\right).} The quantity e y {\displaystyle e^{y}} can be found by taking the limit on both sides as n {\displaystyle n} tends to infinity and using Wallis' product, which shows that e y = 2 π {\displaystyle e^{y}={\sqrt {2\pi }}} . Therefore, one obtains Stirling's formula: n ! = 2 π n ( n e ) n ( 1 + O ( 1 n ) ) . {\displaystyle n!={\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}\left(1+O\left({\frac {1}{n}}\right)\right).} == Alternative derivations == An alternative formula for n ! {\displaystyle n!} using the gamma function is n ! = ∫ 0 ∞ x n e − x d x . {\displaystyle n!=\int _{0}^{\infty }x^{n}e^{-x}\,{\rm {d}}x.} (as can be seen by repeated integration by parts). Rewriting and changing variables x = ny, one obtains n ! = ∫ 0 ∞ e n ln x − x d x = e n ln n n ∫ 0 ∞ e n ( ln y − y ) d y . {\displaystyle n!=\int _{0}^{\infty }e^{n\ln x-x}\,{\rm {d}}x=e^{n\ln n}n\int _{0}^{\infty }e^{n(\ln y-y)}\,{\rm {d}}y.} Applying Laplace's method one has ∫ 0 ∞ e n ( ln y − y ) d y ∼ 2 π n e − n , {\displaystyle \int _{0}^{\infty }e^{n(\ln y-y)}\,{\rm {d}}y\sim {\sqrt {\frac {2\pi }{n}}}e^{-n},} which recovers Stirling's formula: n ! ∼ e n ln n n 2 π n e − n = 2 π n ( n e ) n . {\displaystyle n!\sim e^{n\ln n}n{\sqrt {\frac {2\pi }{n}}}e^{-n}={\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}.} === Higher orders === In fact, further corrections can also be obtained using Laplace's method. From previous result, we know that Γ ( x ) ∼ x x e − x {\displaystyle \Gamma (x)\sim x^{x}e^{-x}} , so we "peel off" this dominant term, then perform two changes of variables, to obtain: x − x e x Γ ( x ) = ∫ R e x ( 1 + t − e t ) d t {\displaystyle x^{-x}e^{x}\Gamma (x)=\int _{\mathbb {R} }e^{x(1+t-e^{t})}dt} To verify this: ∫ R e x ( 1 + t − e t ) d t = t ↦ ln t e x ∫ 0 ∞ t x − 1 e − x t d t = t ↦ t / x x − x e x ∫ 0 ∞ e − t t x − 1 d t = x − x e x Γ ( x ) {\displaystyle \int _{\mathbb {R} }e^{x(1+t-e^{t})}dt{\overset {t\mapsto \ln t}{=}}e^{x}\int _{0}^{\infty }t^{x-1}e^{-xt}dt{\overset {t\mapsto t/x}{=}}x^{-x}e^{x}\int _{0}^{\infty }e^{-t}t^{x-1}dt=x^{-x}e^{x}\Gamma (x)} . Now the function t ↦ 1 + t − e t {\displaystyle t\mapsto 1+t-e^{t}} is unimodal, with maximum value zero. Locally around zero, it looks like − t 2 / 2 {\displaystyle -t^{2}/2} , which is why we are able to perform Laplace's method. In order to extend Laplace's method to higher orders, we perform another change of variables by 1 + t − e t = − τ 2 / 2 {\displaystyle 1+t-e^{t}=-\tau ^{2}/2} . This equation cannot be solved in closed form, but it can be solved by serial expansion, which gives us t = τ − τ 2 / 6 + τ 3 / 36 + a 4 τ 4 + O ( τ 5 ) {\displaystyle t=\tau -\tau ^{2}/6+\tau ^{3}/36+a_{4}\tau ^{4}+O(\tau ^{5})} . Now plug back to the equation to obtain x − x e x Γ ( x ) = ∫ R e − x τ 2 / 2 ( 1 − τ / 3 + τ 2 / 12 + 4 a 4 τ 3 + O ( τ 4 ) ) d τ = 2 π ( x − 1 / 2 + x − 3 / 2 / 12 ) + O ( x − 5 / 2 ) {\displaystyle x^{-x}e^{x}\Gamma (x)=\int _{\mathbb {R} }e^{-x\tau ^{2}/2}(1-\tau /3+\tau ^{2}/12+4a_{4}\tau ^{3}+O(\tau ^{4}))d\tau ={\sqrt {2\pi }}(x^{-1/2}+x^{-3/2}/12)+O(x^{-5/2})} notice how we don't need to actually find a 4 {\displaystyle a_{4}} , since it is cancelled out by the integral. Higher orders can be achieved by computing more terms in t = τ + ⋯ {\displaystyle t=\tau +\cdots } , which can be obtained programmatically. Thus we get Stirling's formula to two orders: n ! = 2 π n ( n e ) n ( 1 + 1 12 n + O ( 1 n 2 ) ) . {\displaystyle n!={\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}\left(1+{\frac {1}{12n}}+O\left({\frac {1}{n^{2}}}\right)\right).} === Complex-analytic version === A complex-analysis version of this method is to consider 1 n ! {\displaystyle {\frac {1}{n!}}} as a Taylor coefficient of the exponential function e z = ∑ n = 0 ∞ z n n ! {\displaystyle e^{z}=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}} , computed by Cauchy's integral formula as 1 n ! = 1 2 π i ∮ | z | = r e z z n + 1 d z . {\displaystyle {\frac {1}{n!}}={\frac {1}{2\pi i}}\oint \limits _{|z|=r}{\frac {e^{z}}{z^{n+1}}}\,\mathrm {d} z.} This line integral can then be approximated using the saddle-point method with an appropriate choice of contour radius r = r n {\displaystyle r=r_{n}} . The dominant portion of the integral near the saddle point is then approximated by a real integral and Laplace's method, while the remaining portion of the integral can be bounded above to give an error term. === Using the Central Limit Theorem and the Poisson distribution === An alternative version uses the fact that the Poisson distribution converges to a normal distribution by the Central Limit Theorem. Since the Poisson distribution with parameter λ {\displaystyle \lambda } converges to a normal distribution with mean λ {\displaystyle \lambda } and variance λ {\displaystyle \lambda } , their density functions will be approximately the same: exp ( − μ ) μ x x ! ≈ 1 2 π μ exp ( − 1 2 ( x − μ μ ) ) {\displaystyle {\frac {\exp(-\mu )\mu ^{x}}{x!}}\approx {\frac {1}{\sqrt {2\pi \mu }}}\exp(-{\frac {1}{2}}({\frac {x-\mu }{\sqrt {\mu }}}))} Evaluating this expression at the mean, at which the approximation is particularly accurate, simplifies this expression to: exp ( − μ ) μ μ μ ! ≈ 1 2 π μ {\displaystyle {\frac {\exp(-\mu )\mu ^{\mu }}{\mu !}}\approx {\frac {1}{\sqrt {2\pi \mu }}}} Taking logs then results in: − μ + μ ln μ − ln μ ! ≈ − 1 2 ln 2 π μ {\displaystyle -\mu +\mu \ln \mu -\ln \mu !\approx -{\frac {1}{2}}\ln 2\pi \mu } which can easily be rearranged to give: ln μ ! ≈ μ ln μ − μ + 1 2 ln 2 π μ {\displaystyle \ln \mu !\approx \mu \ln \mu -\mu +{\frac {1}{2}}\ln 2\pi \mu } Evaluating at μ = n {\displaystyle \mu =n} gives the usual, more precise form of Stirling's approximation. == Speed of convergence and error estimates == Stirling's formula is in fact the first approximation to the following series (now called the Stirling series): n ! ∼ 2 π n ( n e ) n ( 1 + 1 12 n + 1 288 n 2 − 139 51840 n 3 − 571 2488320 n 4 + ⋯ ) . {\displaystyle n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}\left(1+{\frac {1}{12n}}+{\frac {1}{288n^{2}}}-{\frac {139}{51840n^{3}}}-{\frac {571}{2488320n^{4}}}+\cdots \right).} An explicit formula for the coefficients in this series was given by G. Nemes. Further terms are listed in the On-Line Encyclopedia of Integer Sequences as A001163 and A001164. The first graph in this section shows the relative error vs. n {\displaystyle n} , for 1 through all 5 terms listed above. (Bender and Orszag p. 218) gives the asymptotic formula for the coefficients: A 2 j + 1 ∼ ( − 1 ) j 2 ( 2 j ) ! / ( 2 π ) 2 ( j + 1 ) {\displaystyle A_{2j+1}\sim (-1)^{j}2(2j)!/(2\pi )^{2(j+1)}} which shows that it grows superexponentially, and that by the ratio test the radius of convergence is zero. As n → ∞, the error in the truncated series is asymptotically equal to the first omitted term. This is an example of an asymptotic expansion. It is not a convergent series; for any particular value of n {\displaystyle n} there are only so many terms of the series that improve accuracy, after which accuracy worsens. This is shown in the next graph, which shows the relative error versus the number of terms in the series, for larger numbers of terms. More precisely, let S(n, t) be the Stirling series to t {\displaystyle t} terms evaluated at n {\displaystyle n} . The graphs show | ln ( S ( n , t ) n ! ) | , {\displaystyle \left|\ln \left({\frac {S(n,t)}{n!}}\right)\right|,} which, when small, is essentially the relative error. Writing Stirling's series in the form ln ( n ! ) ∼ n ln n − n + 1 2 ln ( 2 π n ) + 1 12 n − 1 360 n 3 + 1 1260 n 5 − 1 1680 n 7 + ⋯ , {\displaystyle \ln(n!)\sim n\ln n-n+{\tfrac {1}{2}}\ln(2\pi n)+{\frac {1}{12n}}-{\frac {1}{360n^{3}}}+{\frac {1}{1260n^{5}}}-{\frac {1}{1680n^{7}}}+\cdots ,} it is known that the error in truncating the series is always of the opposite sign and at most the same magnitude as the first omitted term. Other bounds, due to Robbins, valid for all positive integers n {\displaystyle n} are 2 π n ( n e ) n e 1 12 n + 1 < n ! < 2 π n ( n e ) n e 1 12 n . {\displaystyle {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}e^{\frac {1}{12n+1}}<n!<{\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}e^{\frac {1}{12n}}.} This upper bound corresponds to stopping the above series for ln ( n ! ) {\displaystyle \ln(n!)} after the 1 n {\displaystyle {\frac {1}{n}}} term. The lower bound is weaker than that obtained by stopping the series after the 1 n 3 {\displaystyle {\frac {1}{n^{3}}}} term. A looser version of this bound is that n ! e n n n + 1 2 ∈ ( 2 π , e ] {\displaystyle {\frac {n!e^{n}}{n^{n+{\frac {1}{2}}}}}\in ({\sqrt {2\pi }},e]} for all n ≥ 1 {\displaystyle n\geq 1} . == Stirling's formula for the gamma function == For all positive integers, n ! = Γ ( n + 1 ) , {\displaystyle n!=\Gamma (n+1),} where Γ denotes the gamma function. However, the gamma function, unlike the factorial, is more broadly defined for all complex numbers other than non-positive integers; nevertheless, Stirling's formula may still be applied. If Re(z) > 0, then ln Γ ( z ) = z ln z − z + 1 2 ln 2 π z + ∫ 0 ∞ 2 arctan ( t z ) e 2 π t − 1 d t . {\displaystyle \ln \Gamma (z)=z\ln z-z+{\tfrac {1}{2}}\ln {\frac {2\pi }{z}}+\int _{0}^{\infty }{\frac {2\arctan \left({\frac {t}{z}}\right)}{e^{2\pi t}-1}}\,{\rm {d}}t.} Repeated integration by parts gives ln Γ ( z ) ∼ z ln z − z + 1 2 ln 2 π z + ∑ n = 1 N − 1 B 2 n 2 n ( 2 n − 1 ) z 2 n − 1 = z ln z − z + 1 2 ln 2 π z + 1 12 z − 1 360 z 3 + 1 1260 z 5 + … , {\displaystyle {\begin{aligned}\ln \Gamma (z)\sim z\ln z-z+{\tfrac {1}{2}}\ln {\frac {2\pi }{z}}+\sum _{n=1}^{N-1}{\frac {B_{2n}}{2n(2n-1)z^{2n-1}}}\\=z\ln z-z+{\tfrac {1}{2}}\ln {\frac {2\pi }{z}}+{\frac {1}{12z}}-{\frac {1}{360z^{3}}}+{\frac {1}{1260z^{5}}}+\dots ,\end{aligned}}} where B n {\displaystyle B_{n}} is the n {\displaystyle n} th Bernoulli number (note that the limit of the sum as N → ∞ {\displaystyle N\to \infty } is not convergent, so this formula is just an asymptotic expansion). The formula is valid for z {\displaystyle z} large enough in absolute value, when |arg(z)| < π − ε, where ε is positive, with an error term of O(z−2N+ 1). The corresponding approximation may now be written: Γ ( z ) = 2 π z ( z e ) z ( 1 + O ( 1 z ) ) . {\displaystyle \Gamma (z)={\sqrt {\frac {2\pi }{z}}}\,{\left({\frac {z}{e}}\right)}^{z}\left(1+O\left({\frac {1}{z}}\right)\right).} where the expansion is identical to that of Stirling's series above for n ! {\displaystyle n!} , except that n {\displaystyle n} is replaced with z − 1. A further application of this asymptotic expansion is for complex argument z with constant Re(z). See for example the Stirling formula applied in Im(z) = t of the Riemann–Siegel theta function on the straight line 1/4 + it. == A convergent version of Stirling's formula == Thomas Bayes showed, in a letter to John Canton published by the Royal Society in 1763, that Stirling's formula did not give a convergent series. Obtaining a convergent version of Stirling's formula entails evaluating Binet's formula: ∫ 0 ∞ 2 arctan ( t x ) e 2 π t − 1 d t = ln Γ ( x ) − x ln x + x − 1 2 ln 2 π x . {\displaystyle \int _{0}^{\infty }{\frac {2\arctan \left({\frac {t}{x}}\right)}{e^{2\pi t}-1}}\,{\rm {d}}t=\ln \Gamma (x)-x\ln x+x-{\tfrac {1}{2}}\ln {\frac {2\pi }{x}}.} One way to do this is by means of a convergent series of inverted rising factorials. If z n ¯ = z ( z + 1 ) ⋯ ( z + n − 1 ) , {\displaystyle z^{\bar {n}}=z(z+1)\cdots (z+n-1),} then ∫ 0 ∞ 2 arctan ( t x ) e 2 π t − 1 d t = ∑ n = 1 ∞ c n ( x + 1 ) n ¯ , {\displaystyle \int _{0}^{\infty }{\frac {2\arctan \left({\frac {t}{x}}\right)}{e^{2\pi t}-1}}\,{\rm {d}}t=\sum _{n=1}^{\infty }{\frac {c_{n}}{(x+1)^{\bar {n}}}},} where c n = 1 n ∫ 0 1 x n ¯ ( x − 1 2 ) d x = 1 2 n ∑ k = 1 n k | s ( n , k ) | ( k + 1 ) ( k + 2 ) , {\displaystyle c_{n}={\frac {1}{n}}\int _{0}^{1}x^{\bar {n}}\left(x-{\tfrac {1}{2}}\right)\,{\rm {d}}x={\frac {1}{2n}}\sum _{k=1}^{n}{\frac {k|s(n,k)|}{(k+1)(k+2)}},} where s(n, k) denotes the Stirling numbers of the first kind. From this one obtains a version of Stirling's series ln Γ ( x ) = x ln x − x + 1 2 ln 2 π x + 1 12 ( x + 1 ) + 1 12 ( x + 1 ) ( x + 2 ) + + 59 360 ( x + 1 ) ( x + 2 ) ( x + 3 ) + 29 60 ( x + 1 ) ( x + 2 ) ( x + 3 ) ( x + 4 ) + ⋯ , {\displaystyle {\begin{aligned}\ln \Gamma (x)&=x\ln x-x+{\tfrac {1}{2}}\ln {\frac {2\pi }{x}}+{\frac {1}{12(x+1)}}+{\frac {1}{12(x+1)(x+2)}}+\\&\quad +{\frac {59}{360(x+1)(x+2)(x+3)}}+{\frac {29}{60(x+1)(x+2)(x+3)(x+4)}}+\cdots ,\end{aligned}}} which converges when Re(x) > 0. Stirling's formula may also be given in convergent form as Γ ( x ) = 2 π x x − 1 2 e − x + μ ( x ) {\displaystyle \Gamma (x)={\sqrt {2\pi }}x^{x-{\frac {1}{2}}}e^{-x+\mu (x)}} where μ ( x ) = ∑ n = 0 ∞ ( ( x + n + 1 2 ) ln ( 1 + 1 x + n ) − 1 ) . {\displaystyle \mu \left(x\right)=\sum _{n=0}^{\infty }\left(\left(x+n+{\frac {1}{2}}\right)\ln \left(1+{\frac {1}{x+n}}\right)-1\right).} == Versions suitable for calculators == The approximation Γ ( z ) ≈ 2 π z ( z e z sinh 1 z + 1 810 z 6 ) z {\displaystyle \Gamma (z)\approx {\sqrt {\frac {2\pi }{z}}}\left({\frac {z}{e}}{\sqrt {z\sinh {\frac {1}{z}}+{\frac {1}{810z^{6}}}}}\right)^{z}} and its equivalent form 2 ln Γ ( z ) ≈ ln ( 2 π ) − ln z + z ( 2 ln z + ln ( z sinh 1 z + 1 810 z 6 ) − 2 ) {\displaystyle 2\ln \Gamma (z)\approx \ln(2\pi )-\ln z+z\left(2\ln z+\ln \left(z\sinh {\frac {1}{z}}+{\frac {1}{810z^{6}}}\right)-2\right)} can be obtained by rearranging Stirling's extended formula and observing a coincidence between the resultant power series and the Taylor series expansion of the hyperbolic sine function. This approximation is good to more than 8 decimal digits for z with a real part greater than 8. Robert H. Windschitl suggested it in 2002 for computing the gamma function with fair accuracy on calculators with limited program or register memory. Gergő Nemes proposed in 2007 an approximation which gives the same number of exact digits as the Windschitl approximation but is much simpler: Γ ( z ) ≈ 2 π z ( 1 e ( z + 1 12 z − 1 10 z ) ) z , {\displaystyle \Gamma (z)\approx {\sqrt {\frac {2\pi }{z}}}\left({\frac {1}{e}}\left(z+{\frac {1}{12z-{\frac {1}{10z}}}}\right)\right)^{z},} or equivalently, ln Γ ( z ) ≈ 1 2 ( ln ( 2 π ) − ln z ) + z ( ln ( z + 1 12 z − 1 10 z ) − 1 ) . {\displaystyle \ln \Gamma (z)\approx {\tfrac {1}{2}}\left(\ln(2\pi )-\ln z\right)+z\left(\ln \left(z+{\frac {1}{12z-{\frac {1}{10z}}}}\right)-1\right).} An alternative approximation for the gamma function stated by Srinivasa Ramanujan in Ramanujan's lost notebook is Γ ( 1 + x ) ≈ π ( x e ) x ( 8 x 3 + 4 x 2 + x + 1 30 ) 1 6 {\displaystyle \Gamma (1+x)\approx {\sqrt {\pi }}\left({\frac {x}{e}}\right)^{x}\left(8x^{3}+4x^{2}+x+{\frac {1}{30}}\right)^{\frac {1}{6}}} for x ≥ 0. The equivalent approximation for ln n! has an asymptotic error of 1/1400n3 and is given by ln n ! ≈ n ln n − n + 1 6 ln ( 8 n 3 + 4 n 2 + n + 1 30 ) + 1 2 ln π . {\displaystyle \ln n!\approx n\ln n-n+{\tfrac {1}{6}}\ln(8n^{3}+4n^{2}+n+{\tfrac {1}{30}})+{\tfrac {1}{2}}\ln \pi .} The approximation may be made precise by giving paired upper and lower bounds; one such inequality is π ( x e ) x ( 8 x 3 + 4 x 2 + x + 1 100 ) 1 / 6 < Γ ( 1 + x ) < π ( x e ) x ( 8 x 3 + 4 x 2 + x + 1 30 ) 1 / 6 . {\displaystyle {\sqrt {\pi }}\left({\frac {x}{e}}\right)^{x}\left(8x^{3}+4x^{2}+x+{\frac {1}{100}}\right)^{1/6}<\Gamma (1+x)<{\sqrt {\pi }}\left({\frac {x}{e}}\right)^{x}\left(8x^{3}+4x^{2}+x+{\frac {1}{30}}\right)^{1/6}.} == History == The formula was first discovered by Abraham de Moivre in the form n ! ∼ [ c o n s t a n t ] ⋅ n n + 1 2 e − n . {\displaystyle n!\sim [{\rm {constant}}]\cdot n^{n+{\frac {1}{2}}}e^{-n}.} De Moivre gave an approximate rational-number expression for the natural logarithm of the constant. Stirling's contribution consisted of showing that the constant is precisely 2 π {\displaystyle {\sqrt {2\pi }}} . == See also == Lanczos approximation Spouge's approximation == References == == Further reading == Abramowitz, M. & Stegun, I. (2002), Handbook of Mathematical Functions Paris, R. B. & Kaminski, D. (2001), Asymptotics and Mellin–Barnes Integrals, New York: Cambridge University Press, ISBN 978-0-521-79001-7 Whittaker, E. T. & Watson, G. N. (1996), A Course in Modern Analysis (4th ed.), New York: Cambridge University Press, ISBN 978-0-521-58807-2 Romik, Dan (2000), "Stirling's approximation for n ! {\displaystyle n!} : the ultimate short proof?", The American Mathematical Monthly, 107 (6): 556–557, doi:10.2307/2589351, JSTOR 2589351, MR 1767064 Li, Yuan-Chuan (July 2006), "A note on an identity of the gamma function and Stirling's formula", Real Analysis Exchange, 32 (1): 267–271, MR 2329236 == External links == "Stirling_formula", Encyclopedia of Mathematics, EMS Press, 2001 [1994] Peter Luschny, Approximation formulas for the factorial function n! Weisstein, Eric W., "Stirling's Approximation", MathWorld Stirling's approximation at PlanetMath.
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Wikipedia:Stjepan Gradić#0
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Stjepan Gradić, also known as Stefano Gradi (Latin: Stephanus Gradius; 6 March 1613 – 2 May 1683) was a polymath, philosopher, scientist and a patrician of the Republic of Ragusa. == Biography == Stijepo's parents were Miho Gradi (Gradić) and Marija Benessa (Beneša). He was born in Ragusa (Dubrovnik), Republic of Ragusa, where he was first schooled. He moved to Rome by the order of his uncle, a vicar general of Ragusa, Petar Benessa. In Rome and in Bologna he studied philosophy, theology, law and mathematics. His mathematics professor in Rome was Bonaventura Cavalieri and in Bologna his mathematics professor was Benedetto Castelli. He became a priest in 1643, the year he returned home and soon became abbot of the Benedictine abbey of St. Cosmas and Damian on the island of Pašman, canon of cathedral choir in Ragusa and Ragusan deputy Archbishop. After a private trip to Rome he remained there until his death as the official diplomatic representative of the Republic of Ragusa to the Holy See. Since 1682 he was the head of the Vatican Library. Gradić was a polymath. He cooperated with the historian Joannes Lucius in defending the honor and reputation of their native country of unjust attacks of some Italian and French writers, translated classical authors, wrote a biography of the Ragusan writer Junije Palmotić and a poem about the earthquake in Ragusa. In the literary and scientific circle of pope Alexander VII and Queen Christina of Sweden Gradić discussed scientific and philosophical issues. His philosophical works are written in the spirit of Aristotelianism and scholasticism. Gradić was a member of the Royal Academy in Padua, having correspondence with many notable Europeans. He described the disastrous earthquake in Ragusa in 1667 in Latin verses and organized help from all over Europe for the devastated city. Along with philosophy, he engaged in mathematics, physics, astronomy, literature and diplomacy. In mathematics, he dealt with Galileo's paradox. This work went unnoticed and was even unknown to Roger Joseph Boscovich who was a professor of mathematics at the Collegium Romanum where a century before Gradić had been an alumnus. In his only printed mathematical treatise De loco Galilaei quo punctum lineae aequale pronuntiat published in the collection Dissertationes physico-mathematicae quatuor he disputed the concept of indivisible and developed a series of ideas en route to infinitesimal method. He was solving many mathematical problems which are left in his own inheritance and correspondence with other mathematicians as well, including those of Ghetaldus such as the first problem from Ghetaldus' work Apollonius redivivus. In scientific correspondence with Giovanni Alfonso Borelli and Honoré Fabri he published works dealing with the natural causes of motion and the laws of acceleration and falling bodies. He wrote on the problem of true and apparent position of the polar star. == See also == List of notable Ragusans Dalmatia House of Gradić == References == == External links == Montanari, Tomaso (2002). "GRADI, Stefano". Dizionario Biografico degli Italiani, Volume 58: Gonzales–Graziani (in Italian). Rome: Istituto dell'Enciclopedia Italiana. ISBN 978-8-81200032-6.
