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Wikipedia:Panagiotis E. Souganidis#0
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Panagiotis E. Souganidis (Παναγιώτης E. Σουγανίδης) is an American mathematician, specializing in partial differential equations. == Biography == Souganidis graduated in 1981 with B.A. from the National and Kapodistrian University of Athens. At the University of Wisconsin–Madison he graduated with M.A. in 1981 and Ph.D. in 1984 with thesis under the supervision of Michael G. Crandall. Souganidis was a postdoc in 1984–1985 at the University of Minnesota's Institute for Mathematics and its Applications and was at the Institute for Advanced Study in 1988 and 1990. After holding professorships at Brown University, the University of Wisconsin–Madison, and the University of Texas at Austin, he became in 2008 the Charles H. Swift Distinguished Service Professor in Mathematics at the University of Chicago. He has held visiting positions at academic institutions in Italy, Japan, Greece, France, the UK, and Sweden. He works on non-linear partial differential equations, and stochastic analysis. The main parts of his work are qualitative properties of viscosity and entropy solutions, front propagation and asymptotic behavior of reaction diffusion equations and particle systems, stochastic homogenization, the theory of pathwise solutions for first and second order partial differential equations, including stochastic Hamilton-Jacobi equations and scalar conservation laws. Souganidis is the author or co-author of over 100 publications in refereed journals. His wife is Thaleia Zariphopoulou, a Greek-American mathematician and professor at the University of Texas at Austin. == Awards and honors == 1989 — Sloan Research Fellow 1994 — Invited Speaker of the International Congress of Mathematicians 2003 — Highly Cited Researcher 2012 — Fellow of the American Mathematical Society (Class of 2013) 2015 — Fellow of the Society for Industrial and Applied Mathematics 2017 — Fellow of the American Association for the Advancement of Science 2019 — Invited Speaker of the International Congress on Industrial and Applied Mathematics == Selected publications == Evans, L. C.; Souganidis, P. E. (1984). "Differential Games and Representation Formulas for Solutions of Hamilton-Jacobi-Isaacs Equations". Indiana University Mathematics Journal. 33 (5): 773–797. doi:10.1512/iumj.1984.33.33040. JSTOR 45010271. Souganidis, Panagiotis E. (1985). "Approximation schemes for viscosity solutions of Hamilton-Jacobi equations". Journal of Differential Equations. 59 (1): 1–43. Bibcode:1985JDE....59....1S. doi:10.1016/0022-0396(85)90136-6. Lions, P.-L.; Souganidis, P. E. (1985). "Differential Games, Optimal Control and Directional Derivatives of Viscosity Solutions of Bellman's and Isaacs' Equations". SIAM Journal on Control and Optimization. 23 (4): 566–583. doi:10.1137/0323036. Bona, J. L.; Souganidis, P. E.; Strauss, W. A. (1987). "Stability and Instability of Solitary Waves of Korteweg-de Vries Type". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 411 (1841): 395–412. Bibcode:1987RSPSA.411..395B. doi:10.1098/rspa.1987.0073. S2CID 120894859. Fleming, W. H.; Souganidis, P. E. (1989). "On the Existence of Value Functions of Two-Player, Zero-Sum Stochastic Differential Games". Indiana University Mathematics Journal. 38 (2): 293–314. doi:10.1512/iumj.1989.38.38015. JSTOR 24895386. Bona, J. L.; Souganidis, P. E.; Strauss, W. A. (1987). "Stability and Instability of Solitary Waves of Korteweg-de Vries Type". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 411 (1841): 395–412. Bibcode:1987RSPSA.411..395B. doi:10.1098/rspa.1987.0073. S2CID 120894859. Barles, G.; Soner, H. M.; Souganidis, P. E. (1993). "Front Propagation and Phase Field Theory". SIAM Journal on Control and Optimization. 31 (2): 439–469. doi:10.1137/0331021. S2CID 39091910. Lions, Pierre-Louis; Perthame, Benoît; Souganidis, Panagiotis E. (1998). "Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates". Communications on Pure and Applied Mathematics. 49 (6): 599–638. doi:10.1002/(SICI)1097-0312(199606)49:6<599::AID-CPA2>3.0.CO;2-5. Barles, Guy; Souganidis, Panagiotis E. (1998). "A New Approach to Front Propagation Problems: Theory and Applications". Archive for Rational Mechanics and Analysis. 141 (3): 237–296. Bibcode:1998ArRMA.141..237B. doi:10.1007/s002050050077. S2CID 121972598. Lions, Pierre-Louis; Souganidis, Panagiotis E. (2003). "Correctors for the homogenization of Hamilton-Jacobi equations in the stationary ergodic setting". Communications on Pure and Applied Mathematics. 56 (10): 1501–1524. doi:10.1002/cpa.10101. S2CID 121828182. == References == == External links == International Centre for Theoretical Physics (ICTP) talks by Panagiotis Souganidis, 2018 "Phase-field models for motion by mean curvature - 1". YouTube. ICTP Mathematics. 11 June 2018. "Phase-field models for motion by mean curvature - 2". YouTube. ICTP Mathematics. 12 June 2018. "Phase-field models for motion by mean curvature - 3". YouTube. ICTP Mathematics. 12 June 2018. "Phase-field models for motion by mean curvature - 4". YouTube. ICTP Mathematics. 14 June 2018.
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Wikipedia:Paneitz operator#0
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In the mathematical field of differential geometry, the Paneitz operator is a fourth-order differential operator defined on a Riemannian manifold of dimension n. It is named after Stephen Paneitz, who discovered it in 1983, and whose preprint was later published posthumously in Paneitz 2008. In fact, the same operator was found earlier in the context of conformal supergravity by E. Fradkin and A. Tseytlin in 1982 (Phys Lett B 110 (1982) 117 and Nucl Phys B 1982 (1982) 157 ). It is given by the formula P = Δ 2 − δ { ( n − 2 ) J − 4 V ⋅ } d + ( n − 4 ) Q {\displaystyle P=\Delta ^{2}-\delta \left\{(n-2)J-4V\cdot \right\}d+(n-4)Q} where Δ is the Laplace–Beltrami operator, d is the exterior derivative, δ is its formal adjoint, V is the Schouten tensor, J is the trace of the Schouten tensor, and the dot denotes tensor contraction on either index. Here Q is the scalar invariant ( − 4 | V | 2 + n J 2 + 2 Δ J ) / 4 , {\displaystyle (-4|V|^{2}+nJ^{2}+2\Delta J)/4,} where Δ is the positive Laplacian. In four dimensions this yields the Q-curvature. The operator is especially important in conformal geometry, because in a suitable sense it depends only on the conformal structure. Another operator of this kind is the conformal Laplacian. But, whereas the conformal Laplacian is second-order, with leading symbol a multiple of the Laplace–Beltrami operator, the Paneitz operator is fourth-order, with leading symbol the square of the Laplace–Beltrami operator. The Paneitz operator is conformally invariant in the sense that it sends conformal densities of weight 2 − n/2 to conformal densities of weight −2 − n/2. Concretely, using the canonical trivialization of the density bundles in the presence of a metric, the Paneitz operator P can be represented in terms of a representative the Riemannian metric g as an ordinary operator on functions that transforms according under a conformal change g ↦ Ω2g according to the rule Ω n / 2 + 2 P ( g ) ϕ = P ( Ω 2 g ) Ω n / 2 − 2 ϕ . {\displaystyle \Omega ^{n/2+2}P(g)\phi =P(\Omega ^{2}g)\Omega ^{n/2-2}\phi .\,} The operator was originally derived by working out specifically the lower-order correction terms in order to ensure conformal invariance. Subsequent investigations have situated the Paneitz operator into a hierarchy of analogous conformally invariant operators on densities: the GJMS operators. The Paneitz operator has been most thoroughly studied in dimension four where it appears naturally in connection with extremal problems for the functional determinant of the Laplacian (via the Polyakov formula; see Branson & Ørsted 1991). In dimension four only, the Paneitz operator is the "critical" GJMS operator, meaning that there is a residual scalar piece (the Q curvature) that can only be recovered by asymptotic analysis. The Paneitz operator appears in extremal problems for the Moser–Trudinger inequality in dimension four as well (Chang 1999) == CR Paneitz operator == There is a close connection between 4 dimensional Conformal Geometry and 3 dimensional CR geometry associated with the study of CR manifolds. There is a naturally defined fourth order operator on CR manifolds introduced by C. Robin Graham and John Lee that has many properties similar to the classical Paneitz operator defined on 4 dimensional Riemannian manifolds. This operator in CR geometry is called the CR Paneitz operator. The operator defined by Graham and Lee though defined on all odd dimensional CR manifolds, is not known to be conformally covariant in real dimension 5 and higher. The conformal covariance of this operator has been established in real dimension 3 by Kengo Hirachi. It is always a non-negative operator in real dimension 5 and higher. Here unlike changing the metric by a conformal factor as in the Riemannian case discussed above, one changes the contact form on the CR 3 manifold by a conformal factor. Non-negativity of the CR Paneitz operator in dimension 3 is a CR invariant condition as proved below. This follows by the conformal covariant properties of the CR Paneitz operator first observed by Kengo Hirachi. Furthermore, the CR Paneitz operator plays an important role in obtaining the sharp eigenvalue lower bound for Kohn's Laplacian. This is a result of Sagun Chanillo, Hung-Lin Chiu and Paul C. Yang. This sharp eigenvalue lower bound is the exact analog in CR Geometry of the famous André Lichnerowicz lower bound for the Laplace–Beltrami operator on compact Riemannian manifolds. It allows one to globally embed, compact, strictly pseudoconvex, abstract CR manifolds into C n {\displaystyle C^{n}} . More precisely, the conditions in [3] to embed a CR manifold into C n {\displaystyle C^{n}} are phrased CR invariantly and non-perturbatively. There is also a partial converse of the above result where the authors, J. S. Case, S. Chanillo, P. Yang, obtain conditions that guarantee when embedded, compact CR manifolds have non-negative CR Paneitz operators. The formal definition of the CR Paneitz operator P 4 {\displaystyle P_{4}} on CR manifolds of real dimension three is as follows( the subscript 4 {\displaystyle 4} is to remind the reader that this is a fourth order operator) P 4 ϕ = 1 8 ( ( ◻ b ◻ b ¯ + ◻ b ¯ ◻ b ) ϕ + 8 I m ( A 11 ϕ 1 ) 1 ) {\displaystyle P_{4}\phi ={\frac {1}{8}}((\Box _{b}{\overline {\Box _{b}}}+{\overline {\Box _{b}}}\Box _{b})\phi +8Im(A^{11}\phi _{1})_{1})} ◻ b {\displaystyle \Box _{b}} denotes the Kohn Laplacian which plays a fundamental role in CR Geometry and several complex variables and was introduced by Joseph J. Kohn. One may consult The tangential Cauchy–Riemann complex (Kohn Laplacian, Kohn–Rossi complex) for the definition of the Kohn Laplacian. Further, A 11 {\displaystyle A^{11}} denotes the Webster-Tanaka Torsion tensor and ϕ 1 {\displaystyle \phi _{1}} the covariant derivative of the function ϕ {\displaystyle \phi } with respect to the Webster-Tanaka connection. Accounts of the Webster-Tanaka, connection, Torsion and curvature tensor may be found in articles by John M. Lee and Sidney M. Webster. There is yet another way to view the CR Paneitz operator in dimension 3. John M. Lee constructed a third order operator P 3 {\displaystyle P_{3}} which has the property that the kernel of P 3 {\displaystyle P_{3}} consists of exactly the CR pluriharmonic functions (real parts of CR holomorphic functions). The Paneitz operator displayed above is exactly the divergence of this third order operator P 3 {\displaystyle P_{3}} . The third order operator P 3 {\displaystyle P_{3}} is defined as follows: P 3 ϕ = ( ϕ 1 ¯ 1 ¯ 1 + − 1 A 11 ϕ 1 ) θ 1 {\displaystyle P_{3}\phi =({{\phi _{\bar {1}}}^{\bar {1}}}_{1}+{\sqrt {-1}}A_{11}\phi ^{1})\theta ^{1}} Here A 11 {\displaystyle A_{11}} is the Webster-Tanaka torsion tensor. The derivatives are taken using the Webster-Tanaka connection and θ 1 {\displaystyle \theta ^{1}} is the dual 1-form to the CR-holomorphic tangent vector that defines the CR structure on the compact manifold. Thus P 3 {\displaystyle P_{3}} sends functions to ( 1 , 0 ) {\displaystyle (1,0)} forms. The divergence of such an operator thus will take functions to functions. The third order operator constructed by J. Lee only characterizes CR pluriharmonic functions on CR manifolds of real dimension three. Hirachi's covariant transformation formula for P 4 {\displaystyle P_{4}} on three dimensional CR manifolds is as follows. Let the CR manifold be ( M , θ , J ) {\displaystyle (M,\theta ,J)} , where θ {\displaystyle \theta } is the contact form and J {\displaystyle J} the CR structure on the kernel of θ {\displaystyle \theta } that is on the contact planes. Let us transform the background contact form θ {\displaystyle \theta } by a conformal transformation to θ ~ = e 2 f θ {\displaystyle {\tilde {\theta }}=e^{2f}\theta } . Note this new contact form obtained by a conformal change of the old contact form or background contact form, has not changed the kernel of θ {\displaystyle \theta } . That is θ ~ {\displaystyle {\tilde {\theta }}} and θ {\displaystyle \theta } have the same kernel, i.e. the contact planes have remained unchanged. The CR structure J {\displaystyle J} has been kept unchanged. The CR Paneitz operator P ~ 4 {\displaystyle {\tilde {P}}_{4}} for the new contact form θ ~ {\displaystyle {\tilde {\theta }}} is now seen to be related to the CR Paneitz operator for the contact form θ {\displaystyle \theta } by the formula of Hirachi: P ~ 4 = e − 4 f P 4 {\displaystyle {\tilde {P}}_{4}=e^{-4f}P_{4}} Next note the volume forms on the manifold M {\displaystyle M} satisfy d V ~ = θ ~ ∧ d θ ~ = e 4 f θ ∧ d θ = e 4 f d V {\displaystyle d{\tilde {V}}={\tilde {\theta }}\wedge d{\tilde {\theta }}=e^{4f}\theta \wedge d\theta =e^{4f}dV} Using the transformation formula of Hirachi, it follows that, ∫ M P ~ 4 ϕ ϕ d V ~ = ∫ M P 4 ϕ ϕ d V {\displaystyle \int _{M}{\tilde {P}}_{4}\phi \phi d{\tilde {V}}=\int _{M}P_{4}\phi \phi dV} Thus we easily conclude that: ∫ M P 4 ϕ ϕ d V {\displaystyle \int _{M}P_{4}\phi \phi dV} is a CR invariant. That is the integral displayed above has the same value for different contact forms describing the same CR structure J {\displaystyle J} . The operator P 4 {\displaystyle P_{4}} is a real self-adjoint operator. On CR manifolds like S 3 {\displaystyle S^{3}} where the Webster-Tanaka torsion tensor is zero, it is seen from the formula displayed above that only the leading terms involving the Kohn Laplacian survives. Next from the tensor commutation formulae given in [5], one can easily check that the operators ◻ b , ◻ b ¯ {\displaystyle \Box _{b},{\overline {\Box _{b}}}} commute when the Webster-Tanaka torsion tensor A 11 {\displaystyle A_{11}} vanishes. More precisely one has [ ◻ b , ◻ b ¯ ] = 4 − 1 I m Q {\displaystyle [\Box _{b},{\overline {\Box _{b}}}]=4{\sqrt {-1}}ImQ} where Q ϕ = 2 − 1 ( A 11 ϕ 1 ) 1 {\displaystyle Q\phi =2{\sqrt {-1}}(A_{11}\phi _{1})_{1}} Thus ◻ b , ◻ b ¯ {\displaystyle \Box _{b},{\overline {\Box _{b}}}} are simultaneously diagonalizable under the zero torsion assumption. Next note that ◻ b {\displaystyle \Box _{b}} and ◻ b ¯ {\displaystyle {\overline {\Box _{b}}}} have the same sequence of eigenvalues that are also perforce real. Thus we conclude from the formula for P 4 {\displaystyle P_{4}} that CR structures having zero torsion have CR Paneitz operators that are non-negative. The article [4] among other things shows that real ellipsoids in C 2 {\displaystyle C^{2}} carry a CR structure inherited from the complex structure of C 2 {\displaystyle C^{2}} whose CR Paneitz operator is non-negative. This CR structure on ellipsoids has non-vanishing Webster-Tanaka torsion. Thus [4] provides the first examples of CR manifolds where the CR Paneitz operator is non-negative and the Torsion tensor too does not vanish. Since we have observed above that the CR Paneitz is the divergence of an operator whose kernel is the pluriharmonic functions, it also follows that the kernel of the CR Paneitz operator contains all CR Pluriharmonic functions. So the kernel of the CR Paneitz operator in sharp contrast to the Riemannian case, has an infinite dimensional kernel. Results on when the kernel is exactly the pluriharmonic functions, the nature and role of the supplementary space in the kernel etc., may be found in the article cited as [4] below. One of the principal applications of the CR Paneitz operator and the results in [3] are to the CR analog of the Positive Mass theorem due to Jih-Hsin Cheng, Andrea Malchiodi and Paul C. Yang. This allows one to obtain results on the CR Yamabe problem. More facts related to the role of the CR Paneitz operator in CR geometry can be obtained from the article CR manifold. == See also == Calabi conjecture Monge-Ampere equations Positive mass conjecture Yamabe conjecture == References == Branson, Thomas P.; Ørsted, Bent (1991), "Explicit functional determinants in four dimensions", Proceedings of the American Mathematical Society, 113 (3): 669–682, doi:10.2307/2048601, ISSN 0002-9939, JSTOR 2048601, MR 1050018. Chang, Sun-Yung A. (1999), "A fourth order differential operator in conformal geometry", in M. Christ; C. Kenig; C. Sadorsky (eds.), Harmonic Analysis and Partial Differential Equations; Essays in honor of Alberto P. Calderón, Chicago Lectures in Mathematics, pp. 127–150. Paneitz, Stephen M. (2008), "A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds (summary)", Symmetry, Integrability and Geometry: Methods and Applications, 4: Paper 036, 3, arXiv:0803.4331, Bibcode:2008SIGMA...4..036P, doi:10.3842/SIGMA.2008.036, ISSN 1815-0659, MR 2393291, S2CID 115155901.