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Wikipedia:Stojan Radenović#0
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Stojan Radenović (Serbian Cyrillic: Стојан Раденовић; born 9 March 1948) is a Serbian academic and politician. An internationally respected mathematician, he has served in Serbia's national assembly since 2022 as an independent delegate endorsed by the Serbian Progressive Party (SNS). He was acting president of the assembly from February to March 2024. == Early life and academic career == Radenović was born in the village of Dobra Voda in the municipality of Bojnik, in what was then the People's Republic of Serbia in the Federal People's Republic of Yugoslavia. He received a Ph.D. in 1979 and taught for the next two decades at the University of Kragujevac Faculty of Natural Sciences and Mathematics. From 2000 to 2013, he was a full professor at the University of Belgrade Faculty of Mechanical Engineering. In 2016, Radenović was personally responsible for the University of Belgrade ranking in the world's top three hundred universities for the first time on the prestigious Shanghai list, due to his large number of published works and citations. This accomplishment brought him to the attention of the wider public. == Politician == In the 2022 Serbian parliamentary election, the Serbian Progressive Party reserved the lead positions on its Together We Can Do Everything electoral list for non-party cultural figures and academics. Radenović was given the second position on the list; this was tantamount to election, and he was indeed elected when the list won a plurality victory with 120 out of 250 mandates. During the campaign, he credited the SNS with improving conditions in Serbia over its decade in power. In his first term, Radenović was a member of the education committee and the subcommittee on science and higher education. He was given the ninth position on the SNS's list in the 2023 parliamentary election and was re-elected when the list won a majority victory with 129 seats. As the oldest member of the new parliament, Radenović was chosen as interim president when the assembly convened in February 2024. He served in this role until 20 March, when Ana Brnabić was elected to the position. He is now once again a member of the assembly's education committee. == Notes == == References ==
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Wikipedia:Stokes operator#0
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The Stokes operator, named after George Gabriel Stokes, is an unbounded linear operator used in the theory of partial differential equations, specifically in the fields of fluid dynamics and electromagnetics. == Definition == If we define P σ {\displaystyle P_{\sigma }} as the Leray projection onto divergence free vector fields, then the Stokes Operator A {\displaystyle A} is defined by A := − P σ Δ , {\displaystyle A:=-P_{\sigma }\Delta ,} where Δ ≡ ∇ 2 {\displaystyle \Delta \equiv \nabla ^{2}} is the Laplacian. Since A {\displaystyle A} is unbounded, we must also give its domain of definition, which is defined as D ( A ) = H 2 ∩ V {\displaystyle {\mathcal {D}}(A)=H^{2}\cap V} , where V = { u → ∈ ( H 0 1 ( Ω ) ) n | div u → = 0 } {\displaystyle V=\{{\vec {u}}\in (H_{0}^{1}(\Omega ))^{n}|\operatorname {div} \,{\vec {u}}=0\}} . Here, Ω {\displaystyle \Omega } is a bounded open set in R n {\displaystyle \mathbb {R} ^{n}} (usually n = 2 or 3), H 2 ( Ω ) {\displaystyle H^{2}(\Omega )} and H 0 1 ( Ω ) {\displaystyle H_{0}^{1}(\Omega )} are the standard Sobolev spaces, and the divergence of u → {\displaystyle {\vec {u}}} is taken in the distribution sense. == Properties == For a given domain Ω {\displaystyle \Omega } which is open, bounded, and has C 2 {\displaystyle C^{2}} boundary, the Stokes operator A {\displaystyle A} is a self-adjoint positive-definite operator with respect to the L 2 {\displaystyle L^{2}} inner product. It has an orthonormal basis of eigenfunctions { w k } k = 1 ∞ {\displaystyle \{w_{k}\}_{k=1}^{\infty }} corresponding to eigenvalues { λ k } k = 1 ∞ {\displaystyle \{\lambda _{k}\}_{k=1}^{\infty }} which satisfy 0 < λ 1 < λ 2 ≤ λ 3 ⋯ ≤ λ k ≤ ⋯ {\displaystyle 0<\lambda _{1}<\lambda _{2}\leq \lambda _{3}\cdots \leq \lambda _{k}\leq \cdots } and λ k → ∞ {\displaystyle \lambda _{k}\rightarrow \infty } as k → ∞ {\displaystyle k\rightarrow \infty } . Note that the smallest eigenvalue is unique and non-zero. These properties allow one to define powers of the Stokes operator. Let α > 0 {\displaystyle \alpha >0} be a real number. We define A α {\displaystyle A^{\alpha }} by its action on u → ∈ D ( A ) {\displaystyle {\vec {u}}\in {\mathcal {D}}(A)} : A α u → = ∑ k = 1 ∞ λ k α u k w k → {\displaystyle A^{\alpha }{\vec {u}}=\sum _{k=1}^{\infty }\lambda _{k}^{\alpha }u_{k}{\vec {w_{k}}}} where u k := ( u → , w k → ) {\displaystyle u_{k}:=({\vec {u}},{\vec {w_{k}}})} and ( ⋅ , ⋅ ) {\displaystyle (\cdot ,\cdot )} is the L 2 ( Ω ) {\displaystyle L^{2}(\Omega )} inner product. The inverse A − 1 {\displaystyle A^{-1}} of the Stokes operator is a bounded, compact, self-adjoint operator in the space H := { u → ∈ ( L 2 ( Ω ) ) n | div u → = 0 and γ ( u → ) = 0 } {\displaystyle H:=\{{\vec {u}}\in (L^{2}(\Omega ))^{n}|\operatorname {div} \,{\vec {u}}=0{\text{ and }}\gamma ({\vec {u}})=0\}} , where γ {\displaystyle \gamma } is the trace operator. Furthermore, A − 1 : H → V {\displaystyle A^{-1}:H\rightarrow V} is injective. == References == Temam, Roger (2001), Navier-Stokes Equations: Theory and Numerical Analysis, AMS Chelsea Publishing, ISBN 0-8218-2737-5 Constantin, Peter and Foias, Ciprian. Navier-Stokes Equations, University of Chicago Press, (1988)
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Wikipedia:Stokes phenomenon#0
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In complex analysis the Stokes phenomenon, discovered by G. G. Stokes (1847, 1858), is where the asymptotic behavior of functions can differ in different regions of the complex plane. This seemingly gives rise to a paradox when looking at the asymptotic expansion of an analytic function. Since an analytic function is continuous you would expect the asymptotic expansion to be continuous. This paradox is the subject of Stokes' early research and is known as Stokes phenomenon. The regions in the complex plane with different asymptotic behaviour are bounded by possibly one or two types of curves known as Stokes curves and Anti-Stokes Curves. This apparent paradox has since been resolved and the supposed discontinuous jump in the asymptotic expansions has been shown to be smooth and continuous. In order to resolve this paradox the asymptotic expansion needs to be handled in a careful manner. More specifically the asymptotic expansion must include additional exponentially small terms relative to the usual algebraic terms included in a usual asymptotic expansion. What happens in Stokes phenomenon is that an asymptotic expansion in one region may contain an exponentially small contribution (neglecting this contribution still gives a correct asymptotic expansion for that region). However, this exponentially small term can become exponentially large in another region of the complex plane, this change occurs across the Anti-Stokes curves. Furthermore the exponentially small term may switch on or off other exponentially small terms, this change occurs across a Stokes curve. Including these exponentially small terms allows the asymptotic expansion to be written as a continuous expansion for the entire complex domain which resolves the Stokes Phenomenon paradox. == Stokes Curves and anti-Stokes Curves == Across a Stokes curve, an exponentially small term can switch on or off another exponentially small term. Across an anti-Stokes curve, a subdominant exponentially small term can switch to a dominant exponentially large term or vice versa. This change in behaviour across the Stokes and anti-Stokes curves is directly related to the divergence of the asymptotic expansion. The usual type of divergence seen in an asymptotic series that exhibits Stokes phenomenon is known as factorial-over-power divergence and has the typical form ∑ j ∞ ϵ j A Γ ( j + α ) χ j + α , {\displaystyle \sum _{j}^{\infty }\epsilon ^{j}{\frac {A\Gamma (j+\alpha )}{\chi ^{j+\alpha }}},} Where A {\displaystyle A} is a function known as the prefactor, χ {\displaystyle \chi } is a function known as the Singulant and Γ {\displaystyle \Gamma } is the gamma function. Stokes curves are determined using the condition ℑ { χ } = 0 {\displaystyle \Im \{\chi \}=0} and ℜ { χ } > 0 {\displaystyle \Re \{\chi \}>0} . Anti Stokes curve are determined by the condition ℜ { χ } = 0 {\displaystyle \Re \{\chi \}=0} . == Example: the Airy function == The Airy function Ai(x) is one of two solutions to a simple differential equation y ″ − x y = 0 , {\displaystyle y''-xy=0,\,} which it is often useful to approximate for many values of x – including complex values. For large x of given argument the solution can be approximated by a linear combination of the functions e ± 2 3 x 3 / 2 x 1 / 4 . {\displaystyle {\frac {e^{\pm {\frac {2}{3}}x^{3/2}}}{x^{1/4}}}.} However, the linear combination has to change as the argument of x passes certain values (when x crosses a branch cut) because these approximations contain multi-valued functions. In contrast, the Airy function is single valued and indeed entire and therefore, in order to make sense of the approximation, one has to choose a single value out of the multiple possible values (this imposes a branch cut for the approximation, by implication). For example, if we regard the limit of x as large and real, and would like to approximate the Airy function for both positive and negative values, we would find that A i ( x ) ∼ e − 2 3 x 3 / 2 2 π x 1 / 4 A i ( − x ) ∼ sin ( 2 3 x 3 / 2 + 1 4 π ) π x 1 / 4 {\displaystyle {\begin{aligned}\mathrm {Ai} (x)&{}\sim {\frac {e^{-{\frac {2}{3}}x^{3/2}}}{2{\sqrt {\pi }}\,x^{1/4}}}\\\mathrm {Ai} (-x)&{}\sim {\frac {\sin({\frac {2}{3}}x^{3/2}+{\frac {1}{4}}\pi )}{{\sqrt {\pi }}\,x^{1/4}}}\\\end{aligned}}} which are two very different expressions. What has happened is that as we have increased the argument of x from 0 to pi (rotating it around through the upper half complex plane) we have crossed an anti-Stokes line, which in this case is at arg x = π / 3 {\displaystyle \operatorname {arg} \,x=\pi /3} . At this anti-Stokes line, the coefficient of e − 2 3 x 3 / 2 x 1 / 4 {\displaystyle {\frac {e^{-{\frac {2}{3}}x^{3/2}}}{x^{1/4}}}} is forced to jump. The coefficient of e + 2 3 x 3 / 2 x 1 / 4 {\displaystyle {\frac {e^{+{\frac {2}{3}}x^{3/2}}}{x^{1/4}}}} can jump at this line but is not forced to; it can change gradually as arg x varies from π/3 to π because it is not determined in this region. There are three anti-Stokes lines with arguments π/3, π. –π/3, and three Stokes lines with arguments 2π/3, 0. –2π/3. == Example: second order linear differential equations == The Airy function example can be generalized to a broad class of second order linear differential equations as follows. By standard changes of variables, a second order equation can often be changed to one of the form d 2 w d z 2 = f ( z ) w {\displaystyle {\frac {d^{2}w}{dz^{2}}}=f(z)w} where f is holomorphic in a simply-connected region and w is a solution of the differential equation. Then in some cases the WKB method gives an asymptotic approximation for w as a linear combination of functions of the form e ± ∫ a z f ( z ′ ) d z ′ f ( z ) 1 / 4 {\displaystyle {\frac {e^{\pm \int _{a}^{z}{\sqrt {f(z')}}\,dz'}}{f(z)^{1/4}}}} for some constant a. (Choosing different values of a is equivalent to choosing different coefficients in the linear combination.) The anti-Stokes lines and Stokes lines are then the zeros of the real and imaginary parts, respectively, of ∫ a z f ( z ′ ) d z ′ . {\displaystyle \int _{a}^{z}{\sqrt {f(z')}}\,dz'.} If a is a simple zero of f then locally f looks like f ′ ( a ) ( z − a ) {\displaystyle f'(a)(z-a)} . Solutions will locally behave like the Airy functions; they will have three Stokes lines and three anti-Stokes lines meeting at a. == See also == Borel summation == References == Berry, M. V. (1988), "Stokes' phenomenon; smoothing a Victorian discontinuity.", Inst. Hautes Études Sci. Publ. Math., 68: 211–221, doi:10.1007/bf02698550, MR 1001456, S2CID 121293430 Berry, M. V. (1989), "Uniform asymptotic smoothing of Stokes's discontinuities", Proc. R. Soc. Lond. A, 422 (1862): 7–21, Bibcode:1989RSPSA.422....7B, doi:10.1098/rspa.1989.0018, JSTOR 2398522, MR 0990851, S2CID 122020328 Meyer, R. E. (1989), "A simple explanation of the Stokes phenomenon", SIAM Rev., 31 (3): 435–445, doi:10.1137/1031090, JSTOR 2031404, MR 1012299, archived from the original on September 24, 2017 Olver, Frank William John (1997) [1974], Asymptotics and special functions, AKP Classics, Wellesley, MA: A K Peters Ltd., ISBN 978-1-56881-069-0, MR 1429619 Stokes, G. G. (1847), "On the numerical calculation of a class of definite integrals and infinite series", Transactions of the Cambridge Philosophical Society, IX (I): 166–189 Stokes, G. G. (1858), "On the discontinuity of arbitrary constants which appear in divergent developments", Transactions of the Cambridge Philosophical Society, X (I): 105–128 Witten, Ed (2010). "Analytic Continuation Of Chern-Simons Theory". arXiv:1001.2933v4 [hep-th]. Bender, Carl M.; Orszag, Steven A. (1978), Advanced Mathematical Methods for Scientists and Engineers, International series in pure and applied mathematics, McGraw Hill Inc., ISBN 0-07-004452-X Ablowitz, M. J., & Fokas, A. S. (2003). Complex variables: introduction and applications. Cambridge University Press.
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Wikipedia:Stone algebra#0
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In mathematics, a Stone algebra or Stone lattice is a pseudocomplemented distributive lattice L in which any of the following equivalent statements hold for all x , y ∈ L : {\displaystyle x,y\in L:} (x∧y)* = x* ∨ y*; (x∨y)** = x** ∨ y**; x* ∨ x** = 1. They were introduced by Grätzer & Schmidt (1957) and named after Marshall Harvey Stone. The set S(L) ≝ { x** | x ∈ L } is called the skeleton of L. Then L is a Stone algebra if and only if it's skeleton S(L) is a sublattice of L. Boolean algebras are Stone algebras, and Stone algebras are Ockham algebras. Examples: The open-set lattice of an extremally disconnected space is a Stone algebra. The lattice of positive divisors of a given positive integer is a Stone lattice. == See also == De Morgan algebra Heyting algebra == References == Balbes, Raymond (1970), "A survey of Stone algebras", Proceedings of the Conference on Universal Algebra (Queen's Univ., Kingston, Ont., 1969), Kingston, Ont.: Queen's Univ., pp. 148–170, MR 0260638 Fofanova, T.S. (2001) [1994], "Stone lattice", Encyclopedia of Mathematics, EMS Press Grätzer, George; Schmidt, E. T. (1957), "On a problem of M. H. Stone", Acta Mathematica Academiae Scientiarum Hungaricae, 8 (3–4): 455–460, doi:10.1007/BF02020328, ISSN 0001-5954, MR 0092763 Grätzer, George (1971), Lattice theory. First concepts and distributive lattices, W. H. Freeman and Co., ISBN 978-0-486-47173-0, MR 0321817
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Wikipedia:Stone–Weierstrass theorem#0
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In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on a closed interval [a, b] can be uniformly approximated as closely as desired by a polynomial function. Because polynomials are among the simplest functions, and because computers can directly evaluate polynomials, this theorem has both practical and theoretical relevance, especially in polynomial interpolation. The original version of this result was established by Karl Weierstrass in 1885 using the Weierstrass transform. Marshall H. Stone considerably generalized the theorem and simplified the proof. His result is known as the Stone–Weierstrass theorem. The Stone–Weierstrass theorem generalizes the Weierstrass approximation theorem in two directions: instead of the real interval [a, b], an arbitrary compact Hausdorff space X is considered, and instead of the algebra of polynomial functions, a variety of other families of continuous functions on X {\displaystyle X} are shown to suffice, as is detailed below. The Stone–Weierstrass theorem is a vital result in the study of the algebra of continuous functions on a compact Hausdorff space. Further, there is a generalization of the Stone–Weierstrass theorem to noncompact Tychonoff spaces, namely, any continuous function on a Tychonoff space is approximated uniformly on compact sets by algebras of the type appearing in the Stone–Weierstrass theorem and described below. A different generalization of Weierstrass' original theorem is Mergelyan's theorem, which generalizes it to functions defined on certain subsets of the complex plane. == Weierstrass approximation theorem == The statement of the approximation theorem as originally discovered by Weierstrass is as follows: A constructive proof of this theorem using Bernstein polynomials is outlined on that page. === Degree of approximation === For differentiable functions, Jackson's inequality bounds the error of approximations by polynomials of a given degree: if f {\displaystyle f} has a continuous k-th derivative, then for every n ∈ N {\displaystyle n\in \mathbb {N} } there exists a polynomial p n {\displaystyle p_{n}} of degree at most n {\displaystyle n} such that ‖ f − p n ‖ ≤ π 2 1 ( n + 1 ) k ‖ f ( k ) ‖ {\displaystyle \lVert f-p_{n}\rVert \leq {\frac {\pi }{2}}{\frac {1}{(n+1)^{k}}}\lVert f^{(k)}\rVert } . However, if f {\displaystyle f} is merely continuous, the convergence of the approximations can be arbitrarily slow in the following sense: for any sequence of positive real numbers ( a n ) n ∈ N {\displaystyle (a_{n})_{n\in \mathbb {N} }} decreasing to 0 there exists a function f {\displaystyle f} such that ‖ f − p ‖ > a n {\displaystyle \lVert f-p\rVert >a_{n}} for every polynomial p {\displaystyle p} of degree at most n {\displaystyle n} . === Applications === As a consequence of the Weierstrass approximation theorem, one can show that the space C[a, b] is separable: the polynomial functions are dense, and each polynomial function can be uniformly approximated by one with rational coefficients; there are only countably many polynomials with rational coefficients. Since C[a, b] is metrizable and separable it follows that C[a, b] has cardinality at most 2ℵ0. (Remark: This cardinality result also follows from the fact that a continuous function on the reals is uniquely determined by its restriction to the rationals.) == Stone–Weierstrass theorem, real version == The set C[a, b] of continuous real-valued functions on [a, b], together with the supremum norm ‖f‖ = supa ≤ x ≤ b |f (x)| is a Banach algebra, (that is, an associative algebra and a Banach space such that ‖fg‖ ≤ ‖f‖·‖g‖ for all f, g). The set of all polynomial functions forms a subalgebra of C[a, b] (that is, a vector subspace of C[a, b] that is closed under multiplication of functions), and the content of the Weierstrass approximation theorem is that this subalgebra is dense in C[a, b]. Stone starts with an arbitrary compact Hausdorff space X and considers the algebra C(X, R) of real-valued continuous functions on X, with the topology of uniform convergence. He wants to find subalgebras of C(X, R) which are dense. It turns out that the crucial property that a subalgebra must satisfy is that it separates points: a set A of functions defined on X is said to separate points if, for every two different points x and y in X there exists a function p in A with p(x) ≠ p(y). Now we may state: This implies Weierstrass' original statement since the polynomials on [a, b] form a subalgebra of C[a, b] which contains the constants and separates points. === Locally compact version === A version of the Stone–Weierstrass theorem is also true when X is only locally compact. Let C0(X, R) be the space of real-valued continuous functions on X that vanish at infinity; that is, a continuous function f is in C0(X, R) if, for every ε > 0, there exists a compact set K ⊂ X such that |f| < ε on X \ K. Again, C0(X, R) is a Banach algebra with the supremum norm. A subalgebra A of C0(X, R) is said to vanish nowhere if not all of the elements of A simultaneously vanish at a point; that is, for every x in X, there is some f in A such that f (x) ≠ 0. The theorem generalizes as follows: This version clearly implies the previous version in the case when X is compact, since in that case C0(X, R) = C(X, R). There are also more general versions of the Stone–Weierstrass theorem that weaken the assumption of local compactness. === Applications === The Stone–Weierstrass theorem can be used to prove the following two statements, which go beyond Weierstrass's result. If f is a continuous real-valued function defined on the set [a, b] × [c, d] and ε > 0, then there exists a polynomial function p in two variables such that | f (x, y) − p(x, y) | < ε for all x in [a, b] and y in [c, d]. If X and Y are two compact Hausdorff spaces and f : X × Y → R is a continuous function, then for every ε > 0 there exist n > 0 and continuous functions f1, ..., fn on X and continuous functions g1, ..., gn on Y such that ‖f − Σ fi gi‖ < ε. == Stone–Weierstrass theorem, complex version == Slightly more general is the following theorem, where we consider the algebra C ( X , C ) {\displaystyle C(X,\mathbb {C} )} of complex-valued continuous functions on the compact space X {\displaystyle X} , again with the topology of uniform convergence. This is a C*-algebra with the *-operation given by pointwise complex conjugation. The complex unital *-algebra generated by S {\displaystyle S} consists of all those functions that can be obtained from the elements of S {\displaystyle S} by throwing in the constant function 1 and adding them, multiplying them, conjugating them, or multiplying them with complex scalars, and repeating finitely many times. This theorem implies the real version, because if a net of complex-valued functions uniformly approximates a given function, f n → f {\displaystyle f_{n}\to f} , then the real parts of those functions uniformly approximate the real part of that function, Re f n → Re f {\displaystyle \operatorname {Re} f_{n}\to \operatorname {Re} f} , and because for real subsets, S ⊂ C ( X , R ) ⊂ C ( X , C ) , {\displaystyle S\subset C(X,\mathbb {R} )\subset C(X,\mathbb {C} ),} taking the real parts of the generated complex unital (selfadjoint) algebra agrees with the generated real unital algebra generated. As in the real case, an analog of this theorem is true for locally compact Hausdorff spaces. The following is an application of this complex version. Fourier series: The set of linear combinations of functions en(x) = e2πinx, n ∈ Z is dense in C([0, 1]/{0, 1}), where we identify the endpoints of the interval [0, 1] to obtain a circle. An important consequence of this is that the en are an orthonormal basis of the space L2([0, 1]) of square-integrable functions on [0, 1]. == Stone–Weierstrass theorem, quaternion version == Following Holladay (1957), consider the algebra C(X, H) of quaternion-valued continuous functions on the compact space X, again with the topology of uniform convergence. If a quaternion q is written in the form q = a + i b + j c + k d {\textstyle q=a+ib+jc+kd} its scalar part a is the real number q − i q i − j q j − k q k 4 {\displaystyle {\frac {q-iqi-jqj-kqk}{4}}} . Likewise the scalar part of −qi is b which is the real number − q i − i q + j q k − k q j 4 {\displaystyle {\frac {-qi-iq+jqk-kqj}{4}}} . the scalar part of −qj is c which is the real number − q j − i q k − j q + k q i 4 {\displaystyle {\frac {-qj-iqk-jq+kqi}{4}}} . the scalar part of −qk is d which is the real number − q k + i q j − j q k − k q 4 {\displaystyle {\frac {-qk+iqj-jqk-kq}{4}}} . Then we may state: == Stone–Weierstrass theorem, C*-algebra version == The space of complex-valued continuous functions on a compact Hausdorff space X {\displaystyle X} i.e. C ( X , C ) {\displaystyle C(X,\mathbb {C} )} is the canonical example of a unital commutative C*-algebra A {\displaystyle {\mathfrak {A}}} . The space X may be viewed as the space of pure states on A {\displaystyle {\mathfrak {A}}} , with the weak-* topology. Following the above cue, a non-commutative extension of the Stone–Weierstrass theorem, which remains unsolved, is as follows: In 1960, Jim Glimm proved a weaker version of the above conjecture. == Lattice versions == Let X be a compact Hausdorff space. Stone's original proof of the theorem used the idea of lattices in C(X, R). A subset L of C(X, R) is called a lattice if for any two elements f, g ∈ L, the functions max{ f, g}, min{ f, g} also belong to L. The lattice version of the Stone–Weierstrass theorem states: The above versions of Stone–Weierstrass can be proven from this version once one realizes that the lattice property can also be formulated using the absolute value | f | which in turn can be approximated by polynomials in f . A variant of the theorem applies to linear subspaces of C(X, R) closed under max: More precise information is available: Suppose X is a compact Hausdorff space with at least two points and L is a lattice in C(X, R). The function φ ∈ C(X, R) belongs to the closure of L if and only if for each pair of distinct points x and y in X and for each ε > 0 there exists some f ∈ L for which | f (x) − φ(x)| < ε and | f (y) − φ(y)| < ε. == Bishop's theorem == Another generalization of the Stone–Weierstrass theorem is due to Errett Bishop. Bishop's theorem is as follows: Glicksberg (1962) gives a short proof of Bishop's theorem using the Krein–Milman theorem in an essential way, as well as the Hahn–Banach theorem: the process of Louis de Branges (1959). See also Rudin (1973, §5.7). == Nachbin's theorem == Nachbin's theorem gives an analog for Stone–Weierstrass theorem for algebras of complex valued smooth functions on a smooth manifold. Nachbin's theorem is as follows: == Editorial history == In 1885 it was also published in an English version of the paper whose title was On the possibility of giving an analytic representation to an arbitrary function of real variable. According to the mathematician Yamilet Quintana, Weierstrass "suspected that any analytic functions could be represented by power series". == See also == Müntz–Szász theorem Bernstein polynomial Runge's phenomenon shows that finding a polynomial P such that f (x) = P(x) for some finely spaced x = xn is a bad way to attempt to find a polynomial approximating f uniformly. A better approach, explained e.g. in Rudin (1976), p. 160, eq. (51) ff., is to construct polynomials P uniformly approximating f by taking the convolution of f with a family of suitably chosen polynomial kernels. Mergelyan's theorem, concerning polynomial approximations of complex functions. == Notes == == References == Holladay, John C. (1957), "The Stone–Weierstrass theorem for quaternions" (PDF), Proc. Amer. Math. Soc., 8: 656, doi:10.1090/S0002-9939-1957-0087047-7. Louis de Branges (1959), "The Stone–Weierstrass theorem", Proc. Amer. Math. Soc., 10 (5): 822–824, doi:10.1090/s0002-9939-1959-0113131-7. Jan Brinkhuis & Vladimir Tikhomirov (2005) Optimization: Insights and Applications, Princeton University Press ISBN 978-0-691-10287-0 MR2168305. Glimm, James (1960), "A Stone–Weierstrass Theorem for C*-algebras", Annals of Mathematics, Second Series, 72 (2): 216–244, doi:10.2307/1970133, JSTOR 1970133 Glicksberg, Irving (1962), "Measures Orthogonal to Algebras and Sets of Antisymmetry", Transactions of the American Mathematical Society, 105 (3): 415–435, doi:10.2307/1993729, JSTOR 1993729. Rudin, Walter (1976), Principles of mathematical analysis (3rd ed.), McGraw-Hill, ISBN 978-0-07-054235-8 Rudin, Walter (1973), Functional analysis, McGraw-Hill, ISBN 0-07-054236-8. JG Burkill, Lectures On Approximation By Polynomials (PDF). === Historical works === The historical publication of Weierstrass (in German language) is freely available from the digital online archive of the Berlin Brandenburgische Akademie der Wissenschaften: K. Weierstrass (1885). Über die analytische Darstellbarkeit sogenannter willkürlicher Functionen einer reellen Veränderlichen. Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin, 1885 (II). Erste Mitteilung (part 1) pp. 633–639, Zweite Mitteilung (part 2) pp. 789–805. == External links == "Stone–Weierstrass theorem", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
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Wikipedia:Straightedge and compass construction#0
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In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an idealized ruler and a compass. The idealized ruler, known as a straightedge, is assumed to be infinite in length, have only one edge, and no markings on it. The compass is assumed to have no maximum or minimum radius, and is assumed to "collapse" when lifted from the page, so it may not be directly used to transfer distances. (This is an unimportant restriction since, using a multi-step procedure, a distance can be transferred even with a collapsing compass; see compass equivalence theorem. Note however that whilst a non-collapsing compass held against a straightedge might seem to be equivalent to marking it, the neusis construction is still impermissible and this is what unmarked really means: see Markable rulers below.) More formally, the only permissible constructions are those granted by the first three postulates of Euclid's Elements. It turns out to be the case that every point constructible using straightedge and compass may also be constructed using compass alone, or by straightedge alone if given a single circle and its center. Ancient Greek mathematicians first conceived straightedge-and-compass constructions, and a number of ancient problems in plane geometry impose this restriction. The ancient Greeks developed many constructions, but in some cases were unable to do so. Gauss showed that some polygons are constructible but that most are not. Some of the most famous straightedge-and-compass problems were proved impossible by Pierre Wantzel in 1837 using field theory, namely trisecting an arbitrary angle and doubling the volume of a cube (see § impossible constructions). Many of these problems are easily solvable provided that other geometric transformations are allowed; for example, neusis construction can be used to solve the former two problems. In terms of algebra, a length is constructible if and only if it represents a constructible number, and an angle is constructible if and only if its cosine is a constructible number. A number is constructible if and only if it can be written using the four basic arithmetic operations and the extraction of square roots but of no higher-order roots. == Straightedge and compass tools == The "straightedge" and "compass" of straightedge-and-compass constructions are idealized versions of real-world rulers and compasses. The straightedge is an infinitely long edge with no markings on it. It can only be used to draw a line segment between two points or to extend an existing line segment. The compass can have an arbitrarily large radius with no markings on it (unlike certain real-world compasses). Circles and circular arcs can be drawn starting from two given points: the centre and a point on the circle. The compass may or may not collapse (i.e. fold after being taken off the page, erasing its 'stored' radius). Lines and circles constructed have infinite precision and zero width. Actual compasses do not collapse and modern geometric constructions often use this feature. A 'collapsing compass' would appear to be a less powerful instrument. However, by the compass equivalence theorem in Proposition 2 of Book 1 of Euclid's Elements, no power is lost by using a collapsing compass. Although the proposition is correct, its proofs have a long and checkered history. In any case, the equivalence is why this feature is not stipulated in the definition of the ideal compass. Each construction must be mathematically exact. "Eyeballing" distances (looking at the construction and guessing at its accuracy) or using markings on a ruler are not permitted. Each construction must also terminate. That is, it must have a finite number of steps and not be the limit of ever closer approximations. (If an unlimited number of steps is permitted, some otherwise-impossible constructions become possible by means of infinite sequences converging to a limit.) Stated this way, straightedge-and-compass constructions appear to be a parlour game, rather than a serious practical problem; but the purpose of the restriction is to ensure that constructions can be proven to be exactly correct. == History == The ancient Greek mathematicians first attempted straightedge-and-compass constructions, and they discovered how to construct sums, differences, products, ratios, and square roots of given lengths.: p. 1 They could also construct half of a given angle, a square whose area is twice that of another square, a square having the same area as a given polygon, and regular polygons of 3, 4, or 5 sides: p. xi (or one with twice the number of sides of a given polygon: pp. 49–50 ). But they could not construct one third of a given angle except in particular cases, or a square with the same area as a given circle, or regular polygons with other numbers of sides.: p. xi Nor could they construct the side of a cube whose volume is twice the volume of a cube with a given side.: p. 29 Hippocrates and Menaechmus showed that the volume of the cube could be doubled by finding the intersections of hyperbolas and parabolas, but these cannot be constructed by straightedge and compass.: p. 30 In the fifth century BCE, Hippias used a curve that he called a quadratrix to both trisect the general angle and square the circle, and Nicomedes