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Wikipedia:Paola Antonietti#0
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Paola F. Antonietti (born 1980) is an Italian applied mathematician and numerical analyst whose research concerns numerical methods for partial differential equations, and particularly domain decomposition methods, with applications in scientific computing and Applied Sciences such as the computational modelling of neurodegenerative disorders and simulating the propagation of seismic waves. She is Professor of Numerical Analysis at the Polytechnic University of Milan. == Education and career == Antonietti was born in 1980 in Milan. She earned a laurea cum laude from the University of Pavia in 2003, and continued there for a Ph.D. in 2007. Her dissertation, Domain decomposition, spectral correctness and numerical testing of discontinuous Galerkin methods, was jointly supervised by Annalisa Buffa and Ilaria Perugia, and included time as a visiting student at the University of Oxford. After postdoctoral research at the University of Nottingham, she returned to Italy as a tenure-track assistant professor in the Polytechnic University of Milan in 2008. She was given tenure as an associate professor in 2015, promoted to full professor in 2019, and was appointed head of the Laboratory of Modeling and Scientific Computing (MOX) by Irene Sabadini in 2023. == Recognition == As a student, Antonietti won the S. Cinquini and M. Cinquini Cibrario Prize of the University of Pavia. In 2008, the Italian Society for Industrial Applications of Mathematics (SIMAI) gave her their F. Saleri prize, and she was the winner of the 2015 SIMAI prize. In 2020, the European Community on Computational Methods in Applied Sciences (ECCOMAS) gave her their Jacques-Louis Lions award. She is the recipient of the “NEMESIS” ERC SYNERGY grant 2023. == References == == External links == Home page Paola Antonietti publications indexed by Google Scholar
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Wikipedia:Paola Loreti#0
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Paola Loreti is an Italian mathematician, and a professor of mathematical analysis at Sapienza University of Rome. She is known for her research on Fourier analysis, control theory, and non-integer representations. The Komornik–Loreti constant, the smallest non-integer base for which the representation of 1 is unique, is named after her and Vilmos Komornik. Loreti earned a laurea from Sapienza University in 1984. Her dissertation, Programmazione dinamica ed equazione di Bellman [dynamic programming and the Bellman equation] was supervised by Italo Capuzzo-Dolcetta. With Vilmos Komornik, Loreti is the author of the book Fourier Series in Control Theory (Springer, 2005). == References == == External links == Home page
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Wikipedia:Paolo Ruffini#0
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Paolo Ruffini (22 September 1765 – 10 May 1822) was an Italian mathematician and philosopher. == Education and career == By 1788 he had earned university degrees in philosophy, medicine/surgery and mathematics. His works include developments in algebra: an incomplete proof (Abel–Ruffini theorem) that quintic (and higher-order) equations cannot be solved by radicals (1799). Abel would complete the proof in 1824. Ruffini's rule, which is a quick method for polynomial division. contributions to group theory. He also wrote on probability and the quadrature of the circle. He was a professor of mathematics at the University of Modena and a medical doctor including scientific work on typhus. == Group theory == In 1799 Ruffini marked a major improvement for group theory, developing Joseph-Louis Lagrange's work on permutation theory ("Réflexions sur la théorie algébrique des équations", 1770–1771). Lagrange's work was largely ignored until Ruffini established strong connections between permutations and the solvability of algebraic equations. Ruffini was the first to assert, controversially, the unsolvability by radicals of algebraic equations higher than quartics, which angered many members of the community such as Gian Francesco Malfatti (1731–1807). Work in that area was later carried on by those such as Abel and Galois, who succeeded in such a proof. == Publications == 1799: Teoria Generale delle Equazioni, in cui si dimostra impossibile la soluzione algebraica delle equazioni generali di grado superiore al quarto ("General Theory of equations, in which the algebraic solution of general equations of degree higher than four is proven impossible") 1802: Riflessioni intorno alla rettificazione ed alla quadratura del circulo ("reflections on the rectification and the squaring of the circle") 1802: Della soluzione delle equazioni algebraiche determinate particolari di grado superiore al quarto ("on the solution of certain determined algebraic equations of degree higher than four") 1804: Sopra la determinazione delle radici nelle equazioni numeriche di qualunque grado ("about the determination of the roots in the numerical equations of any degree") 1806: Della immortalità dell’anima ("on the immortality of the soul") 1807: Algebra elementare ("elementary algebra") 1813: Riflessioni intorno alla soluzione delle equazioni algebraiche generali ("reflections on the algebraic solutions of equations") 1820: Memoria sul tifo contagioso ("essay on contagious typhoid") 1821: Riflessioni critiche sopra il saggio filosofico intorno alle probabilità del signor conte Laplace ("critical reflections on the philosophical essay about probability by Count Laplace") == See also == 8524 Paoloruffini, asteroid named after him == References == == External links == This article incorporates text from a publication now in the public domain: Herbermann, Charles, ed. (1913). "Paolo Ruffini". Catholic Encyclopedia. New York: Robert Appleton Company. Pepe, Luigi (2017). "RUFFINI, Paolo". Dizionario Biografico degli Italiani, Volume 89: Rovereto–Salvemini (in Italian). Rome: Istituto dell'Enciclopedia Italiana. ISBN 978-8-81200032-6.
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Wikipedia:Pappus chain#0
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In geometry, the Pappus chain is a ring of circles between two tangent circles investigated by Pappus of Alexandria in the 3rd century AD. == Construction == The arbelos is defined by two circles, CU and CV, which are tangent at the point A and where CU is enclosed by CV. Let the radii of these two circles be denoted as rU, rV, respectively, and let their respective centers be the points U, V. The Pappus chain consists of the circles in the shaded grey region, which are externally tangent to CU (the inner circle) and internally tangent to CV (the outer circle). Let the radius, diameter and center point of the nth circle of the Pappus chain be denoted as rn, dn, Pn, respectively. == Properties == === Centers of the circles === ==== Ellipse ==== All the centers of the circles in the Pappus chain are located on a common ellipse, for the following reason. The sum of the distances from the nth circle of the Pappus chain to the two centers U, V of the arbelos circles equals a constant P n U ¯ + P n V ¯ = ( r U + r n ) + ( r V − r n ) = r U + r V {\displaystyle {\overline {P_{n}U}}+{\overline {P_{n}V}}=(r_{U}+r_{n})+(r_{V}-r_{n})=r_{U}+r_{V}} Thus, the foci of this ellipse are U, V, the centers of the two circles that define the arbelos; these points correspond to the midpoints of the line segments AB, AC, respectively. ==== Coordinates ==== If r = A C ¯ A B ¯ , {\displaystyle r={\tfrac {\overline {AC}}{\overline {AB}}},} then the center of the nth circle in the chain is: ( x n , y n ) = ( r ( 1 + r ) 2 [ n 2 ( 1 − r ) 2 + r ] , n r ( 1 − r ) n 2 ( 1 − r ) 2 + r ) {\displaystyle (x_{n},y_{n})=\left({\frac {r(1+r)}{2[n^{2}(1-r)^{2}+r]}}~,~{\frac {nr(1-r)}{n^{2}(1-r)^{2}+r}}\right)} === Radii of the circles === If r = A C ¯ A B ¯ , {\displaystyle r={\tfrac {\overline {AC}}{\overline {AB}}},} then the radius of the nth circle in the chain is: r n = ( 1 − r ) r 2 [ n 2 ( 1 − r ) 2 + r ] {\displaystyle r_{n}={\frac {(1-r)r}{2[n^{2}(1-r)^{2}+r]}}} === Circle inversion === The height hn of the center of the nth circle above the base diameter ACB equals n times dn. This may be shown by inverting in a circle centered on the tangent point A. The circle of inversion is chosen to intersect the nth circle perpendicularly, so that the nth circle is transformed into itself. The two arbelos circles, CU and CV, are transformed into parallel lines tangent to and sandwiching the nth circle; hence, the other circles of the Pappus chain are transformed into similarly sandwiched circles of the same diameter. The initial circle C0 and the final circle Cn each contribute 1/2dn to the height hn, whereas the circles C1 to Cn−1 each contribute dn. Adding these contributions together yields the equation hn = ndn. The same inversion can be used to show that the points where the circles of the Pappus chain are tangent to one another lie on a common circle. As noted above, the inversion centered at point A transforms the arbelos circles CU, CV into two parallel lines, and the circles of the Pappus chain into a stack of equally sized circles sandwiched between the two parallel lines. Hence, the points of tangency between the transformed circles lie on a line midway between the two parallel lines. Undoing the inversion in the circle, this line of tangent points is transformed back into a circle. === Steiner chain === In these properties of having centers on an ellipse and tangencies on a circle, the Pappus chain is analogous to the Steiner chain, in which finitely many circles are tangent to two circles. == References == == Bibliography == Ogilvy, C. S. (1990). Excursions in Geometry. Dover. pp. 54–55. ISBN 0-486-26530-7. Bankoff, L. (1981). "How did Pappus do it?". In Klarner, D. A. (ed.). The Mathematical Gardner. Boston: Prindle, Weber, & Schmidt. pp. 112–118. Johnson, R. A. (1960). Advanced Euclidean Geometry: An elementary treatise on the geometry of the triangle and the circle (reprint of 1929 edition by Houghton Mifflin ed.). New York: Dover Publications. pp. 116–117. ISBN 978-0-486-46237-0. {{cite book}}: ISBN / Date incompatibility (help) Wells, D. (1991). The Penguin Dictionary of Curious and Interesting Geometry. New York: Penguin Books. pp. 5–6. ISBN 0-14-011813-6. == External links == Floer van Lamoen and Eric W. Weisstein. "Pappus Chain". MathWorld. Tan, Stephen. "Arbelos".