in the second century BCE showed how to use a conchoid to trisect an arbitrary angle;: p. 37 but these methods also cannot be followed with just straightedge and compass. No progress on the unsolved problems was made for two millennia, until in 1796 Gauss showed that a regular polygon with 17 sides could be constructed; five years later he showed the sufficient criterion for a regular polygon of n sides to be constructible.: pp. 51 ff. In 1837 Pierre Wantzel published a proof of the impossibility of trisecting an arbitrary angle or of doubling the volume of a cube, based on the impossibility of constructing cube roots of lengths. He also showed that Gauss's sufficient constructibility condition for regular polygons is also necessary. Then in 1882 Lindemann showed that π {\displaystyle \pi } is a transcendental number, and thus that it is impossible by straightedge and compass to construct a square with the same area as a given circle.: p. 47 == The basic constructions == All straightedge-and-compass constructions consist of repeated application of five basic constructions using the points, lines and circles that have already been constructed. These are: Creating the line through two points Creating the circle that contains one point and has a center at another point Creating the point at the intersection of two (non-parallel) lines Creating the one point or two points in the intersection of a line and a circle (if they intersect) Creating the one point or two points in the intersection of two circles (if they intersect). For example, starting with just two distinct points, we can create a line or either of two circles (in turn, using each point as centre and passing through the other point). If we draw both circles, two new points are created at their intersections. Drawing lines between the two original points and one of these new points completes the construction of an equilateral triangle. Therefore, in any geometric problem we have an initial set of symbols (points and lines), an algorithm, and some results. From this perspective, geometry is equivalent to an axiomatic algebra, replacing its elements by symbols. Probably Gauss first realized this, and used it to prove the impossibility of some constructions; only much later did Hilbert find a complete set of axioms for geometry. == Common straightedge-and-compass constructions == The most-used straightedge-and-compass constructions include: Constructing the perpendicular bisector from a segment Finding the midpoint of a segment. Drawing a perpendicular line from a point to a line. Bisecting an angle Mirroring a point in a line Constructing a line through a point tangent to a circle Constructing a circle through 3 noncollinear points Drawing a line through a given point parallel to a given line. == Constructible points == One can associate an algebra to our geometry using a Cartesian coordinate system made of two lines, and represent points of our plane by vectors. Finally we can write these vectors as complex numbers. Using the equations for lines and circles, one can show that the points at which they intersect lie in a quadratic extension of the smallest field F containing two points on the line, the center of the circle, and the radius of the circle. That is, they are of the form x + y = k {\displaystyle x+y={\sqrt {k}}} , where x, y, and k are in F. Since the field of constructible points is closed under square roots, it contains all points that can be obtained by a finite sequence of quadratic extensions of the field of complex numbers with rational coefficients. By the above paragraph, one can show that any constructible point can be obtained by such a sequence of extensions. As a corollary of this, one finds that the degree of the minimal polynomial for a constructible point (and therefore of any constructible length) is a power of 2. In particular, any constructible point (or length) is an algebraic number, though not every algebraic number is constructible; for example, 3√2 is algebraic but not constructible. === Constructible angles === There is a bijection between the angles that are constructible and the points that are constructible on any constructible circle. The angles that are constructible form an abelian group under addition modulo 2π (which corresponds to multiplication of the points on the unit circle viewed as complex numbers). The angles that are constructible are exactly those whose tangent (or equivalently, sine or cosine) is constructible as a number. For example, the regular heptadecagon (the seventeen-sided regular polygon) is constructible because cos ( 2 π 17 ) = − 1 16 + 1 16 17 + 1 16 34 − 2 17 + 1 8 17 + 3 17 − 34 − 2 17 − 2 34 + 2 17 {\displaystyle {\begin{aligned}\cos {\left({\frac {2\pi }{17}}\right)}&=\,-{\frac {1}{16}}\,+\,{\frac {1}{16}}{\sqrt {17}}\,+\,{\frac {1}{16}}{\sqrt {34-2{\sqrt {17}}}}\\[5mu]&\qquad +\,{\frac {1}{8}}{\sqrt {17+3{\sqrt {17}}-{\sqrt {34-2{\sqrt {17}}}}-2{\sqrt {34+2{\sqrt {17}}}}}}\end{aligned}}} as discovered by Gauss. The group of constructible angles is closed under the operation that halves angles (which corresponds to taking square roots in the complex numbers). The only angles of finite order that may be constructed starting with two points are those whose order is either a power of two, or a product of a power of two and a set of distinct Fermat primes. In addition there is a dense set of constructible angles of infinite order. === Relation to complex arithmetic === Given a set of points in the Euclidean plane, selecting any one of them to be called 0 and another to be called 1, together with an arbitrary choice of orientation allows us to consider the points as a set of complex numbers. Given any such interpretation of a set of points as complex numbers, the points constructible using valid straightedge-and-compass constructions alone are precisely the elements of the smallest field containing the original set of points and closed under the complex conjugate and square root operations (to avoid ambiguity, we can specify the square root with complex argument less than π). The elements of this field are precisely those that may be expressed as a formula in the original points using only the operations of addition, subtraction, multiplication, division, complex conjugate, and square root, which is easily seen to be a countable dense subset of the plane. Each of these six operations corresponding to a simple straightedge-and-compass construction. From such a formula it is straightforward to produce a construction of the corresponding point by combining the constructions for each of the arithmetic operations. More efficient constructions of a particular set of points correspond to shortcuts in such calculations. Equivalently (and with no need to arbitrarily choose two points) we can say that, given an arbitrary choice of orientation, a set of points determines a set of complex ratios given by the ratios of the differences between any two pairs of points. The set of ratios constructible using straightedge and compass from such a set of ratios is precisely the smallest field containing the original ratios and closed under taking complex conjugates and square roots. For example, the real part, imaginary part and modulus of a point or ratio z (taking one of the two viewpoints above) are constructible as these may be expressed as R e ( z ) = z + z ¯ 2 {\displaystyle \mathrm {Re} (z)={\frac {z+{\bar {z}}}{2}}\;} I m ( z ) = z − z ¯ 2 i {\displaystyle \mathrm {Im} (z)={\frac {z-{\bar {z}}}{2i}}\;} | z | = z z ¯ . {\displaystyle \left|z\right|={\sqrt {z{\bar {z}}}}.\;} Doubling the cube and trisection of an angle (except for special angles such as any φ such that φ/(2π)) is a rational number with denominator not divisible by 3) require ratios which are the solution to cubic equations, while squaring the circle requires a transcendental ratio. None of these are in the fields described, hence no straightedge-and-compass construction for these exists. == Impossible constructions == The ancient Greeks thought that the construction problems they could not solve were simply obstinate, not unsolvable. With modern methods, however, these straightedge-and-compass constructions have been shown to be logically impossible to perform. (The problems themselves, however, are solvable, and the Greeks knew how to solve them without the constraint of working only with straightedge and compass.) === Squaring the circle === The most famous of these problems, squaring the circle, otherwise known as the quadrature of the circle, involves constructing a square with the same area as a given circle using only straightedge and compass. Squaring the circle has been proved impossible, as it involves generating a transcendental number, that is, √π. Only certain algebraic numbers can be constructed with ruler and compass alone, namely those constructed from the integers with a finite sequence of operations of addition, subtraction, multiplication, division, and taking square roots. The phrase "squaring the circle" is often used to mean "doing the impossible" for this reason. Without the constraint of requiring solution by ruler and compass alone, the problem is easily solvable by a wide variety of geometric and algebraic means, and was solved many times in antiquity. A method which comes very close to approximating the "quadrature of the circle" can be achieved using a Kepler triangle. === Doubling the cube === Doubling the cube is the construction, using only a straightedge and compass, of the edge of a cube that has twice the volume of a cube with a given edge. This is impossible because the cube root of 2, though algebraic, cannot be computed from integers by addition, subtraction, multiplication, division, and taking square roots. This follows because its minimal polynomial over the rationals has degree 3. This construction is possible using a straightedge with two marks on it and a compass. === Angle trisection === Angle trisection is the construction, using only a straightedge and a compass, of an angle that is one-third of a given arbitrary angle. This is impossible in the general case. For example, the angle 2π/5 radians (72° = 360°/5) can be trisected, but the angle of π/3 radians (60°) cannot be trisected. The general trisection problem is also easily solved when a straightedge with two marks on it is allowed (a neusis construction). === Distance to an ellipse === The line segment from any point in the plane to the nearest point on a circle can be constructed, but the segment from any point in the plane to the nearest point on an ellipse of positive eccentricity cannot in general be constructed. See Note that results proven here are mostly a consequence of the non-constructivity of conics. If the initial conic is considered as a given, then the proof must be reviewed to check if other distinct conic needs to be generated. As an example, constructions for normals of a parabola are known, but they need to use an intersection between circle and the parabola itself. So they are not constructible in the sense that the parabola is not constructible. === Alhazen's problem === In 1997, the Oxford mathematician Peter M. Neumann proved the theorem that there is no ruler-and-compass construction for the general solution of the ancient Alhazen's problem (billiard problem or reflection from a spherical mirror). == Constructing regular polygons == Some regular polygons (e.g. a pentagon) are easy to construct with straightedge and compass; others are not. This led to the question: Is it possible to construct all regular polygons with straightedge and compass? Carl Friedrich Gauss in 1796 showed that a regular 17-sided polygon can be constructed, and five years later showed that a regular n-sided polygon can be constructed with straightedge and compass if the odd prime factors of n are distinct Fermat primes. Gauss conjectured that this condition was also necessary; the conjecture was proven by Pierre Wantzel in 1837. The first few constructible regular polygons have the following numbers of sides: 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40, 48, 51, 60, 64, 68, 80, 85, 96, 102, 120, 128, 136, 160, 170, 192, 204, 240, 255, 256, 257, 272... (sequence A003401 in the OEIS) There are known to be an infinitude of constructible regular polygons with an even number of sides (because if a regular n-gon is constructible, then so is a regular 2n-gon and hence a regular 4n-gon, 8n-gon, etc.). However, there are only 5 known Fermat primes, giving only 31 known constructible regular n-gons with an odd number of sides. == Constructing a triangle from three given characteristic points or lengths == Sixteen key points of a triangle are its vertices, the midpoints of its sides, the feet of its altitudes, the feet of its internal angle bisectors, and its circumcenter, centroid, orthocenter, and incenter. These can be taken three at a time to yield 139 distinct nontrivial problems of constructing a triangle from three points. Of these problems, three involve a point that can be uniquely constructed from the other two points; 23 can be non-uniquely constructed (in fact for infinitely many solutions) but only if the locations of the points obey certain constraints; in 74 the problem is constructible in the general case; and in 39 the required triangle exists but is not constructible. Twelve key lengths of a triangle are the three side lengths, the three altitudes, the three medians, and the three angle bisectors. Together with the three angles, these give 95 distinct combinations, 63 of which give rise to a constructible triangle, 30 of which do not, and two of which are underdefined.: pp. 201–203 == Restricted constructions == Various attempts have been made to restrict the allowable tools for constructions under various rules, in order to determine what is still constructible and how it may be constructed, as well as determining the minimum criteria necessary to still be able to construct everything that compass and straightedge can. === Constructing with only ruler or only compass === It is possible (according to the Mohr–Mascheroni theorem) to construct anything with just a compass if it can be constructed with a ruler and compass, provided that the given data and the data to be found consist of discrete points (not lines or circles). The truth of this theorem depends on the truth of Archimedes' axiom, which is not first-order in nature. Examples of compass-only constructions include Napoleon's problem. It is impossible to take a square root with just a ruler, so some things that cannot be constructed with a ruler can be constructed with a compass; but (by the Poncelet–Steiner theorem) given a single circle and its center, they can be constructed. == Extended constructions == The ancient Greeks classified constructions into three major categories, depending on the complexity of the tools required for their solution. If a construction used only a straightedge and compass, it was called planar; if it also required one or more conic sections (other than the circle), then it was called solid; the third category included all constructions that did not fall into either of the other two categories. This categorization meshes nicely with the modern algebraic point of view. A complex number that can be expressed using only the field operations and square roots (as described above) has a planar construction. A complex number that includes also the extraction of cube roots has a solid construction. In the language of fields, a complex number that is planar has degree a power of two, and lies in a field extension that can be broken down into a tower of fields where each extension has degree two. A complex number that has a solid construction has degree with prime factors of only two and three, and lies in a field extension that is at the top of a tower of fields where each extension has degree 2 or 3. === Solid constructions === A point has a solid construction if it can be constructed using a straightedge, compass, and a (possibly hypothetical) conic drawing tool that can draw any conic with already constructed focus, directrix, and eccentricity. The same set of points can often be constructed using a smaller set of tools. For example, using a compass, straightedge, and a piece of paper on which we have the parabola y=x2 together with the points (0,0) and (1,0), one can construct any complex number that has a solid construction. Likewise, a tool that can draw any ellipse with already constructed foci and major axis (think two pins and a piece of string) is just as powerful. The ancient Greeks knew that doubling the cube and trisecting an arbitrary angle both had solid constructions. Archimedes gave a neusis construction of the regular heptagon, which was interpreted by medieval Arabic commentators, Bartel Leendert van der Waerden, and others as being based on a solid construction, but this has been disputed, as other interpretations are possible. The quadrature of the circle does not have a solid construction. A regular n-gon has a solid construction if and only if n=2a3bm where a and b are some non-negative integers and m is a product of zero or more distinct Pierpont primes (primes of the form 2r3s+1). Therefore, regular n-gon admits a solid, but not planar, construction if and only if n is in the sequence 7, 9, 13, 14, 18, 19, 21, 26, 27, 28, 35, 36, 37, 38, 39, 42, 45, 52, 54, 56, 57, 63, 65, 70, 72, 73, 74, 76, 78, 81, 84, 90, 91, 95, 97... (sequence A051913 in the OEIS) The set of n for which a regular n-gon has no solid construction is the sequence 11, 22, 23, 25, 29, 31, 33, 41, 43, 44, 46, 47, 49, 50, 53, 55, 58, 59, 61, 62, 66, 67, 69, 71, 75, 77, 79, 82, 83, 86, 87, 88, 89, 92, 93, 94, 98, 99, 100... (sequence A048136 in the OEIS) Like the question with Fermat primes, it is an open question as to whether there are an infinite number of Pierpont primes. === Angle trisection === What if, together with the straightedge and compass, we had a tool that could (only) trisect an arbitrary angle? Such constructions are solid constructions, but there exist numbers with solid constructions that cannot be constructed using such a tool. For example, we cannot double the cube with such a tool. On the other hand, every regular n-gon that has a solid construction can be constructed using such a tool. === Origami === The mathematical theory of origami is more powerful than straightedge-and-compass construction. Folds satisfying the Huzita–Hatori axioms can construct exactly the same set of points as the extended constructions using a compass and conic drawing tool. Therefore, origami can also be used to solve cubic equations (and hence quartic equations), and thus solve two of the classical problems. === Markable rulers === Archimedes, Nicomedes and Apollonius gave constructions involving the use of a markable ruler. This would permit them, for example, to take a line segment, two lines (or circles), and a point; and then draw a line which passes through the given point and intersects the two given lines, such that the distance between the points of intersection equals the given segment. This the Greeks called neusis ("inclination", "tendency" or "verging"), because the new line tends to the point. In this expanded scheme, we can trisect an arbitrary angle (see Archimedes' trisection) or extract an arbitrary cube root (due to Nicomedes). Hence, any distance whose ratio to an existing distance is the solution of a cubic or a quartic equation is constructible. Using a markable ruler, regular polygons with solid constructions, like the heptagon, are constructible; and John H. Conway and Richard K. Guy give constructions for several of them. The neusis construction is more powerful than a conic drawing tool, as one can construct complex numbers that do not have solid constructions. In fact, using this tool one can solve some quintics that are not solvable using radicals. It is known that one cannot solve an irreducible polynomial of prime degree greater or equal to 7 using the neusis construction, so it is not possible to construct a regular 23-gon or 29-gon using this tool. Benjamin and Snyder proved that it is possible to construct the regular 11-gon, but did not give a construction. It is still open as to whether a regular 25-gon or 31-gon is constructible using this tool. === Trisect a straight segment === Given a straight line segment called AB, could this be divided in three new equal segments and in many parts required by the use of intercept theorem. == Computation of binary digits == In 1998 Simon Plouffe gave a ruler-and-compass algorithm that can be used to compute binary digits of certain numbers. The algorithm involves the repeated doubling of an angle and becomes physically impractical after about 20 binary digits. == See also == Carlyle circle Geometric cryptography Geometrography List of interactive geometry software, most of them show straightedge-and-compass constructions Mathematics of paper folding Underwood Dudley, a mathematician who has made a sideline of collecting false straightedge-and-compass proofs. == References == == External links == Regular polygon constructions by Dr. Math at The Math Forum @ Drexel Construction with the Compass Only at cut-the-knot Angle Trisection by Hippocrates at cut-the-knot Weisstein, Eric W. "Angle Trisection". MathWorld.
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Wikipedia:Straightening theorem for vector fields#0
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In differential calculus, the domain-straightening theorem states that, given a vector field X {\displaystyle X} on a manifold, there exist local coordinates y 1 , … , y n {\displaystyle y_{1},\dots ,y_{n}} such that X = ∂ / ∂ y 1 {\displaystyle X=\partial /\partial y_{1}} in a neighborhood of a point where X {\displaystyle X} is nonzero. The theorem is also known as straightening out of a vector field. The Frobenius theorem in differential geometry can be considered as a higher-dimensional generalization of this theorem. == Proof == It is clear that we only have to find such coordinates at 0 in R n {\displaystyle \mathbb {R} ^{n}} . First we write X = ∑ j f j ( x ) ∂ ∂ x j {\displaystyle X=\sum _{j}f_{j}(x){\partial \over \partial x_{j}}} where x {\displaystyle x} is some coordinate system at 0 , {\displaystyle 0,} and f 1 , f 2 , … , f n {\displaystyle f_{1},f_{2},\dots ,f_{n}} are the component function of X {\displaystyle X} relative to x . {\displaystyle x.} Let f = ( f 1 , … , f n ) {\displaystyle f=(f_{1},\dots ,f_{n})} . By linear change of coordinates, we can assume f ( 0 ) = ( 1 , 0 , … , 0 ) . {\displaystyle f(0)=(1,0,\dots ,0).} Let Φ ( t , p ) {\displaystyle \Phi (t,p)} be the solution of the initial value problem x ˙ = f ( x ) , x ( 0 ) = p {\displaystyle {\dot {x}}=f(x),x(0)=p} and let ψ ( x 1 , … , x n ) = Φ ( x 1 , ( 0 , x 2 , … , x n ) ) . {\displaystyle \psi (x_{1},\dots ,x_{n})=\Phi (x_{1},(0,x_{2},\dots ,x_{n})).} Φ {\displaystyle \Phi } (and thus ψ {\displaystyle \psi } ) is smooth by smooth dependence on initial conditions in ordinary differential equations. It follows that ∂ ∂ x 1 ψ ( x ) = f ( ψ ( x ) ) {\displaystyle {\partial \over \partial x_{1}}\psi (x)=f(\psi (x))} , and, since ψ ( 0 , x 2 , … , x n ) = Φ ( 0 , ( 0 , x 2 , … , x n ) ) = ( 0 , x 2 , … , x n ) {\displaystyle \psi (0,x_{2},\dots ,x_{n})=\Phi (0,(0,x_{2},\dots ,x_{n}))=(0,x_{2},\dots ,x_{n})} , the differential d ψ {\displaystyle d\psi } is the identity at 0 {\displaystyle 0} . Thus, y = ψ − 1 ( x ) {\displaystyle y=\psi ^{-1}(x)} is a coordinate system at 0 {\displaystyle 0} . Finally, since x = ψ ( y ) {\displaystyle x=\psi (y)} , we have: ∂ x j ∂ y 1 = f j ( ψ ( y ) ) = f j ( x ) {\displaystyle {\partial x_{j} \over \partial y_{1}}=f_{j}(\psi (y))=f_{j}(x)} and so ∂ ∂ y 1 = X {\displaystyle {\partial \over \partial y_{1}}=X} as required. == References ==
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Wikipedia:Stress majorization#0
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Stress majorization is an optimization strategy used in multidimensional scaling (MDS) where, for a set of n {\displaystyle n} m {\displaystyle m} -dimensional data items, a configuration X {\displaystyle X} of n {\displaystyle n} points in r {\displaystyle r} ( ≪ m ) {\displaystyle (\ll m)} -dimensional space is sought that minimizes the so-called stress function σ ( X ) {\displaystyle \sigma (X)} . Usually r {\displaystyle r} is 2 {\displaystyle 2} or 3 {\displaystyle 3} , i.e. the ( n × r ) {\displaystyle (n\times r)} matrix X {\displaystyle X} lists points in 2 − {\displaystyle 2-} or 3 − {\displaystyle 3-} dimensional Euclidean space so that the result may be visualised (i.e. an MDS plot). The function σ {\displaystyle \sigma } is a cost or loss function that measures the squared differences between ideal ( m {\displaystyle m} -dimensional) distances and actual distances in r-dimensional space. It is defined as: σ ( X ) = ∑ i < j ≤ n w i j ( d i j ( X ) − δ i j ) 2 {\displaystyle \sigma (X)=\sum _{i<j\leq n}w_{ij}(d_{ij}(X)-\delta _{ij})^{2}} where w i j ≥ 0 {\displaystyle w_{ij}\geq 0} is a weight for the measurement between a pair of points ( i , j ) {\displaystyle (i,j)} , d i j ( X ) {\displaystyle d_{ij}(X)} is the euclidean distance between i {\displaystyle i} and j {\displaystyle j} and δ i j {\displaystyle \delta _{ij}} is the ideal distance between the points (their separation) in the m {\displaystyle m} -dimensional data space. Note that w i j {\displaystyle w_{ij}} can be used to specify a degree of confidence in the similarity between points (e.g. 0 can be specified if there is no information for a particular pair). A configuration X {\displaystyle X} which minimizes σ ( X ) {\displaystyle \sigma (X)} gives a plot in which points that are close together correspond to points that are also close together in the original m {\displaystyle m} -dimensional data space. There are many ways that σ ( X ) {\displaystyle \sigma (X)} could be minimized. For example, Kruskal recommended an iterative steepest descent approach. However, a significantly better (in terms of guarantees on, and rate of, convergence) method for minimizing stress was introduced by Jan de Leeuw. De Leeuw's iterative majorization method at each step minimizes a simple convex function which both bounds σ {\displaystyle \sigma } from above and touches the surface of σ {\displaystyle \sigma } at a point Z {\displaystyle Z} , called the supporting point. In convex analysis such a function is called a majorizing function. This iterative majorization process is also referred to as the SMACOF algorithm ("Scaling by MAjorizing a COmplicated Function"). == The SMACOF algorithm == The stress function σ {\displaystyle \sigma } can be expanded as follows: σ ( X ) = ∑ i < j ≤ n w i j ( d i j ( X ) − δ i j ) 2 = ∑ i < j w i j δ i j 2 + ∑ i < j w i j d i j 2 ( X ) − 2 ∑ i < j w i j δ i j d i j ( X ) {\displaystyle \sigma (X)=\sum _{i<j\leq n}w_{ij}(d_{ij}(X)-\delta _{ij})^{2}=\sum _{i<j}w_{ij}\delta _{ij}^{2}+\sum _{i<j}w_{ij}d_{ij}^{2}(X)-2\sum _{i<j}w_{ij}\delta _{ij}d_{ij}(X)} Note that the first term is a constant C {\displaystyle C} and the second term is quadratic in X {\displaystyle X} (i.e. for the Hessian matrix V {\displaystyle V} the second term is equivalent to tr X ′ V X {\displaystyle X'VX} ) and therefore relatively easily solved. The third term is bounded by: ∑ i < j w i j δ i j d i j ( X ) = tr X ′ B ( X ) X ≥ tr X ′ B ( Z ) Z {\displaystyle \sum _{i<j}w_{ij}\delta _{ij}d_{ij}(X)=\,\operatorname {tr} \,X'B(X)X\geq \,\operatorname {tr} \,X'B(Z)Z} where B ( Z ) {\displaystyle B(Z)} has: b i j = − w i j δ i j d i j ( Z ) {\displaystyle b_{ij}=-{\frac {w_{ij}\delta _{ij}}{d_{ij}(Z)}}} for d i j ( Z ) ≠ 0 , i ≠ j {\displaystyle d_{ij}(Z)\neq 0,i\neq j} and b i j = 0 {\displaystyle b_{ij}=0} for d i j ( Z ) = 0 , i ≠ j {\displaystyle d_{ij}(Z)=0,i\neq j} and b i i = − ∑ j = 1 , j ≠ i n b i j {\displaystyle b_{ii}=-\sum _{j=1,j\neq i}^{n}b_{ij}} . Proof of this inequality is by the Cauchy-Schwarz inequality, see Borg (pp. 152–153). Thus, we have a simple quadratic function τ ( X , Z ) {\displaystyle \tau (X,Z)} that majorizes stress: σ ( X ) = C + tr X ′ V X − 2 tr X ′ B ( X ) X {\displaystyle \sigma (X)=C+\,\operatorname {tr} \,X'VX-2\,\operatorname {tr} \,X'B(X)X} ≤ C + tr X ′ V X − 2 tr X ′ B ( Z ) Z = τ ( X , Z ) {\displaystyle \leq C+\,\operatorname {tr} \,X'VX-2\,\operatorname {tr} \,X'B(Z)Z=\tau (X,Z)} The iterative minimization procedure is then: at the k t h {\displaystyle k^{th}} step we set Z ← X k − 1 {\displaystyle Z\leftarrow X^{k-1}} X k ← min X τ ( X , Z ) {\displaystyle X^{k}\leftarrow \min _{X}\tau (X,Z)} stop if σ ( X k − 1 ) − σ ( X k ) < ϵ {\displaystyle \sigma (X^{k-1})-\sigma (X^{k})<\epsilon } otherwise repeat. This algorithm has been shown to decrease stress monotonically (see de Leeuw). == Use in graph drawing == Stress majorization and algorithms similar to SMACOF also have application in the field of graph drawing. That is, one can find a reasonably aesthetically appealing layout for a network or graph by minimizing a stress function over the positions of the nodes in the graph. In this case, the δ i j {\displaystyle \delta _{ij}} are usually set to the graph-theoretic distances between nodes i {\displaystyle i} and j {\displaystyle j} and the weights w i j {\displaystyle w_{ij}} are taken to be δ i j − α {\displaystyle \delta _{ij}^{-\alpha }} . Here, α {\displaystyle \alpha } is chosen as a trade-off between preserving long- or short-range ideal distances. Good results have been shown for α = 2 {\displaystyle \alpha =2} . == References ==
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Wikipedia:Strichartz estimate#0
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In mathematical analysis, Strichartz estimates are a family of inequalities for linear dispersive partial differential equations. These inequalities establish size and decay of solutions in mixed norm Lebesgue spaces. They were first noted by Robert Strichartz and arose out of connections to the Fourier restriction problem. == Examples == Consider the linear Schrödinger equation in R d {\displaystyle \mathbb {R} ^{d}} with h = m = 1. Then the solution for initial data u 0 {\displaystyle u_{0}} is given by e i t Δ / 2 u 0 {\displaystyle e^{it\Delta /2}u_{0}} . Let q and r be real numbers satisfying 2 ≤ q , r ≤ ∞ {\displaystyle 2\leq q,r\leq \infty } ; 2 q + d r = d 2 {\displaystyle {\frac {2}{q}}+{\frac {d}{r}}={\frac {d}{2}}} ; and ( q , r , d ) ≠ ( 2 , ∞ , 2 ) {\displaystyle (q,r,d)\neq (2,\infty ,2)} . In this case the homogeneous Strichartz estimates take the form: ‖ e i t Δ / 2 u 0 ‖ L t q L x r ≤ C d , q , r ‖ u 0 ‖ L x 2 . {\displaystyle \|e^{it\Delta /2}u_{0}\|_{L_{t}^{q}L_{x}^{r}}\leq C_{d,q,r}\|u_{0}\|_{L_{x}^{2}}.} Further suppose that q ~ , r ~ {\displaystyle {\tilde {q}},{\tilde {r}}} satisfy the same restrictions as q , r {\displaystyle q,r} and q ~ ′ , r ~ ′ {\displaystyle {\tilde {q}}',{\tilde {r}}'} are their dual exponents, then the dual homogeneous Strichartz estimates take the form: ‖ ∫ R e − i s Δ / 2 F ( s ) d s ‖ L x 2 ≤ C d , q ~ , r ~ ‖ F ‖ L t q ~ ′ L x r ~ ′ . {\displaystyle \left\|\int _{\mathbb {R} }e^{-is\Delta /2}F(s)\,ds\right\|_{L_{x}^{2}}\leq C_{d,{\tilde {q}},{\tilde {r}}}\|F\|_{L_{t}^{{\tilde {q}}'}L_{x}^{{\tilde {r}}'}}.} The inhomogeneous Strichartz estimates are: ‖ ∫ s < t e i ( t − s ) Δ / 2 F ( s ) d s ‖ L t q L x r ≤ C d , q , r , q ~ , r ~ ‖ F ‖ L t q ~ ′ L x r ~ ′ . {\displaystyle \left\|\int _{s<t}e^{i(t-s)\Delta /2}F(s)\,ds\right\|_{L_{t}^{q}L_{x}^{r}}\leq C_{d,q,r,{\tilde {q}},{\tilde {r}}}\|F\|_{L_{t}^{{\tilde {q}}'}L_{x}^{{\tilde {r}}'}}.} == References ==
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Wikipedia:Strip packing problem#0