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Wikipedia:Pappus configuration#0
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In geometry, the Pappus configuration is a configuration of nine points and nine lines in the Euclidean plane, with three points per line and three lines through each point. == History and construction == This configuration is named after Pappus of Alexandria. Pappus's hexagon theorem states that every two triples of collinear points ABC and abc (none of which lie on the intersection of the two lines) can be completed to form a Pappus configuration, by adding the six lines Ab, aB, Ac, aC, Bc, and bC, and their three intersection points X = Ab · aB, Y = Ac · aC, and Z = Bc · bC. These three points are the intersection points of the "opposite" sides of the hexagon AbCaBc. According to Pappus' theorem, the resulting system of nine points and eight lines always has a ninth line containing the three intersection points X, Y, and Z, called the Pappus line. The Pappus configuration can also be derived from two triangles △XcC and △YbB that are in perspective with each other (the three lines through corresponding pairs of points meet at a single crossing point) in three different ways, together with their three centers of perspectivity Z, a, and A. The points of the configuration are the points of the triangles and centers of perspectivity, and the lines of the configuration are the lines through corresponding pairs of points. == Related constructions == The Levi graph of the Pappus configuration is known as the Pappus graph. It is a bipartite symmetric cubic graph with 18 vertices and 27 edges. Adding three more parallel lines to the Pappus configuration, through each triple of points that are not already connected by lines of the configuration, produces the Hesse configuration. Like the Pappus configuration, the Desargues configuration can be defined in terms of perspective triangles, and the Reye configuration can be defined analogously from two tetrahedra that are in perspective with each other in four different ways, forming a desmic system of tetrahedra. For any nonsingular cubic plane curve in the Euclidean plane, three real inflection points of the curve, and a fourth point on the curve, there is a unique way of completing these four points to form a Pappus configuration in such a way that all nine points lie on the curve. == Applications == A variant of the Pappus configuration provides a solution to the orchard-planting problem, the problem of finding sets of points that have the largest possible number of lines through three points. The nine points of the Pappus configuration form only nine three-point lines. However, they can be arranged so that there is another three-point line, making a total of ten. This is the maximum possible number of three-point lines through nine points. == References == == External links == Weisstein, Eric W., "Pappus Configuration", MathWorld
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Wikipedia:Pappus's area theorem#0
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Pappus's area theorem describes the relationship between the areas of three parallelograms attached to three sides of an arbitrary triangle. The theorem, which can also be thought of as a generalization of the Pythagorean theorem, is named after the Greek mathematician Pappus of Alexandria (4th century AD), who discovered it. == Theorem == Given an arbitrary triangle with two arbitrary parallelograms attached to two of its sides the theorem tells how to construct a parallelogram over the third side, such that the area of the third parallelogram equals the sum of the areas of the other two parallelograms. Let ABC be the arbitrary triangle and ABDE and ACFG the two arbitrary parallelograms attached to the triangle sides AB and AC. The extended parallelogram sides DE and FG intersect at H. The line segment AH now "becomes" the side of the third parallelogram BCML attached to the triangle side BC, i.e., one constructs line segments BL and CM over BC, such that BL and CM are a parallel and equal in length to AH. The following identity then holds for the areas (denoted by A) of the parallelograms: A A B D E + A A C F G = A B C M L {\displaystyle {\text{A}}_{ABDE}+{\text{A}}_{ACFG}={\text{A}}_{BCML}} The theorem generalizes the Pythagorean theorem twofold. Firstly it works for arbitrary triangles rather than only for right angled ones and secondly it uses parallelograms rather than squares. For squares on two sides of an arbitrary triangle it yields a parallelogram of equal area over the third side and if the two sides are the legs of a right angle the parallelogram over the third side will be square as well. For a right-angled triangle, two parallelograms attached to the legs of the right angle yield a rectangle of equal area on the third side and again if the two parallelograms are squares then the rectangle on the third side will be a square as well. == Proof == Due to having the same base length and height the parallelograms ABDE and ABUH have the same area, the same argument applying to the parallelograms ACFG and ACVH, ABUH and BLQR, ACVH and RCMQ. This already yields the desired result, as we have: A A B D E + A A C F G = A A B U H + A A C V H = A B L Q R + A R C M Q = A B C M L {\displaystyle {\begin{aligned}{\text{A}}_{ABDE}+{\text{A}}_{ACFG}&={\text{A}}_{ABUH}+{\text{A}}_{ACVH}\\&={\text{A}}_{BLQR}+{\text{A}}_{RCMQ}\\&={\text{A}}_{BCML}\end{aligned}}} == References == Howard Eves: Pappus's Extension of the Pythagorean Theorem.The Mathematics Teacher, Vol. 51, No. 7 (November 1958), pp. 544–546 (JSTOR) Howard Eves: Great Moments in Mathematics (before 1650). Mathematical Association of America, 1983, ISBN 9780883853108, p. 37 (excerpt, p. 37, at Google Books) Eli Maor: The Pythagorean Theorem: A 4,000-year History. Princeton University Press, 2007, ISBN 9780691125268, pp. 58–59 (excerpt, p. 58, at Google Books) Claudi Alsina, Roger B. Nelsen: Charming Proofs: A Journey Into Elegant Mathematics. MAA, 2010, ISBN 9780883853481, pp. 77–78 (excerpt, p. 77, at Google Books) == External links == The Pappus Area Theorem Pappus theorem
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Wikipedia:Pappus's centroid theorem#0
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In mathematics, Pappus's centroid theorem (also known as the Guldinus theorem, Pappus–Guldinus theorem or Pappus's theorem) is either of two related theorems dealing with the surface areas and volumes of surfaces and solids of revolution. The theorems are attributed to Pappus of Alexandria and Paul Guldin. Pappus's statement of this theorem appears in print for the first time in 1659, but it was known before, by Kepler in 1615 and by Guldin in 1640. == The first theorem == The first theorem states that the surface area A of a surface of revolution generated by rotating a plane curve C about an axis external to C and on the same plane is equal to the product of the arc length s of C and the distance d traveled by the geometric centroid of C: A = s d . {\displaystyle A=sd.} For example, the surface area of the torus with minor radius r and major radius R is A = ( 2 π r ) ( 2 π R ) = 4 π 2 R r . {\displaystyle A=(2\pi r)(2\pi R)=4\pi ^{2}Rr.} === Proof === A curve given by the positive function f ( x ) {\displaystyle f(x)} is bounded by two points given by: a ≥ 0 {\displaystyle a\geq 0} and b ≥ a {\displaystyle b\geq a} If d L {\displaystyle dL} is an infinitesimal line element tangent to the curve, the length of the curve is given by: L = ∫ a b d L = ∫ a b d x 2 + d y 2 = ∫ a b 1 + ( d y d x ) 2 d x {\displaystyle L=\int _{a}^{b}dL=\int _{a}^{b}{\sqrt {dx^{2}+dy^{2}}}=\int _{a}^{b}{\sqrt {1+\left({\frac {dy}{dx}}\right)^{2}}}\,dx} The y {\displaystyle y} component of the centroid of this curve is: y ¯ = 1 L ∫ a b y d L = 1 L ∫ a b y 1 + ( d y d x ) 2 d x {\displaystyle {\bar {y}}={\frac {1}{L}}\int _{a}^{b}y\,dL={\frac {1}{L}}\int _{a}^{b}y{\sqrt {1+\left({\frac {dy}{dx}}\right)^{2}}}\,dx} The area of the surface generated by rotating the curve around the x-axis is given by: A = 2 π ∫ a b y d L = 2 π ∫ a b y 1 + ( d y d x ) 2 d x {\displaystyle A=2\pi \int _{a}^{b}y\,dL=2\pi \int _{a}^{b}y{\sqrt {1+\left({\frac {dy}{dx}}\right)^{2}}}\,dx} Using the last two equations to eliminate the integral we have: A = 2 π y ¯ L {\displaystyle A=2\pi {\bar {y}}L} == The second theorem == The second theorem states that the volume V of a solid of revolution generated by rotating a plane figure F about an external axis is equal to the product of the area A of F and the distance d traveled by the geometric centroid of F. (The centroid of F is usually different from the centroid of its boundary curve C.) That is: V = A d . {\displaystyle V=Ad.} For example, the volume of the torus with minor radius r and major radius R is V = ( π r 2 ) ( 2 π R ) = 2 π 2 R r 2 . {\displaystyle V=(\pi r^{2})(2\pi R)=2\pi ^{2}Rr^{2}.} This special case was derived by Johannes Kepler using infinitesimals. === Proof 1 === The area bounded by the two functions: y = f ( x ) , y ≥ 0 {\displaystyle y=f(x),\,\qquad y\geq 0} y = g ( x ) , f ( x ) ≥ g ( x ) {\displaystyle y=g(x),\,\qquad f(x)\geq g(x)} and bounded by the two lines: x = a ≥ 0 {\displaystyle x=a\geq 0} and x = b ≥ a {\displaystyle x=b\geq a} is given by: A = ∫ a b d A = ∫ a b [ f ( x ) − g ( x ) ] d x {\displaystyle A=\int _{a}^{b}dA=\int _{a}^{b}[f(x)-g(x)]\,dx} The x {\displaystyle x} component of the centroid of this area is given by: x ¯ = 1 A ∫ a b x [ f ( x ) − g ( x ) ] d x {\displaystyle {\bar {x}}={\frac {1}{A}}\,\int _{a}^{b}x\,[f(x)-g(x)]\,dx} If this area is rotated about the y-axis, the volume generated can be calculated using the shell method. It is given by: V = 2 π ∫ a b x [ f ( x ) − g ( x ) ] d x {\displaystyle V=2\pi \int _{a}^{b}x\,[f(x)-g(x)]\,dx} Using the last two equations to eliminate the integral we have: V = 2 π x ¯ A {\displaystyle V=2\pi {\bar {x}}A} === Proof 2 === Let A {\displaystyle A} be the area of F {\displaystyle F} , W {\displaystyle W} the solid of revolution of F {\displaystyle F} , and V {\displaystyle V} the volume of W {\displaystyle W} . Suppose F {\displaystyle F} starts in the x z {\displaystyle xz} -plane and rotates around the z {\displaystyle z} -axis. The distance of the centroid of F {\displaystyle F} from the z {\displaystyle z} -axis is its x {\displaystyle x} -coordinate R = ∫ F x d A A , {\displaystyle R={\frac {\int _{F}x\,dA}{A}},} and the theorem states that V = A d = A ⋅ 2 π R = 2 π ∫ F x d A . {\displaystyle V=Ad=A\cdot 2\pi R=2\pi \int _{F}x\,dA.} To show this, let F {\displaystyle F} be in the xz-plane, parametrized by Φ ( u , v ) = ( x ( u , v ) , 0 , z ( u , v ) ) {\displaystyle \mathbf {\Phi } (u,v)=(x(u,v),0,z(u,v))} for ( u , v ) ∈ F ∗ {\displaystyle (u,v)\in F^{*}} , a parameter region. Since Φ {\displaystyle {\boldsymbol {\Phi }}} is essentially a mapping from R 2 {\displaystyle \mathbb {R} ^{2}} to R 2 {\displaystyle \mathbb {R} ^{2}} , the area of F {\displaystyle F} is given by the change of variables formula: A = ∫ F d A = ∬ F ∗ | ∂ ( x , z ) ∂ ( u , v ) | d u d v = ∬ F ∗ | ∂ x ∂ u ∂ z ∂ v − ∂ x ∂ v ∂ z ∂ u | d u d v , {\displaystyle A=\int _{F}dA=\iint _{F^{*}}\left|{\frac {\partial (x,z)}{\partial (u,v)}}\right|\,du\,dv=\iint _{F^{*}}\left|{\frac {\partial x}{\partial u}}{\frac {\partial z}{\partial v}}-{\frac {\partial x}{\partial v}}{\frac {\partial z}{\partial u}}\right|\,du\,dv,} where | ∂ ( x , z ) ∂ ( u , v ) | {\displaystyle \left|{\tfrac {\partial (x,z)}{\partial (u,v)}}\right|} is the determinant of the Jacobian matrix of the change of variables. The solid W {\displaystyle W} has the toroidal parametrization Φ ( u , v , θ ) = ( x ( u , v ) cos θ , x ( u , v ) sin θ , z ( u , v ) ) {\displaystyle {\boldsymbol {\Phi }}(u,v,\theta )=(x(u,v)\cos \theta ,x(u,v)\sin \theta ,z(u,v))} for ( u , v , θ ) {\displaystyle (u,v,\theta )} in the parameter region W ∗ = F ∗ × [ 0 , 2 π ] {\displaystyle W^{*}=F^{*}\times [0,2\pi ]} ; and its volume is V = ∫ W d V = ∭ W ∗ | ∂ ( x , y , z ) ∂ ( u , v , θ ) | d u d v d θ . {\displaystyle V=\int _{W}dV=\iiint _{W^{*}}\left|{\frac {\partial (x,y,z)}{\partial (u,v,\theta )}}\right|\,du\,dv\,d\theta .} Expanding, | ∂ ( x , y , z ) ∂ ( u , v , θ ) | = | det [ ∂ x ∂ u cos θ ∂ x ∂ v cos θ − x sin θ ∂ x ∂ u sin θ ∂ x ∂ v sin θ x cos θ ∂ z ∂ u ∂ z ∂ v 0 ] | = | − ∂ z ∂ v ∂ x ∂ u x + ∂ z ∂ u ∂ x ∂ v x | = | − x ∂ ( x , z ) ∂ ( u , v ) | = x | ∂ ( x , z ) ∂ ( u , v ) | . {\displaystyle {\begin{aligned}\left|{\frac {\partial (x,y,z)}{\partial (u,v,\theta )}}\right|&=\left|\det {\begin{bmatrix}{\frac {\partial x}{\partial u}}\cos \theta &{\frac {\partial x}{\partial v}}\cos \theta &-x\sin \theta \\[6pt]{\frac {\partial x}{\partial u}}\sin \theta &{\frac {\partial x}{\partial v}}\sin \theta &x\cos \theta \\[6pt]{\frac {\partial z}{\partial u}}&{\frac {\partial z}{\partial v}}&0\end{bmatrix}}\right|\\[5pt]&=\left|-{\frac {\partial z}{\partial v}}{\frac {\partial x}{\partial u}}\,x+{\frac {\partial z}{\partial u}}{\frac {\partial x}{\partial v}}\,x\right|=\ \left|-x\,{\frac {\partial (x,z)}{\partial (u,v)}}\right|=x\left|{\frac {\partial (x,z)}{\partial (u,v)}}\right|.\end{aligned}}} The last equality holds because the axis of rotation must be external to F {\displaystyle F} , meaning x ≥ 0 {\displaystyle x\geq 0} . Now, V = ∭ W ∗ | ∂ ( x , y , z ) ∂ ( u , v , θ ) | d u d v d θ = ∫ 0 2 π ∬ F ∗ x ( u , v ) | ∂ ( x , z ) ∂ ( u , v ) | d u d v d θ = 2 π ∬ F ∗ x ( u , v ) | ∂ ( x , z ) ∂ ( u , v ) | d u d v = 2 π ∫ F x d A {\displaystyle {\begin{aligned}V&=\iiint _{W^{*}}\left|{\frac {\partial (x,y,z)}{\partial (u,v,\theta )}}\right|\,du\,dv\,d\theta \\[1ex]&=\int _{0}^{2\pi }\!\!\!\!\iint _{F^{*}}x(u,v)\left|{\frac {\partial (x,z)}{\partial (u,v)}}\right|du\,dv\,d\theta \\[6pt]&=2\pi \iint _{F^{*}}x(u,v)\left|{\frac {\partial (x,z)}{\partial (u,v)}}\right|\,du\,dv\\[1ex]&=2\pi \int _{F}x\,dA\end{aligned}}} by change of variables. == Generalizations == The theorems can be generalized for arbitrary curves and shapes, under appropriate conditions. Goodman & Goodman generalize the second theorem as follows. If the figure F moves through space so that it remains perpendicular to the curve L traced by the centroid of F, then it sweeps out a solid of volume V = Ad, where A is the area of F and d is the length of L. (This assumes the solid does not intersect itself.) In particular, F may rotate about its centroid during the motion. However, the corresponding generalization of the first theorem is only true if the curve L traced by the centroid lies in a plane perpendicular to the plane of C. == In n-dimensions == In general, one can generate an n {\displaystyle n} dimensional solid by rotating an n − p {\displaystyle n-p} dimensional solid F {\displaystyle F} around a p {\displaystyle p} dimensional sphere. This is called an n {\displaystyle n} -solid of revolution of species p {\displaystyle p} . Let the p {\displaystyle p} -th centroid of F {\displaystyle F} be defined by R = ∬ F x p d A A , {\displaystyle R={\frac {\iint _{F}x^{p}\,dA}{A}},} Then Pappus' theorems generalize to: Volume of n {\displaystyle n} -solid of revolution of species p {\displaystyle p} = (Volume of generating ( n − p ) {\displaystyle (n{-}p)} -solid) × {\displaystyle \times } (Surface area of p {\displaystyle p} -sphere traced by the p {\displaystyle p} -th centroid of the generating solid) and Surface area of n {\displaystyle n} -solid of revolution of species p {\displaystyle p} = (Surface area of generating ( n − p ) {\displaystyle (n{-}p)} -solid) × {\displaystyle \times } (Surface area of p {\displaystyle p} -sphere traced by the p {\displaystyle p} -th centroid of the generating solid) The original theorems are the case with n = 3 , p = 1 {\displaystyle n=3,\,p=1} . == Footnotes == == References == == External links == Weisstein, Eric W. "Pappus's Centroid Theorem". MathWorld.
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Wikipedia:Pappus's hexagon theorem#0
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In mathematics, Pappus's hexagon theorem (attributed to Pappus of Alexandria) states that given one set of collinear points A , B , C , {\displaystyle A,B,C,} and another set of collinear points a , b , c , {\displaystyle a,b,c,} then the intersection points X , Y , Z {\displaystyle X,Y,Z} of line pairs A b {\displaystyle Ab} and a B , A c {\displaystyle aB,Ac} and a C , B c {\displaystyle aC,Bc} and b C {\displaystyle bC} are collinear, lying on the Pappus line. These three points are the points of intersection of the "opposite" sides of the hexagon A b C a B c {\displaystyle AbCaBc} . It holds in a projective plane over any field, but fails for projective planes over any noncommutative division ring. Projective planes in which the "theorem" is valid are called pappian planes. If one considers a pappian plane containing a hexagon as just described but with sides A b {\displaystyle Ab} and a B {\displaystyle aB} parallel and also sides B c {\displaystyle Bc} and b C {\displaystyle bC} parallel (so that the Pappus line u {\displaystyle u} is the line at infinity), one gets the affine version of Pappus's theorem shown in the second diagram. If the Pappus line u {\displaystyle u} and the lines g , h {\displaystyle g,h} have a point in common, one gets the so-called little version of Pappus's theorem. The dual of this incidence theorem states that given one set of concurrent lines A , B , C {\displaystyle A,B,C} , and another set of concurrent lines a , b , c {\displaystyle a,b,c} , then the lines x , y , z {\displaystyle x,y,z} defined by pairs of points resulting from pairs of intersections A ∩ b {\displaystyle A\cap b} and a ∩ B , A ∩ c {\displaystyle a\cap B,\;A\cap c} and a ∩ C , B ∩ c {\displaystyle a\cap C,\;B\cap c} and b ∩ C {\displaystyle b\cap C} are concurrent. (Concurrent means that the lines pass through one point.) Pappus's theorem is a special case of Pascal's theorem for a conic—the limiting case when the conic degenerates into 2 straight lines. Pascal's theorem is in turn a special case of the Cayley–Bacharach theorem. The Pappus configuration is the configuration of 9 lines and 9 points that occurs in Pappus's theorem, with each line meeting 3 of the points and each point meeting 3 lines. In general, the Pappus line does not pass through the point of intersection of A B C {\displaystyle ABC} and a b c {\displaystyle abc} . This configuration is self dual. Since, in particular, the lines B c , b C , X Y {\displaystyle Bc,bC,XY} have the properties of the lines x , y , z {\displaystyle x,y,z} of the dual theorem, and collinearity of X , Y , Z {\displaystyle X,Y,Z} is equivalent to concurrence of B c , b C , X Y {\displaystyle Bc,bC,XY} , the dual theorem is therefore just the same as the theorem itself. The Levi graph of the Pappus configuration is the Pappus graph, a bipartite distance-regular graph with 18 vertices and 27 edges. == Proof: affine form == If the affine form of the statement can be proven, then the projective form of Pappus's theorem is proven, as the extension of a pappian plane to a projective plane is unique. Because of the parallelity in an affine plane one has to distinct two cases: g ∦ h {\displaystyle g\not \parallel h} and g ∥ h {\displaystyle g\parallel h} . The key for a simple proof is the possibility for introducing a "suitable" coordinate system: Case 1: The lines g , h {\displaystyle g,h} intersect at point S = g ∩ h {\displaystyle S=g\cap h} . In this case coordinates are introduced, such that S = ( 0 , 0 ) , A = ( 0 , 1 ) , c = ( 1 , 0 ) {\displaystyle \;S=(0,0),\;A=(0,1),\;c=(1,0)\;} (see diagram). B , C {\displaystyle B,C} have the coordinates B = ( 0 , γ ) , C = ( 0 , δ ) , γ , δ ∉ { 0 , 1 } {\displaystyle \;B=(0,\gamma ),\;C=(0,\delta ),\;\gamma ,\delta \notin \{0,1\}} . From the parallelity of the lines B c , C b {\displaystyle Bc,\;Cb} one gets b = ( δ γ , 0 ) {\displaystyle b=({\tfrac {\delta }{\gamma }},0)} and the parallelity of the lines A b , B a {\displaystyle Ab,Ba} yields a = ( δ , 0 ) {\displaystyle a=(\delta ,0)} . Hence line C a {\displaystyle Ca} has slope − 1 {\displaystyle -1} and is parallel line A c {\displaystyle Ac} . Case 2: g ∥ h {\displaystyle g\parallel h\ } (little theorem). In this case the coordinates are chosen such that c = ( 0 , 0 ) , b = ( 1 , 0 ) , A = ( 0 , 1 ) , B = ( γ , 1 ) , γ ≠ 0 {\displaystyle \;c=(0,0),\;b=(1,0),\;A=(0,1),\;B=(\gamma ,1),\;\gamma \neq 0} . From the parallelity of A b ∥ B a {\displaystyle Ab\parallel Ba} and c B ∥ b C {\displaystyle cB\parallel bC} one gets C = ( γ + 1 , 1 ) {\displaystyle \;C=(\gamma +1,1)\;} and a = ( γ + 1 , 0 ) {\displaystyle \;a=(\gamma +1,0)\;} , respectively, and at least the parallelity A c ∥ C a {\displaystyle \;Ac\parallel Ca\;} . == Proof with homogeneous coordinates == Choose homogeneous coordinates with C = ( 1 , 0 , 0 ) , c = ( 0 , 1 , 0 ) , X = ( 0 , 0 , 1 ) , A = ( 1 , 1 , 1 ) {\displaystyle C=(1,0,0),\;c=(0,1,0),\;X=(0,0,1),\;A=(1,1,1)} . On the lines A C , A c , A X {\displaystyle AC,Ac,AX} , given by x 2 = x 3 , x 1 = x 3 , x 2 = x 1 {\displaystyle x_{2}=x_{3},\;x_{1}=x_{3},\;x_{2}=x_{1}} , take the points B , Y , b {\displaystyle B,Y,b} to be B = ( p , 1 , 1 ) , Y = ( 1 , q , 1 ) , b = ( 1 , 1 , r ) {\displaystyle B=(p,1,1),\;Y=(1,q,1),\;b=(1,1,r)} for some p , q , r {\displaystyle p,q,r} . The three lines X B , C Y , c b {\displaystyle XB,CY,cb} are x 1 = x 2 p , x 2 = x 3 q , x 3 = x 1 r {\displaystyle x_{1}=x_{2}p,\;x_{2}=x_{3}q,\;x_{3}=x_{1}r} , so they pass through the same point a {\displaystyle a} if and only if r q p = 1 {\displaystyle rqp=1} . The condition for the three lines C b , c B {\displaystyle Cb,cB} and X Y {\displaystyle XY} with equations x 2 = x 1 q , x 1 = x 3 p , x 3 = x 2 r {\displaystyle x_{2}=x_{1}q,\;x_{1}=x_{3}p,\;x_{3}=x_{2}r} to pass through the same point Z {\displaystyle Z} is r p q = 1 {\displaystyle rpq=1} . So this last set of three lines is concurrent if all the other eight sets are because multiplication is commutative, so p q = q p {\displaystyle pq=qp} . Equivalently, X , Y , Z {\displaystyle X,Y,Z} are collinear. The proof above also shows that for Pappus's theorem to hold for a projective space over a division ring it is both sufficient and necessary that the division ring is a (commutative) field. German mathematician Gerhard Hessenberg proved that Pappus's theorem implies Desargues's theorem. In general, Pappus's theorem holds for some projective plane if and only if it is a projective plane over a commutative field. The projective planes in which Pappus's theorem does not hold are Desarguesian projective planes over noncommutative division rings, and non-Desarguesian planes. The proof is invalid if C , c , X {\displaystyle C,c,X} happen to be collinear. In that case an alternative proof can be provided, for example, using a different projective reference. == Dual theorem == Because of the principle of duality for projective planes the dual theorem of Pappus is true: If 6 lines A , b , C , a , B , c {\displaystyle A,b,C,a,B,c} are chosen alternately from two pencils with centers G , H {\displaystyle G,H} , the lines X := ( A ∩ b ) ( a ∩ B ) , {\displaystyle X:=(A\cap b)(a\cap B),} Y := ( c ∩ A ) ( C ∩ a ) , {\displaystyle Y:=(c\cap A)(C\cap a),} Z := ( b ∩ C ) ( B ∩ c ) {\displaystyle Z:=(b\cap C)(B\cap c)} are concurrent, that means: they have a point U {\displaystyle U} in common. The left diagram shows the projective version, the right one an affine version, where the points G , H {\displaystyle G,H} are points at infinity. If point U {\displaystyle U} is on the line G H {\displaystyle GH} than one gets the "dual little theorem" of Pappus' theorem. If in the affine version of the dual "little theorem" point U {\displaystyle U} is a point at infinity too, one gets Thomsen's theorem, a statement on 6 points on the sides of a triangle (see diagram). The Thomsen figure plays an essential role coordinatising an axiomatic defined projective plane. The proof of the closure of Thomsen's figure is covered by the proof for the "little theorem", given above. But there exists a simple direct proof, too: Because the statement of Thomsen's theorem (the closure of the figure) uses only the terms connect, intersect and parallel, the statement is affinely invariant, and one can introduce coordinates such that P = ( 0 , 0 ) , Q = ( 1 , 0 ) , R = ( 0 , 1 ) {\displaystyle P=(0,0),\;Q=(1,0),\;R=(0,1)} (see right diagram). The starting point of the sequence of chords is ( 0 , λ ) . {\displaystyle (0,\lambda ).} One easily verifies the coordinates of the points given in the diagram, which shows: the last point coincides with the first point. == Other statements of the theorem == In addition to the above characterizations of Pappus's theorem and its dual, the following are equivalent statements: If the six vertices of a hexagon lie alternately on two lines, then the three points of intersection of pairs of opposite sides are collinear. Arranged in a matrix of nine points (as in the figure and description above) and thought of as evaluating a permanent, if the first two rows and the six "diagonal" triads are collinear, then the third row is collinear. | A B C a b c X Y Z | {\displaystyle \left|{\begin{matrix}A&B&C\\a&b&c\\X&Y&Z\end{matrix}}\right|} That is, if A B C , a b c , A b Z , B c X , C a Y , X b C , Y c A , Z a B {\displaystyle \ ABC,abc,AbZ,BcX,CaY,XbC,YcA,ZaB\ } are lines, then Pappus's theorem states that X Y Z {\displaystyle XYZ} must be a line. Also, note that the same matrix formulation applies to the dual form of the theorem when ( A , B , C ) {\displaystyle (A,B,C)} etc. are triples of concurrent lines. Given three distinct points on each of two distinct lines, pair each point on one of the lines with one from the other line, then the joins of points not paired will meet in (opposite) pairs at points along a line. If two triangles are perspective in at least two different ways, then they are perspective in three ways. If A B , C D , {\displaystyle \;AB,CD,\;} and E F {\displaystyle EF} are concurrent and D E , F A , {\displaystyle DE,FA,} and B C {\displaystyle BC} are concurrent, then A D , B E , {\displaystyle AD,BE,} and C F {\displaystyle CF} are concurrent. == Origins == In its earliest known form, Pappus's Theorem is Propositions 138, 139, 141, and 143 of Book VII of Pappus's Collection. These are Lemmas XII, XIII, XV, and XVII in the part of Book VII consisting of lemmas to the first of the three books of Euclid's Porisms. The lemmas are proved in terms of what today is known as the cross ratio of four collinear points. Three earlier lemmas are used. The first of these, Lemma III, has the diagram below (which uses Pappus's lettering, with G for Γ, D for Δ, J for Θ, and L for Λ). Here three concurrent straight lines, AB, AG, and AD, are crossed by two lines, JB and JE, which concur at J. Also KL is drawn parallel to AZ. Then KJ : JL :: (KJ : AG & AG : JL) :: (JD : GD & BG : JB). These proportions might be written today as equations: KJ/JL = (KJ/AG)(AG/JL) = (JD/GD)(BG/JB). The last compound