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The strip packing problem is a 2-dimensional geometric minimization problem. Given a set of axis-aligned rectangles and a strip of bounded width and infinite height, determine an overlapping-free packing of the rectangles into the strip, minimizing its height. This problem is a cutting and packing problem and is classified as an Open Dimension Problem according to Wäscher et al. This problem arises in the area of scheduling, where it models jobs that require a contiguous portion of the memory over a given time period. Another example is the area of industrial manufacturing, where rectangular pieces need to be cut out of a sheet of material (e.g., cloth or paper) that has a fixed width but infinite length, and one wants to minimize the wasted material. This problem was first studied in 1980. It is strongly-NP hard and there exists no polynomial-time approximation algorithm with a ratio smaller than 3 / 2 {\displaystyle 3/2} unless P = N P {\displaystyle P=NP} . However, the best approximation ratio achieved so far (by a polynomial time algorithm by Harren et al.) is ( 5 / 3 + ε ) {\displaystyle (5/3+\varepsilon )} , imposing an open question of whether there is an algorithm with approximation ratio 3 / 2 {\displaystyle 3/2} . == Definition == An instance I = ( I , W ) {\displaystyle I=({\mathcal {I}},W)} of the strip packing problem consists of a strip with width W = 1 {\displaystyle W=1} and infinite height, as well as a set I {\displaystyle {\mathcal {I}}} of rectangular items. Each item i ∈ I {\displaystyle i\in {\mathcal {I}}} has a width w i ∈ ( 0 , 1 ] ∩ Q {\displaystyle w_{i}\in (0,1]\cap \mathbb {Q} } and a height h i ∈ ( 0 , 1 ] ∩ Q {\displaystyle h_{i}\in (0,1]\cap \mathbb {Q} } . A packing of the items is a mapping that maps each lower-left corner of an item i ∈ I {\displaystyle i\in {\mathcal {I}}} to a position ( x i , y i ) ∈ ( [ 0 , 1 − w i ] ∩ Q ) × Q ≥ 0 {\displaystyle (x_{i},y_{i})\in ([0,1-w_{i}]\cap \mathbb {Q} )\times \mathbb {Q} _{\geq 0}} inside the strip. An inner point of a placed item i ∈ I {\displaystyle i\in {\mathcal {I}}} is a point from the set i n n ( i ) = { ( x , y ) ∈ Q × Q | x i < x < x i + w i , y i < y < y i + h i } {\displaystyle \mathrm {inn} (i)=\{(x,y)\in \mathbb {Q} \times \mathbb {Q} |x_{i}<x<x_{i}+w_{i},y_{i}<y<y_{i}+h_{i}\}} . Two (placed) items overlap if they share an inner point. The height of the packing is defined as max { y i + h i | i ∈ I } {\displaystyle \max\{y_{i}+h_{i}|i\in {\mathcal {I}}\}} . The objective is to find an overlapping-free packing of the items inside the strip while minimizing the height of the packing. This definition is used for all polynomial time algorithms. For pseudo-polynomial time and FPT-algorithms, the definition is slightly changed for the simplification of notation. In this case, all appearing sizes are integral. Especially the width of the strip is given by an arbitrary integer number larger than 1. Note that these two definitions are equivalent. == Variants == There are several variants of the strip packing problem that have been studied. These variants concern the objects' geometry, the problem's dimension, the rotateability of the items, and the structure of the packing. Geometry: In the standard variant of this problem, the set of given items consists of rectangles. In an often considered subcase, all the items have to be squares. This variant was already considered in the first paper about strip packing. Additionally, variants where the shapes are circular or even irregular have been studied. In the latter case, it is referred to as irregular strip packing. Dimension: When not mentioned differently, the strip packing problem is a 2-dimensional problem. However, it also has been studied in three or even more dimensions. In this case, the objects are hyperrectangles, and the strip is open-ended in one dimension and bounded in the residual ones. Rotation: In the classical strip packing problem, the items are not allowed to be rotated. However, variants have been studied where rotating by 90 degrees or even an arbitrary angle is allowed. Structure: In the general strip packing problem, the structure of the packing is irrelevant. However, there are applications that have explicit requirements on the structure of the packing. One of these requirements is to be able to cut the items from the strip by horizontal or vertical edge-to-edge cuts. Packings that allow this kind of cutting are called guillotine packing. == Hardness == The strip packing problem contains the bin packing problem as a special case when all the items have the same height 1. For this reason, it is strongly NP-hard, and there can be no polynomial time approximation algorithm that has an approximation ratio smaller than 3 / 2 {\displaystyle 3/2} unless P = N P {\displaystyle P=NP} . Furthermore, unless P = N P {\displaystyle P=NP} , there cannot be a pseudo-polynomial time algorithm that has an approximation ratio smaller than 5 / 4 {\displaystyle 5/4} , which can be proven by a reduction from the strongly NP-complete 3-partition problem. Note that both lower bounds 3 / 2 {\displaystyle 3/2} and 5 / 4 {\displaystyle 5/4} also hold for the case that a rotation of the items by 90 degrees is allowed. Additionally, it was proven by Ashok et al. that strip packing is W[1]-hard when parameterized by the height of the optimal packing. == Properties of optimal solutions == There are two trivial lower bounds on optimal solutions. The first is the height of the largest item. Define h max ( I ) := max { h ( i ) | i ∈ I } {\displaystyle h_{\max }(I):=\max\{h(i)|i\in {\mathcal {I}}\}} . Then it holds that O P T ( I ) ≥ h max ( I ) {\displaystyle OPT(I)\geq h_{\max }(I)} . Another lower bound is given by the total area of the items. Define A R E A ( I ) := ∑ i ∈ I h ( i ) w ( i ) {\displaystyle \mathrm {AREA} ({\mathcal {I}}):=\sum _{i\in {\mathcal {I}}}h(i)w(i)} then it holds that O P T ( I ) ≥ A R E A ( I ) / W {\displaystyle OPT(I)\geq \mathrm {AREA} ({\mathcal {I}})/W} . The following two lower bounds take notice of the fact that certain items cannot be placed next to each other in the strip, and can be computed in O ( n log ( n ) ) {\displaystyle {\mathcal {O}}(n\log(n))} . For the first lower bound assume that the items are sorted by non-increasing height. Define k := max { i : ∑ j = 1 k w ( j ) ≤ W } {\displaystyle k:=\max\{i:\sum _{j=1}^{k}w(j)\leq W\}} . For each l > k {\displaystyle l>k} define i ( l ) ≤ k {\displaystyle i(l)\leq k} the first index such that w ( l ) + ∑ j = 1 i ( l ) w ( j ) > W {\displaystyle w(l)+\sum _{j=1}^{i(l)}w(j)>W} . Then it holds that O P T ( I ) ≥ max { h ( l ) + h ( i ( l ) ) | l > k ∧ w ( l ) + ∑ j = 1 i ( l ) w ( j ) > W } {\displaystyle OPT(I)\geq \max\{h(l)+h(i(l))|l>k\wedge w(l)+\sum _{j=1}^{i(l)}w(j)>W\}} . For the second lower bound, partition the set of items into three sets. Let α ∈ [ 1 , W / 2 ] ∩ N {\displaystyle \alpha \in [1,W/2]\cap \mathbb {N} } and define I 1 ( α ) := { i ∈ I | w ( i ) > W − α } {\displaystyle {\mathcal {I}}_{1}(\alpha ):=\{i\in {\mathcal {I}}|w(i)>W-\alpha \}} , I 2 ( α ) := { i ∈ I | W − α ≥ w ( i ) > W / 2 } {\displaystyle {\mathcal {I}}_{2}(\alpha ):=\{i\in {\mathcal {I}}|W-\alpha \geq w(i)>W/2\}} , and I 3 ( α ) := { i ∈ I | W / 2 ≥ w ( i ) > α } {\displaystyle {\mathcal {I}}_{3}(\alpha ):=\{i\in {\mathcal {I}}|W/2\geq w(i)>\alpha \}} . Then it holds that O P T ( I ) ≥ max α ∈ [ 1 , W / 2 ] ∩ N { ∑ i ∈ I 1 ( α ) ∪ I 2 ( α ) h ( i ) + ( ∑ i ∈ I 3 ( α ) h ( i ) w ( i ) − ∑ i ∈ I 2 ( α ) ( W − w ( i ) ) h ( i ) W ) + } {\displaystyle OPT(I)\geq \max _{\alpha \in [1,W/2]\cap \mathbb {N} }{\Bigg \{}\sum _{i\in {\mathcal {I}}_{1}(\alpha )\cup {\mathcal {I}}_{2}(\alpha )}h(i)+\left({\frac {\sum _{i\in {\mathcal {I}}_{3}(\alpha )h(i)w(i)-\sum _{i\in {\mathcal {I}}_{2}(\alpha )}(W-w(i))h(i)}}{W}}\right)_{+}{\Bigg \}}} , where ( x ) + := max { x , 0 } {\displaystyle (x)_{+}:=\max\{x,0\}} for each x ∈ R {\displaystyle x\in \mathbb {R} } . On the other hand, Steinberg has shown that the height of an optimal solution can be upper bounded by O P T ( I ) ≤ 2 max { h max ( I ) , A R E A ( I ) / W } . {\displaystyle OPT(I)\leq 2\max\{h_{\max }(I),\mathrm {AREA} ({\mathcal {I}})/W\}.} More precisely he showed that given a W ≥ w max ( I ) {\displaystyle W\geq w_{\max }({\mathcal {I}})} and a H ≥ h max ( I ) {\displaystyle H\geq h_{\max }(I)} then the items I {\displaystyle {\mathcal {I}}} can be placed inside a box with width W {\displaystyle W} and height H {\displaystyle H} if W H ≥ 2 A R E A ( I ) + ( 2 w max ( I ) − W ) + ( 2 h max ( I ) − H ) + {\displaystyle WH\geq 2\mathrm {AREA} ({\mathcal {I}})+(2w_{\max }({\mathcal {I}})-W)_{+}(2h_{\max }(I)-H)_{+}} , where ( x ) + := max { x , 0 } {\displaystyle (x)_{+}:=\max\{x,0\}} . == Polynomial time approximation algorithms == Since this problem is NP-hard, approximation algorithms have been studied for this problem. Most of the heuristic approaches have an approximation ratio between 3 {\displaystyle 3} and 2 {\displaystyle 2} . Finding an algorithm with a ratio below 2 {\displaystyle 2} seems complicated, and the complexity of the corresponding algorithms increases regarding their running time and their descriptions. The smallest approximation ratio achieved so far is ( 5 / 3 + ε ) {\displaystyle (5/3+\varepsilon )} . === Bottom-up left-justified (BL) === This algorithm was first described by Baker et al. It works as follows: Let L {\displaystyle L} be a sequence of rectangular items. The algorithm iterates the sequence in the given order. For each considered item r ∈ L {\displaystyle r\in L} , it searches for the bottom-most position to place it and then shifts it as far to the left as possible. Hence, it places r {\displaystyle r} at the bottom-most left-most possible coordinate ( x , y ) {\displaystyle (x,y)} in the strip. This algorithm has the following properties: The approximation ratio of this algorithm cannot be bounded by a constant. More precisely they showed that for each M > 0 {\displaystyle M>0} there exists a list L {\displaystyle L} of rectangular items ordered by increasing width such that B L ( L ) / O P T ( L ) > M {\displaystyle BL(L)/OPT(L)>M} , where B L ( L ) {\displaystyle BL(L)} is the height of the packing created by the BL algorithm and O P T ( L ) {\displaystyle OPT(L)} is the height of the optimal solution for L {\displaystyle L} . If the items are ordered by decreasing widths, then B L ( L ) / O P T ( L ) ≤ 3 {\displaystyle BL(L)/OPT(L)\leq 3} . If the item are all squares and are ordered by decreasing widths, then B L ( L ) / O P T ( L ) ≤ 2 {\displaystyle BL(L)/OPT(L)\leq 2} . For any δ > 0 {\displaystyle \delta >0} , there exists a list L {\displaystyle L} of rectangles ordered by decreasing widths such that B L ( L ) / O P T ( L ) > 3 − δ {\displaystyle BL(L)/OPT(L)>3-\delta } . For any δ > 0 {\displaystyle \delta >0} , there exists a list L {\displaystyle L} of squares ordered by decreasing widths such that B L ( L ) / O P T ( L ) > 2 − δ {\displaystyle BL(L)/OPT(L)>2-\delta } . For each ε ∈ ( 0 , 1 ] {\displaystyle \varepsilon \in (0,1]} , there exists an instance containing only squares where each order of the squares L {\displaystyle L} has a ratio of B L ( L ) / O P T ( L ) > 12 11 + ε {\displaystyle BL(L)/OPT(L)>{\frac {12}{11+\varepsilon }}} , i.e., there exist instances where BL does not find the optimum even when iterating all possible orders of the items. In 2024 this lower bound has been improved by Hougardy and Zondervan to B L ( L ) / O P T ( L ) > 4 3 + ε {\displaystyle BL(L)/OPT(L)>{\frac {4}{3+\varepsilon }}} . === Next-fit decreasing-height (NFDH) === This algorithm was first described by Coffman et al. in 1980 and works as follows: Let I {\displaystyle {\mathcal {I}}} be the given set of rectangular items. First, the algorithm sorts the items by order of nonincreasing height. Then, starting at position ( 0 , 0 ) {\displaystyle (0,0)} , the algorithm places the items next to each other in the strip until the next item will overlap the right border of the strip. At this point, the algorithm defines a new level at the top of the tallest item in the current level and places the items next to each other in this new level. This algorithm has the following properties: The running time can be bounded by O ( | I | log ( | I | ) ) {\displaystyle {\mathcal {O}}(|{\mathcal {I}}|\log(|{\mathcal {I}}|))} and if the items are already sorted even by O ( | I | ) {\displaystyle {\mathcal {O}}(|{\mathcal {I}}|)} . For every set of items I {\displaystyle {\mathcal {I}}} , it produces a packing of height N F D H ( I ) ≤ 2 O P T ( I ) + h max ≤ 3 O P T ( I ) {\displaystyle NFDH({\mathcal {I}})\leq 2OPT({\mathcal {I}})+h_{\max }\leq 3OPT({\mathcal {I}})} , where h max {\displaystyle h_{\max }} is the largest height of an item in I {\displaystyle {\mathcal {I}}} . For every ε > 0 {\displaystyle \varepsilon >0} there exists a set of rectangles I {\displaystyle {\mathcal {I}}} such that N F D H ( I | ) > ( 2 − ε ) O P T ( I ) . {\displaystyle NFDH({\mathcal {I}}|)>(2-\varepsilon )OPT({\mathcal {I}}).} The packing generated is a guillotine packing. This means the items can be obtained through a sequence of horizontal or vertical edge-to-edge cuts. === First-fit decreasing-height (FFDH) === This algorithm, first described by Coffman et al. in 1980, works similar to the NFDH algorithm. However, when placing the next item, the algorithm scans the levels from bottom to top and places the item in the first level on which it will fit. A new level is only opened if the item does not fit in any previous ones. This algorithm has the following properties: The running time can be bounded by O ( | I | 2 ) {\displaystyle {\mathcal {O}}(|{\mathcal {I}}|^{2})} , since there are at most | I | {\displaystyle |{\mathcal {I}}|} levels. For every set of items I {\displaystyle {\mathcal {I}}} it produces a packing of height F F D H ( I ) ≤ 1.7 O P T ( I ) + h max ≤ 2.7 O P T ( I ) {\displaystyle FFDH({\mathcal {I}})\leq 1.7OPT({\mathcal {I}})+h_{\max }\leq 2.7OPT({\mathcal {I}})} , where h max {\displaystyle h_{\max }} is the largest height of an item in I {\displaystyle {\mathcal {I}}} . Let m ≥ 2 {\displaystyle m\geq 2} . For any set of items I {\displaystyle {\mathcal {I}}} and strip with width W {\displaystyle W} such that w ( i ) ≤ W / m {\displaystyle w(i)\leq W/m} for each i ∈ I {\displaystyle i\in {\mathcal {I}}} , it holds that F F D H ( I ) ≤ ( 1 + 1 / m ) O P T ( I ) + h max {\displaystyle FFDH({\mathcal {I}})\leq \left(1+1/m\right)OPT({\mathcal {I}})+h_{\max }} . Furthermore, for each ε > 0 {\displaystyle \varepsilon >0} , there exists such a set of items I {\displaystyle {\mathcal {I}}} with F F D H ( I ) > ( 1 + 1 / m − ε ) O P T ( I ) {\displaystyle FFDH({\mathcal {I}})>\left(1+1/m-\varepsilon \right)OPT({\mathcal {I}})} . If all the items in I {\displaystyle {\mathcal {I}}} are squares, it holds that F F D H ( I ) ≤ ( 3 / 2 ) O P T ( I ) + h max {\displaystyle FFDH({\mathcal {I}})\leq (3/2)OPT({\mathcal {I}})+h_{\max }} . Furthermore, for each ε > 0 {\displaystyle \varepsilon >0} , there exists a set of squares I {\displaystyle {\mathcal {I}}} such that F F D H ( I ) > ( 3 / 2 − ε ) O P T ( I ) {\displaystyle FFDH({\mathcal {I}})>\left(3/2-\varepsilon \right)OPT({\mathcal {I}})} . The packing generated is a guillotine packing. This means the items can be obtained through a sequence of horizontal or vertical edge-to-edge cuts. === The split-fit algorithm (SF) === This algorithm was first described by Coffman et al. For a given set of items I {\displaystyle {\mathcal {I}}} and strip with width W {\displaystyle W} , it works as follows: Determinate m ∈ N {\displaystyle m\in \mathbb {N} } , the largest integer such that the given rectangles have width W / m {\displaystyle W/m} or less. Divide I {\displaystyle {\mathcal {I}}} into two sets I w i d e {\displaystyle {\mathcal {I}}_{wide}} and I n a r r o w {\displaystyle {\mathcal {I}}_{narrow}} , such that I w i d e {\displaystyle {\mathcal {I}}_{wide}} contains all the items i ∈ I {\displaystyle i\in {\mathcal {I}}} with a width w ( i ) > W / ( m + 1 ) {\displaystyle w(i)>W/(m+1)} while I n a r r o w {\displaystyle {\mathcal {I}}_{narrow}} contains all the items with w ( i ) ≤ W / ( m + 1 ) {\displaystyle w(i)\leq W/(m+1)} . Order I w i d e {\displaystyle {\mathcal {I}}_{wide}} and I n a r r o w {\displaystyle {\mathcal {I}}_{narrow}} by nonincreasing height. Pack the items in I w i d e {\displaystyle {\mathcal {I}}_{wide}} with the FFDH algorithm. Reorder the levels/shelves constructed by FFDH such that all the shelves with a total width larger than W ( m + 1 ) / ( m + 2 ) {\displaystyle W(m+1)/(m+2)} are below the more narrow ones. This leaves a rectangular area R {\displaystyle R} of with W / ( m + 2 ) {\displaystyle W/(m+2)} , next to more narrow levels/shelves, that contains no item. Use the FFDH algorithm to pack the items in I n a r r o w {\displaystyle {\mathcal {I}}_{narrow}} using the area R {\displaystyle R} as well. This algorithm has the following properties: For every set of items I {\displaystyle {\mathcal {I}}} and the corresponding m {\displaystyle m} , it holds that S F ( I ) ≤ ( m + 2 ) / ( m + 1 ) O P T ( I ) + 2 h max {\displaystyle SF({\mathcal {I}})\leq (m+2)/(m+1)OPT({\mathcal {I}})+2h_{\max }} . Note that for m = 1 {\displaystyle m=1} , it holds that S F ( I ) ≤ ( 3 / 2 ) O P T ( I ) + 2 h max {\displaystyle SF({\mathcal {I}})\leq (3/2)OPT({\mathcal {I}})+2h_{\max }} For each ε > 0 {\displaystyle \varepsilon >0} , there is a set of items I {\displaystyle {\mathcal {I}}} such that S F ( I ) > ( ( m + 2 ) / ( m + 1 ) − ε ) O P T ( I ) {\displaystyle SF({\mathcal {I}})>\left((m+2)/(m+1)-\varepsilon \right)OPT({\mathcal {I}})} . === Sleator's algorithm === For a given set of items I {\displaystyle {\mathcal {I}}} and strip with width W {\displaystyle W} , it works as follows: Find all the items with a width larger than W / 2 {\displaystyle W/2} and stack them at the bottom of the strip (in random order). Call the total height of these items h 0 {\displaystyle h_{0}} . All the other items will be placed above h 0 {\displaystyle h_{0}} . Sort all the remaining items in nonincreasing order of height. The items will be placed in this order. Consider the horizontal line at h 0 {\displaystyle h_{0}} as a shelf. The algorithm places the items on this shelf in nonincreasing order of height until no item is left or the next one does not fit. Draw a vertical line at W / 2 {\displaystyle W/2} , which cuts the strip into two equal halves. Let h l {\displaystyle h_{l}} be the highest point covered by any item in the left half and h r {\displaystyle h_{r}} the corresponding point on the right half. Draw two horizontal line segments of length W / 2 {\displaystyle W/2} at h l {\displaystyle h_{l}} and h r {\displaystyle h_{r}} across the left and the right half of the strip. These two lines build new shelves on which the algorithm will place the items, as in step 3. Choose the half which has the lower shelf and place the items on this shelf until no other item fits. Repeat this step until no item is left. This algorithm has the following properties: The running time can be bounded by O ( | I | log ( | I | ) ) {\displaystyle {\mathcal {O}}(|{\mathcal {I}}|\log(|{\mathcal {I}}|))} and if the items are already sorted even by O ( | I | ) {\displaystyle {\mathcal {O}}(|{\mathcal {I}}|)} . For every set of items I {\displaystyle {\mathcal {I}}} it produces a packing of height A ( I ) ≤ 2 O P T ( I ) + h max / 2 ≤ 2.5 O P T ( I ) {\displaystyle A({\mathcal {I}})\leq 2OPT({\mathcal {I}})+h_{\max }/2\leq 2.5OPT({\mathcal {I}})} , where h max {\displaystyle h_{\max }} is the largest height of an item in I {\displaystyle {\mathcal {I}}} . === The split algorithm (SP) === This algorithm is an extension of Sleator's approach and was first described by Golan. It places the items in nonincreasing order of width. The intuitive idea is to split the strip into sub-strips while placing some items. Whenever possible, the algorithm places the current item i {\displaystyle i} side-by-side of an already placed item j {\displaystyle j} . In this case, it splits the corresponding sub-strip into two pieces: one containing the first item j {\displaystyle j} and the other containing the current item i {\displaystyle i} . If this is not possible, it places i {\displaystyle i} on top of an already placed item and does not split the sub-strip. This algorithm creates a set S of sub-strips. For each sub-strip s ∈ S we know its lower left corner s.xposition and s.yposition, its width s.width, the horizontal lines parallel to the upper and lower border of the item placed last inside this sub-strip s.upper and s.lower, as well as the width of it s.itemWidth. function Split Algorithm (SP) is input: items I, width of the strip W output: A packing of the items Sort I in nonincreasing order of widths; Define empty list S of sub-strips; Define a new sub-strip s with s.xposition = 0, s.yposition = 0, s.width = W, s.lower = 0, s.upper = 0, s.itemWidth = W; Add s to S; while I not empty do i := I.pop(); Removes widest item from I Define new list S_2 containing all the substrips with s.width - s.itemWidth ≥ i.width; S_2 contains all sub-strips where i fits next to the already placed item if S_2 is empty then In this case, place the item on top of another one. Find the sub-strip s in S with smallest s.upper; i.e. the least filled sub-strip Place i at position (s.xposition, s.upper); Update s: s.lower := s.upper; s.upper := s.upper+i.height; s.itemWidth := i.width; else In this case, place the item next to another one at the same level and split the corresponding sub-strip at this position. Find s ∈ S_2 with the smallest s.lower; Place i at position (s.xposition + s.itemWidth, s.lower); Remove s from S; Define two new sub-strips s1 and s2 with s1.xposition = s.xposition, s1.yposition = s.upper, s1.width = s.itemWidth, s1.lower = s.upper, s1.upper = s.upper, s1.itemWidth = s.itemWidth; s2.xposition = s.xposition+s.itemWidth, s2.yposition = s.lower, s2.width = s.width - s.itemWidth, s2.lower = s.lower, s2.upper = s.lower + i.height, s2.itemWidth = i.width; S.add(s1,s2); return end function This algorithm has the following properties: The running time can be bounded by O ( | I | 2 ) {\displaystyle {\mathcal {O}}(|{\mathcal {I}}|^{2})} since the number of substrips is bounded by | I | {\displaystyle |{\mathcal {I}}|} . For any set of items I {\displaystyle {\mathcal {I}}} it holds that S P ( I ) ≤ 2 O P T ( I ) + h max ≤ 3 O P T ( I ) {\displaystyle SP({\mathcal {I}})\leq 2OPT({\mathcal {I}})+h_{\max }\leq 3OPT({\mathcal {I}})} . For any ε > 0 {\displaystyle \varepsilon >0} , there exists a set of items I {\displaystyle {\mathcal {I}}} such that S P ( I ) > ( 3 − ε ) O P T ( I ) {\displaystyle SP({\mathcal {I}})>(3-\varepsilon )OPT({\mathcal {I}})} . For any ε > 0 {\displaystyle \varepsilon >0} and C > 0 {\displaystyle C>0} , there exists a set of items I {\displaystyle {\mathcal {I}}} such that S P ( I ) > ( 2 − ε ) O P T ( I ) + C {\displaystyle SP({\mathcal {I}})>(2-\varepsilon )OPT({\mathcal {I}})+C} . === Reverse-fit (RF) === This algorithm was first described by Schiermeyer. The description of this algorithm needs some additional notation. For a placed item i ∈ I {\displaystyle i\in {\mathcal {I}}} , its lower left corner is denoted by ( a i , c i ) {\displaystyle (a_{i},c_{i})} and its upper right corner by ( b i , d i ) {\displaystyle (b_{i},d_{i})} . Given a set of items I {\displaystyle {\mathcal {I}}} and a strip of width W {\displaystyle W} , it works as follows: Stack all the rectangles of width greater than W / 2 {\displaystyle W/2} on top of each other (in random order) at the bottom of the strip. Denote by H 0 {\displaystyle H_{0}} the height of this stack. All other items will be packed above H 0 {\displaystyle H_{0}} . Sort the remaining items in order of nonincreasing height and consider the items in this order in the following steps. Let h max {\displaystyle h_{\max }} be the height of the tallest of these remaining items. Place the items one by one left aligned on a shelf defined by H 0 {\displaystyle H_{0}} until no other item fit on this shelf or there is no item left. Call this shelf the first level. Let h 1 {\displaystyle h_{1}} be the height of the tallest unpacked item. Define a new shelf at H 0 + h max + h 1 {\displaystyle H_{0}+h_{\max }+h_{1}} . The algorithm will fill this shelf from right to left, aligning the items to the right, such that the items touch this shelf with their top. Call this shelf the second reverse-level. Place the items into the two shelves due to First-Fit, i.e., placing the items in the first level where they fit and in the second one otherwise. Proceed until there are no items left, or the total width of the items in the second shelf is at least W / 2 {\displaystyle W/2} . Shift the second reverse-level down until an item from it touches an item from the first level. Define H 1 {\displaystyle H_{1}} as the new vertical position of the shifted shelf. Let f {\displaystyle f} and s {\displaystyle s} be the right most pair of touching items with f {\displaystyle f} placed on the first level and s {\displaystyle s} on the second reverse-level. Define x r := max ( b f , b s ) {\displaystyle x_{r}:=\max(b_{f},b_{s})} . If x r < W / 2 {\displaystyle x_{r}<W/2} then s {\displaystyle s} is the last rectangle placed in the second reverse-level. Shift all the other items from this level further down (all the same amount) until the first one touches an item from the first level. Again the algorithm determines the rightmost pair of touching items f ′ {\displaystyle f'} and s ′ {\displaystyle s'} . Define h 2 {\displaystyle h_{2}} as the amount by which the shelf was shifted down. If h 2 ≤ h ( s ) {\displaystyle h_{2}\leq h(s)} then shift s {\displaystyle s} to the left until it touches another item or the border of the strip. Define the third level at the top of s ′ {\displaystyle s'} . If h 2 > h ( s ) {\displaystyle h_{2}>h(s)} then shift s {\displaystyle s} define the third level at the top of s ′ {\displaystyle s'} . Place s {\displaystyle s} left-aligned in this third level, such that it touches an item from the first level on its left. Continue packing the items using the First-Fit heuristic. Each following level (starting at level three) is defined by a horizontal line through the top of the largest item on the previous level. Note that the first item placed in the next level might not touch the border of the strip with their left side, but an item from the first level or the item s {\displaystyle s} . This algorithm has the following properties: The running time can be bounded by O ( | I | 2 ) {\displaystyle {\mathcal {O}}(|{\mathcal {I}}|^{2})} , since there are at most | I | {\displaystyle |{\mathcal {I}}|} levels. For every set of items I {\displaystyle {\mathcal {I}}} it produces a packing of height R F ( I ) ≤ 2 O P T ( I ) {\displaystyle RF({\mathcal {I}})\leq 2OPT({\mathcal {I}})} . === Steinberg's algorithm (ST) === Steinbergs algorithm is a recursive one. Given a set of rectangular items I {\displaystyle {\mathcal {I}}} and a rectangular target region with width W {\displaystyle W} and height H {\displaystyle H} , it proposes four reduction rules, that place some of the items and leaves a smaller rectangular region with the same properties as before regarding of the residual items. Consider the following notations: Given a set of items I {\displaystyle {\mathcal {I}}} we denote by h max ( I ) {\displaystyle h_{\max }({\mathcal {I}})} the tallest item height in I {\displaystyle {\mathcal {I}}} , w max ( I ) {\displaystyle w_{\max }({\mathcal {I}})} the largest item width appearing in I {\displaystyle {\mathcal {I}}} and by A R E A ( I ) := ∑ i ∈ I w ( i ) h ( i ) {\displaystyle \mathrm {AREA} ({\mathcal {I}}):=\sum _{i\in {\mathcal {I}}}w(i)h(i)} the total area of these items. Steinbergs shows that if h max ( I ) ≤ H {\displaystyle h_{\max }({\mathcal {I}})\leq H} , w max ( I ) ≤ W {\displaystyle w_{\max }({\mathcal {I}})\leq W} , and A R E A ( I ) ≤ W ⋅ H − ( 2 h max ( I ) − h ) + ( 2 w max ( I ) − W ) + {\displaystyle \mathrm {AREA} ({\mathcal {I}})\leq W\cdot H-(2h_{\max }({\mathcal {I}})-h)_{+}(2w_{\max }({\mathcal {I}})-W)_{+}} , where ( a ) + := max { 0 , a } {\displaystyle (a)_{+}:=\max\{0,a\}} , then all the items can be placed inside the target region of size W × H {\displaystyle W\times H} . Each reduction rule will produce a smaller target area and a subset of items that have to be placed. When the condition from above holds before the procedure started, then the created subproblem will have this property as well. Procedure 1: It can be applied if w max ( I ′ ) ≥ W / 2 {\displaystyle w_{\max }({\mathcal {I}}')\geq W/2} . Find all the items i ∈ I {\displaystyle i\in {\mathcal {I}}} with width w ( i ) ≥ W / 2 {\displaystyle w(i)\geq W/2} and remove them from I {\displaystyle {\mathcal {I}}} . Sort them by nonincreasing width and place them left-aligned at the bottom of the target region. Let h 0 {\displaystyle h_{0}} be their total height. Find all the items i ∈ I {\displaystyle i\in {\mathcal {I}}} with width h ( i ) > H − h 0 {\displaystyle h(i)>H-h_{0}} . Remove them from I {\displaystyle {\mathcal {I}}} and place them in a new set I H {\displaystyle {\mathcal {I}}_{H}} . If I H {\displaystyle {\mathcal {I}}_{H}} is empty, define the new target region as the area above h 0 {\displaystyle h_{0}} , i.e. it has height H − h 0 {\displaystyle H-h_{0}} and width W {\displaystyle W} . Solve the problem consisting of this new target region and the reduced set of items with one of the procedures. If I H {\displaystyle {\mathcal {I}}_{H}} is not empty, sort it by nonincreasing height and place the items right allinged one by one in the upper right corner of the target area. Let w 0 {\displaystyle w_{0}} be the total width of these items. Define a new target area with width W − w 0 {\displaystyle W-w_{0}} and height H − h 0 {\displaystyle H-h_{0}} in the upper left corner. Solve the problem consisting of this new target region and the reduced set of items with one of the procedures. Procedure 2: It can be applied if the following conditions hold: w max ( I ) ≤ W / 2 {\displaystyle w_{\max }({\mathcal {I}})\leq W/2} , h max ( I ) ≤ H / 2 {\displaystyle h_{\max }({\mathcal {I}})\leq H/2} , and there exist two different items i , i ′ ∈ I {\displaystyle i,i'\in {\mathcal {I}}} with w ( i ) ≥ W / 4 {\displaystyle w(i)\geq W/4} , w ( i ′ ) ≥ W / 4 {\displaystyle w(i')\geq W/4} , h ( i ) ≥ H / 4 {\displaystyle h(i)\geq H/4} , h ( i ′ ) ≥ H / 4 {\displaystyle h(i')\geq H/4} and 2 ( A R E A ( I ) − w ( i ) h ( i ) − w ( i ′ ) h ( i ′ ) ) ≤ ( W − max { w ( i ) , w ( i ′ ) } ) H {\displaystyle 2(\mathrm {AREA} ({\mathcal {I}})-w(i)h(i)-w(i')h(i'))\leq (W-\max\{w(i),w(i')\})H} . Find i {\displaystyle i} and i ′ {\displaystyle i'} and remove them from I {\displaystyle {\mathcal {I}}} . Place the wider one in the lower-left corner of the target area and the more narrow one left-aligned on the top of the first. Define a new target area on the right of these both items, such that it has the width W − max { w ( i ) , w ( i ′ ) } {\displaystyle W-\max\{w(i),w(i')\}} and height H {\displaystyle H} . Place the residual items in I {\displaystyle {\mathcal {I}}} into the new target area using one of the procedures. Procedure 3: It can be applied if the following conditions hold: w max ( I ) ≤ W / 2 {\displaystyle w_{\max }({\mathcal {I}})\leq W/2} , h max ( I ) ≤ H / 2 {\displaystyle h_{\max }({\mathcal {I}})\leq H/2} , | I | > 1 {\displaystyle |{\mathcal {I}}|>1} , and when sorting the items by decreasing width there exist an index m {\displaystyle m} such that when defining I ′ {\displaystyle {\mathcal {I'}}} as the first m {\displaystyle m} items it holds that A R E A ( I ) − W H / 4 ≤ A R E A ( I ′ ) ≤ 3 W H / 8 {\displaystyle \mathrm {AREA} ({\mathcal {I}})-WH/4\leq \mathrm {AREA} ({\mathcal {I'}})\leq 3WH/8} as well as w ( i m + 1 ) ≤ W / 4 {\displaystyle w(i_{m+1})\leq W/4} Set W 1 := max W / 2 , 2 A R E A ( I ′ ) / H {\displaystyle W_{1}:=\max {W/2,2\mathrm {AREA} ({\mathcal {I'}})/H}} . Define two new rectangular target areas one at the lower-left corner of the original one with height H {\displaystyle H} and width W 1 {\displaystyle W_{1}} and the other left of it with height H {\displaystyle H} and width W − W 1 {\displaystyle W-W_{1}} . Use one of the procedures to place the items in I ′ {\displaystyle {\mathcal {I'}}} into the first new target area and the items in I ∖ I ′ {\displaystyle {\mathcal {I}}\setminus {\mathcal {I'}}} into the second one. Note that procedures 1 to 3 have a symmetric version when swapping the height and the width of the items and the target region. Procedure 4: It can be applied if the following conditions hold: w max ( I ) ≤ W / 2 {\displaystyle w_{\max }({\mathcal {I}})\leq W/2} , h max ( I ) ≤ H / 2 {\displaystyle h_{\max }({\mathcal {I}})\leq H/2} , and there exists an item i ∈ I {\displaystyle i\in {\mathcal {I}}} such that w ( i ) h ( i ) ≥ A R E A ( I ) − W H / 4 {\displaystyle w(i)h(i)\geq \mathrm {AREA} ({\mathcal {I}})-WH/4} . Place the item i {\displaystyle i} in the lower-left corner of the target area and remove it from I {\displaystyle {\mathcal {I}}} . Define a new target area right of this item such that it has the width W − w ( i ) {\displaystyle W-w(i)} and height H {\displaystyle H} and place the residual items inside this area using one of the procedures. This algorithm has the following properties: The running time can be bounded by O ( | I | log ( | I | ) 2 / log ( log ( | I | ) ) ) {\displaystyle {\mathcal {O}}(|{\mathcal {I}}|\log(|{\mathcal {I}}|)^{2}/\log(\log(|{\mathcal {I}}|)))} . For every set of items I {\displaystyle {\mathcal {I}}} it produces a packing of height S T ( I ) ≤ 2 O P T ( I ) {\displaystyle ST({\mathcal {I}})\leq 2OPT({\mathcal {I}})} . == Pseudo-polynomial time approximation algorithms == To improve upon the lower bound of 3 / 2 {\displaystyle 3/2} for polynomial-time algorithms, pseudo-polynomial time algorithms for the strip packing problem have been considered. When considering this type of algorithms, all the sizes of the items and the strip are given as integrals. Furthermore, the width of the strip W {\displaystyle W} is allowed to appear polynomially in the running time. Note that this is no longer considered as a polynomial running time since, in the given instance, the width of the strip needs an encoding size of log ( W ) {\displaystyle \log(W)} . The pseudo-polynomial time algorithms that have been developed mostly use the same approach. It is shown that each optimal solution can be simplified and transformed into one that has one of a constant number of structures. The algorithm then iterates all these structures and places the items inside using linear and dynamic programming. The best ratio accomplished so far is ( 5 / 4 + ε ) O P T ( I ) {\displaystyle (5/4+\varepsilon )OPT(I)} . while there cannot be a pseudo-polynomial time algorithm with ratio better than 5 / 4 {\displaystyle 5/4} unless P = N P {\displaystyle P=NP} == Online algorithms == In the online variant of strip packing, the items arrive over time. When an item arrives, it has to be placed immediately before the next item is known. There are two types of online algorithms that have been considered. In the first variant, it is not allowed to alter the packing once an item is placed. In the second, items may be repacked when another item arrives. This variant is called the migration model. The quality of an online algorithm is measured by the (absolute) competitive ratio s u p I A ( I ) / O P T ( I ) {\displaystyle \mathrm {sup} _{I}A(I)/OPT(I)} , where A ( I ) {\displaystyle A(I)} corresponds to the solution generated by the online algorithm and O P T ( I ) {\displaystyle OPT(I)} corresponds to the size of the optimal solution. In addition to the absolute competitive ratio, the asymptotic competitive ratio of online algorithms has been studied. For instances I {\displaystyle I} with h max ( I ) ≤ 1 {\displaystyle h_{\max }(I)\leq 1} it is defined as lim s u p O P T ( I ) → ∞ A ( I ) / O P T ( I ) {\displaystyle \lim \mathrm {sup} _{OPT(I)\rightarrow \infty }A(I)/OPT(I)} . Note that all the instances can be scaled such that h max ( I ) ≤ 1 {\displaystyle h_{\max }(I)\leq 1} . The framework of Han et al. is applicable in the online setting if the online bin packing algorithm belongs to the class Super Harmonic. Thus, Seiden's online bin packing algorithm Harmonic++ implies an algorithm for online strip packing with asymptotic ratio 1.58889. == References ==
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Wikipedia:Strongly measurable function#0
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Strong measurability has a number of different meanings, some of which are explained below. == Values in Banach spaces == For a function f with values in a Banach space (or Fréchet space), strong measurability usually means Bochner measurability. However, if the values of f lie in the space L ( X , Y ) {\displaystyle {\mathcal {L}}(X,Y)} of continuous linear operators from X to Y, then often strong measurability means that the operator f(x) is Bochner measurable for each fixed x in the domain of f, whereas the Bochner measurability of f is called uniform measurability (cf. "uniformly continuous" vs. "strongly continuous"). == Bounded operators == A family of bounded linear operators combined with the direct integral is strongly measurable, when each of the individual operators is strongly measurable. == Semigroups == A semigroup of linear operators can be strongly measurable yet not strongly continuous. It is uniformly measurable if and only if it is uniformly continuous, i.e., if and only if its generator is bounded. == References ==
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Wikipedia:Strongly regular graph#0
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In graph theory, a strongly regular graph (SRG) is a regular graph G = (V, E) with v vertices and degree k such that for some given integers λ , μ ≥ 0 {\displaystyle \lambda ,\mu \geq 0} every two adjacent vertices have λ common neighbours, and every two non-adjacent vertices have μ common neighbours. Such a strongly regular graph is denoted by srg(v, k, λ, μ). Its complement graph is also strongly regular: it is an srg(v, v − k − 1, v − 2 − 2k + μ, v − 2k + λ). A strongly regular graph is a distance-regular graph with diameter 2 whenever μ is non-zero. It is a locally linear graph whenever λ = 1. == Etymology == A strongly regular graph is denoted as an srg(v, k, λ, μ) in the literature. By convention, graphs which satisfy the definition trivially are excluded from detailed studies and lists of strongly regular graphs. These include the disjoint union of one or more equal-sized complete graphs, and their complements, the complete multipartite graphs with equal-sized independent sets. Andries Brouwer and Hendrik van Maldeghem (see #References) use an alternate but fully equivalent definition of a strongly regular graph based on spectral graph theory: a strongly regular graph is a finite regular graph that has exactly three eigenvalues, only one of which is equal to the degree k, of multiplicity 1. This automatically rules out fully connected graphs (which have only two distinct eigenvalues, not three) and disconnected graphs (for which the multiplicity of the degree k is equal to the number of different connected components, which would therefore exceed one). Much of the literature, including Brouwer, refers to the larger eigenvalue as r (with multiplicity f) and the smaller one as s (with multiplicity g). == History == Strongly regular graphs were introduced by R.C. Bose in 1963. They built upon earlier work in the 1950s in the then-new field of spectral graph theory. == Examples == The cycle of length 5 is an srg(5, 2, 0, 1). The Petersen graph is an srg(10, 3, 0, 1). The Clebsch graph is an srg(16, 5, 0, 2). The Shrikhande graph is an srg(16, 6, 2, 2) which is not a distance-transitive graph. The n × n square rook's graph, i.e., the line graph of a balanced complete bipartite graph Kn,n, is an srg(n2, 2n − 2, n − 2, 2). The parameters for n = 4 coincide with those of the Shrikhande graph, but the two graphs are not isomorphic. (The vertex neighborhood for the Shrikhande graph is a hexagon, while that for the rook graph is two triangles.) The line graph of a complete graph Kn is an srg ( ( n 2 ) , 2 ( n − 2 ) , n − 2 , 4 ) {\textstyle \operatorname {srg} \left({\binom {n}{2}},2(n-2),n-2,4\right)} . The three Chang graphs are srg(28, 12, 6, 4), the same as the line graph of K8, but these four graphs are not isomorphic. Every generalized quadrangle of order (s, t) gives an srg((s + 1)(st + 1), s(t + 1), s − 1, t + 1) as its line graph. For example, GQ(2, 4) gives srg(27, 10, 1, 5) as its line graph. The Schläfli graph is an srg(27, 16, 10, 8) and is the complement of the aforementioned line graph on GQ(2, 4). The Hoffman–Singleton graph is an srg(50, 7, 0, 1). The Gewirtz graph is an srg(56, 10, 0, 2). The M22 graph aka the Mesner graph is an srg(77, 16, 0, 4). The Brouwer–Haemers graph is an srg(81, 20, 1, 6). The Higman–Sims graph is an srg(100, 22, 0, 6). The Local McLaughlin graph is an srg(162, 56, 10, 24). The Cameron graph is an srg(231, 30, 9, 3). The Berlekamp–van Lint–Seidel graph is an srg(243, 22, 1, 2). The McLaughlin graph is an srg(275, 112, 30, 56). The Paley graph of order q is an srg(q, (q − 1)/2, (q − 5)/4, (q − 1)/4). The smallest Paley graph, with q = 5, is the 5-cycle (above). Self-complementary arc-transitive graphs are strongly regular. A strongly regular graph is called primitive if both the graph and its complement are connected. All the above graphs are primitive, as otherwise μ = 0 or λ = k. Conway's 99-graph problem asks for the construction of an srg(99, 14, 1, 2). It is unknown whether a graph with these parameters exists, and John Horton Conway offered a $1000 prize for the solution to this problem. === Triangle-free graphs === The strongly regular graphs with λ = 0 are triangle free. Apart from the complete graphs on fewer than 3 vertices and all complete bipartite graphs, the seven listed earlier (pentagon, Petersen, Clebsch, Hoffman-Singleton, Gewirtz, Mesner-M22, and Higman-Sims) are the only known ones. === Geodetic graphs === Every strongly regular graph with μ = 1 {\displaystyle \mu =1} is a geodetic graph, a graph in which every two vertices have a unique unweighted shortest path. The only known strongly regular graphs with μ = 1 {\displaystyle \mu =1} are those where λ {\displaystyle \lambda } is 0, therefore triangle-free as well. These are called the Moore graphs and are explored below in more detail. Other combinations of parameters such as (400, 21, 2, 1) have not yet been ruled out. Despite ongoing research on the properties that a strongly regular graph with μ = 1 {\displaystyle \mu =1} would have, it is not known whether any more exist or even whether their number is finite. Only the elementary result is known, that λ {\displaystyle \lambda } cannot be 1 for such a graph. == Algebraic properties of strongly regular graphs == === Basic relationship between parameters === The four parameters in an srg(v, k, λ, μ) are not independent: In order for an srg(v, k, λ, μ) to exist, the parameters must obey the following relation: ( v − k − 1 ) μ = k ( k − λ − 1 ) {\displaystyle (v-k-1)\mu =k(k-\lambda -1)} The above relation is derived through a counting argument as follows: Imagine the vertices of the graph to lie in three levels. Pick any vertex as the root, in Level 0. Then its k neighbors lie in Level 1, and all other vertices lie in Level 2. Vertices in Level 1 are directly connected to the root, hence they must have λ other neighbors in common with the root, and these common neighbors must also be in Level 1. Since each vertex has degree k, there are k − λ − 1 {\displaystyle k-\lambda -1} edges remaining for each Level 1 node to connect to vertices in Level 2. Therefore, there are k ( k − λ − 1 ) {\displaystyle k(k-\lambda -1)} edges between Level 1 and Level 2. Vertices in Level 2 are not directly connected to the root, hence they must have μ common neighbors with the root, and these common neighbors must all be in Level 1. There are ( v − k − 1 ) {\displaystyle (v-k-1)} vertices in Level 2, and each is connected to μ vertices in Level 1. Therefore the number of edges between Level 1 and Level 2 is ( v − k − 1 ) μ {\displaystyle (v-k-1)\mu } . Equating the two expressions for the edges between Level 1 and Level 2, the relation follows. This relation is a necessary condition for the existence of a strongly regular graph, but not a sufficient condition. For instance, the quadruple (21,10,4,5) obeys this relation, but there does not exist a strongly regular graph with these parameters. === Adjacency matrix equations === Let I denote the identity matrix and let J denote the matrix of ones, both matrices of order v. The adjacency matrix A of a strongly regular graph satisfies two equations. First: A J = J A = k J , {\displaystyle AJ=JA=kJ,} which is a restatement of the regularity requirement. This shows that k is an eigenvalue of the adjacency matrix with the all-ones eigenvector. Second: A 2 = k I + λ A + μ ( J − I − A ) {\displaystyle A^{2}=kI+\lambda {A}+\mu (J-I-A)} which expresses strong regularity. The ij-th element of the left hand side gives the number of two-step paths from i to j. The first term of the right hand side gives the number of two-step paths from i back to i, namely k edges out and back in. The second term gives the number of two-step paths when i and j are directly connected. The third term gives the corresponding value when i and j are not connected. Since the three cases are mutually exclusive and collectively exhaustive, the simple additive equality follows. Conversely, a graph whose adjacency matrix satisfies both of the above conditions and which is not a complete or null graph is a strongly regular graph. === Eigenvalues and graph spectrum === Since the adjacency matrix A is symmetric, it follows that its eigenvectors are orthogonal. We already observed one eigenvector above which is made of all ones, corresponding to the eigenvalue k. Therefore the other eigenvectors x must all satisfy J x = 0 {\displaystyle Jx=0} where J is the all-ones matrix as before. Take the previously established equation: A 2 = k I + λ A + μ ( J − I − A ) {\displaystyle A^{2}=kI+\lambda {A}+\mu (J-I-A)} and multiply the above equation by eigenvector x: A 2 x = k I x + λ A x + μ ( J − I − A ) x {\displaystyle A^{2}x=kIx+\lambda {A}x+\mu (J-I-A)x} Call the corresponding eigenvalue p (not to be confused with λ {\displaystyle \lambda } the graph parameter) and substitute A x = p x {\displaystyle Ax=px} , J x = 0 {\displaystyle Jx=0} and I x = x {\displaystyle Ix=x} : p 2 x = k x + λ p x − μ x − μ p x {\displaystyle p^{2}x=kx+\lambda px-\mu x-\mu px} Eliminate x and rearrange to get a quadratic: p 2 + ( μ − λ ) p − ( k − μ ) = 0 {\displaystyle p^{2}+(\mu -\lambda )p-(k-\mu )=0} This gives the two additional eigenvalues 1 2 [ ( λ − μ ) ± ( λ − μ ) 2 + 4 ( k − μ ) ] {\displaystyle {\frac {1}{2}}\left[(\lambda -\mu )\pm {\sqrt {(\lambda -\mu )^{2}+4(k-\mu )}}\,\right]} . There are thus exactly three eigenvalues for a strongly regular matrix. Conversely, a connected regular graph with only three eigenvalues is strongly regular. Following the terminology in much of the strongly regular graph literature, the larger eigenvalue is called r with multiplicity f and the smaller one is called s with multiplicity g. Since the sum of all the eigenvalues is the trace of the adjacency matrix, which is zero in this case, the respective multiplicities f and g can be calculated: Eigenvalue k has multiplicity 1. Eigenvalue r = 1 2 [ ( λ − μ ) + ( λ − μ ) 2 + 4 ( k − μ ) ] {\displaystyle r={\frac {1}{2}}\left[(\lambda -\mu )+{\sqrt {(\lambda -\mu )^{2}+4(k-\mu )}}\,\right]} has multiplicity f = 1 2 [ ( v − 1 ) − 2 k + ( v − 1 ) ( λ − μ ) ( λ − μ ) 2 + 4 ( k − μ ) ] {\displaystyle f={\frac {1}{2}}\left[(v-1)-{\frac {2k+(v-1)(\lambda -\mu )}{\sqrt {(\lambda -\mu )^{2}+4(k-\mu )}}}\right]} Eigenvalue s = 1 2 [ ( λ − μ ) − ( λ − μ ) 2 + 4 ( k − μ ) ] {\displaystyle s={\frac {1}{2}}\left[(\lambda -\mu )-{\sqrt {(\lambda -\mu )^{2}+4(k-\mu )}}\,\right]} has multiplicity g = 1 2 [ ( v − 1 ) + 2 k + ( v − 1 ) ( λ − μ ) ( λ − μ ) 2 + 4 ( k − μ ) ] {\displaystyle g={\frac {1}{2}}\left[(v-1)+{\frac {2k+(v-1)(\lambda -\mu )}{\sqrt {(\lambda -\mu )^{2}+4(k-\mu )}}}\right]} As the multiplicities must be integers, their expressions provide further constraints on the values of v, k, μ, and λ. Strongly regular graphs for which 2 k + ( v − 1 ) ( λ − μ ) ≠ 0 {\displaystyle 2k+(v-1)(\lambda -\mu )\neq 0} have integer eigenvalues with unequal multiplicities. Strongly regular graphs for which 2 k + ( v − 1 ) ( λ − μ ) = 0 {\displaystyle 2k+(v-1)(\lambda -\mu )=0} are called conference graphs because of their connection with symmetric conference matrices. Their parameters reduce to srg ( v , 1 2 ( v − 1 ) , 1 4 ( v − 5 ) , 1 4 ( v − 1 ) ) . {\displaystyle \operatorname {srg} \left(v,{\frac {1}{2}}(v-1),{\frac {1}{4}}(v-5),{\frac {1}{4}}(v-1)\right).} Their eigenvalues are r = − 1 + v 2 {\displaystyle r={\frac {-1+{\sqrt {v}}}{2}}} and s = − 1 − v 2 {\displaystyle s={\frac {-1-{\sqrt {v}}}{2}}} , both of whose multiplicities are equal to v − 1 2 {\displaystyle {\frac {v-1}{2}}} . Further, in this case, v must equal the sum of two squares, related to the Bruck–Ryser–Chowla theorem. Further properties of the eigenvalues and their multiplicities are: ( A − r I ) × ( A − s I ) = μ . J {\displaystyle (A-rI)\times (A-sI)=\mu .J} , therefore ( k − r ) . ( k − s ) = μ v {\displaystyle (k-r).(k-s)=\mu v} λ − μ = r + s {\displaystyle \lambda -\mu =r+s} k − μ = − r × s {\displaystyle k-\mu =-r\times s} k ≥ r {\displaystyle k\geq r} Given an srg(v, k, λ, μ) with eigenvalues r and s, its complement srg(v, v − k − 1, v − 2 − 2k + μ, v − 2k + λ) has eigenvalues -1-s and -1-r. Alternate equations for the multiplicities are f = ( s + 1 ) k ( k − s ) μ ( s − r ) {\displaystyle f={\frac {(s+1)k(k-s)}{\mu (s-r)}}} and g = ( r + 1 ) k ( k − r ) μ ( r − s ) {\displaystyle g={\frac {(r+1)k(k-r)}{\mu (r-s)}}} The frame quotient condition: v k ( v − k − 1 ) = f g ( r − s ) 2 {\displaystyle vk(v-k-1)=fg(r-s)^{2}} . As a corollary, v = ( r − s ) 2 {\displaystyle v=(r-s)^{2}} if and only if f , g = k , v − k − 1 {\displaystyle {f,g}={k,v-k-1}} in some order. Krein conditions: ( v − k − 1 ) 2 ( k 2 + r 3 ) ≥ ( r + 1 ) 3 k 2 {\displaystyle (v-k-1)^{2}(k^{2}+r^{3})\geq (r+1)^{3}k^{2}} and ( v − k − 1 ) 2 ( k 2 + s 3 ) ≥ ( s + 1 ) 3 k 2 {\displaystyle (v-k-1)^{2}(k^{2}+s^{3})\geq (s+1)^{3}k^{2}} Absolute bound: v ≤ f ( f + 3 ) 2 {\displaystyle v\leq {\frac {f(f+3)}{2}}} and v ≤ g ( g + 3 ) 2 {\displaystyle v\leq {\frac {g(g+3)}{2}}} . Claw bound: if r + 1 > s ( s + 1 ) ( μ + 1 ) 2 {\displaystyle r+1>{\frac {s(s+1)(\mu +1)}{2}}} , then μ = s 2 {\displaystyle \mu =s^{2}} or μ = s ( s + 1 ) {\displaystyle \mu =s(s+1)} . If the above condition(s) are violated for any set of parameters, then there exists no strongly regular graph for those parameters. Brouwer has compiled such lists of existence or non-existence here with reasons for non-existence if any. === The Hoffman–Singleton theorem === As noted above, the multiplicities of the eigenvalues are given by M ± = 1 2 [ ( v − 1 ) ± 2 k + ( v − 1 ) ( λ − μ ) ( λ − μ ) 2 + 4 ( k − μ ) ] {\displaystyle M_{\pm }={\frac {1}{2}}\left[(v-1)\pm {\frac {2k+(v-1)(\lambda -\mu )}{\sqrt {(\lambda -\mu )^{2}+4(k-\mu )}}}\right]} which must be integers. In 1960, Alan Hoffman and Robert Singleton examined those expressions when applied on Moore graphs that have λ = 0 and μ = 1. Such graphs are free of triangles (otherwise λ would exceed zero) and quadrilaterals (otherwise μ would exceed 1), hence they have a girth (smallest cycle length) of 5. Substituting the values of λ and μ in the equation ( v − k − 1 ) μ = k ( k − λ − 1 ) {\displaystyle (v-k-1)\mu =k(k-\lambda -1)} , it can be seen that v = k 2 + 1 {\displaystyle v=k^{2}+1} , and the eigenvalue multiplicities reduce to M ± = 1 2 [ k 2 ± 2 k − k 2 4 k − 3 ] {\displaystyle M_{\pm }={\frac {1}{2}}\left[k^{2}\pm {\frac {2k-k^{2}}{\sqrt {4k-3}}}\right]} For the multiplicities to be integers, the quantity 2 k − k 2 4 k − 3 {\displaystyle {\frac {2k-k^{2}}{\sqrt {4k-3}}}} must be rational, therefore either the numerator 2 k − k 2 {\displaystyle 2k-k^{2}} is zero or the denominator 4 k − 3 {\displaystyle {\sqrt {4k-3}}} is an integer. If the numerator 2 k − k 2 {\displaystyle 2k-k^{2}} is zero, the possibilities are: k = 0 and v = 1 yields a trivial graph with one vertex and no edges, and k = 2 and v = 5 yields the 5-vertex cycle graph C 5 {\displaystyle C_{5}} , usually drawn as a regular pentagon. If the denominator 4 k − 3 {\displaystyle {\sqrt {4k-3}}} is an integer t, then 4 k − 3 {\displaystyle 4k-3} is a perfect square t 2 {\displaystyle t^{2}} , so k = t 2 + 3 4 {\displaystyle k={\frac {t^{2}+3}{4}}} . Substituting: M ± = 1 2 [ ( t 2 + 3 4 ) 2 ± t 2 + 3 2 − ( t 2 + 3 4 ) 2 t ] 32 M ± = ( t 2 + 3 ) 2 ± 8 ( t 2 + 3 ) − ( t 2 + 3 ) 2 t = t 4 + 6 t 2 + 9 ± − t 4 + 2 t 2 + 15 t = t 4 + 6 t 2 + 9 ± ( − t 3 + 2 t + 15 t ) {\displaystyle {\begin{aligned}M_{\pm }&={\frac {1}{2}}\left[\left({\frac {t^{2}+3}{4}}\right)^{2}\pm {\frac {{\frac {t^{2}+3}{2}}-\left({\frac {t^{2}+3}{4}}\right)^{2}}{t}}\right]\\32M_{\pm }&=(t^{2}+3)^{2}\pm {\frac {8(t^{2}+3)-(t^{2}+3)^{2}}{t}}\\&=t^{4}+6t^{2}+9\pm {\frac {-t^{4}+2t^{2}+15}{t}}\\&=t^{4}+6t^{2}+9\pm \left(-t^{3}+2t+{\frac {15}{t}}\right)\end{aligned}}} Since both sides are integers, 15 t {\displaystyle {\frac {15}{t}}} must be an integer, therefore t is a factor of 15, namely t ∈ { ± 1 , ± 3 , ± 5 , ± 15 } {\displaystyle t\in \{\pm 1,\pm 3,\pm 5,\pm 15\}} , therefore k ∈ { 1 , 3 , 7 , 57 } {\displaystyle k\in \{1,3,7,57\}} . In turn: k = 1 and v = 2 yields a trivial graph of two vertices joined by an edge, k = 3 and v = 10 yields the Petersen graph, k = 7 and v = 50 yields the Hoffman–Singleton graph, discovered by Hoffman and Singleton in the course of this analysis, and k = 57 and v = 3250 predicts a famous graph that has neither been discovered since 1960, nor has its existence been disproven. The Hoffman-Singleton theorem states that there are no strongly regular girth-5 Moore graphs except the ones listed above. == See also == Partial geometry Seidel adjacency matrix Two-graph == Notes == == References == Andries Brouwer and Hendrik van Maldeghem (2022), Strongly Regular Graphs. Cambridge: Cambridge University Press. ISBN 1316512037. ISBN 978-1316512036 A.E. Brouwer, A.M. Cohen, and A. Neumaier (1989), Distance Regular Graphs. Berlin, New York: Springer-Verlag. ISBN 3-540-50619-5, ISBN 0-387-50619-5 Chris Godsil and Gordon Royle (2004), Algebraic Graph Theory. New York: Springer-Verlag. ISBN 0-387-95241-1 == External links == Eric W. Weisstein, Mathworld article with numerous examples. Gordon Royle, List of larger graphs and families. Andries E. Brouwer, Parameters of Strongly Regular Graphs. Brendan McKay, Some collections of graphs. Ted Spence, Strongly regular graphs on at most 64 vertices.
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Wikipedia:Strongly unimodal#0
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In mathematics, unimodality means possessing a unique mode. More generally, unimodality means there is only a single highest value, somehow defined, of some mathematical object. == Unimodal probability distribution == In statistics, a unimodal probability distribution or unimodal distribution is a probability distribution which has a single peak. The term "mode" in this context refers to any peak of the distribution, not just to the strict definition of mode which is usual in statistics. If there is a single mode, the distribution function is called "unimodal". If it has more modes it is "bimodal" (2), "trimodal" (3), etc., or in general, "multimodal". Figure 1 illustrates normal distributions, which are unimodal. Other examples of unimodal distributions include Cauchy distribution, Student's t-distribution, chi-squared distribution and exponential distribution. Among discrete distributions, the binomial distribution and Poisson distribution can be seen as unimodal, though for some parameters they can have two adjacent values with the same probability. Figure 2 and Figure 3 illustrate bimodal distributions. === Other definitions === Other definitions of unimodality in distribution functions also exist. In continuous distributions, unimodality can be defined through the behavior of the cumulative distribution function (cdf). If the cdf is convex for x < m and concave for x > m, then the distribution is unimodal, m being the mode. Note that under this definition the uniform distribution is unimodal, as well as any other distribution in which the maximum distribution is achieved for a range of values, e.g. trapezoidal distribution. Usually this definition allows for a discontinuity at the mode; usually in a continuous distribution the probability of any single value is zero, while this definition allows for a non-zero probability, or an "atom of probability", at the mode. Criteria for unimodality can also be defined through the characteristic function of the distribution or through its Laplace–Stieltjes transform. Another way to define a unimodal discrete distribution is by the occurrence of sign changes in the sequence of differences of the probabilities. A discrete distribution with a probability mass function, { p n : n = … , − 1 , 0 , 1 , … } {\displaystyle \{p_{n}:n=\dots ,-1,0,1,\dots \}} , is called unimodal if the sequence … , p − 2 − p − 1 , p − 1 − p 0 , p 0 − p 1 , p 1 − p 2 , … {\displaystyle \dots ,p_{-2}-p_{-1},p_{-1}-p_{0},p_{0}-p_{1},p_{1}-p_{2},\dots } has exactly one sign change (when zeroes don't count). === Uses and results === One reason for the importance of distribution unimodality is that it allows for several important results. Several inequalities are given below which are only valid for unimodal distributions. Thus, it is important to assess whether or not a given data set comes from a unimodal distribution. Several tests for unimodality are given in the article on multimodal distribution. === Inequalities === ==== Gauss's inequality ==== A first important result is Gauss's inequality. Gauss's inequality gives an upper bound on the probability that a value lies more than any given distance from its mode. This inequality depends on unimodality. ==== Vysochanskiï–Petunin inequality ==== A second is the Vysochanskiï–Petunin inequality, a refinement of the Chebyshev inequality. The Chebyshev inequality guarantees that in any probability distribution, "nearly all" the values are "close to" the mean value. The Vysochanskiï–Petunin inequality refines this to even nearer values, provided that the distribution function is continuous and unimodal. Further results were shown by Sellke and Sellke. ==== Mode, median and mean ==== Gauss also showed in 1823 that for a unimodal distribution σ ≤ ω ≤ 2 σ {\displaystyle \sigma \leq \omega \leq 2\sigma } and | ν − μ | ≤ 3 4 ω , {\displaystyle |\nu -\mu |\leq {\sqrt {\frac {3}{4}}}\omega ,} where the median is ν, the mean is μ and ω is the root mean square deviation from the mode. It can be shown for a unimodal distribution that the median ν and the mean μ lie within (3/5)1/2 ≈ 0.7746 standard deviations of each other. In symbols, | ν − μ | σ ≤ 3 5 {\displaystyle {\frac {|\nu -\mu |}{\sigma }}\leq {\sqrt {\frac {3}{5}}}} where | . | is the absolute value. In 2020, Bernard, Kazzi, and Vanduffel generalized the previous inequality by deriving the maximum distance between the symmetric quantile average q α + q ( 1 − α ) 2 {\displaystyle {\frac {q_{\alpha }+q_{(1-\alpha )}}{2}}} and the mean, | q α + q ( 1 − α ) 2 − μ | σ ≤ { 4 9 ( 1 − α ) − 1 + 1 − α 1 / 3 + α 2 for α ∈ [ 5 6 , 1 ) , 3 α 4 − 3 α + 1 − α 1 / 3 + α 2 for α ∈ ( 1 6 , 5 6 ) , 3 α 4 − 3 α + 4 9 α − 1 2 for α ∈ ( 0 , 1 6 ] . {\displaystyle {\frac {\left|{\frac {q_{\alpha }+q_{(1-\alpha )}}{2}}-\mu \right|}{\sigma }}\leq \left\{{\begin{array}{cl}{\frac {{\sqrt[{}]{{\frac {4}{9(1-\alpha )}}-1}}{\text{ }}+{\text{ }}{\sqrt[{}]{\frac {1-\alpha }{1/3+\alpha }}}}{2}}&{\text{for }}\alpha \in \left[{\frac {5}{6}},1\right)\!,\\{\frac {{\sqrt[{}]{\frac {3\alpha }{4-3\alpha }}}{\text{ }}+{\text{ }}{\sqrt[{}]{\frac {1-\alpha }{1/3+\alpha }}}}{2}}&{\text{for }}\alpha \in \left({\frac {1}{6}},{\frac {5}{6}}\right)\!,\\{\frac {{\sqrt[{}]{\frac {3\alpha }{4-3\alpha }}}{\text{ }}+{\text{ }}{\sqrt[{}]{{\frac {4}{9\alpha }}-1}}}{2}}&{\text{for }}\alpha \in \left(0,{\frac {1}{6}}\right]\!.\end{array}}\right.} The maximum distance is minimized at α = 0.5 {\displaystyle \alpha =0.5} (i.e., when the symmetric quantile average is equal to q 0.5 = ν {\displaystyle q_{0.5}=\nu } ), which indeed motivates the common choice of the median as a robust estimator for the mean. Moreover, when α = 0.5 {\displaystyle \alpha =0.5} , the bound is equal to 3 / 5 {\displaystyle {\sqrt {3/5}}} , which is the maximum distance between the median and the mean of a unimodal distribution. A similar relation holds between the median and the mode θ: they lie within 31/2 ≈ 1.732 standard deviations of each other: | ν − θ | σ ≤ 3 . {\displaystyle {\frac {|\nu -\theta |}{\sigma }}\leq {\sqrt {3}}.} It can also be shown that the mean and the mode lie within 31/2 of each other: | μ − θ | σ ≤ 3 . {\displaystyle {\frac {|\mu -\theta |}{\sigma }}\leq {\sqrt {3}}.} ==== Skewness and kurtosis ==== Rohatgi and Szekely claimed that the skewness and kurtosis of a unimodal distribution are related by the inequality: γ 2 − κ ≤ 6 5 = 1.2 {\displaystyle \gamma ^{2}-\kappa \leq {\frac {6}{5}}=1.2} where κ is the kurtosis and γ is the skewness. Klaassen, Mokveld, and van Es showed that this only applies in certain settings, such as the set of unimodal distributions where the mode and mean coincide. They derived a weaker inequality which applies to all unimodal distributions: γ 2 − κ ≤ 186 125 = 1.488 {\displaystyle \gamma ^{2}-\kappa \leq {\frac {186}{125}}=1.488} This bound is sharp, as it is reached by the equal-weights mixture of the uniform distribution on [0,1] and the discrete distribution at {0}. == Unimodal function == As the term "modal" applies to data sets and probability distribution, and not in general to functions, the definitions above do not apply. The definition of "unimodal" was extended to functions of real numbers as well. A common definition is as follows: a function f(x) is a unimodal function if for some value m, it is monotonically increasing for x ≤ m and monotonically decreasing for x ≥ m. In that case, the maximum value of f(x) is f(m) and there are no other local maxima. Proving unimodality is often hard. One way consists in using the definition of that property, but it turns out to be suitable for simple functions only. A general method based on derivatives exists, but it does not succeed for every function despite its simplicity. Examples of unimodal functions include quadratic polynomial functions with a negative quadratic coefficient, tent map functions, and more. The above is sometimes related to as strong unimodality, from the fact that the monotonicity implied is strong monotonicity. A function f(x) is a weakly unimodal function if there exists a value m for which it is weakly monotonically increasing for x ≤ m and weakly monotonically decreasing for x ≥ m. In that case, the maximum value f(m) can be reached for a continuous range of values of x. An example of a weakly unimodal function which is not strongly unimodal is every other row in Pascal's triangle. Depending on context, unimodal function may also refer to a function that has only one local minimum, rather than maximum. For example, local unimodal sampling, a method for doing numerical optimization, is often demonstrated with such a function. It can be said that a unimodal function under this extension is a function with a single local extremum. One important property of unimodal functions is that the extremum can be found using search algorithms such as golden section search, ternary search or successive parabolic interpolation. == Other extensions == A function f(x) is "S-unimodal" (often referred to as "S-unimodal map") if its Schwarzian derivative is negative for all x ≠ c {\displaystyle x\neq c} , where c {\displaystyle c} is the critical point. In computational geometry if a function is unimodal it permits the design of efficient algorithms for finding the extrema of the function. A more general definition, applicable to a function f(X) of a vector variable X is that f is unimodal if there is a one-to-one differentiable mapping X = G(Z) such that f(G(Z)) is convex. Usually one would want G(Z) to be continuously differentiable with nonsingular Jacobian matrix. Quasiconvex functions and quasiconcave functions extend the concept of unimodality to functions whose arguments belong to higher-dimensional Euclidean spaces. == See also == Bimodal distribution Read's conjecture == References ==
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Wikipedia:Studia Mathematica#0
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Studia Mathematica is a triannual peer-reviewed scientific journal of mathematics published by the Polish Academy of Sciences. Papers are written in English, French, German, or Russian, primarily covering functional analysis, abstract methods of mathematical analysis, and probability theory. The editor-in-chief is Adam Skalski. == History == The journal was established in 1929 by Stefan Banach and Hugo Steinhaus and its first editors were Banach, Steinhaus and Herman Auerbach. Due to the Second World War publication stopped after volume 9 (1940) and was not resumed until volume 10 in 1948. == Abstracting and indexing == The journal is abstracted and indexed in: Current Contents/Physical, Chemical & Earth Sciences MathSciNet Science Citation Index Scopus Zentralblatt MATH According to the Journal Citation Reports, the journal has a 2018 impact factor of 0.617. == References == == External links == Official website
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Wikipedia:Sturm separation theorem#0
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In mathematics, in the field of ordinary differential equations, Sturm separation theorem, named after Jacques Charles François Sturm, describes the location of roots of solutions of homogeneous second order linear differential equations. Basically the theorem states that given two linear independent solutions of such an equation the zeros of the two solutions are alternating. == Sturm separation theorem == If u(x) and v(x) are two non-trivial continuous linearly independent solutions to a homogeneous second order linear differential equation with x0 and x1 being successive roots of u(x), then v(x) has exactly one root in the open interval (x0, x1). It is a special case of the Sturm-Picone comparison theorem. == Proof == Since u {\displaystyle \displaystyle u} and v {\displaystyle \displaystyle v} are linearly independent it follows that the Wronskian W [ u , v ] {\displaystyle \displaystyle W[u,v]} must satisfy W [ u , v ] ( x ) ≡ W ( x ) ≠ 0 {\displaystyle W[u,v](x)\equiv W(x)\neq 0} for all x {\displaystyle \displaystyle x} where the differential equation is defined, say I {\displaystyle \displaystyle I} . Without loss of generality, suppose that W ( x ) < 0 ∀ x ∈ I {\displaystyle W(x)<0{\mbox{ }}\forall {\mbox{ }}x\in I} . Then u ( x ) v ′ ( x ) − u ′ ( x ) v ( x ) ≠ 0. {\displaystyle u(x)v'(x)-u'(x)v(x)\neq 0.} So at x = x 0 {\displaystyle \displaystyle x=x_{0}} W ( x 0 ) = − u ′ ( x 0 ) v ( x 0 ) {\displaystyle W(x_{0})=-u'\left(x_{0}\right)v\left(x_{0}\right)} and either u ′ ( x 0 ) {\displaystyle u'\left(x_{0}\right)} and v ( x 0 ) {\displaystyle v\left(x_{0}\right)} are both positive or both negative. Without loss of generality, suppose that they are both positive. Now, at x = x 1 {\displaystyle \displaystyle x=x_{1}} W ( x 1 ) = − u ′ ( x 1 ) v ( x 1 ) {\displaystyle W(x_{1})=-u'\left(x_{1}\right)v\left(x_{1}\right)} and since x = x 0 {\displaystyle \displaystyle x=x_{0}} and x = x 1 {\displaystyle \displaystyle x=x_{1}} are successive zeros of u ( x ) {\displaystyle \displaystyle u(x)} it causes u ′ ( x 1 ) < 0 {\displaystyle u'\left(x_{1}\right)<0} . Thus, to keep W ( x ) < 0 {\displaystyle \displaystyle W(x)<0} we must have v ( x 1 ) < 0 {\displaystyle v\left(x_{1}\right)<0} . We see this by observing that if u ′ ( x ) > 0 ∀ x ∈ ( x 0 , x 1 ] {\displaystyle \displaystyle u'(x)>0{\mbox{ }}\forall {\mbox{ }}x\in \left(x_{0},x_{1}\right]} then u ( x ) {\displaystyle \displaystyle u(x)} would be increasing (away from the x {\displaystyle \displaystyle x} -axis), which would never lead to a zero at x = x 1 {\displaystyle \displaystyle x=x_{1}} . So for a zero to occur at x = x 1 {\displaystyle \displaystyle x=x_{1}} at most u ′ ( x 1 ) = 0 {\displaystyle u'\left(x_{1}\right)=0} (i.e., u ′ ( x 1 ) ≤ 0 {\displaystyle u'\left(x_{1}\right)\leq 0} and it turns out, by our result from the Wronskian that u ′ ( x 1 ) ≤ 0 {\displaystyle u'\left(x_{1}\right)\leq 0} ). So somewhere in the interval ( x 0 , x 1 ) {\displaystyle \left(x_{0},x_{1}\right)} the sign of v ( x ) {\displaystyle \displaystyle v(x)} changed. By the Intermediate Value Theorem there exists x ∗ ∈ ( x 0 , x 1 ) {\displaystyle x^{*}\in \left(x_{0},x_{1}\right)} such that v ( x ∗ ) = 0 {\displaystyle v\left(x^{*}\right)=0} . On the other hand, there can be only one zero in ( x 0 , x 1 ) {\displaystyle \left(x_{0},x_{1}\right)} , because otherwise v {\displaystyle v} would have two zeros and there would be no zeros of u {\displaystyle u} in between, and it was just proved that this is impossible. == References == Teschl, G. (2012). Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society. ISBN 978-0-8218-8328-0.