ratio (namely JD : GD & BG : JB) is what is known today as the cross ratio of the collinear points J, G, D, and B in that order; it is denoted today by (J, G; D, B). So we have shown that this is independent of the choice of the particular straight line JD that crosses the three straight lines that concur at A. In particular (J, G; D, B) = (J, Z; H, E). It does not matter on which side of A the straight line JE falls. In particular, the situation may be as in the next diagram, which is the diagram for Lemma X. Just as before, we have (J, G; D, B) = (J, Z; H, E). Pappus does not explicitly prove this; but Lemma X is a converse, namely that if these two cross ratios are the same, and the straight lines BE and DH cross at A, then the points G, A, and Z must be collinear. What we showed originally can be written as (J, ∞; K, L) = (J, G; D, B), with ∞ taking the place of the (nonexistent) intersection of JK and AG. Pappus shows this, in effect, in Lemma XI, whose diagram, however, has different lettering: What Pappus shows is DE.ZH : EZ.HD :: GB : BE, which we may write as (D, Z; E, H) = (∞, B; E, G). The diagram for Lemma XII is: The diagram for Lemma XIII is the same, but BA and DG, extended, meet at N. In any case, considering straight lines through G as cut by the three straight lines through A, (and accepting that equations of cross ratios remain valid after permutation of the entries,) we have by Lemma III or XI (G, J; E, H) = (G, D; ∞ Z). Considering straight lines through D as cut by the three straight lines through B, we have (L, D; E, K) = (G, D; ∞ Z). Thus (E, H; J, G) = (E, K; D, L), so by Lemma X, the points H, M, and K are collinear. That is, the points of intersection of the pairs of opposite sides of the hexagon ADEGBZ are collinear. Lemmas XV and XVII are that, if the point M is determined as the intersection of HK and BG, then the points A, M, and D are collinear. That is, the points of intersection of the pairs of opposite sides of the hexagon BEKHZG are collinear. == Notes == == References == Coxeter, Harold Scott MacDonald (1969), Introduction to Geometry (2nd ed.), New York: John Wiley & Sons, ISBN 978-0-471-50458-0, MR 0123930 Cronheim, A. (1953), "A proof of Hessenberg's theorem", Proceedings of the American Mathematical Society, 4 (2): 219–221, doi:10.2307/2031794, JSTOR 2031794 Dembowski, Peter (1968), Finite Geometries, Berlin: Springer-Verlag Heath, Thomas (1981) [1921], A History of Greek Mathematics, New York: Dover Publications Hessenberg, Gerhard (1905), "Beweis des Desarguesschen Satzes aus dem Pascalschen", Mathematische Annalen, 61 (2), Berlin / Heidelberg: Springer: 161–172, doi:10.1007/BF01457558, ISSN 1432-1807, S2CID 120456855 Hultsch, Fridericus (1877), Pappi Alexandrini Collectionis Quae Supersunt, Berlin{{citation}}: CS1 maint: location missing publisher (link) Kline, Morris (1972), Mathematical Thought From Ancient to Modern Times, New York: Oxford University Press Pambuccian, Victor; Schacht, Celia (2019), "The axiomatic destiny of the theorems of Pappus and Desargues", in Dani, S. G.; Papadopoulos, A. (eds.), Geometry in history, Springer, pp. 355–399, ISBN 978-3-030-13611-6 Whicher, Olive (1971), Projective Geometry, Rudolph Steiner Press, ISBN 0-85440-245-4 == External links == Pappus's hexagon theorem at cut-the-knot Dual to Pappus's hexagon theorem at cut-the-knot Pappus’s Theorem: Nine proofs and three variations
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Wikipedia:Parabolic Hausdorff dimension#0
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In fractal geometry, the parabolic Hausdorff dimension is a restricted version of the genuine Hausdorff dimension. Only parabolic cylinders, i. e. rectangles with a distinct non-linear scaling between time and space are permitted as covering sets. It is useful to determine the Hausdorff dimension of self-similar stochastic processes, such as the geometric Brownian motion or stable Lévy processes plus Borel measurable drift function f {\displaystyle f} . == Definitions == We define the α {\displaystyle \alpha } -parabolic β {\displaystyle \beta } -Hausdorff outer measure for any set A ⊆ R d + 1 {\displaystyle A\subseteq \mathbb {R} ^{d+1}} as P α − H β ( A ) := lim δ ↓ 0 inf { ∑ k = 1 ∞ | P k | β : A ⊆ ⋃ k = 1 ∞ P k , P k ∈ P α , | P k | ≤ δ } . {\displaystyle {\mathcal {P}}^{\alpha }-{\mathcal {H}}^{\beta }(A):=\lim _{\delta \downarrow 0}\inf \left\{\sum _{k=1}^{\infty }\left|P_{k}\right|^{\beta }:A\subseteq \bigcup _{k=1}^{\infty }P_{k},P_{k}\in {\mathcal {P}}^{\alpha },\left|P_{k}\right|\leq \delta \right\}.} where the α {\displaystyle \alpha } -parabolic cylinders ( P k ) k ∈ N {\displaystyle \left(P_{k}\right)_{k\in \mathbb {N} }} are contained in P α := { [ t , t + c ] × ∏ i = 1 d [ x i , x i + c 1 / α ] ; t , x i ∈ R , c ∈ ( 0 , 1 ] } . {\displaystyle {\mathcal {P}}^{\alpha }:=\left\{[t,t+c]\times \prod _{i=1}^{d}\left[x_{i},x_{i}+c^{1/\alpha }\right];t,x_{i}\in \mathbb {R} ,c\in (0,1]\right\}.} We define the α {\displaystyle \alpha } -parabolic Hausdorff dimension of A {\displaystyle A} as P α − dim A := inf { β ≥ 0 : P α − H β ( A ) = 0 } . {\displaystyle {\mathcal {P}}^{\alpha }-\dim A:=\inf \left\{\beta \geq 0:{\mathcal {P}}^{\alpha }-{\mathcal {H}}^{\beta }(A)=0\right\}.} The case α = 1 {\displaystyle \alpha =1} equals the genuine Hausdorff dimension dim {\displaystyle \dim } . == Application == Let φ α := P α − dim G T ( f ) {\displaystyle \varphi _{\alpha }:={\mathcal {P}}^{\alpha }-\dim {\mathcal {G}}_{T}(f)} . We can calculate the Hausdorff dimension of the fractional Brownian motion B H {\displaystyle B^{H}} of Hurst index 1 / α = H ∈ ( 0 , 1 ] {\displaystyle 1/\alpha =H\in (0,1]} plus some measurable drift function f {\displaystyle f} . We get dim G T ( B H + f ) = φ α ∧ 1 α ⋅ φ α + ( 1 − 1 α ) ⋅ d {\displaystyle \dim {\mathcal {G}}_{T}\left(B^{H}+f\right)=\varphi _{\alpha }\wedge {\frac {1}{\alpha }}\cdot \varphi _{\alpha }+\left(1-{\frac {1}{\alpha }}\right)\cdot d} and dim R T ( B H + f ) = φ α ∧ d . {\displaystyle \dim {\mathcal {R}}_{T}\left(B^{H}+f\right)=\varphi _{\alpha }\wedge d.} For an isotropic α {\displaystyle \alpha } -stable Lévy process X {\displaystyle X} for α ∈ ( 0 , 2 ] {\displaystyle \alpha \in (0,2]} plus some measurable drift function f {\displaystyle f} we get dim G T ( X + f ) = { φ 1 , α ∈ ( 0 , 1 ] , φ α ∧ 1 α ⋅ φ α + ( 1 − 1 α ) ⋅ d , α ∈ [ 1 , 2 ] {\displaystyle \dim {\mathcal {G}}_{T}(X+f)={\begin{cases}\varphi _{1},&\alpha \in (0,1],\\\varphi _{\alpha }\wedge {\frac {1}{\alpha }}\cdot \varphi _{\alpha }+\left(1-{\frac {1}{\alpha }}\right)\cdot d,&\alpha \in [1,2]\end{cases}}} and dim R T ( X + f ) = { α ⋅ φ α ∧ d , α ∈ ( 0 , 1 ] , φ α ∧ d , α ∈ [ 1 , 2 ] . {\displaystyle \dim {\mathcal {R}}_{T}\left(X+f\right)={\begin{cases}\alpha \cdot \varphi _{\alpha }\wedge d,&\alpha \in (0,1],\\\varphi _{\alpha }\wedge d,&\alpha \in [1,2].\end{cases}}} == Inequalities and identities == For ϕ α := P α − dim A {\displaystyle \phi _{\alpha }:={\mathcal {P}}^{\alpha }-\dim A} one has dim A ≤ { ϕ α ∧ α ⋅ ϕ α + 1 − α , α ∈ ( 0 , 1 ] , ϕ α ∧ 1 α ⋅ α + ( 1 − 1 α ) ⋅ d , α ∈ [ 1 , ∞ ) {\displaystyle \dim A\leq {\begin{cases}\phi _{\alpha }\wedge \alpha \cdot \phi _{\alpha }+1-\alpha ,&\alpha \in (0,1],\\\phi _{\alpha }\wedge {\frac {1}{\alpha }}\cdot \alpha +\left(1-{\frac {1}{\alpha }}\right)\cdot d,&\alpha \in [1,\infty )\end{cases}}} and dim A ≥ { α ⋅ ϕ α ∨ ϕ α + ( 1 − 1 α ) ⋅ d , α ∈ ( 0 , 1 ] , ϕ α + 1 − α , α ∈ [ 1 , ∞ ) . {\displaystyle \dim A\geq {\begin{cases}\alpha \cdot \phi _{\alpha }\vee \phi _{\alpha }+\left(1-{\frac {1}{\alpha }}\right)\cdot d,&\alpha \in (0,1],\\\phi _{\alpha }+1-\alpha ,&\alpha \in [1,\infty ).\end{cases}}} Further, for the fractional Brownian motion B H {\displaystyle B^{H}} of Hurst index 1 / α = H ∈ ( 0 , 1 ] {\displaystyle 1/\alpha =H\in (0,1]} one has P α − dim G T ( B H ) = α ⋅ dim T {\displaystyle {\mathcal {P}}^{\alpha }-\dim {\mathcal {G}}_{T}\left(B^{H}\right)=\alpha \cdot \dim T} and for an isotropic α {\displaystyle \alpha } -stable Lévy process X {\displaystyle X} for α ∈ ( 0 , 2 ] {\displaystyle \alpha \in (0,2]} one has P α − dim G T ( X ) = ( α ∨ 1 ) ⋅ dim T {\displaystyle {\mathcal {P}}^{\alpha }-\dim {\mathcal {G}}_{T}\left(X\right)=(\alpha \vee 1)\cdot \dim T} and dim R T ( X ) = α ⋅ dim T ∧ d . {\displaystyle \dim {\mathcal {R}}_{T}(X)=\alpha \cdot \dim T\wedge d.} For constant functions f C {\displaystyle f_{C}} we get P α − dim G T ( f C ) = ( α ∨ 1 ) ⋅ dim T . {\displaystyle {\mathcal {P}}^{\alpha }-\dim {\mathcal {G}}_{T}\left(f_{C}\right)=(\alpha \vee 1)\cdot \dim T.} If f ∈ C β ( T , R d ) {\displaystyle f\in C^{\beta }(T,\mathbb {R} ^{d})} , i. e. f {\displaystyle f} is β {\displaystyle \beta } -Hölder continuous, for φ α = P α − dim G T ( f ) {\displaystyle \varphi _{\alpha }={\mathcal {P}}^{\alpha }-\dim {\mathcal {G}}_{T}(f)} the estimates φ α ≤ { dim T + ( 1 α − β ) ⋅ d ∧ dim T α ⋅ β ∧ d + 1 , α ∈ ( 0 , 1 ] , α ⋅ dim T + ( 1 − α ⋅ β ) ⋅ d ∧ dim T β ∧ d + 1 , α ∈ [ 1 , 1 β ] , α ⋅ dim T + 1 β ( dim T − 1 ) + α ∧ d + 1 , α ∈ [ 1 β , ∞ ) ] {\displaystyle \varphi _{\alpha }\leq {\begin{cases}\dim T+\left({\frac {1}{\alpha }}-\beta \right)\cdot d\wedge {\frac {\dim T}{\alpha \cdot \beta }}\wedge d+1,&\alpha \in (0,1],\\\alpha \cdot \dim T+(1-\alpha \cdot \beta )\cdot d\wedge {\frac {\dim T}{\beta }}\wedge d+1,&\alpha \in \left[1,{\frac {1}{\beta }}\right],\\\alpha \cdot \dim T+{\frac {1}{\beta }}(\dim T-1)+\alpha \wedge d+1,&\alpha \in \left[{\frac {1}{\beta }},\infty )\right]\end{cases}}} hold. Finally, for the Brownian motion B {\displaystyle B} and f ∈ C β ( T , R d ) {\displaystyle f\in C^{\beta }\left(T,\mathbb {R} ^{d}\right)} we get dim G T ( B + f ) ≤ { d + 1 2 , β ≤ dim T d − 1 2 d , dim T + ( 1 − β ) ⋅ d , dim T d − 1 2 d ≤ β ≤ dim T d ∧ 1 2 , dim T β , dim T d ≤ β ≤ 1 2 , 2 ⋅ dim T ∧ dim T + d 2 , else {\displaystyle \dim {\mathcal {G}}_{T}(B+f)\leq {\begin{cases}d+{\frac {1}{2}},&\beta \leq {\frac {\dim T}{d}}-{\frac {1}{2d}},\\\dim T+(1-\beta )\cdot d,&{\frac {\dim T}{d}}-{\frac {1}{2d}}\leq \beta \leq {\frac {\dim T}{d}}\wedge {\frac {1}{2}},\\{\frac {\dim T}{\beta }},&{\frac {\dim T}{d}}\leq \beta \leq {\frac {1}{2}},\\2\cdot \dim T\wedge \dim T+{\frac {d}{2}},&{\text{ else}}\end{cases}}} and dim R T ( B + f ) ≤ { dim T β , dim T d ≤ β ≤ 1 2 , 2 ⋅ dim T ∧ d , dim T d ≤ 1 2 ≤ β , d , else . {\displaystyle \dim {\mathcal {R}}_{T}(B+f)\leq {\begin{cases}{\frac {\dim T}{\beta }},&{\frac {\dim T}{d}}\leq \beta \leq {\frac {1}{2}},\\2\cdot \dim T\wedge d,&{\frac {\dim T}{d}}\leq {\frac {1}{2}}\leq \beta ,\\d,&{\text{ else}}.\end{cases}}} == References == == Sources == Kern, Peter; Pleschberger, Leonard (2024). "Parabolic Fractal Geometry of Stable Lévy Processes with Drift". arXiv:2312.13800 [math.PR]. Peres, Yuval; Sousi, Perla (2016). "Dimension of fractional Brownian motion with variable drift". Probability Theory and Related Fields. 165 (3–4): 771–794. arXiv:1310.7002. doi:10.1007/s00440-015-0645-5. Taylor, S. J.; Watson, N. A. (1985). "A Hausdorff measure classification of polar sets for the heat equation". Mathematical Proceedings of the Cambridge Philosophical Society. 97 (2): 325–344. doi:10.1017/S0305004100062873.