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Wikipedia:Sturm's theorem#0
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In mathematics, the Sturm sequence of a univariate polynomial p is a sequence of polynomials associated with p and its derivative by a variant of Euclid's algorithm for polynomials. Sturm's theorem expresses the number of distinct real roots of p located in an interval in terms of the number of changes of signs of the values of the Sturm sequence at the bounds of the interval. Applied to the interval of all the real numbers, it gives the total number of real roots of p. Whereas the fundamental theorem of algebra readily yields the overall number of complex roots, counted with multiplicity, it does not provide a procedure for calculating them. Sturm's theorem counts the number of distinct real roots and locates them in intervals. By subdividing the intervals containing some roots, it can isolate the roots into arbitrarily small intervals, each containing exactly one root. This yields the oldest real-root isolation algorithm, and arbitrary-precision root-finding algorithm for univariate polynomials. For computing over the reals, Sturm's theorem is less efficient than other methods based on Descartes' rule of signs. However, it works on every real closed field, and, therefore, remains fundamental for the theoretical study of the computational complexity of decidability and quantifier elimination in the first order theory of real numbers. The Sturm sequence and Sturm's theorem are named after Jacques Charles François Sturm, who discovered the theorem in 1829. == The theorem == The Sturm chain or Sturm sequence of a univariate polynomial P(x) with real coefficients is the sequence of polynomials P 0 , P 1 , … , {\displaystyle P_{0},P_{1},\ldots ,} such that P 0 = P , P 1 = P ′ , P i + 1 = − rem ( P i − 1 , P i ) , {\displaystyle {\begin{aligned}P_{0}&=P,\\P_{1}&=P',\\P_{i+1}&=-\operatorname {rem} (P_{i-1},P_{i}),\end{aligned}}} for i ≥ 1, where P' is the derivative of P, and rem ( P i − 1 , P i ) {\displaystyle \operatorname {rem} (P_{i-1},P_{i})} is the remainder of the Euclidean division of P i − 1 {\displaystyle P_{i-1}} by P i . {\displaystyle P_{i}.} The length of the Sturm sequence is at most the degree of P. The number of sign variations at ξ of the Sturm sequence of P is the number of sign changes (ignoring zeros) in the sequence of real numbers P 0 ( ξ ) , P 1 ( ξ ) , P 2 ( ξ ) , … . {\displaystyle P_{0}(\xi ),P_{1}(\xi ),P_{2}(\xi ),\ldots .} This number of sign variations is denoted here V(ξ). Sturm's theorem states that, if P is a square-free polynomial, the number of distinct real roots of P in the half-open interval (a, b] is V(a) − V(b) (here, a and b are real numbers such that a < b). The theorem extends to unbounded intervals by defining the sign at +∞ of a polynomial as the sign of its leading coefficient (that is, the coefficient of the term of highest degree). At –∞ the sign of a polynomial is the sign of its leading coefficient for a polynomial of even degree, and the opposite sign for a polynomial of odd degree. In the case of a non-square-free polynomial, if neither a nor b is a multiple root of p, then V(a) − V(b) is the number of distinct real roots of P. The proof of the theorem is as follows: when the value of x increases from a to b, it may pass through a zero of some P i {\displaystyle P_{i}} (i > 0); when this occurs, the number of sign variations of ( P i − 1 , P i , P i + 1 ) {\displaystyle (P_{i-1},P_{i},P_{i+1})} does not change. When x passes through a root of P 0 = P , {\displaystyle P_{0}=P,} the number of sign variations of ( P 0 , P 1 ) {\displaystyle (P_{0},P_{1})} decreases from 1 to 0. These are the only values of x where some sign may change. == Example == Suppose we wish to find the number of roots in some range for the polynomial p ( x ) = x 4 + x 3 − x − 1 {\displaystyle p(x)=x^{4}+x^{3}-x-1} . So p 0 ( x ) = p ( x ) = x 4 + x 3 − x − 1 p 1 ( x ) = p ′ ( x ) = 4 x 3 + 3 x 2 − 1 {\displaystyle {\begin{aligned}p_{0}(x)&=p(x)=x^{4}+x^{3}-x-1\\p_{1}(x)&=p'(x)=4x^{3}+3x^{2}-1\end{aligned}}} The remainder of the Euclidean division of p0 by p1 is − 3 16 x 2 − 3 4 x − 15 16 ; {\displaystyle -{\tfrac {3}{16}}x^{2}-{\tfrac {3}{4}}x-{\tfrac {15}{16}};} multiplying it by −1 we obtain p 2 ( x ) = 3 16 x 2 + 3 4 x + 15 16 {\displaystyle p_{2}(x)={\tfrac {3}{16}}x^{2}+{\tfrac {3}{4}}x+{\tfrac {15}{16}}} . Next dividing p1 by p2 and multiplying the remainder by −1, we obtain p 3 ( x ) = − 32 x − 64 {\displaystyle p_{3}(x)=-32x-64} . Now dividing p2 by p3 and multiplying the remainder by −1, we obtain p 4 ( x ) = − 3 16 {\displaystyle p_{4}(x)=-{\tfrac {3}{16}}} . As this is a constant, this finishes the computation of the Sturm sequence. To find the number of real roots of p 0 {\displaystyle p_{0}} one has to evaluate the sequences of the signs of these polynomials at −∞ and ∞, which are respectively (+, −, +, +, −) and (+, +, +, −, −). Thus V ( − ∞ ) − V ( + ∞ ) = 3 − 1 = 2 , {\displaystyle V(-\infty )-V(+\infty )=3-1=2,} where V denotes the number of sign changes in the sequence, which shows that p has two real roots. This can be verified by noting that p(x) can be factored as (x2 − 1)(x2 + x + 1), where the first factor has the roots −1 and 1, and second factor has no real roots. This last assertion results from the quadratic formula, and also from Sturm's theorem, which gives the sign sequences (+, –, –) at −∞ and (+, +, –) at +∞. == Generalization == Sturm sequences have been generalized in two directions. To define each polynomial in the sequence, Sturm used the negative of the remainder of the Euclidean division of the two preceding ones. The theorem remains true if one replaces the negative of the remainder by its product or quotient by a positive constant or the square of a polynomial. It is also useful (see below) to consider sequences where the second polynomial is not the derivative of the first one. A generalized Sturm sequence is a finite sequence of polynomials with real coefficients P 0 , P 1 , … , P m {\displaystyle P_{0},P_{1},\dots ,P_{m}} such that the degrees are decreasing after the first one: deg P i < deg P i − 1 {\displaystyle \deg P_{i}<\deg P_{i-1}} for i = 2, ..., m; P m {\displaystyle P_{m}} does not have any real root or has no sign changes near its real roots. if Pi(ξ) = 0 for 0 < i < m and ξ a real number, then Pi −1 (ξ) Pi + 1(ξ) < 0. The last condition implies that two consecutive polynomials do not have any common real root. In particular the original Sturm sequence is a generalized Sturm sequence, if (and only if) the polynomial has no multiple real root (otherwise the first two polynomials of its Sturm sequence have a common root). When computing the original Sturm sequence by Euclidean division, it may happen that one encounters a polynomial that has a factor that is never negative, such a x 2 {\displaystyle x^{2}} or x 2 + 1 {\displaystyle x^{2}+1} . In this case, if one continues the computation with the polynomial replaced by its quotient by the nonnegative factor, one gets a generalized Sturm sequence, which may also be used for computing the number of real roots, since the proof of Sturm's theorem still applies (because of the third condition). This may sometimes simplify the computation, although it is generally difficult to find such nonnegative factors, except for even powers of x. == Use of pseudo-remainder sequences == In computer algebra, the polynomials that are considered have integer coefficients or may be transformed to have integer coefficients. The Sturm sequence of a polynomial with integer coefficients generally contains polynomials whose coefficients are not integers (see above example). To avoid computation with rational numbers, a common method is to replace Euclidean division by pseudo-division for computing polynomial greatest common divisors. This amounts to replacing the remainder sequence of the Euclidean algorithm by a pseudo-remainder sequence, a pseudo remainder sequence being a sequence p 0 , … , p k {\displaystyle p_{0},\ldots ,p_{k}} of polynomials such that there are constants a i {\displaystyle a_{i}} and b i {\displaystyle b_{i}} such that b i p i + 1 {\displaystyle b_{i}p_{i+1}} is the remainder of the Euclidean division of a i p i − 1 {\displaystyle a_{i}p_{i-1}} by p i . {\displaystyle p_{i}.} (The different kinds of pseudo-remainder sequences are defined by the choice of a i {\displaystyle a_{i}} and b i ; {\displaystyle b_{i};} typically, a i {\displaystyle a_{i}} is chosen for not introducing denominators during Euclidean division, and b i {\displaystyle b_{i}} is a common divisor of the coefficients of the resulting remainder; see Pseudo-remainder sequence for details.) For example, the remainder sequence of the Euclidean algorithm is a pseudo-remainder sequence with a i = b i = 1 {\displaystyle a_{i}=b_{i}=1} for every i, and the Sturm sequence of a polynomial is a pseudo-remainder sequence with a i = 1 {\displaystyle a_{i}=1} and b i = − 1 {\displaystyle b_{i}=-1} for every i. Various pseudo-remainder sequences have been designed for computing greatest common divisors of polynomials with integer coefficients without introducing denominators (see Pseudo-remainder sequence). They can all be made generalized Sturm sequences by choosing the sign of the b i {\displaystyle b_{i}} to be the opposite of the sign of the a i . {\displaystyle a_{i}.} This allows the use of Sturm's theorem with pseudo-remainder sequences. == Root isolation == For a polynomial with real coefficients, root isolation consists of finding, for each real root, an interval that contains this root, and no other roots. This is useful for root finding, allowing the selection of the root to be found and providing a good starting point for fast numerical algorithms such as Newton's method; it is also useful for certifying the result, as if Newton's method converge outside the interval one may immediately deduce that it converges to the wrong root. Root isolation is also useful for computing with algebraic numbers. For computing with algebraic numbers, a common method is to represent them as a pair of a polynomial to which the algebraic number is a root, and an isolation interval. For example 2 {\displaystyle {\sqrt {2}}} may be unambiguously represented by ( x 2 − 2 , [ 0 , 2 ] ) . {\displaystyle (x^{2}-2,[0,2]).} Sturm's theorem provides a way for isolating real roots that is less efficient (for polynomials with integer coefficients) than other methods involving Descartes' rule of signs. However, it remains useful in some circumstances, mainly for theoretical purposes, for example for algorithms of real algebraic geometry that involve infinitesimals. For isolating the real roots, one starts from an interval ( a , b ] {\displaystyle (a,b]} containing all the real roots, or the roots of interest (often, typically in physical problems, only positive roots are interesting), and one computes V ( a ) {\displaystyle V(a)} and V ( b ) . {\displaystyle V(b).} For defining this starting interval, one may use bounds on the size of the roots (see Properties of polynomial roots § Bounds on (complex) polynomial roots). Then, one divides this interval in two, by choosing c in the middle of ( a , b ] . {\displaystyle (a,b].} The computation of V ( c ) {\displaystyle V(c)} provides the number of real roots in ( a , c ] {\displaystyle (a,c]} and ( c , b ] , {\displaystyle (c,b],} and one may repeat the same operation on each subinterval. When one encounters, during this process an interval that does not contain any root, it may be suppressed from the list of intervals to consider. When one encounters an interval containing exactly one root, one may stop dividing it, as it is an isolation interval. The process stops eventually, when only isolating intervals remain. This isolating process may be used with any method for computing the number of real roots in an interval. Theoretical complexity analysis and practical experiences show that methods based on Descartes' rule of signs are more efficient. It follows that, nowadays, Sturm sequences are rarely used for root isolation. == Application == Generalized Sturm sequences allow counting the roots of a polynomial where another polynomial is positive (or negative), without computing these root explicitly. If one knows an isolating interval for a root of the first polynomial, this allows also finding the sign of the second polynomial at this particular root of the first polynomial, without computing a better approximation of the root. Let P(x) and Q(x) be two polynomials with real coefficients such that P and Q have no common root and P has no multiple roots. In other words, P and P' Q are coprime polynomials. This restriction does not really affect the generality of what follows as GCD computations allows reducing the general case to this case, and the cost of the computation of a Sturm sequence is the same as that of a GCD. Let W(a) denote the number of sign variations at a of a generalized Sturm sequence starting from P and P' Q. If a < b are two real numbers, then W(a) – W(b) is the number of roots of P in the interval ( a , b ] {\displaystyle (a,b]} such that Q(a) > 0 minus the number of roots in the same interval such that Q(a) < 0. Combined with the total number of roots of P in the same interval given by Sturm's theorem, this gives the number of roots of P such that Q(a) > 0 and the number of roots of P such that Q(a) < 0. == See also == == References == Basu, Saugata; Pollack, Richard; Roy, Marie-Françoise (2006). "Section 2.2.2". Algorithms in real algebraic geometry (2nd ed.). Springer. pp. 52–57. ISBN 978-3-540-33098-1. Sturm, Jacques Charles François (1829). "Mémoire sur la résolution des équations numériques". Bulletin des Sciences de Férussac. 11: 419–425. Sylvester, J. J. (1853). "On a theory of the syzygetic relations of two rational integral functions, comprising an application to the theory of Sturm's functions, and that of the greatest algebraical common measure". Phil. Trans. R. Soc. Lond. 143: 407–548. doi:10.1098/rstl.1853.0018. JSTOR 108572. Thomas, Joseph Miller (1941). "Sturm's theorem for multiple roots". National Mathematics Magazine. 15 (8): 391–394. doi:10.2307/3028551. JSTOR 3028551. MR 0005945. Heindel, Lee E. (1971). Integer arithmetic algorithms for polynomial real zero determination. Proc. SYMSAC '71. p. 415. doi:10.1145/800204.806312. MR 0300434. S2CID 9971778. de Moura, Leonardo; Passmore, Grant Olney (2013). "Computation in Real Closed Infinitesimal and Transcendental Extensions of the Rationals". Automated Deduction – CADE-24. Lecture Notes in Computer Science. Vol. 7898. p. 178-192. doi:10.1007/978-3-642-38574-2_12. ISBN 978-3-642-38573-5. S2CID 9308312. Panton, Don B.; Verdini, William A. (1981). "A fortran program for applying Sturm's theorem in counting internal rates of return". J. Financ. Quant. Anal. 16 (3): 381–388. doi:10.2307/2330245. JSTOR 2330245. S2CID 154334522. Akritas, Alkiviadis G. (1982). "Reflections on a pair of theorems by Budan and Fourier". Math. Mag. 55 (5): 292–298. doi:10.2307/2690097. JSTOR 2690097. MR 0678195. Pedersen, Paul (1991). "Multivariate Sturm theory". In Mattson, Harold F.; Mora, Teo; Rao, T. R. N. (eds.). Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, 9th International Symposium, AAECC-9, New Orleans, LA, USA, October 7–11, 1991, Proceedings. Lecture Notes in Computer Science. Vol. 539. Berlin: Springer. pp. 318–332. doi:10.1007/3-540-54522-0_120. ISBN 978-3-540-54522-4. MR 1229329. Yap, Chee (2000). Fundamental Problems in Algorithmic Algebra. Oxford University Press. ISBN 0-19-512516-9. Rahman, Q. I.; Schmeisser, G. (2002). Analytic theory of polynomials. London Mathematical Society Monographs. New Series. Vol. 26. Oxford: Oxford University Press. ISBN 0-19-853493-0. Zbl 1072.30006. Baumol, William. Economic Dynamics, chapter 12, Section 3, "Qualitative information on real roots" D.G. Hook and P. R. McAree, "Using Sturm Sequences To Bracket Real Roots of Polynomial Equations" in Graphic Gems I (A. Glassner ed.), Academic Press, pp. 416–422, 1990.
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Wikipedia:Sturm–Picone comparison theorem#0
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In mathematics, in the field of ordinary differential equations, the Sturm–Picone comparison theorem, named after Jacques Charles François Sturm and Mauro Picone, is a classical theorem which provides criteria for the oscillation and non-oscillation of solutions of certain linear differential equations in the real domain. Let pi, qi for i = 1, 2 be real-valued continuous functions on the interval [a, b] and let ( p 1 ( x ) y ′ ) ′ + q 1 ( x ) y = 0 {\displaystyle (p_{1}(x)y^{\prime })^{\prime }+q_{1}(x)y=0} ( p 2 ( x ) y ′ ) ′ + q 2 ( x ) y = 0 {\displaystyle (p_{2}(x)y^{\prime })^{\prime }+q_{2}(x)y=0} be two homogeneous linear second order differential equations in self-adjoint form with 0 < p 2 ( x ) ≤ p 1 ( x ) {\displaystyle 0<p_{2}(x)\leq p_{1}(x)} and q 1 ( x ) ≤ q 2 ( x ) . {\displaystyle q_{1}(x)\leq q_{2}(x).} Let u be a non-trivial solution of (1) with successive roots at z1 and z2 and let v be a non-trivial solution of (2). Then one of the following properties holds. There exists an x in (z1, z2) such that v(x) = 0; or there exists a λ in R such that v(x) = λ u(x). The first part of the conclusion is due to Sturm (1836), while the second (alternative) part of the theorem is due to Picone (1910) whose simple proof was given using his now famous Picone identity. In the special case where both equations are identical one obtains the Sturm separation theorem. == Notes == == References == Diaz, J. B.; McLaughlin, Joyce R. Sturm comparison theorems for ordinary and partial differential equations. Bull. Amer. Math. Soc. 75 1969 335–339 [1] Heinrich Guggenheimer (1977) Applicable Geometry, page 79, Krieger, Huntington ISBN 0-88275-368-1 . Teschl, G. (2012). Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society. ISBN 978-0-8218-8328-0.
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Wikipedia:Stylianos Pichorides#0
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Stylianos Konstantinos Pichorides (Στυλιανός Κωνσταντίνος Πιχωρίδης, 18 October 1940, Athens – 18 June 1992, Madrid) was a Greek mathematician, specializing in harmonic analysis. After graduating from secondary school in Athens, Pichorides matriculated at the National Technical University of Athens, where he graduated in 1963 with a degree in electrical engineering. He then worked as an electrical engineer in Athens, but also studied mathematics and received in 1968 a scholarship to study at the University of Chicago. There in 1971 he received his Ph.D. with thesis On the best values of the constants in the theorems of M. Riesz, Zygmund and Kolmogorov written under the supervision of Antoni Zygmund. In 1972 Pichorides returned to Athens and worked at the National Centre of Scientific Research "Demokritos", where he was employed until 1983, with interruptions by leave of absence. He was from 1974 to 1979 an attaché de recherché of the CNRS in Orsay, a visiting professor from 1979 to 1980 at the Paris-Sud University in Orsay, and from 1980 to 1981 a visiting professor at the University of California, Los Angeles. He organized, with Nicholas Petridis and Nicholas Varopoulos, a successful conference on harmonic analysis in Iraklion in 1978. From 1983 until his death in 1992, Pichorides was a professor at the University of Crete's mathematics department, which he co-founded. He held visiting professorships at Paris-Sud University in Orsay, Caltech, and the University of Chicago. For the academic year 1991–1992 he was a visiting professor at the University of Cyprus. He had short stays at the Mittag-Leffler Institute, the University of Cambridge, Brown University, and the University of Chicago. He died unexpectedly while attending a conference in Spain in 1992. Pichorides is known for results on inequalities in the theory of Fourier series. In 1980 he received the Salem Prize for his research on Littlewood's conjecture on a lower bound for averaged exponential sums. Research by Pichorides and others provided the basis for the 1981 proof by Sergei Vladimirovich Konyagin of Littlewood's conjecture on the lower bound. The Foundation for Research & Technology – Hellas (FORTH) has funded the Pichorides Postgraduate Scholarship and the Pichorides Distinguished Lectureship. == Selected publications == Pichorides, S. K. (1974). "A lower bound for the L1 norm of exponential sums". Mathematika. 21 (2): 155–159. doi:10.1112/S0025579300008536. Pichorides, S. K. (March 1977). "A remark on exponential sums" (PDF). Bulletin of the American Mathematical Society. 83 (2): 283–285. doi:10.1090/S0002-9904-1977-14308-5. Pichorides, S. K. (1977). Norms of exponential sums. Publications mathématiques d'Orsay 73. Paris: Université de Paris-Sud. pp. iv+65 pages. catalog entry, Universiteits bibliotheek, Ghent, Belgium Pichorides, S.K. (1977). "On a Conjecture of Littlewood Concerning Exponentials Sums, I.". Δελτίο της Ελληνικής Μαθηματικής Εταιρίας. 18: 8–16. (Bulletin of the Hellenic Mathematical Society) Pichorides, S.K. (1978). "On a Conjecture of Littlewood Concerning Exponentials Sums, II". Δελτίο της Ελληνικής Μαθηματικής Εταιρίας. 19: 274–277. Pichorides, Stylianos K. (1980). "On the L1 norm of exponential sums" (PDF). Annales de l'Institut Fourier. 30 (2): 79–89. doi:10.5802/aif.785. Pichorides, Stylianos K. (1992). "A remark on the constants of the Littlewood-Paley inequality" (PDF). Proceedings of the American Mathematical Society. 114 (3): 787–789. doi:10.1090/S0002-9939-1992-1088445-6. == References ==
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Wikipedia:Stål Aanderaa#0
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Stål Aanderaa (born 1 February 1931) is a Norwegian mathematician. == Biography == Aanderaa was born in Beitstad. He completed the mag.scient. degree in 1959 and his doctorate at Harvard University in 1966. He was a professor at the University of Oslo from 1978 to his retirement in 2001. Aanderaa is a member of the Norwegian Academy of Science and Letters. == Work == Aanderaa is one of the namesakes of the Aanderaa–Karp–Rosenberg conjecture. == References ==
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Wikipedia:Suanfa tongzong#0
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Suanfa tongzong (Chinese: 算法統宗) is a mathematical text written by sixteenth century Chinese mathematician Cheng Dawei (1533–1606) and published in the year 1592. The book contains 595 problems divided into 17 chapters. The book is essentially general arithmetic for the abacus. The book was the main source available to scholars concerning mathematics as it developed in China's tradition. Six years after the publication of Suanfa Tongzong, Cheng Dawei published another book titled Suanfa Zuanyao (A Compendium of calculating Methods). About 90% of the content of the new book came from the contents of four chapters of the first book with some rearrangement. It is said that when Suanfa Tongzong was first published, it sold so many copies that the cost of paper went up and the lucrative sales resulted in unscrupulous people beginning to print pirated copies of the book with many errors. It was this that forced the author to print an abridged version. == Some features == Suanfa Tongzong has some noteworthy features. As Jean-Claude Martzloff, a historian of Chinese mathematics has observed, it is an encyclopedic hotch-potch of ideas which contains everything from A to Z relating to the Chinese mystique of numbers. There are sections in the book which explains how computation should be taught and studied. The book is considered as an authoritative text on Chinese Zhusuan which is the knowledge and practices of arithmetic calculation through the abacus. There are descriptions of topics generally thought of as mathematical recreations and mathematical curiosities of different types. In particular, the book contains descriptions of several different types of magic circles. There is a collection of problems without solutions given as challenges to the readers. Also, some of the formulas and some of the problems are presented in verse for easy remembrance. == Some sample problems == The following is a list of sample problems appearing in the book: "Boy shepherd B with his one sheep behind him asked shepherd A "Are there 100 sheep in your flock?". Shepherd A replies "Yet add the same flock, the same flock again, half, one quarter flock and your sheep. There are then 100 sheep altogether." "Now a pile of rice is against the wall with a base circumference 60 chi and an altitude of 12 chi. What is the volume? Another pile is at an inner corner, with a base circumference of 30 chi and an altitude of 12 chi. What is the volume? Another pile is at an outer corner, with base circumference of 90 chi and an altitude of 12 chi. What is the volume?" "A small river cuts right across a circular field whose area is unknown. Given the diameter of the field and the breadth of the river find the area of the non-flooded part of the field." "In the right-angled triangle with sides of length a, b, and c with a > b > c, we know that a + b = 81 and a + c = 72. Find a, b, and c." == Popularity of the book == After its first publication in 1592, it was republished several times later and it became widely popular. Practically everybody who is involved in mathematics had a copy of the book. It was popular even beyond the limited circle of people interested in mathematics. Even in the mid-20th century (1964), the well-known historians of Chinese mathematics Li Yan and Du Shiran remarked that: "Nowadays, various editions of the Suanfa Tongzong can still be found throughout China and some old people still recite the versified formulae and talk to each other about its difficult problems." == References ==
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Wikipedia:Subadditivity#0
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In mathematics, subadditivity is a property of a function that states, roughly, that evaluating the function for the sum of two elements of the domain always returns something less than or equal to the sum of the function's values at each element. There are numerous examples of subadditive functions in various areas of mathematics, particularly norms and square roots. Additive maps are special cases of subadditive functions. == Definitions == A subadditive function is a function f : A → B {\displaystyle f\colon A\to B} , having a domain A and an ordered codomain B that are both closed under addition, with the following property: ∀ x , y ∈ A , f ( x + y ) ≤ f ( x ) + f ( y ) . {\displaystyle \forall x,y\in A,f(x+y)\leq f(x)+f(y).} An example is the square root function, having the non-negative real numbers as domain and codomain: since ∀ x , y ≥ 0 {\displaystyle \forall x,y\geq 0} we have: x + y ≤ x + y . {\displaystyle {\sqrt {x+y}}\leq {\sqrt {x}}+{\sqrt {y}}.} A sequence { a n } n ≥ 1 {\displaystyle \left\{a_{n}\right\}_{n\geq 1}} is called subadditive if it satisfies the inequality a n + m ≤ a n + a m {\displaystyle a_{n+m}\leq a_{n}+a_{m}} for all m and n. This is a special case of subadditive function, if a sequence is interpreted as a function on the set of natural numbers. Note that while a concave sequence is subadditive, the converse is false. For example, arbitrarily assign a 1 , a 2 , . . . {\displaystyle a_{1},a_{2},...} with values in [ 0.5 , 1 ] {\displaystyle [0.5,1]} ; then the sequence is subadditive but not concave. == Properties == === Sequences === A useful result pertaining to subadditive sequences is the following lemma due to Michael Fekete. The analogue of Fekete's lemma holds for superadditive sequences as well, that is: a n + m ≥ a n + a m . {\displaystyle a_{n+m}\geq a_{n}+a_{m}.} (The limit then may be positive infinity: consider the sequence a n = log n ! {\displaystyle a_{n}=\log n!} .) There are extensions of Fekete's lemma that do not require the inequality a n + m ≤ a n + a m {\displaystyle a_{n+m}\leq a_{n}+a_{m}} to hold for all m and n, but only for m and n such that 1 2 ≤ m n ≤ 2. {\textstyle {\frac {1}{2}}\leq {\frac {m}{n}}\leq 2.} Moreover, the condition a n + m ≤ a n + a m {\displaystyle a_{n+m}\leq a_{n}+a_{m}} may be weakened as follows: a n + m ≤ a n + a m + ϕ ( n + m ) {\displaystyle a_{n+m}\leq a_{n}+a_{m}+\phi (n+m)} provided that ϕ {\displaystyle \phi } is an increasing function such that the integral ∫ ϕ ( t ) t − 2 d t {\textstyle \int \phi (t)t^{-2}\,dt} converges (near the infinity). There are also results that allow one to deduce the rate of convergence to the limit whose existence is stated in Fekete's lemma if some kind of both superadditivity and subadditivity is present. Besides, analogues of Fekete's lemma have been proved for subadditive real maps (with additional assumptions) from finite subsets of an amenable group , and further, of a cancellative left-amenable semigroup. === Functions === If f is a subadditive function, and if 0 is in its domain, then f(0) ≥ 0. To see this, take the inequality at the top. f ( x ) ≥ f ( x + y ) − f ( y ) {\displaystyle f(x)\geq f(x+y)-f(y)} . Hence f ( 0 ) ≥ f ( 0 + y ) − f ( y ) = 0 {\displaystyle f(0)\geq f(0+y)-f(y)=0} A concave function f : [ 0 , ∞ ) → R {\displaystyle f:[0,\infty )\to \mathbb {R} } with f ( 0 ) ≥ 0 {\displaystyle f(0)\geq 0} is also subadditive. To see this, one first observes that f ( x ) ≥ y x + y f ( 0 ) + x x + y f ( x + y ) {\displaystyle f(x)\geq \textstyle {\frac {y}{x+y}}f(0)+\textstyle {\frac {x}{x+y}}f(x+y)} . Then looking at the sum of this bound for f ( x ) {\displaystyle f(x)} and f ( y ) {\displaystyle f(y)} , will finally verify that f is subadditive. The negative of a subadditive function is superadditive. == Examples in various domains == === Entropy === Entropy plays a fundamental role in information theory and statistical physics, as well as in quantum mechanics in a generalized formulation due to von Neumann. Entropy appears always as a subadditive quantity in all of its formulations, meaning the entropy of a supersystem or a set union of random variables is always less or equal than the sum of the entropies of its individual components. Additionally, entropy in physics satisfies several more strict inequalities such as the Strong Subadditivity of Entropy in classical statistical mechanics and its quantum analog. === Economics === Subadditivity is an essential property of some particular cost functions. It is, generally, a necessary and sufficient condition for the verification of a natural monopoly. It implies that production from only one firm is socially less expensive (in terms of average costs) than production of a fraction of the original quantity by an equal number of firms. Economies of scale are represented by subadditive average cost functions. Except in the case of complementary goods, the price of goods (as a function of quantity) must be subadditive. Otherwise, if the sum of the cost of two items is cheaper than the cost of the bundle of two of them together, then nobody would ever buy the bundle, effectively causing the price of the bundle to "become" the sum of the prices of the two separate items. Thus proving that it is not a sufficient condition for a natural monopoly; since the unit of exchange may not be the actual cost of an item. This situation is familiar to everyone in the political arena where some minority asserts that the loss of some particular freedom at some particular level of government means that many governments are better; whereas the majority assert that there is some other correct unit of cost. === Finance === Subadditivity is one of the desirable properties of coherent risk measures in risk management. The economic intuition behind risk measure subadditivity is that a portfolio risk exposure should, at worst, simply equal the sum of the risk exposures of the individual positions that compose the portfolio. The lack of subadditivity is one of the main critiques of VaR models which do not rely on the assumption of normality of risk factors. The Gaussian VaR ensures subadditivity: for example, the Gaussian VaR of a two unitary long positions portfolio V {\displaystyle V} at the confidence level 1 − p {\displaystyle 1-p} is, assuming that the mean portfolio value variation is zero and the VaR is defined as a negative loss, VaR p ≡ z p σ Δ V = z p σ x 2 + σ y 2 + 2 ρ x y σ x σ y {\displaystyle {\text{VaR}}_{p}\equiv z_{p}\sigma _{\Delta V}=z_{p}{\sqrt {\sigma _{x}^{2}+\sigma _{y}^{2}+2\rho _{xy}\sigma _{x}\sigma _{y}}}} where z p {\displaystyle z_{p}} is the inverse of the normal cumulative distribution function at probability level p {\displaystyle p} , σ x 2 , σ y 2 {\displaystyle \sigma _{x}^{2},\sigma _{y}^{2}} are the individual positions returns variances and ρ x y {\displaystyle \rho _{xy}} is the linear correlation measure between the two individual positions returns. Since variance is always positive, σ x 2 + σ y 2 + 2 ρ x y σ x σ y ≤ σ x + σ y {\displaystyle {\sqrt {\sigma _{x}^{2}+\sigma _{y}^{2}+2\rho _{xy}\sigma _{x}\sigma _{y}}}\leq \sigma _{x}+\sigma _{y}} Thus the Gaussian VaR is subadditive for any value of ρ x y ∈ [ − 1 , 1 ] {\displaystyle \rho _{xy}\in [-1,1]} and, in particular, it equals the sum of the individual risk exposures when ρ x y = 1 {\displaystyle \rho _{xy}=1} which is the case of no diversification effects on portfolio risk. === Thermodynamics === Subadditivity occurs in the thermodynamic properties of non-ideal solutions and mixtures like the excess molar volume and heat of mixing or excess enthalpy. === Combinatorics on words === A factorial language L {\displaystyle L} is one where if a word is in L {\displaystyle L} , then all factors of that word are also in L {\displaystyle L} . In combinatorics on words, a common problem is to determine the number A ( n ) {\displaystyle A(n)} of length- n {\displaystyle n} words in a factorial language. Clearly A ( m + n ) ≤ A ( m ) A ( n ) {\displaystyle A(m+n)\leq A(m)A(n)} , so log A ( n ) {\displaystyle \log A(n)} is subadditive, and hence Fekete's lemma can be used to estimate the growth of A ( n ) {\displaystyle A(n)} . For every k ≥ 1 {\displaystyle k\geq 1} , sample two strings of length n {\displaystyle n} uniformly at random on the alphabet 1 , 2 , . . . , k {\displaystyle 1,2,...,k} . The expected length of the longest common subsequence is a super-additive function of n {\displaystyle n} , and thus there exists a number γ k ≥ 0 {\displaystyle \gamma _{k}\geq 0} , such that the expected length grows as ∼ γ k n {\displaystyle \sim \gamma _{k}n} . By checking the case with n = 1 {\displaystyle n=1} , we easily have 1 k < γ k ≤ 1 {\displaystyle {\frac {1}{k}}<\gamma _{k}\leq 1} . The exact value of even γ 2 {\displaystyle \gamma _{2}} , however, is only known to be between 0.788 and 0.827. == See also == Apparent molar property – Difference in properties of one mole of substance in a mixture vs. an ideal solution Choquet integral – Subadditive or superadditive integral Superadditivity – Property of a function Triangle inequality – Property of geometry, also used to generalize the notion of "distance" in metric spaces == Notes == == References == György Pólya and Gábor Szegő. Problems and Theorems in Analysis, vol. 1. Springer-Verlag, New York (1976). ISBN 0-387-05672-6. Einar Hille. "Functional analysis and semi-groups". American Mathematical Society, New York (1948). N.H. Bingham, A.J. Ostaszewski. "Generic subadditive functions." Proceedings of American Mathematical Society, vol. 136, no. 12 (2008), pp. 4257–4266. == External links == This article incorporates material from subadditivity on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
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Wikipedia:Subfield of an algebra#0
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In algebra, a subfield of an algebra A over a field F is an F-subalgebra that is also a field. A maximal subfield is a subfield that is not contained in a strictly larger subfield of A. If A is a finite-dimensional central simple algebra, then a subfield E of A is called a strictly maximal subfield if [ E : F ] = ( dim F A ) 1 / 2 {\displaystyle [E:F]=(\dim _{F}A)^{1/2}} . == References == Richard S. Pierce. Associative algebras. Graduate texts in mathematics, Vol. 88, Springer-Verlag, 1982, ISBN 978-0-387-90693-5
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Wikipedia:Sublinear function#0
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In linear algebra, a sublinear function (or functional as is more often used in functional analysis), also called a quasi-seminorm or a Banach functional, on a vector space X {\displaystyle X} is a real-valued function with only some of the properties of a seminorm. Unlike seminorms, a sublinear function does not have to be nonnegative-valued and also does not have to be absolutely homogeneous. Seminorms are themselves abstractions of the more well known notion of norms, where a seminorm has all the defining properties of a norm except that it is not required to map non-zero vectors to non-zero values. In functional analysis the name Banach functional is sometimes used, reflecting that they are most commonly used when applying a general formulation of the Hahn–Banach theorem. The notion of a sublinear function was introduced by Stefan Banach when he proved his version of the Hahn-Banach theorem. There is also a different notion in computer science, described below, that also goes by the name "sublinear function." == Definitions == Let X {\displaystyle X} be a vector space over a field K , {\displaystyle \mathbb {K} ,} where K {\displaystyle \mathbb {K} } is either the real numbers R {\displaystyle \mathbb {R} } or complex numbers C . {\displaystyle \mathbb {C} .} A real-valued function p : X → R {\displaystyle p:X\to \mathbb {R} } on X {\displaystyle X} is called a sublinear function (or a sublinear functional if K = R {\displaystyle \mathbb {K} =\mathbb {R} } ), and also sometimes called a quasi-seminorm or a Banach functional, if it has these two properties: Positive homogeneity/Nonnegative homogeneity: p ( r x ) = r p ( x ) {\displaystyle p(rx)=rp(x)} for all real r ≥ 0 {\displaystyle r\geq 0} and all x ∈ X . {\displaystyle x\in X.} This condition holds if and only if p ( r x ) = r p ( x ) {\displaystyle p(rx)=rp(x)} for all positive real r > 0 {\displaystyle r>0} and all x ∈ X . {\displaystyle x\in X.} Subadditivity/Triangle inequality: p ( x + y ) ≤ p ( x ) + p ( y ) {\displaystyle p(x+y)\leq p(x)+p(y)} for all x , y ∈ X . {\displaystyle x,y\in X.} This subadditivity condition requires p {\displaystyle p} to be real-valued. A function p : X → R {\displaystyle p:X\to \mathbb {R} } is called positive or nonnegative if p ( x ) ≥ 0 {\displaystyle p(x)\geq 0} for all x ∈ X , {\displaystyle x\in X,} although some authors define positive to instead mean that p ( x ) ≠ 0 {\displaystyle p(x)\neq 0} whenever x ≠ 0 ; {\displaystyle x\neq 0;} these definitions are not equivalent. It is a symmetric function if p ( − x ) = p ( x ) {\displaystyle p(-x)=p(x)} for all x ∈ X . {\displaystyle x\in X.} Every subadditive symmetric function is necessarily nonnegative. A sublinear function on a real vector space is symmetric if and only if it is a seminorm. A sublinear function on a real or complex vector space is a seminorm if and only if it is a balanced function or equivalently, if and only if p ( u x ) ≤ p ( x ) {\displaystyle p(ux)\leq p(x)} for every unit length scalar u {\displaystyle u} (satisfying | u | = 1 {\displaystyle |u|=1} ) and every x ∈ X . {\displaystyle x\in X.} The set of all sublinear functions on X , {\displaystyle X,} denoted by X # , {\displaystyle X^{\#},} can be partially ordered by declaring p ≤ q {\displaystyle p\leq q} if and only if p ( x ) ≤ q ( x ) {\displaystyle p(x)\leq q(x)} for all x ∈ X . {\displaystyle x\in X.} A sublinear function is called minimal if it is a minimal element of X # {\displaystyle X^{\#}} under this order. A sublinear function is minimal if and only if it is a real linear functional. == Examples and sufficient conditions == Every norm, seminorm, and real linear functional is a sublinear function. The identity function R → R {\displaystyle \mathbb {R} \to \mathbb {R} } on X := R {\displaystyle X:=\mathbb {R} } is an example of a sublinear function (in fact, it is even a linear functional) that is neither positive nor a seminorm; the same is true of this map's negation x ↦ − x . {\displaystyle x\mapsto -x.} More generally, for any real a ≤ b , {\displaystyle a\leq b,} the map S a , b : R → R x ↦ { a x if x ≤ 0 b x if x ≥ 0 {\displaystyle {\begin{alignedat}{4}S_{a,b}:\;&&\mathbb {R} &&\;\to \;&\mathbb {R} \\[0.3ex]&&x&&\;\mapsto \;&{\begin{cases}ax&{\text{ if }}x\leq 0\\bx&{\text{ if }}x\geq 0\\\end{cases}}\\\end{alignedat}}} is a sublinear function on X := R {\displaystyle X:=\mathbb {R} } and moreover, every sublinear function p : R → R {\displaystyle p:\mathbb {R} \to \mathbb {R} } is of this form; specifically, if a := − p ( − 1 ) {\displaystyle a:=-p(-1)} and b := p ( 1 ) {\displaystyle b:=p(1)} then a ≤ b {\displaystyle a\leq b} and p = S a , b . {\displaystyle p=S_{a,b}.} If p {\displaystyle p} and q {\displaystyle q} are sublinear functions on a real vector space X {\displaystyle X} then so is the map x ↦ max { p ( x ) , q ( x ) } . {\displaystyle x\mapsto \max\{p(x),q(x)\}.} More generally, if P {\displaystyle {\mathcal {P}}} is any non-empty collection of sublinear functionals on a real vector space X {\displaystyle X} and if for all x ∈ X , {\displaystyle x\in X,} q ( x ) := sup { p ( x ) : p ∈ P } , {\displaystyle q(x):=\sup\{p(x):p\in {\mathcal {P}}\},} then q {\displaystyle q} is a sublinear functional on X . {\displaystyle X.} A function p : X → R {\displaystyle p:X\to \mathbb {R} } which is subadditive, convex, and satisfies p ( 0 ) ≤ 0 {\displaystyle p(0)\leq 0} is also positively homogeneous (the latter condition p ( 0 ) ≤ 0 {\displaystyle p(0)\leq 0} is necessary as the example of p ( x ) := x 2 + 1 {\displaystyle p(x):={\sqrt {x^{2}+1}}} on X := R {\displaystyle X:=\mathbb {R} } shows). If p {\displaystyle p} is positively homogeneous, it is convex if and only if it is subadditive. Therefore, assuming p ( 0 ) ≤ 0 {\displaystyle p(0)\leq 0} , any two properties among subadditivity, convexity, and positive homogeneity implies the third. == Properties == Every sublinear function is a convex function: For 0 ≤ t ≤ 1 , {\displaystyle 0\leq t\leq 1,} p ( t x + ( 1 − t ) y ) ≤ p ( t x ) + p ( ( 1 − t ) y ) subadditivity = t p ( x ) + ( 1 − t ) p ( y ) nonnegative homogeneity {\displaystyle {\begin{alignedat}{3}p(tx+(1-t)y)&\leq p(tx)+p((1-t)y)&&\quad {\text{ subadditivity}}\\&=tp(x)+(1-t)p(y)&&\quad {\text{ nonnegative homogeneity}}\\\end{alignedat}}} If p : X → R {\displaystyle p:X\to \mathbb {R} } is a sublinear function on a vector space X {\displaystyle X} then p ( 0 ) = 0 ≤ p ( x ) + p ( − x ) , {\displaystyle p(0)~=~0~\leq ~p(x)+p(-x),} for every x ∈ X , {\displaystyle x\in X,} which implies that at least one of p ( x ) {\displaystyle p(x)} and p ( − x ) {\displaystyle p(-x)} must be nonnegative; that is, for every x ∈ X , {\displaystyle x\in X,} 0 ≤ max { p ( x ) , p ( − x ) } . {\displaystyle 0~\leq ~\max\{p(x),p(-x)\}.} Moreover, when p : X → R {\displaystyle p:X\to \mathbb {R} } is a sublinear function on a real vector space then the map q : X → R {\displaystyle q:X\to \mathbb {R} } defined by q ( x ) = def max { p ( x ) , p ( − x ) } {\displaystyle q(x)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\max\{p(x),p(-x)\}} is a seminorm. Subadditivity of p : X → R {\displaystyle p:X\to \mathbb {R} } guarantees that for all vectors x , y ∈ X , {\displaystyle x,y\in X,} p ( x ) − p ( y ) ≤ p ( x − y ) , {\displaystyle p(x)-p(y)~\leq ~p(x-y),} − p ( x ) ≤ p ( − x ) , {\displaystyle -p(x)~\leq ~p(-x),} so if p {\displaystyle p} is also symmetric then the reverse triangle inequality will hold for all vectors x , y ∈ X , {\displaystyle x,y\in X,} | p ( x ) − p ( y ) | ≤ p ( x − y ) . {\displaystyle |p(x)-p(y)|~\leq ~p(x-y).} Defining ker p = def p − 1 ( 0 ) , {\displaystyle \ker p~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~p^{-1}(0),} then subadditivity also guarantees that for all x ∈ X , {\displaystyle x\in X,} the value of p {\displaystyle p} on the set x + ( ker p ∩ − ker p ) = { x + k : p ( k ) = 0 = p ( − k ) } {\displaystyle x+(\ker p\cap -\ker p)=\{x+k:p(k)=0=p(-k)\}} is constant and equal to p ( x ) . {\displaystyle p(x).} In particular, if ker p = p − 1 ( 0 ) {\displaystyle \ker p=p^{-1}(0)} is a vector subspace of X {\displaystyle X} then − ker p = ker p {\displaystyle -\ker p=\ker p} and the assignment x + ker p ↦ p ( x ) , {\displaystyle x+\ker p\mapsto p(x),} which will be denoted by p ^ , {\displaystyle {\hat {p}},} is a well-defined real-valued sublinear function on the quotient space X / ker p {\displaystyle X\,/\,\ker p} that satisfies p ^ − 1 ( 0 ) = ker p . {\displaystyle {\hat {p}}^{-1}(0)=\ker p.} If p {\displaystyle p} is a seminorm then p ^ {\displaystyle {\hat {p}}} is just the usual canonical norm on the quotient space X / ker p . {\displaystyle X\,/\,\ker p.} Adding b c {\displaystyle bc} to both sides of the hypothesis p ( x ) + a c < inf p ( x + a K ) {\textstyle p(x)+ac\,<\,\inf _{}p(x+aK)} (where p ( x + a K ) = def { p ( x + a k ) : k ∈ K } {\displaystyle p(x+aK)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\{p(x+ak):k\in K\}} ) and combining that with the conclusion gives p ( x ) + a c + b c < inf p ( x + a K ) + b c ≤ p ( x + a z ) + b c < inf p ( x + a z + b K ) {\displaystyle p(x)+ac+bc~<~\inf _{}p(x+aK)+bc~\leq ~p(x+a\mathbf {z} )+bc~<~\inf _{}p(x+a\mathbf {z} +bK)} which yields many more inequalities, including, for instance, p ( x ) + a c + b c < p ( x + a z ) + b c < p ( x + a z + b z ) {\displaystyle p(x)+ac+bc~<~p(x+a\mathbf {z} )+bc~<~p(x+a\mathbf {z} +b\mathbf {z} )} in which an expression on one side of a strict inequality < {\displaystyle \,<\,} can be obtained from the other by replacing the symbol c {\displaystyle c} with z {\displaystyle \mathbf {z} } (or vice versa) and moving the closing parenthesis to the right (or left) of an adjacent summand (all other symbols remain fixed and unchanged). === Associated seminorm === If p : X → R {\displaystyle p:X\to \mathbb {R} } is a real-valued sublinear function on a real vector space X {\displaystyle X} (or if X {\displaystyle X} is complex, then when it is considered as a real vector space) then the map q ( x ) = def max { p ( x ) , p ( − x ) } {\displaystyle q(x)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\max\{p(x),p(-x)\}} defines a seminorm on the real vector space X {\displaystyle X} called the seminorm associated with p . {\displaystyle p.} A sublinear function p {\displaystyle p} on a real or complex vector space is a symmetric function if and only if p = q {\displaystyle p=q} where q ( x ) = def max { p ( x ) , p ( − x ) } {\displaystyle q(x)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\max\{p(x),p(-x)\}} as before. More generally, if p : X → R {\displaystyle p:X\to \mathbb {R} } is a real-valued sublinear function on a (real or complex) vector space X {\displaystyle X} then q ( x ) = def sup | u | = 1 p ( u x ) = sup { p ( u x ) : u is a unit scalar } {\displaystyle q(x)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\sup _{|u|=1}p(ux)~=~\sup\{p(ux):u{\text{ is a unit scalar }}\}} will define a seminorm on X {\displaystyle X} if this supremum is always a real number (that is, never equal to ∞ {\displaystyle \infty } ). === Relation to linear functionals === If p {\displaystyle p} is a sublinear function on a real vector space X {\displaystyle X} then the following are equivalent: p {\displaystyle p} is a linear functional. for every x ∈ X , {\displaystyle x\in X,} p ( x ) + p ( − x ) ≤ 0. {\displaystyle p(x)+p(-x)\leq 0.} for every x ∈ X , {\displaystyle x\in X,} p ( x ) + p ( − x ) = 0. {\displaystyle p(x)+p(-x)=0.} p {\displaystyle p} is a minimal sublinear function. If p {\displaystyle p} is a sublinear function on a real vector space X {\displaystyle X} then there exists a linear functional f {\displaystyle f} on X {\displaystyle X} such that f ≤ p . {\displaystyle f\leq p.} If X {\displaystyle X} is a real vector space, f {\displaystyle f} is a linear functional on X , {\displaystyle X,} and p {\displaystyle p} is a positive sublinear function on X , {\displaystyle X,} then f ≤ p {\displaystyle f\leq p} on X {\displaystyle X} if and only if f − 1 ( 1 ) ∩ { x ∈ X : p ( x ) < 1 } = ∅ . {\displaystyle f^{-1}(1)\cap \{x\in X:p(x)<1\}=\varnothing .} ==== Dominating a linear functional ==== A real-valued function f {\displaystyle f} defined on a subset of a real or complex vector space X {\displaystyle X} is said to be dominated by a sublinear function p {\displaystyle p} if f ( x ) ≤ p ( x ) {\displaystyle f(x)\leq p(x)} for every x {\displaystyle x} that belongs to the domain of f . {\displaystyle f.} If f : X → R {\displaystyle f:X\to \mathbb {R} } is a real linear functional on X {\displaystyle X} then f {\displaystyle f} is dominated by p {\displaystyle p} (that is, f ≤ p {\displaystyle f\leq p} ) if and only if − p ( − x ) ≤ f ( x ) ≤ p ( x ) for every x ∈ X . {\displaystyle -p(-x)\leq f(x)\leq p(x)\quad {\text{ for every }}x\in X.} Moreover, if p {\displaystyle p} is a seminorm or some other symmetric map (which by definition means that p ( − x ) = p ( x ) {\displaystyle p(-x)=p(x)} holds for all x {\displaystyle x} ) then f ≤ p {\displaystyle f\leq p} if and only if | f | ≤ p . {\displaystyle |f|\leq p.} === Continuity === Suppose X {\displaystyle X} is a topological vector space (TVS) over the real or complex numbers and p {\displaystyle p} is a sublinear function on X . {\displaystyle X.} Then the following are equivalent: p {\displaystyle p} is continuous; p {\displaystyle p} is continuous at 0; p {\displaystyle p} is uniformly continuous on X {\displaystyle X} ; and if p {\displaystyle p} is positive then this list may be extended to include: { x ∈ X : p ( x ) < 1 } {\displaystyle \{x\in X:p(x)<1\}} is open in X . {\displaystyle X.} If X {\displaystyle X} is a real TVS, f {\displaystyle f} is a linear functional on X , {\displaystyle X,} and p {\displaystyle p} is a continuous sublinear function on X , {\displaystyle X,} then f ≤ p {\displaystyle f\leq p} on X {\displaystyle X} implies that f {\displaystyle f} is continuous. === Relation to Minkowski functions and open convex sets === ==== Relation to open convex sets ==== == Operators == The concept can be extended to operators that are homogeneous and subadditive. This requires only that the codomain be, say, an ordered vector space to make sense of the conditions. == Computer science definition == In computer science, a function f : Z + → R {\displaystyle f:\mathbb {Z} ^{+}\to \mathbb {R} } is called sublinear if lim n → ∞ f ( n ) n = 0 , {\displaystyle \lim _{n\to \infty }{\frac {f(n)}{n}}=0,} or f ( n ) ∈ o ( n ) {\displaystyle f(n)\in o(n)} in asymptotic notation (notice the small o {\displaystyle o} ). Formally, f ( n ) ∈ o ( n ) {\displaystyle f(n)\in o(n)} if and only if, for any given c > 0 , {\displaystyle c>0,} there exists an N {\displaystyle N} such that f ( n ) < c n {\displaystyle f(n)<cn} for n ≥ N . {\displaystyle n\geq N.} That is, f {\displaystyle f} grows slower than any linear function. The two meanings should not be confused: while a Banach functional is convex, almost the opposite is true for functions of sublinear growth: every function f ( n ) ∈ o ( n ) {\displaystyle f(n)\in o(n)} can be upper-bounded by a concave function of sublinear growth. == See also == Asymmetric norm – Generalization of the concept of a norm Auxiliary normed space Hahn-Banach theorem – Theorem on extension of bounded linear functionalsPages displaying short descriptions of redirect targets Linear functional – Linear map from a vector space to its field of scalarsPages displaying short descriptions of redirect targets Minkowski functional – Function made from a set Norm (mathematics) – Length in a vector space Seminorm – Mathematical function Superadditivity – Property of a function == Notes == Proofs == References == == Bibliography == Kubrusly, Carlos S. (2011). The Elements of Operator Theory (Second ed.). Boston: Birkhäuser. ISBN 978-0-8176-4998-2. OCLC 710154895. Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. Schechter, Eric (1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. ISBN 978-0-12-622760-4. OCLC 175294365. Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
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Wikipedia:Subquotient#0
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In the mathematical fields of category theory and abstract algebra, a subquotient is a quotient object of a subobject. Subquotients are particularly important in abelian categories, and in group theory, where they are also known as sections, though this conflicts with a different meaning in category theory. So in the algebraic structure of groups, H {\displaystyle H} is a subquotient of G {\displaystyle G} if there exists a subgroup G ′ {\displaystyle G'} of G {\displaystyle G} and a normal subgroup G ″ {\displaystyle G''} of G ′ {\displaystyle G'} so that H {\displaystyle H} is isomorphic to G ′ / G ″ {\displaystyle G'/G''} . In the literature about sporadic groups wordings like “ H {\displaystyle H} is involved in G {\displaystyle G} “ can be found with the apparent meaning of “ H {\displaystyle H} is a subquotient of G {\displaystyle G} “. As in the context of subgroups, in the context of subquotients the term trivial may be used for the two subquotients G {\displaystyle G} and { 1 } {\displaystyle \{1\}} which are present in every group G {\displaystyle G} . A quotient of a subrepresentation of a representation (of, say, a group) might be called a subquotient representation; e. g., Harish-Chandra's subquotient theorem. == Example == There are subquotients of groups which are neither subgroup nor quotient of it. E. g. according to article Sporadic group, Fi22 has a double cover which is a subgroup of Fi23, so it is a subquotient of Fi23 without being a subgroup or quotient of it. == Order relation == The relation subquotient of is an order relation – which shall be denoted by ⪯ {\displaystyle \preceq } . It shall be proved for groups. Notation For group G {\displaystyle G} , subgroup G ′ {\displaystyle G'} of G {\displaystyle G} ( ⇔: G ′ ≤ G ) {\displaystyle (\Leftrightarrow :G'\leq G)} and normal subgroup G ″ {\displaystyle G''} of G ′ {\displaystyle G'} ( ⇔: G ″ ⊲ G ′ ) {\displaystyle (\Leftrightarrow :G''\vartriangleleft G')} the quotient group H := G ′ / G ″ {\displaystyle H:=G'/G''} is a subquotient of G {\displaystyle G} , i. e. H ⪯ G {\displaystyle H\preceq G} . Reflexivity: G ⪯ G {\displaystyle G\preceq G} , i. e. every element is related to itself. Indeed, G {\displaystyle G} is isomorphic to the subquotient G / { 1 } {\displaystyle G/\{1\}} of G {\displaystyle G} . Antisymmetry: if G ⪯ H {\displaystyle G\preceq H} and H ⪯ G {\displaystyle H\preceq G} then G ≅ H {\displaystyle G\cong H} , i. e. no two distinct elements precede each other. Indeed, a comparison of the group orders of G {\displaystyle G} and H {\displaystyle H} then yields | G | = | H | {\displaystyle |G|=|H|} from which G ≅ H {\displaystyle G\cong H} . Transitivity: if H ′ / H ″ ⪯ H {\displaystyle H'/H''\preceq H} and H ⪯ G {\displaystyle H\preceq G} then H ′ / H ″ ⪯ G {\displaystyle H'/H''\preceq G} . === Proof of transitivity for groups === Let H ′ / H ″ {\displaystyle H'/H''} be subquotient of H {\displaystyle H} , furthermore H := G ′ / G ″ {\displaystyle H:=G'/G''} be subquotient of G {\displaystyle G} and φ : G ′ → H {\displaystyle \varphi \colon G'\to H} be the canonical homomorphism. Then all vertical ( ↓ {\displaystyle \downarrow } ) maps φ : X → Y , x ↦ x G ″ {\displaystyle \varphi \colon X\to Y,\;x\mapsto x\,G''} are surjective for the respective pairs The preimages φ − 1 ( H ′ ) {\displaystyle \varphi ^{-1}\left(H'\right)} and φ − 1 ( H ″ ) {\displaystyle \varphi ^{-1}\left(H''\right)} are both subgroups of G ′ {\displaystyle G'} containing G ″ , {\displaystyle G'',} and it is φ ( φ − 1 ( H ′ ) ) = H ′ {\displaystyle \varphi \left(\varphi ^{-1}\left(H'\right)\right)=H'} and φ ( φ − 1 ( H ″ ) ) = H ″ , {\displaystyle \varphi \left(\varphi ^{-1}\left(H''\right)\right)=H'',} because every h ∈ H {\displaystyle h\in H} has a preimage g ∈ G ′ {\displaystyle g\in G'} with φ ( g ) = h . {\displaystyle \varphi (g)=h.} Moreover, the subgroup φ − 1 ( H ″ ) {\displaystyle \varphi ^{-1}\left(H''\right)} is normal in φ − 1 ( H ′ ) . {\displaystyle \varphi ^{-1}\left(H'\right).} As a consequence, the subquotient H ′ / H ″ {\displaystyle H'/H''} of H {\displaystyle H} is a subquotient of G {\displaystyle G} in the form H ′ / H ″ ≅ φ − 1 ( H ′ ) / φ − 1 ( H ″ ) . {\displaystyle H'/H''\cong \varphi ^{-1}\left(H'\right)/\varphi ^{-1}\left(H''\right).} === Relation to cardinal order === In constructive set theory, where the law of excluded middle does not necessarily hold, one can consider the relation subquotient of as replacing the usual order relation(s) on cardinals. When one has the law of the excluded middle, then a subquotient Y {\displaystyle Y} of X {\displaystyle X} is either the empty set or there is an onto function X → Y {\displaystyle X\to Y} . This order relation is traditionally denoted ≤ ∗ . {\displaystyle \leq ^{\ast }.} If additionally the axiom of choice holds, then Y {\displaystyle Y} has a one-to-one function to X {\displaystyle X} and this order relation is the usual ≤ {\displaystyle \leq } on corresponding cardinals. == See also == Homological algebra Subcountable == References ==
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Wikipedia:Sue Ann Campbell#0
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Sue Ann Campbell is a Canadian applied mathematician and computational neuroscientist known for her work on dynamical systems, delay differential equations, and their applications in modeling neural networks, population dynamics, and balance. She is a professor of applied mathematics at the University of Waterloo, former chair of the Department of Applied Mathematics, associate dean for research and international in the university's Faculty of Mathematics, and president of the Canadian Applied and Industrial Mathematical Society. == Education and career == Campbell has a bachelor's degree in mathematics (B.Math.) from the University of Waterloo, earned in 1986. She completed her Ph.D. in 1991 at Cornell University. Her dissertation, The Effects of Symmetry on the Dynamics of Low-Dimensional Modal Interactions, was supervised by Philip Holmes. She was a postdoctoral researcher at the University of Montreal before taking an assistant professor position at Concordia University. She returned to the University of Waterloo as an assistant professor in 1994. She was elected as president of the Canadian Applied and Industrial Mathematical Society in 2021, for a term beginning in 2023. == Recognition == Campbell was the 2005 winner of the Arthur Beaumont Distinguished Service Award of the Canadian Applied and Industrial Mathematics Society. == References == == External links == Home page Sue Ann Campbell publications indexed by Google Scholar
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Wikipedia:Sue Chandler#0
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== F S Chandler (1940-2023) == F S Chandler, aka Suzanne or Sue Chandler, was a British schoolteacher and textbook writer, who, together with Linda Bostock, wrote the "Bostock and Chandler" series of textbooks for advanced level mathematics in the UK. At the time she began the series, she was a full-time mathematics teacher at Southgate Technical College, London. She eventually stopped teaching courses and focused on textbook writing. Her books have sold more than 6 million copies. == Life == Sue Chandler became the author of Mathematics textbooks when such books were the exclusive realm of men; the books became staples of Mathematics teaching throughout the world. Sue and her colleague Linda Bostock, when teaching at Southgate Technical College in the 1970s, could find no suitable textbook to supply their needs, so they decided to write their own. Judiciously, they chose to write under the names of L Bostock and F S Chandler. In 1984, they were not surprised when Lord Rothschild, having named one of their books as his Desert Island choice, described it as having been written by 'two wonderful young men’. Sue was the daughter of Francis Rourke, a quantity surveyor, and Paula Ley, a key member of her brother Stanley's West End tailoring business. Paula and her sister had married brothers, both of whom left their young families. The sisters were helped considerably by their six brothers. Paula found a flat overlooking Hampstead Heath and Sue went to school at the local convent. Despite failing the 11-plus, she achieved good A-level results and gained a Maths degree at Sir John Cass College, followed by a PGCE. Six months after meeting him, Sue married Derek Chandler, an engineer who became a patent agent. They formed a successful partnership, despite their opposing political views, and are remembered by their nephews and nieces for their generosity and moral support. Apart from the time spent in the north of England whilst Francis was recuperating from injuries acquired at Dunkirk, Sue lived her whole life in London. Sue enjoyed entertaining family and friends. After her mother’s death, she sought out her half-brother Michael and arranged for him to meet up with her part of the Rourke family. She also took on her mother’s role of keeping the Ley family together and was instrumental in organising regular ‘cousins events’. The last of these was held in September of 2023. Knowing herself to be too frail to travel, Sue insisted on hosting this event at her home: she died as the guests arrived. Sue was survived by her brother Colin; two daughters; three grandchildren and many cousins. Derek died in 2018, Michael in 2022 and Linda in 2012. == Selected publications == Textbooks L Bostock & F S Chandler (1975), Applied Mathematics, Vol. 1, Stanley Thornes L Bostock & F S Chandler (1976), Applied Mathematics, Vol. 2, Stanley Thornes L Bostock & F S Chandler (1978), Pure Mathematics, Vol. 1, Stanley Thornes L Bostock & F S Chandler (1979), Pure Mathematics, Vol. 2, Stanley Thornes L Bostock & F S Chandler (1981), Mathematics – The Core Course for A Level (2nd ed.), Stanley Thornes L Bostock, F S Chandler & C P Rourke (1982), Further Pure Mathematics, Stanley Thornes L Bostock & F S Chandler (1984), Mathematics – Mechanics and Probability, Stanley Thornes L Bostock & F S Chandler (1985), Further Mechanics and Probability, Stanley Thornes L Bostock & F S Chandler (1994), Core Maths for 'A' Level (2nd ed.), Nelson Thornes L Bostock & F S Chandler (2000), Core Maths for 'A' Level (3rd ed.), Nelson Thornes Other Chandler, Sue (1997), "A-level mathematics examinations as a fair assessment of the needs of students post-GCSE intermediate and higher tiers", Teaching Mathematics and Its Applications, 16 (4): 157–159, doi:10.1093/teamat/16.4.157. == References ==