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Wikipedia:Parabolic Lie algebra#0
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In algebra, a parabolic Lie algebra p {\displaystyle {\mathfrak {p}}} is a subalgebra of a semisimple Lie algebra g {\displaystyle {\mathfrak {g}}} satisfying one of the following two conditions: p {\displaystyle {\mathfrak {p}}} contains a maximal solvable subalgebra (a Borel subalgebra) of g {\displaystyle {\mathfrak {g}}} ; the orthogonal complement with respect to the Killing form of p {\displaystyle {\mathfrak {p}}} in g {\displaystyle {\mathfrak {g}}} is isomorphic to the nilradical of p {\displaystyle {\mathfrak {p}}} . These conditions are equivalent over an algebraically closed field of characteristic zero, such as the complex numbers. If the field F {\displaystyle \mathbb {F} } is not algebraically closed, then the first condition is replaced by the assumption that p ⊗ F F ¯ {\displaystyle {\mathfrak {p}}\otimes _{\mathbb {F} }{\overline {\mathbb {F} }}} contains a Borel subalgebra of g ⊗ F F ¯ {\displaystyle {\mathfrak {g}}\otimes _{\mathbb {F} }{\overline {\mathbb {F} }}} where F ¯ {\displaystyle {\overline {\mathbb {F} }}} is the algebraic closure of F {\displaystyle \mathbb {F} } . == Examples == For the general linear Lie algebra g = g l n ( F ) {\displaystyle {\mathfrak {g}}={\mathfrak {gl}}_{n}(\mathbb {F} )} , a parabolic subalgebra is the stabilizer of a partial flag of F n {\displaystyle \mathbb {F} ^{n}} , i.e. a sequence of nested linear subspaces. For a complete flag, the stabilizer gives a Borel subalgebra. For a single linear subspace F k ⊂ F n {\displaystyle \mathbb {F} ^{k}\subset \mathbb {F} ^{n}} , one gets a maximal parabolic subalgebra p {\displaystyle {\mathfrak {p}}} , and the space of possible choices is the Grassmannian G r ( k , n ) {\displaystyle \mathrm {Gr} (k,n)} . In general, for a complex simple Lie algebra g {\displaystyle {\mathfrak {g}}} , parabolic subalgebras are in bijection with subsets of simple roots, i.e. subsets of the nodes of the Dynkin diagram. == See also == Generalized flag variety Parabolic subgroup of a reflection group == Bibliography == Baston, Robert J.; Eastwood, Michael G. (2016) [1989], The Penrose Transform: its Interaction with Representation Theory, Dover, ISBN 9780486816623 Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103. Grothendieck, Alexander (1957), "Sur la classification des fibrés holomorphes sur la sphère de Riemann", Amer. J. Math., 79 (1): 121–138, doi:10.2307/2372388, JSTOR 2372388. Humphreys, J. (1972), Linear Algebraic Groups, Springer, ISBN 978-0-387-90108-4
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Wikipedia:Parahita#0
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Parahita is a system of astronomy prevalent in Kerala and Tamil Nadu, India. It was introduced by the Kerala astronomer Haridatta, (c. 683 AD). Nilakantha Somayaji (1444–1544), in his Dr̥kkaraṇa, relates how Parahita was created based on the combined observations of a group of scholars who had gathered for a festival at Tirunāvāy on the banks of the Bhāratappuzha River. The Sanskrit etymology literally means "for the benefit of the common man", and the intention was to simplify astronomical computations so that everyone could do it. Parahita is a significant step in the simplification of the siddhantic tradition. Of the two texts of the system, Grahacāranibandhana and Mahāmārganibandhana, only the former is known. The system simplified the computational cycle of the Aryabhatiya by introducing a sub-aeon of 576 years and introduced a zero correction called Vāgbhāva based on which the system worked accurately around the time of Haridatta. Also Haridatta simplified the representation of numerals from the cumbersome notation of Aryabhata to the katapayadi system which gained wide currency in later Kerala mathematics. In the katapayadi system, numerals may be represented by various letters so that the large numerical tables required for astronomical computations could be represented as verses and memorized. The work Grahacāranibandhana-sangraha (932 AD) gives further details of the parahita technique. The methods were retained but some of the constants downgraded by Parameshvara in his DrggaNita (1483), and also by Achyuta Pisharati in his rAsigolasphuTanIti (1600). == See also == Drigganita == References ==
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Wikipedia:Parallel (operator)#0
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The parallel operator ‖ {\displaystyle \|} (pronounced "parallel", following the parallel lines notation from geometry; also known as reduced sum, parallel sum or parallel addition) is a binary operation which is used as a shorthand in electrical engineering, but is also used in kinetics, fluid mechanics and financial mathematics. The name parallel comes from the use of the operator computing the combined resistance of resistors in parallel. == Overview == The parallel operator represents the reciprocal value of a sum of reciprocal values (sometimes also referred to as the "reciprocal formula" or "harmonic sum") and is defined by: a ∥ b := 1 1 a + 1 b = a b a + b , {\displaystyle a\parallel b\mathrel {:=} {\frac {1}{{\dfrac {1}{a}}+{\dfrac {1}{b}}}}={\frac {ab}{a+b}},} where a, b, and a ∥ b {\displaystyle a\parallel b} are elements of the extended complex numbers C ¯ = C ∪ { ∞ } . {\displaystyle {\overline {\mathbb {C} }}=\mathbb {C} \cup \{\infty \}.} The operator gives half of the harmonic mean of two numbers a and b. As a special case, for any number a ∈ C ¯ {\displaystyle a\in {\overline {\mathbb {C} }}} : a ∥ a = 1 2 / a = 1 2 a . {\displaystyle a\parallel a={\frac {1}{2/a}}={\tfrac {1}{2}}a.} Further, for all distinct numbers a ≠ b {\displaystyle a\neq b} : | a ∥ b | > 1 2 min ( | a | , | b | ) , {\displaystyle {\big |}\,a\parallel b\,{\big |}>{\tfrac {1}{2}}\min {\bigl (}|a|,|b|{\bigr )},} with | a ∥ b | {\displaystyle {\big |}\,a\parallel b\,{\big |}} representing the absolute value of a ∥ b {\displaystyle a\parallel b} , and min ( x , y ) {\displaystyle \min(x,y)} meaning the minimum (least element) among x and y. If a {\displaystyle a} and b {\displaystyle b} are distinct positive real numbers then 1 2 min ( a , b ) < | a ∥ b | < min ( a , b ) . {\displaystyle {\tfrac {1}{2}}\min(a,b)<{\big |}\,a\parallel b\,{\big |}<\min(a,b).} The concept has been extended from a scalar operation to matrices and further generalized. == Notation == The operator was originally introduced as reduced sum by Sundaram Seshu in 1956, studied as operator ∗ by Kent E. Erickson in 1959, and popularized by Richard James Duffin and William Niles Anderson, Jr. as parallel addition or parallel sum operator : in mathematics and network theory since 1966. While some authors continue to use this symbol up to the present, for example, Sujit Kumar Mitra used ∙ as a symbol in 1970. In applied electronics, a ∥ sign became more common as the operator's symbol around 1974. This was often written as doubled vertical line (||) available in most character sets (sometimes italicized as //), but now can be represented using Unicode character U+2225 ( ∥ ) for "parallel to". In LaTeX and related markup languages, the macros \| and \parallel are often used (and rarely \smallparallel is used) to denote the operator's symbol. == Properties == Let C ~ {\displaystyle {\widetilde {\mathbb {C} }}} represent the extended complex plane excluding zero, C ~ := C ∪ { ∞ } ∖ { 0 } , {\displaystyle {\widetilde {\mathbb {C} }}:=\mathbb {C} \cup \{\infty \}\smallsetminus \{0\},} and φ {\displaystyle \varphi } the bijective function from C {\displaystyle \mathbb {C} } to C ~ {\displaystyle {\widetilde {\mathbb {C} }}} such that φ ( z ) = 1 / z . {\displaystyle \varphi (z)=1/z.} One has identities φ ( z t ) = φ ( z ) φ ( t ) , {\displaystyle \varphi (zt)=\varphi (z)\varphi (t),} and φ ( z + t ) = φ ( z ) ∥ φ ( t ) {\displaystyle \varphi (z+t)=\varphi (z)\parallel \varphi (t)} This implies immediately that C ~ {\displaystyle {\widetilde {\mathbb {C} }}} is a field where the parallel operator takes the place of the addition, and that this field is isomorphic to C . {\displaystyle \mathbb {C} .} The following properties may be obtained by translating through φ {\displaystyle \varphi } the corresponding properties of the complex numbers. === Field properties === As for any field, ( C ~ , ∥ , ⋅ ) {\displaystyle ({\widetilde {\mathbb {C} }},\,\parallel \,,\,\cdot \,)} satisfies a variety of basic identities. It is commutative under parallel and multiplication: a ∥ b = b ∥ a a b = b a {\displaystyle {\begin{aligned}a\parallel b&=b\parallel a\\[3mu]ab&=ba\end{aligned}}} It is associative under parallel and multiplication: ( a ∥ b ) ∥ c = a ∥ ( b ∥ c ) = a ∥ b ∥ c = 1 1 a + 1 b + 1 c = a b c a b + a c + b c , ( a b ) c = a ( b c ) = a b c . {\displaystyle {\begin{aligned}&(a\parallel b)\parallel c=a\parallel (b\parallel c)=a\parallel b\parallel c={\frac {1}{{\dfrac {1}{a}}+{\dfrac {1}{b}}+{\dfrac {1}{c}}}}={\frac {abc}{ab+ac+bc}},\\&(ab)c=a(bc)=abc.\end{aligned}}} Both operations have an identity element; for parallel the identity is ∞ {\displaystyle \infty } while for multiplication the identity is 1: a ∥ ∞ = ∞ ∥ a = 1 1 a + 0 = a , 1 ⋅ a = a ⋅ 1 = a . {\displaystyle {\begin{aligned}&a\parallel \infty =\infty \parallel a={\frac {1}{{\dfrac {1}{a}}+0}}=a,\\&1\cdot a=a\cdot 1=a.\end{aligned}}} Every element a {\displaystyle a} of C ~ {\displaystyle {\widetilde {\mathbb {C} }}} has an inverse under parallel, equal to − a , {\displaystyle -a,} the additive inverse under addition. (But 0 has no inverse under parallel.) a ∥ ( − a ) = 1 1 a − 1 a = ∞ . {\displaystyle a\parallel (-a)={\frac {1}{{\dfrac {1}{a}}-{\dfrac {1}{a}}}}=\infty .} The identity element ∞ {\displaystyle \infty } is its own inverse, ∞ ∥ ∞ = ∞ . {\displaystyle \infty \parallel \infty =\infty .} Every element a ≠ ∞ {\displaystyle a\neq \infty } of C ~ {\displaystyle {\widetilde {\mathbb {C} }}} has a multiplicative inverse a − 1 = 1 / a {\displaystyle a^{-1}=1/a} : a ⋅ 1 a = 1. {\displaystyle a\cdot {\frac {1}{a}}=1.} Multiplication is distributive over parallel: k ( a ∥ b ) = k 1 a + 1 b = 1 1 k a + 1 k b = k a ∥ k b . {\displaystyle k(a\parallel b)={\frac {k}{{\dfrac {1}{a}}+{\dfrac {1}{b}}}}={\frac {1}{{\dfrac {1}{ka}}+{\dfrac {1}{kb}}}}=ka\parallel kb.} === Repeated parallel === Repeated parallel is equivalent to division, a ∥ a ∥ ⋯ ∥ a ⏟ n times = 1 1 a + 1 a + ⋯ + 1 a ⏟ n times = a n . {\displaystyle \underbrace {a\parallel a\parallel \cdots \parallel a} _{n{\text{ times}}}={\frac {1}{\underbrace {{\dfrac {1}{a}}+{\dfrac {1}{a}}+\cdots +{\dfrac {1}{a}}} _{n{\text{ times}}}}}={\frac {a}{n}}.} Or, multiplying both sides by n, n ( a ∥ a ∥ ⋯ ∥ a ⏟ n times ) = a . {\displaystyle n(\underbrace {a\parallel a\parallel \cdots \parallel a} _{n{\text{ times}}})=a.} Unlike for repeated addition, this does not commute: a b ≠ b a implies a ∥ a ∥ ⋯ ∥ a ⏟ b times ≠ b ∥ b ∥ ⋯ ∥ b ⏟ a times . {\displaystyle {\frac {a}{b}}\neq {\frac {b}{a}}\quad {\text{implies}}\quad \underbrace {a\parallel a\parallel \cdots \parallel a} _{b{\text{ times}}}\,\neq \,\underbrace {b\parallel b\parallel \cdots \parallel b} _{a{\text{ times}}}~\!.} === Binomial expansion === Using the distributive property twice, the product of two parallel binomials can be expanded as ( a ∥ b ) ( c ∥ d ) = a ( c ∥ d ) ∥ b ( c ∥ d ) = a c ∥ a d ∥ b c ∥ b d . {\displaystyle {\begin{aligned}(a\parallel b)(c\parallel d)&=a(c\parallel d)\parallel b(c\parallel d)\\[3mu]&=ac\parallel ad\parallel bc\parallel bd.\end{aligned}}} The square of a binomial is ( a ∥ b ) 2 = a 2 ∥ a b ∥ b a ∥ b 2 = a 2 ∥ 1 2 a b ∥ b 2 . {\displaystyle {\begin{aligned}(a\parallel b)^{2}&=a^{2}\parallel ab\parallel ba\parallel b^{2}\\[3mu]&=a^{2}\parallel {\tfrac {1}{2}}ab\parallel b^{2}.\end{aligned}}} The cube of a binomial is ( a ∥ b ) 3 = a 3 ∥ 1 3 a 2 b ∥ 1 3 a b 2 ∥ b 3 . {\displaystyle (a\parallel b)^{3}=a^{3}\parallel {\tfrac {1}{3}}a^{2}b\parallel {\tfrac {1}{3}}ab^{2}\parallel b^{3}.} In general, the nth power of a binomial can be expanded using binomial coefficients which are the reciprocal of those under addition, resulting in an analog of the binomial formula: ( a ∥ b ) n = a n ( n 0 ) ∥ a n − 1 b ( n 1 ) ∥ ⋯ ∥ a n − k b k ( n k ) ∥ ⋯ ∥ b n ( n n ) . {\displaystyle (a\parallel b)^{n}={\frac {a^{n}}{\binom {n}{0}}}\parallel {\frac {a^{n-1}b}{\binom {n}{1}}}\parallel \cdots \parallel {\frac {a^{n-k}b^{k}}{\binom {n}{k}}}\parallel \cdots \parallel {\frac {b^{n}}{\binom {n}{n}}}.