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Wikipedia:Sue Singer#0
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Peggy Sue Wright (née Webb; born March 25, 1943) is a country music singer and songwriter, who had brief success as a country singer in the late 1960s. She is the middle sister of two popular country performers, Loretta Lynn and Crystal Gayle. Her older brother Willie "Jay" Lee Webb was a country music singer/songwriter in the 1960s. == Biography == Peggy Sue Wright was born Peggy Sue Webb in a cabin in Butcher Hollow, Kentucky, on March 25, 1943. She is the second daughter and the sixth child born to Clara Marie "Clary" (née Ramey; May 5, 1912 – November 24, 1981) and Melvin Theodore "Ted" Webb (June 6, 1906 – February 22, 1959). Mr. Webb was a coal miner and subsistence farmer. The family was poor; living hand-to-mouth and relying on her father's meager income. The seven Webb siblings in addition to Wright: Melvin "Junior" Webb (December 4, 1929 – July 1, 1993) Loretta Lynn (née Webb; April 14, 1932 – October 4, 2022) Herman Webb (September 3, 1934 – July 28, 2018) Willie "Jay" Lee Webb (February 12, 1937 – July 31, 1996) Donald Ray Webb (April 2, 1941 – October 13, 2017) Betty Ruth Hopkins (née Webb; born January 5, 1946) Crystal Gayle (born Brenda Gail Webb; January 9, 1951) The family moved to Wabash, Indiana, in 1955 due to her father's illness from working in the coal mines; he would die in 1959 of black lung disease. She began performing with Loretta and her brothers at venues around Wabash. Wright then became a featured act in Loretta's early shows in the 1960s. She also helped write a few of Loretta's compositions, including "Don't Come Home A' Drinkin' (With Lovin' on Your Mind)." In 1969, she signed with Decca Records and released her debut single, "I'm Dynamite," which went into the country top 30. That same year, she released an album of the same name. The second single from that album titled, "I'm Gettin' Tired of Babyin' You" also reached the top 30. After Peggy Sue had a hit with her most successful single, "All-American Husband", she left Decca Records after releasing two albums. Next, Wright recorded two albums in the 1970s for two small labels. Peggy Sue was married twice. Her first marriage was to Douglas Wells (m. 1964-div. 1968); the second was to Sonny Wright (m. 1970-). From her first marriage, Peggy had one daughter: Doyletta Gayle; born May 30, 1967. Doyletta Gayle was named after Doyle Wilburn and Wright's sisters: Loretta Lynn and Crystal Gayle. Doyletta became a victim of spousal abuse when she was killed by her spouse on February 22, 1991. After 1970, she did not appear on the Billboard country charts until 1977. Beginning then, she had a small string of minor hits on her second husband Sonny Wright's label, Doorknob. In 1986, she began performing as a background singer and designing stage costumes for her younger sister, Crystal Gayle. She continues to perform with Gayle today. Occasionally, they would both join with older sister Loretta Lynn for a concert at her Hurricane Mills, Tennessee, ranch. == Discography == === Albums === === Singles === A"I'm Gettin' Tired of Babyin' You" also peaked at number 27 on the RPM Country Tracks chart in Canada. == References == == External links == Peggy Sue discography at Slipcue.com
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Wikipedia:Suely Druck#0
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Suely Druck is a Brazilian mathematician and two-time president of the Brazilian Mathematical Society. == Life and work == Suely Druck holds a degree in Mathematics from the Federal University of Rio de Janeiro (1970) and a master's degree in Mathematics from the National Institute of Pure and Applied Mathematics Association (1977). She earned her D.Sc. at Pontifical Catholic University of Rio de Janeiro in 1984 under the supervision of Paul Alexander Schweitzer with a dissertation titled, Estabilidade de folhas compactas em folheações dadas por fibrados. Druck was elected president of the Brazilian Mathematical Society twice, in 2001 and 2003, to two-year terms. == Selected publications == Druck, Suely, Fuquan Fang, and Sebastiao Firmo. "Fixed points of discrete nilpotent group actions on $ S^ 2$." In Annales de l'institut Fourier, vol. 52, no. 4, pp. 1075-1091. 2002. Druck, Suely, and Sebastiao Firmo. "Periodic leaves for diffeomorphisms preserving codimension one foliations." Journal of the Mathematical Society of Japan 55, no. 1 (2003): 13-37. Druck, Suely, Ana Catarina Pontone Hellmeister, and Deborah Martins Raphael. "Explorando o ensino da matemática." (2004). Druck, Suely. "Educação científica no Brasil: uma urgência." WERTHEIN, Jorge; CUNHA (2005). == References ==
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Wikipedia:Sullivan conjecture#0
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In mathematics, Sullivan conjecture or Sullivan's conjecture on maps from classifying spaces can refer to any of several results and conjectures prompted by homotopy theory work of Dennis Sullivan. A basic theme and motivation concerns the fixed point set in group actions of a finite group G {\displaystyle G} . The most elementary formulation, however, is in terms of the classifying space B G {\displaystyle BG} of such a group. Roughly speaking, it is difficult to map such a space B G {\displaystyle BG} continuously into a finite CW complex X {\displaystyle X} in a non-trivial manner. Such a version of the Sullivan conjecture was first proved by Haynes Miller. Specifically, in 1984, Miller proved that the function space, carrying the compact-open topology, of base point-preserving mappings from B G {\displaystyle BG} to X {\displaystyle X} is weakly contractible. This is equivalent to the statement that the map X {\displaystyle X} → F ( B G , X ) {\displaystyle F(BG,X)} from X to the function space of maps B G {\displaystyle BG} → X {\displaystyle X} , not necessarily preserving the base point, given by sending a point x {\displaystyle x} of X {\displaystyle X} to the constant map whose image is x {\displaystyle x} is a weak equivalence. The mapping space F ( B G , X ) {\displaystyle F(BG,X)} is an example of a homotopy fixed point set. Specifically, F ( B G , X ) {\displaystyle F(BG,X)} is the homotopy fixed point set of the group G {\displaystyle G} acting by the trivial action on X {\displaystyle X} . In general, for a group G {\displaystyle G} acting on a space X {\displaystyle X} , the homotopy fixed points are the fixed points F ( E G , X ) G {\displaystyle F(EG,X)^{G}} of the mapping space F ( E G , X ) {\displaystyle F(EG,X)} of maps from the universal cover E G {\displaystyle EG} of B G {\displaystyle BG} to X {\displaystyle X} under the G {\displaystyle G} -action on F ( E G , X ) {\displaystyle F(EG,X)} given by g {\displaystyle g} in G {\displaystyle G} acts on a map f {\displaystyle f} in F ( E G , X ) {\displaystyle F(EG,X)} by sending it to g f g − 1 {\displaystyle gfg^{-1}} . The G {\displaystyle G} -equivariant map from E G {\displaystyle EG} to a single point ∗ {\displaystyle *} induces a natural map η: X G = F ( ∗ , X ) G {\displaystyle X^{G}=F(*,X)^{G}} → F ( E G , X ) G {\displaystyle F(EG,X)^{G}} from the fixed points to the homotopy fixed points of G {\displaystyle G} acting on X {\displaystyle X} . Miller's theorem is that η is a weak equivalence for trivial G {\displaystyle G} -actions on finite-dimensional CW complexes. An important ingredient and motivation for his proof is a result of Gunnar Carlsson on the homology of B Z / 2 {\displaystyle BZ/2} as an unstable module over the Steenrod algebra. Miller's theorem generalizes to a version of Sullivan's conjecture in which the action on X {\displaystyle X} is allowed to be non-trivial. In, Sullivan conjectured that η is a weak equivalence after a certain p-completion procedure due to A. Bousfield and D. Kan for the group G = Z / 2 {\displaystyle G=Z/2} . This conjecture was incorrect as stated, but a correct version was given by Miller, and proven independently by Dwyer-Miller-Neisendorfer, Carlsson, and Jean Lannes, showing that the natural map ( X G ) p {\displaystyle (X^{G})_{p}} → F ( E G , ( X ) p ) G {\displaystyle F(EG,(X)_{p})^{G}} is a weak equivalence when the order of G {\displaystyle G} is a power of a prime p, and where ( X ) p {\displaystyle (X)_{p}} denotes the Bousfield-Kan p-completion of X {\displaystyle X} . Miller's proof involves an unstable Adams spectral sequence, Carlsson's proof uses his affirmative solution of the Segal conjecture and also provides information about the homotopy fixed points F ( E G , X ) G {\displaystyle F(EG,X)^{G}} before completion, and Lannes's proof involves his T-functor. == References == == External links == Gottlieb, Daniel H. (2001) [1994], "Sullivan conjecture", Encyclopedia of Mathematics, EMS Press Book extract J. Lurie's course notes
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Wikipedia:Sum of radicals#0
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In mathematics, a sum of radicals is defined as a finite linear combination of nth roots: ∑ i = 1 n k i x i r i , {\displaystyle \sum _{i=1}^{n}k_{i}{\sqrt[{r_{i}}]{x_{i}}},} where n , r i {\displaystyle n,r_{i}} are natural numbers and k i , x i {\displaystyle k_{i},x_{i}} are real numbers. A particular special case arising in computational complexity theory is the square-root sum problem, asking whether it is possible to determine the sign of a sum of square roots, with integer coefficients, in polynomial time. This is of importance for many problems in computational geometry, since the computation of the Euclidean distance between two points in the general case involves the computation of a square root, and therefore the perimeter of a polygon or the length of a polygonal chain takes the form of a sum of radicals. In 1991, Blömer proposed a polynomial time Monte Carlo algorithm for determining whether a sum of radicals is zero, or more generally whether it represents a rational number. Blömer's result applies more generally than the square-root sum problem, to sums of radicals that are not necessarily square roots. However, his algorithm does not solve the problem, because it does not determine the sign of a non-zero sum of radicals. == See also == Nested radicals Abel–Ruffini theorem == References ==
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Wikipedia:Sum of two cubes#0
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In mathematics, the sum of two cubes is a cubed number added to another cubed number. == Factorization == Every sum of cubes may be factored according to the identity a 3 + b 3 = ( a + b ) ( a 2 − a b + b 2 ) {\displaystyle a^{3}+b^{3}=(a+b)(a^{2}-ab+b^{2})} in elementary algebra. Binomial numbers generalize this factorization to higher odd powers. === Proof === Starting with the expression, a 2 − a b + b 2 {\displaystyle a^{2}-ab+b^{2}} and multiplying by a + b ( a + b ) ( a 2 − a b + b 2 ) = a ( a 2 − a b + b 2 ) + b ( a 2 − a b + b 2 ) . {\displaystyle (a+b)(a^{2}-ab+b^{2})=a(a^{2}-ab+b^{2})+b(a^{2}-ab+b^{2}).} distributing a and b over a 2 − a b + b 2 {\displaystyle a^{2}-ab+b^{2}} , a 3 − a 2 b + a b 2 + a 2 b − a b 2 + b 3 {\displaystyle a^{3}-a^{2}b+ab^{2}+a^{2}b-ab^{2}+b^{3}} and canceling the like terms, a 3 + b 3 . {\displaystyle a^{3}+b^{3}.} Similarly for the difference of cubes, ( a − b ) ( a 2 + a b + b 2 ) = a ( a 2 + a b + b 2 ) − b ( a 2 + a b + b 2 ) = a 3 + a 2 b + a b 2 − a 2 b − a b 2 − b 3 = a 3 − b 3 . {\displaystyle {\begin{aligned}(a-b)(a^{2}+ab+b^{2})&=a(a^{2}+ab+b^{2})-b(a^{2}+ab+b^{2})\\&=a^{3}+a^{2}b+ab^{2}\;-a^{2}b-ab^{2}-b^{3}\\&=a^{3}-b^{3}.\end{aligned}}} === "SOAP" mnemonic === The mnemonic "SOAP", short for "Same, Opposite, Always Positive", helps recall of the signs: == Fermat's last theorem == Fermat's last theorem in the case of exponent 3 states that the sum of two non-zero integer cubes does not result in a non-zero integer cube. The first recorded proof of the exponent 3 case was given by Euler. == Taxicab and Cabtaxi numbers == A Taxicab number is the smallest positive number that can be expressed as a sum of two positive integer cubes in n distinct ways. The smallest taxicab number after Ta(1) = 1, is Ta(2) = 1729 (the Ramanujan number), expressed as 1 3 + 12 3 {\displaystyle 1^{3}+12^{3}} or 9 3 + 10 3 {\displaystyle 9^{3}+10^{3}} Ta(3), the smallest taxicab number expressed in 3 different ways, is 87,539,319, expressed as 436 3 + 167 3 {\displaystyle 436^{3}+167^{3}} , 423 3 + 228 3 {\displaystyle 423^{3}+228^{3}} or 414 3 + 255 3 {\displaystyle 414^{3}+255^{3}} A Cabtaxi number is the smallest positive number that can be expressed as a sum of two integer cubes in n ways, allowing the cubes to be negative or zero as well as positive. The smallest cabtaxi number after Cabtaxi(1) = 0, is Cabtaxi(2) = 91, expressed as: 3 3 + 4 3 {\displaystyle 3^{3}+4^{3}} or 6 3 − 5 3 {\displaystyle 6^{3}-5^{3}} Cabtaxi(3), the smallest Cabtaxi number expressed in 3 different ways, is 4104, expressed as 16 3 + 2 3 {\displaystyle 16^{3}+2^{3}} , 15 3 + 9 3 {\displaystyle 15^{3}+9^{3}} or − 12 3 + 18 3 {\displaystyle -12^{3}+18^{3}} == See also == Difference of two squares Binomial number Sophie Germain's identity Aurifeuillean factorization Fermat's last theorem == References == == Further reading == Broughan, Kevin A. (January 2003). "Characterizing the Sum of Two Cubes" (PDF). Journal of Integer Sequences. 6 (4): 46. Bibcode:2003JIntS...6...46B.
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Wikipedia:Summability kernel#0
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In mathematics, a summability kernel is a family or sequence of periodic integrable functions satisfying a certain set of properties, listed below. Certain kernels, such as the Fejér kernel, are particularly useful in Fourier analysis. Summability kernels are related to approximation of the identity; definitions of an approximation of identity vary, but sometimes the definition of an approximation of the identity is taken to be the same as for a summability kernel. == Definition == Let T := R / Z {\displaystyle \mathbb {T} :=\mathbb {R} /\mathbb {Z} } . A summability kernel is a sequence ( k n ) {\displaystyle (k_{n})} in L 1 ( T ) {\displaystyle L^{1}(\mathbb {T} )} that satisfies ∫ T k n ( t ) d t = 1 {\displaystyle \int _{\mathbb {T} }k_{n}(t)\,dt=1} ∫ T | k n ( t ) | d t ≤ M {\displaystyle \int _{\mathbb {T} }|k_{n}(t)|\,dt\leq M} (uniformly bounded) ∫ δ ≤ | t | ≤ 1 2 | k n ( t ) | d t → 0 {\displaystyle \int _{\delta \leq |t|\leq {\frac {1}{2}}}|k_{n}(t)|\,dt\to 0} as n → ∞ {\displaystyle n\to \infty } , for every δ > 0 {\displaystyle \delta >0} . Note that if k n ≥ 0 {\displaystyle k_{n}\geq 0} for all n {\displaystyle n} , i.e. ( k n ) {\displaystyle (k_{n})} is a positive summability kernel, then the second requirement follows automatically from the first. With the more usual convention T = R / 2 π Z {\displaystyle \mathbb {T} =\mathbb {R} /2\pi \mathbb {Z} } , the first equation becomes 1 2 π ∫ T k n ( t ) d t = 1 {\displaystyle {\frac {1}{2\pi }}\int _{\mathbb {T} }k_{n}(t)\,dt=1} , and the upper limit of integration on the third equation should be extended to π {\displaystyle \pi } , so that the condition 3 above should be ∫ δ ≤ | t | ≤ π | k n ( t ) | d t → 0 {\displaystyle \int _{\delta \leq |t|\leq \pi }|k_{n}(t)|\,dt\to 0} as n → ∞ {\displaystyle n\to \infty } , for every δ > 0 {\displaystyle \delta >0} . This expresses the fact that the mass concentrates around the origin as n {\displaystyle n} increases. One can also consider R {\displaystyle \mathbb {R} } rather than T {\displaystyle \mathbb {T} } ; then (1) and (2) are integrated over R {\displaystyle \mathbb {R} } , and (3) over | t | > δ {\displaystyle |t|>\delta } . == Examples == The Fejér kernel The Poisson kernel (continuous index) The Landau kernel The Dirichlet kernel is not a summability kernel, since it fails the second requirement. == Convolutions == Let ( k n ) {\displaystyle (k_{n})} be a summability kernel, and ∗ {\displaystyle *} denote the convolution operation. If ( k n ) , f ∈ C ( T ) {\displaystyle (k_{n}),f\in {\mathcal {C}}(\mathbb {T} )} (continuous functions on T {\displaystyle \mathbb {T} } ), then k n ∗ f → f {\displaystyle k_{n}*f\to f} in C ( T ) {\displaystyle {\mathcal {C}}(\mathbb {T} )} , i.e. uniformly, as n → ∞ {\displaystyle n\to \infty } . In the case of the Fejer kernel this is known as Fejér's theorem. If ( k n ) , f ∈ L 1 ( T ) {\displaystyle (k_{n}),f\in L^{1}(\mathbb {T} )} , then k n ∗ f → f {\displaystyle k_{n}*f\to f} in L 1 ( T ) {\displaystyle L^{1}(\mathbb {T} )} , as n → ∞ {\displaystyle n\to \infty } . If ( k n ) {\displaystyle (k_{n})} is radially decreasing symmetric and f ∈ L 1 ( T ) {\displaystyle f\in L^{1}(\mathbb {T} )} , then k n ∗ f → f {\displaystyle k_{n}*f\to f} pointwise a.e., as n → ∞ {\displaystyle n\to \infty } . This uses the Hardy–Littlewood maximal function. If ( k n ) {\displaystyle (k_{n})} is not radially decreasing symmetric, but the decreasing symmetrization k ~ n ( x ) := sup | y | ≥ | x | k n ( y ) {\displaystyle {\widetilde {k}}_{n}(x):=\sup _{|y|\geq |x|}k_{n}(y)} satisfies sup n ∈ N ‖ k ~ n ‖ 1 < ∞ {\displaystyle \sup _{n\in \mathbb {N} }\|{\widetilde {k}}_{n}\|_{1}<\infty } , then a.e. convergence still holds, using a similar argument. == References == Katznelson, Yitzhak (2004), An introduction to Harmonic Analysis, Cambridge University Press, ISBN 0-521-54359-2
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Wikipedia:Sumset#0
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In additive combinatorics, the sumset (also called the Minkowski sum) of two subsets A {\displaystyle A} and B {\displaystyle B} of an abelian group G {\displaystyle G} (written additively) is defined to be the set of all sums of an element from A {\displaystyle A} with an element from B {\displaystyle B} . That is, A + B = { a + b : a ∈ A , b ∈ B } . {\displaystyle A+B=\{a+b:a\in A,b\in B\}.} The n {\displaystyle n} -fold iterated sumset of A {\displaystyle A} is n A = A + ⋯ + A , {\displaystyle nA=A+\cdots +A,} where there are n {\displaystyle n} summands. Many of the questions and results of additive combinatorics and additive number theory can be phrased in terms of sumsets. For example, Lagrange's four-square theorem can be written succinctly in the form 4 ◻ = N , {\displaystyle 4\,\Box =\mathbb {N} ,} where ◻ {\displaystyle \Box } is the set of square numbers. A subject that has received a fair amount of study is that of sets with small doubling, where the size of the set A + A {\displaystyle A+A} is small (compared to the size of A {\displaystyle A} ); see for example Freiman's theorem. == See also == == References == Henry Mann (1976). Addition Theorems: The Addition Theorems of Group Theory and Number Theory (Corrected reprint of 1965 Wiley ed.). Huntington, New York: Robert E. Krieger Publishing Company. ISBN 0-88275-418-1. Nathanson, Melvyn B. (1990). "Best possible results on the density of sumsets". In Berndt, Bruce C.; Diamond, Harold G.; Halberstam, Heini; et al. (eds.). Analytic number theory. Proceedings of a conference in honor of Paul T. Bateman, held on April 25-27, 1989, at the University of Illinois, Urbana, IL (USA). Progress in Mathematics. Vol. 85. Boston: Birkhäuser. pp. 395–403. ISBN 0-8176-3481-9. Zbl 0722.11007. Nathanson, Melvyn B. (1996). Additive Number Theory: Inverse Problems and the Geometry of Sumsets. Graduate Texts in Mathematics. Vol. 165. Springer-Verlag. ISBN 0-387-94655-1. Zbl 0859.11003. Terence Tao and Van Vu, Additive Combinatorics, Cambridge University Press 2006. == External links == Sloman, Leila (2022-12-06). "From Systems in Motion, Infinite Patterns Appear". Quanta Magazine.
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Wikipedia:Sunday Iyahen#0
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Sunday Osarumwense Iyahen (3 October 1937 – 28 January 2018) was a Nigerian mathematician and politician, recognised for his contributions to the field of topological vector spaces and his service as a senator representing Bendel Central Senatorial District. Born in Benin City, Edo State, Nigeria, Iyahen was the eldest of at least seventeen children and embarked on an academic journey that led him to earn a first-class honours degree in mathematics from the University of Ibadan and later a Ph.D. and D.Sc. from the University of Keele. Iyahen's academic career was marked by his tenure as a professor of mathematics at several universities in Nigeria and abroad. He served as the Head of the Department of Mathematics and Dean of the Faculty of Science at the University of Ibadan before joining the Institute of Technology, Benin (now known as the University of Benin), where he became the founding dean of the Faculty of Physical Sciences. His scholarly work includes over 100 published papers and contributions as editor-in-chief for mathematical journals. He was honoured with fellowships from the Nigerian Academy of Science and the Mathematical Association of Nigeria. As a politician, he was elected as a senator, where he contributed to national policy and development. == Early life and education == Iyahen was born on 3 October 1937 in Benin City, Edo State, Nigeria. He was the eldest of at least seventeen children of Solomon Igbinuwen Iyahen and his wife Aiwekhoe. Iyahen attended Saint Matthew's Primary school (1944–45), and Saint Peter's School (1945–51) in Benin City. Both schools were under the administration of the Church Mission Society, a London-based organisation established in 1799. He then attended Edo College. In 1956, he passed in the Cambridge school certificate examination, earning a Division One. He studied at Government College, Ibadan, for his Cambridge Higher School Certificate in 1957–1958. In 1959, he enrolled at University College, Ibadan, and graduated with a first class honours degree in mathematics in 1963. He then proceeded to the University of Keele, where he obtained his Ph.D. in mathematics in 1967. He later obtained his D.Sc. in mathematics from the same university in 1987. == Academic career == Iyahen commenced his academic journey as a mathematics lecturer at the University of Ibadan in 1965. He progressed through the ranks, achieving senior lecturer status in 1969 and professorship in 1974. He was the Head of Department for Mathematics from 1976 to 1978 and Dean of the Faculty of Science from 1978 to 1980. In 1980, he joined the Institute of Technology, Benin (later renamed the University of Benin), where he served as the founding dean of the Faculty of Physical Sciences and director of the Centre for Mathematical Sciences, Abuja, Nigeria. He contributed as a visiting professor to various institutions, including the University of Lagos, University of Jos, University of Port Harcourt, University of Ilorin, University of Nigeria, Nsukka, University of Cape Coast, University of Khartoum, and the University of Waterloo. He published over 100 mathematics-related papers in international journals, served as the editor-in-chief for Afrika Mathematika and Journal of the Nigerian Mathematical Society, and chaired the board of Federal Polytechnic, Idah. He was a fellow of the Nigerian Academy of Science and the Mathematical Association of Nigeria, Iyahen was also a member of the London Mathematical Society, the American Mathematical Society, and the International Mathematical Union. == Political career == Iyahen was a two-time senator of the Federal Republic of Nigeria. He represented Bendel Central Senatorial District under the platform of the National Party of Nigeria (NPN) in the second republic (October to December 1983) and the Social Democratic Party (SDP) in the third republic (August 1992 to November 1993). He served in different capacities in the senate, such as the chairman of the Committee on Education, Science and Technology, and the vice-chairman of the Committee on Finance and Appropriation. == Personal life and death == Iyahen married Veronica Aigboduwa Osagie on 25 September 1967. They had six children and eleven grandchildren. Iyahen died on 28 January 2018 in Benin City, Edo State, Nigeria. He was buried on 16 February 2018 at his residence in Benin City. == Selected publications == ——— (1967). "Some remarks on countably barrelled and countably quasibarrelled spaces". Proceedings of the Edinburgh Mathematical Society. 15 (4). Cambridge University Press (CUP): 295–296. doi:10.1017/s0013091500011949. ISSN 0013-0915. ——— (30 September 2017). "Mathematics, Science, And Cultural Change". Proceedings of the Nigerian Academy of Science. 10 (1). Nigerian Academy of Science. doi:10.57046/eqbu2235. ISSN 0794-7976. ——— (11 August 2010). "Boundedly barrelled spaces and the open mapping theorem". Portugaliae Mathematica. 46 (3). Sociedade Portuguesa de Matemática: 305–312. ISSN 0032-5155. ——— (1978). "The range space in a closed graph theorem". Acta Mathematica Academiae Scientiarum Hungaricae. 32 (1–2). Springer Science and Business Media LLC: 17–20. doi:10.1007/bf01902196. ISSN 0001-5954. ——— (1968). "Semiconvex spaces". Glasgow Mathematical Journal. 9 (2). Cambridge University Press (CUP): 111–118. doi:10.1017/s0017089500000380. ISSN 0017-0895. ——— (1969). "Semiconvex spaces II". Glasgow Mathematical Journal. 10 (2). Cambridge University Press (CUP): 103–105. doi:10.1017/s001708950000063x. ISSN 0017-0895. ——— (1972). "A note on the filter condition". Acta Mathematica Academiae Scientiarum Hungaricae. 23 (3–4): 271–274. doi:10.1007/BF01896945. ISSN 0001-5954. ——— (1968). "-spaces and the closed-graph theorem". Proceedings of the Edinburgh Mathematical Society. 16 (2). Cambridge University Press (CUP): 89–99. doi:10.1017/s0013091500012463. ISSN 0013-0915. ———; Popoola, J. O. (1973). "A generalized inductive limit topology for linear spaces". Glasgow Mathematical Journal. 14 (2). Cambridge University Press (CUP): 105–110. doi:10.1017/s001708950000183x. ISSN 0017-0895. ——— (1971). "Bounded generators in linear topological spaces". Glasgow Mathematical Journal. 12 (2). Cambridge University Press (CUP): 105–109. doi:10.1017/s001708950000121x. ISSN 0017-0895. ——— (1969). "Ultrabarrelled groups and the closed graph theorem". Mathematical Proceedings of the Cambridge Philosophical Society. 65 (1). Cambridge University Press (CUP): 53–58. Bibcode:1969PCPS...65...53I. doi:10.1017/s0305004100044078. ISSN 0305-0041. == References == === Citations === === Sources === == External links == Sunday Iyahen at the Mathematics Genealogy Project O'Connor, John J.; Robertson, Edmund F., "Sunday Iyahen", MacTutor History of Mathematics Archive, University of St Andrews
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Wikipedia:Sunzi Suanjing#0
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Sunzi Suanjing (Chinese: 孫子算經; pinyin: Sūnzǐ Suànjīng; Wade–Giles: Sun Tzu Suan Ching; lit. 'The Mathematical Classic of Master Sun/Master Sun's Mathematical Manual') was a mathematical treatise written during 3rd to 5th centuries CE which was listed as one of the Ten Computational Canons during the Tang dynasty. The specific identity of its author Sunzi (lit. "Master Sun") is still unknown but he lived much later than his namesake Sun Tzu, author of The Art of War. From the textual evidence in the book, some scholars concluded that the work was completed during the Southern and Northern Dynasties. Besides describing arithmetic methods and investigating Diophantine equations, the treatise touches upon astronomy and attempts to develop a calendar. == Contents == The book is divided into three chapters. === Chapter 1 === Chapter 1 discusses measurement units of length, weight and capacity, and the rules of counting rods. Although counting rods were in use in the Spring and Autumn period and there were many ancient books on mathematics such as Book on Numbers and Computation and The Nine Chapters on the Mathematical Art, no detailed account of the rules was given. For the first time, The Mathematical Classic of Sun Zi provided a detail description of the rules of counting rods: "one must know the position of the counting rods, the units are vertical, the tens horizontal, the hundreds stand, the thousands prostrate", followed by the detailed layout and rules for manipulation of the counting rods in addition, subtraction, multiplication, and division with ample examples. === Chapter 2 === Chapter 2 deals with operational rules for fractions with rod numerals: the reduction, addition, subtraction, and division of fractions, followed by mechanical algorithm for the extraction of square roots. === Chapter 3 === Chapter 3 contains the earliest example of the Chinese remainder theorem, a key tool to understanding and resolving Diophantine equations. == Bibliography == Researchers have published a full English translation of the Sūnzĭ Suànjīng: Fleeting Footsteps; Tracing the Conception of Arithmetic and Algebra in Ancient China, by Lam Lay Yong and Ang Tian Se, Part Two, pp 149–182. World Scientific Publishing Company; June 2004 ISBN 981-238-696-3 The original Chinese text is available on Wikisource. == External links == Sun Zi at MacTutor == References ==
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