} === Logarithm and exponential === The following identities hold: 1 log ( a b ) = 1 log ( a ) ∥ 1 log ( b ) , {\displaystyle {\frac {1}{\log(ab)}}={\frac {1}{\log(a)}}\parallel {\frac {1}{\log(b)}},} exp ( 1 a ∥ b ) = exp ( 1 a ) exp ( 1 b ) {\displaystyle \exp \left({\frac {1}{a\parallel b}}\right)=\exp \left({\frac {1}{a}}\right)\exp \left({\frac {1}{b}}\right)} === Factoring parallel polynomials === As with a polynomial under addition, a parallel polynomial with coefficients a k {\displaystyle a_{k}} in C ~ {\textstyle {\widetilde {\mathbb {C} }}} (with a 0 ≠ ∞ {\displaystyle a_{0}\neq \infty } ) can be factored into a product of monomials: a 0 x n ∥ a 1 x n − 1 ∥ ⋯ ∥ a n = a 0 ( x ∥ − r 1 ) ( x ∥ − r 2 ) ⋯ ( x ∥ − r n ) {\displaystyle {\begin{aligned}&a_{0}x^{n}\parallel a_{1}x^{n-1}\parallel \cdots \parallel a_{n}=a_{0}(x\parallel -r_{1})(x\parallel -r_{2})\cdots (x\parallel -r_{n})\end{aligned}}} for some roots r k {\displaystyle r_{k}} (possibly repeated) in C ~ . {\textstyle {\widetilde {\mathbb {C} }}.} Analogous to polynomials under addition, the polynomial equation ( x ∥ − r 1 ) ( x ∥ − r 2 ) ⋯ ( x ∥ − r n ) = ∞ {\displaystyle (x\parallel -r_{1})(x\parallel -r_{2})\cdots (x\parallel -r_{n})=\infty } implies that x = r k {\textstyle x=r_{k}} for some k. === Quadratic formula === A linear equation can be easily solved via the parallel inverse: a x ∥ b = ∞ ⟹ x = − b a . {\displaystyle {\begin{aligned}ax\parallel b&=\infty \\[3mu]\implies x&=-{\frac {b}{a}}.\end{aligned}}} To solve a parallel quadratic equation, complete the square to obtain an analog of the quadratic formula a x 2 ∥ b x ∥ c = ∞ x 2 ∥ b a x = − c a x 2 ∥ b a x ∥ 4 b 2 a 2 = ( − c a ) ∥ 4 b 2 a 2 ( x ∥ 2 b a ) 2 = b 2 ∥ − 1 4 a c 1 4 a 2 ⟹ x = ( − b ) ∥ ± b 2 ∥ − 1 4 a c 1 2 a . {\displaystyle {\begin{aligned}ax^{2}\parallel bx\parallel c&=\infty \\[5mu]x^{2}\parallel {\frac {b}{a}}x&=-{\frac {c}{a}}\\[5mu]x^{2}\parallel {\frac {b}{a}}x\parallel {\frac {4b^{2}}{a^{2}}}&=\left(-{\frac {c}{a}}\right)\parallel {\frac {4b^{2}}{a^{2}}}\\[5mu]\left(x\parallel {\frac {2b}{a}}\right)^{2}&={\frac {b^{2}\parallel -{\tfrac {1}{4}}ac}{{\tfrac {1}{4}}a^{2}}}\\[5mu]\implies x&={\frac {(-b)\parallel \pm {\sqrt {b^{2}\parallel -{\tfrac {1}{4}}ac}}}{{\tfrac {1}{2}}a}}.\end{aligned}}} === Including zero === The extended complex numbers including zero, C ¯ := C ∪ ∞ , {\displaystyle {\overline {\mathbb {C} }}:=\mathbb {C} \cup \infty ,} is no longer a field under parallel and multiplication, because 0 has no inverse under parallel. (This is analogous to the way ( C ¯ , + , ⋅ ) {\displaystyle {\bigl (}{\overline {\mathbb {C} }},{+},{\cdot }{\bigr )}} is not a field because ∞ {\displaystyle \infty } has no additive inverse.) For every non-zero a, a ∥ 0 = 1 1 a + 1 0 = 0 {\displaystyle a\parallel 0={\frac {1}{{\dfrac {1}{a}}+{\dfrac {1}{0}}}}=0} The quantity 0 ∥ ( − 0 ) = 0 ∥ 0 {\displaystyle 0\parallel (-0)=0\parallel 0} can either be left undefined (see indeterminate form) or defined to equal 0. === Precedence === In the absence of parentheses, the parallel operator is defined as taking precedence over addition or subtraction, similar to multiplication. == Applications == There are applications of the parallel operator in mechanics, electronics, optics, and study of periodicity: === Reduced mass === Given masses m and M, the reduced mass μ = m M m + M = m ∥ M {\displaystyle \mu ={\frac {mM}{m+M}}=m\parallel M} is frequently applied in mechanics. For instance, when the masses orbit each other, the moment of inertia is their reduced mass times the distance between them. === Circuit analysis === In electrical engineering, the parallel operator can be used to calculate the total impedance of various serial and parallel electrical circuits. There is a duality between the usual (series) sum and the parallel sum. For instance, the total resistance of resistors connected in parallel is the reciprocal of the sum of the reciprocals of the individual resistors. 1 R eq = 1 R 1 + 1 R 2 + ⋯ + 1 R n R eq = R 1 ∥ R 2 ∥ ⋯ ∥ R n . {\displaystyle {\begin{aligned}{\frac {1}{R_{\text{eq}}}}&={\frac {1}{R_{1}}}+{\frac {1}{R_{2}}}+\cdots +{\frac {1}{R_{n}}}\\[5mu]R_{\text{eq}}&=R_{1}\parallel R_{2}\parallel \cdots \parallel R_{n}.\end{aligned}}} Likewise for the total capacitance of serial capacitors. === Coalescence of independent probability density functions. === The coalesced density function fcoalesced(x) of n independent probability density functions f1(x), f2(x), …, fn(x), is equal to the reciprocal of the sum of the reciprocal densities. 1 f c o a l e s c e d ( x ) = 1 f 1 ( x ) + 1 f 2 ( x ) + ⋯ + 1 f n ( x ) {\displaystyle {\begin{aligned}{\frac {1}{f_{coalesced}(x)}}&={\frac {1}{f_{1}(x)}}+{\frac {1}{f_{2}(x)}}+\cdots +{\frac {1}{f_{n}(x)}}\\[5mu]\end{aligned}}} === Lens equation === In geometric optics the thin lens approximation to the lens maker's equation. f = ρ v i r t u a l ∥ ρ o b j e c t {\displaystyle f=\rho _{virtual}\parallel \rho _{object}} === Synodic period === The time between conjunctions of two orbiting bodies is called the synodic period. If the period of the slower body is T2, and the period of the faster is T1, then the synodic period is T s y n = T 1 ∥ ( − T 2 ) . {\displaystyle T_{syn}=T_{1}\parallel (-T_{2}).} == Examples == Question: Three resistors R 1 = 270 k Ω {\displaystyle R_{1}=270\,\mathrm {k\Omega } } , R 2 = 180 k Ω {\displaystyle R_{2}=180\,\mathrm {k\Omega } } and R 3 = 120 k Ω {\displaystyle R_{3}=120\,\mathrm {k\Omega } } are connected in parallel. What is their resulting resistance? Answer: R 1 ∥ R 2 ∥ R 3 = 270 k Ω ∥ 180 k Ω ∥ 120 k Ω = 1 1 270 k Ω + 1 180 k Ω + 1 120 k Ω ≈ 56.84 k Ω {\displaystyle {\begin{aligned}R_{1}\parallel R_{2}\parallel R_{3}&=270\,\mathrm {k\Omega } \parallel 180\,\mathrm {k\Omega } \parallel 120\,\mathrm {k\Omega } \\[5mu]&={\frac {1}{{\dfrac {1}{270\,\mathrm {k\Omega } }}+{\dfrac {1}{180\,\mathrm {k\Omega } }}+{\dfrac {1}{120\,\mathrm {k\Omega } }}}}\\[5mu]&\approx 56.84\,\mathrm {k\Omega } \end{aligned}}} The effectively resulting resistance is ca. 57 kΩ. Question: A construction worker raises a wall in 5 hours. Another worker would need 7 hours for the same work. How long does it take to build the wall if both workers work in parallel? Answer: t 1 ∥ t 2 = 5 h ∥ 7 h = 1 1 5 h + 1 7 h ≈ 2.92 h {\displaystyle t_{1}\parallel t_{2}=5\,\mathrm {h} \parallel 7\,\mathrm {h} ={\frac {1}{{\dfrac {1}{5\,\mathrm {h} }}+{\dfrac {1}{7\,\mathrm {h} }}}}\approx 2.92\,\mathrm {h} } They will finish in close to 3 hours. == Implementation == Suggested already by Kent E. Erickson as a subroutine in digital computers in 1959, the parallel operator is implemented as a keyboard operator on the Reverse Polish Notation (RPN) scientific calculators WP 34S since 2008 as well as on the WP 34C and WP 43S since 2015, allowing to solve even cascaded problems with few keystrokes like 270↵ Enter180∥120∥. == Projective view == Given a field F there are two embeddings of F into the projective line P(F): z → [z : 1] and z → [1 : z]. These embeddings overlap except for [0:1] and [1:0]. The parallel operator relates the addition operation between the embeddings. In fact, the homographies on the projective line are represented by 2 x 2 matrices M(2,F), and the field operations (+ and ×) are extended to homographies. Each embedding has its addition a + b represented by the following matrix multiplications in M(2,A): ( 1 0 a 1 ) ( 1 0 b 1 ) = ( 1 0 a + b 1 ) , ( 1 a 0 1 ) ( 1 b 0 1 ) = ( 1 a + b 0 1 ) . {\displaystyle {\begin{aligned}{\begin{pmatrix}1&0\\a&1\end{pmatrix}}{\begin{pmatrix}1&0\\b&1\end{pmatrix}}&={\begin{pmatrix}1&0\\a+b&1\end{pmatrix}},\\[10mu]{\begin{pmatrix}1&a\\0&1\end{pmatrix}}{\begin{pmatrix}1&b\\0&1\end{pmatrix}}&={\begin{pmatrix}1&a+b\\0&1\end{pmatrix}}.\end{aligned}}} The two matrix products show that there are two subgroups of M(2,F) isomorphic to (F,+), the additive group of F. Depending on which embedding is used, one operation is +, the other is ∥ . {\displaystyle \parallel .} == Notes == == References == == Further reading == Pekarev, Èdward L.; Šmul'jan, Ju. L. (1976-04-30). "Parallel Addition and Parallel Subtraction of Operators". Mathematics of the USSR-Izvestiya. 10 (2). American Mathematical Society: 351–370. Bibcode:1976IzMat..10..351P. doi:10.1070/IM1976v010n02ABEH001694. Duffin, Richard James; Morley, Tom D. (July 1978). "Almost Definite Operators and Electro-mechanical Systems". SIAM Journal on Applied Mathematics. 35 (1). Society for Industrial and Applied Mathematics (SIAM): 21–30. doi:10.1137/0135003. JSTOR 2101028. (10 pages) Morley, Tom D. (July 1979). "Parallel Summation, Maxwell's Principle and the Infimum of Projections" (PDF). Journal of Mathematical Analysis and Applications. 70 (1). Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA: 33–41. doi:10.1016/0022-247X(79)90073-8. Archived from the original on 2020-08-20. Retrieved 2020-08-20. Seeger, Alberto (May 1990) [1988-03-22]. "Direct and Inverse Addition in Convex Analysis and Applications" (PDF). Journal of Mathematical Analysis and Applications. 148 (2). Department of Mathematics, University of Washington, Seattle, Washington, USA: Academic Press, Inc.: 317–349. doi:10.1016/0022-247X(90)90004-Y. Archived (PDF) from the original on 2020-08-20. Retrieved 2020-08-20. (33 pages) Bryant, Randal E.; Tygar, J. Doug; Huang, Lawrence P. (1994). "Geometric characterization of series-parallel variable resistor networks" (PDF). IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications. 41 (11): 686–698. doi:10.1109/81.331520. Archived from the original (PDF) on 2017-08-14. Antezana, Jorge; Corach, Gustavo; Stojanoff, Demetrio (April 2006) [2005-09-14]. "Bilateral Shorted Operators and Parallel Sums" (PDF). Linear Algebra and Its Applications. 414 (2–3). La Plata, Argentina & Buenos Aires, Argentina: 570–588. arXiv:math/0509327. doi:10.1016/j.laa.2005.10.039. Archived (PDF) from the original on 2017-08-09. Retrieved 2020-08-20. [14] (19 pages) Chansangiam, Pattrawut (February 2016) [August 2015, July 2015]. "Mathematical aspects for electrical network connections". KKU Engineering Journal. 43 (1): 47–54. doi:10.14456/kkuenj.2016.8. Archived (PDF) from the original on 2020-08-20. Retrieved 2020-08-20. Besenyei, Ádám (2016-09-01). "The irresistible inequality of Milne" (PDF). Budapest: Department of Applied Analysis and Computational Mathematics, Eötvös Loránd University. CIA2016. Archived (PDF) from the original on 2019-08-08. Retrieved 2019-08-11. "7.5 Electrical Characteristics: VCC = 5 V / 7.6 Electrical Characteristics: VCC = 2.7 V / 9.1.2.1 Inverting Comparator with Hysteresis". TLV3201, TLV3202: TLV320x 40-ns, microPOWER, Push-Pull Output Comparators (PDF). Revision B. Dallas, Texas, USA: Texas Instruments Incorporated. 2022-06-03 [2016, 2012]. pp. 5, 6, 13–14 [13]. SBOS561B. Archived (PDF) from the original on 2022-08-17. Retrieved 2022-08-18. p. 5: PARAMETER […] TYP […] UNIT […] INPUT IMPEDANCE […] Common mode […] 1013 ∥ 2 […] Ω ∥ pF […] Differential […] 1013 ∥ 4 […] Ω ∥ pF […] (37 pages) (NB. Unusual usage of ∥ for both values and units.) == External links == https://github.com/microsoftarchive/edx-platform-1/blob/master/common/lib/calc/calc/calc.py
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Wikipedia:Parallel postulate#0
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In geometry, the parallel postulate is the fifth postulate in Euclid's Elements and a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry: If a line segment intersects two straight lines forming two interior angles on the same side that are less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles. This postulate does not specifically talk about parallel lines; it is only a postulate related to parallelism. Euclid gave the definition of parallel lines in Book I, Definition 23 just before the five postulates. Euclidean geometry is the study of geometry that satisfies all of Euclid's axioms, including the parallel postulate. The postulate was long considered to be obvious or inevitable, but proofs were elusive. Eventually, it was discovered that inverting the postulate gave valid, albeit different geometries. A geometry where the parallel postulate does not hold is known as a non-Euclidean geometry. Geometry that is independent of Euclid's fifth postulate (i.e., only assumes the modern equivalent of the first four postulates) is known as absolute geometry (or sometimes "neutral geometry"). == Equivalent properties == Probably the best-known equivalent of Euclid's parallel postulate, contingent on his other postulates, is Playfair's axiom, named after the Scottish mathematician John Playfair, which states: In a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point. This axiom by itself is not logically equivalent to the Euclidean parallel postulate since there are geometries in which one is true and the other is not. However, in the presence of the remaining axioms which give Euclidean geometry, one can be used to prove the other, so they are equivalent in the context of absolute geometry. Many other statements equivalent to the parallel postulate have been suggested, some of them appearing at first to be unrelated to parallelism, and some seeming so self-evident that they were unconsciously assumed by people who claimed to have proven the parallel postulate from Euclid's other postulates. These equivalent statements include: There is at most one line that can be drawn parallel to another given one through an external point. (Playfair's axiom) The sum of the angles in every triangle is 180° (triangle postulate). There exists a triangle whose angles add up to 180°. The sum of the angles is the same for every triangle. There exists a pair of similar, but not congruent, triangles. Every triangle can be circumscribed. If three angles of a quadrilateral are right angles, then the fourth angle is also a right angle. There exists a quadrilateral in which all angles are right angles, that is, a rectangle. There exists a pair of straight lines that are at constant distance from each other. Two lines that are parallel to the same line are also parallel to each other. In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides (Pythagoras' theorem). The law of cosines, a generalization of Pythagoras' theorem. There is no upper limit to the area of a triangle. (Wallis axiom) The summit angles of the Saccheri quadrilateral are 90°. If a line intersects one of two parallel lines, both of which are coplanar with the original line, then it also intersects the other. (Proclus' axiom) However, the alternatives which employ the word "parallel" cease appearing so simple when one is obliged to explain which of the four common definitions of "parallel" is meant – constant separation, never meeting, same angles where crossed by some third line, or same angles where crossed by any third line – since the equivalence of these four is itself one of the unconsciously obvious assumptions equivalent to Euclid's fifth postulate. In the list above, it is always taken to refer to non-intersecting lines. For example, if the word "parallel" in Playfair's axiom is taken to mean 'constant separation' or 'same angles where crossed by any third line', then it is no longer equivalent to Euclid's fifth postulate, and is provable from the first four (the axiom says 'There is at most one line...', which is consistent with there being no such lines). However, if the definition is taken so that parallel lines are lines that do not intersect, or that have some line intersecting them in the same angles, Playfair's axiom is contextually equivalent to Euclid's fifth postulate and is thus logically independent of the first four postulates. Note that the latter two definitions are not equivalent, because in hyperbolic geometry the second definition holds only for ultraparallel lines. == History == From the beginning, the postulate came under attack as being provable, and therefore not a postulate, and for more than two thousand years, many attempts were made to prove (derive) the parallel postulate using Euclid's first four postulates. The main reason that such a proof was so highly sought after was that, unlike the first four postulates, the parallel postulate is not self-evident. If the order in which the postulates were listed in the Elements is significant, it indicates that Euclid included this postulate only when he realised he could not prove it or proceed without it. Many attempts were made to prove the fifth postulate from the other four, many of them being accepted as proofs for long periods until the mistake was found. Invariably the mistake was assuming some 'obvious' property which turned out to be equivalent to the fifth postulate (Playfair's axiom). Although known from the time of Proclus, this became known as Playfair's Axiom after John Playfair wrote a famous commentary on Euclid in 1795 in which he proposed replacing Euclid's fifth postulate by his own axiom. Today, over two thousand two hundred years later, Euclid's fifth postulate remains a postulate. Proclus (410–485) wrote a commentary on The Elements where he comments on attempted proofs to deduce the fifth postulate from the other four; in particular, he notes that Ptolemy had produced a false 'proof'. Proclus then goes on to give a false proof of his own. However, he did give a postulate which is equivalent to the fifth postulate. Ibn al-Haytham (Alhazen) (965–1039), an Arab mathematician, made an attempt at proving the parallel postulate using a proof by contradiction, in the course of which he introduced the concept of motion and transformation into geometry. He formulated the Lambert quadrilateral, which Boris Abramovich Rozenfeld names the "Ibn al-Haytham–Lambert quadrilateral", and his attempted proof contains elements similar to those found in Lambert quadrilaterals and Playfair's axiom. The Persian mathematician, astronomer, philosopher, and poet Omar Khayyám (1050–1123), attempted to prove the fifth postulate from another explicitly given postulate (based on the fourth of the five principles due to the Philosopher (Aristotle), namely, "Two convergent straight lines intersect and it is impossible for two convergent straight lines to diverge in the direction in which they converge." He derived some of the earlier results belonging to elliptical geometry and hyperbolic geometry, though his postulate excluded the latter possibility. The Saccheri quadrilateral was also first considered by Omar Khayyám in the late 11th century in Book I of Explanations of the Difficulties in the Postulates of Euclid. Unlike many commentators on Euclid before and after him (including Giovanni Girolamo Saccheri), Khayyám was not trying to prove the parallel postulate as such but to derive it from his equivalent postulate. He recognized that three possibilities arose from omitting Euclid's fifth postulate; if two perpendiculars to one line cross another line, judicious choice of the last can make the internal angles where it meets the two perpendiculars equal (it is then parallel to the first line). If those equal internal angles are right angles, we get Euclid's fifth postulate, otherwise, they must be either acute or obtuse. He showed that the acute and obtuse cases led to contradictions using his postulate, but his postulate is now known to be equivalent to the fifth postulate. Nasir al-Din al-Tusi (1201–1274), in his Al-risala al-shafiya'an al-shakk fi'l-khutut al-mutawaziya (Discussion Which Removes Doubt about Parallel Lines) (1250), wrote detailed critiques of the parallel postulate and on Khayyám's attempted proof a century earlier. Nasir al-Din attempted to derive a proof by contradiction of the parallel postulate. He also considered the cases of what are now known as elliptical and hyperbolic geometry, though he ruled out both of them. Nasir al-Din's son, Sadr al-Din (sometimes known as "Pseudo-Tusi"), wrote a book on the subject in 1298, based on his father's later thoughts, which presented one of the earliest arguments for a non-Euclidean hypothesis equivalent to the parallel postulate. "He essentially revised both the Euclidean system of axioms and postulates and the proofs of many propositions from the Elements." His work was published in Rome in 1594 and was studied by European geometers. This work marked the starting point for Saccheri's work on the subject which opened with a criticism of Sadr al-Din's work and the work of Wallis. Giordano Vitale (1633–1711), in his book Euclide restituo (1680, 1686), used the Khayyam-Saccheri quadrilateral to prove that if three points are equidistant on the base AB and the summit CD, then AB and CD are everywhere equidistant. Girolamo Saccheri (1667–1733) pursued the same line of reasoning more thoroughly, correctly obtaining absurdity from the obtuse case (proceeding, like Euclid, from the implicit assumption that lines can be extended indefinitely and have infinite length), but failing to refute the acute case (although he managed to wrongly persuade himself that he had). In 1766 Johann Lambert wrote, but did not publish, Theorie der Parallellinien in which he attempted, as Saccheri did, to prove the fifth postulate. He worked with a figure that today we call a Lambert quadrilateral, a quadrilateral with three right angles (can be considered half of a Saccheri quadrilateral). He quickly eliminated the possibility that the fourth angle is obtuse, as had Saccheri and Khayyám, and then proceeded to prove many theorems under the assumption of an acute angle. Unlike Saccheri, he never felt that he had reached a contradiction with this assumption. He had proved the non-Euclidean result that the sum of the angles in a triangle increases as the area of the triangle decreases, and this led him to speculate on the possibility of a model of the acute case on a sphere of imaginary radius. He did not carry this idea any further. Where Khayyám and Saccheri had attempted to prove Euclid's fifth by disproving the only possible alternatives, the nineteenth century finally saw mathematicians exploring those alternatives and discovering the logically consistent geometries that result. In 1829, Nikolai Ivanovich Lobachevsky published an account of acute geometry in an obscure Russian journal (later re-published in 1840 in German). In 1831, János Bolyai included, in a book by his father, an appendix describing acute geometry, which, doubtlessly, he had developed independently of Lobachevsky. Carl Friedrich Gauss had also studied the problem, but he did not publish any of his results. Upon hearing of Bolyai's results in a letter from Bolyai's father, Farkas Bolyai, Gauss stated: If I commenced by saying that I am unable to praise this work, you would certainly be surprised for a moment. But I cannot say otherwise. To praise it would be to praise myself. Indeed the whole contents of the work, the path taken by your son, the results to which he is led, coincide almost entirely with my meditations, which have occupied my mind partly for the last thirty or thirty-five years. The resulting geometries were later developed by Lobachevsky, Riemann and Poincaré into hyperbolic geometry (the acute case) and elliptic geometry (the obtuse case). The independence of the parallel postulate from Euclid's other axioms was finally demonstrated by Eugenio Beltrami in 1868. == Converse of Euclid's parallel postulate == Euclid did not postulate the converse of his fifth postulate, which is one way to distinguish Euclidean geometry from elliptic geometry. The Elements contains the proof of an equivalent statement (Book I, Proposition 27): If a straight line falling on two straight lines make the alternate angles equal to one another, the straight lines will be parallel to one another. As De Morgan pointed out, this is logically equivalent to (Book I, Proposition 16). These results do not depend upon the fifth postulate, but they do require the second postulate which is violated in elliptic geometry. == Criticism == Attempts to logically prove the parallel postulate, rather than the eighth axiom, were criticized by Arthur Schopenhauer in The World as Will and Idea. However, the argument used by Schopenhauer was that the postulate is evident by perception, not that it was not a logical consequence of the other axioms. == Decomposition of the parallel postulate == The parallel postulate is equivalent to the conjunction of the Lotschnittaxiom and of Aristotle's axiom. The former states that the perpendiculars to the sides of a right angle intersect, while the latter states that there is no upper bound for the lengths of the distances from the leg of an angle to the other leg. As shown in, the parallel postulate is equivalent to the conjunction of the following incidence-geometric forms of the Lotschnittaxiom and of Aristotle's axiom: Given three parallel lines, there is a line that intersects all three of them. Given a line a and two distinct intersecting lines m and n, each different from a, there exists a line g which intersects a and m, but not n. The splitting of the parallel postulate into the conjunction of these incidence-geometric axioms is possible only in the presence of absolute geometry. == See also == Line at infinity Non-Euclidean geometry == Notes == == References == Carroll, Lewis, Euclid and His Modern Rivals, Dover, ISBN 0-486-22968-8 Faber, Richard L. (1983), Foundations of Euclidean and Non-Euclidean Geometry, New York: Marcel Dekker Inc., ISBN 0-8247-1748-1 Henderson, David W.; Taimiņa, Daina (2005), Experiencing Geometry: Euclidean and Non-Euclidean with History (3rd ed.), Upper Saddle River, NJ: Pearson Prentice Hall, ISBN 0-13-143748-8 Katz, Victor J. (1998), History of Mathematics: An Introduction, Addison-Wesley, ISBN 0-321-01618-1, OCLC 38199387 Rozenfeld, Boris A. (1988), A History of Non-Euclidean Geometry: Evolution of the Concept of a Geometric Space, Springer Science+Business Media, ISBN 0-387-96458-4, OCLC 15550634 Smith, John D. (1992), "The Remarkable Ibn al-Haytham", The Mathematical Gazette, 76 (475), Mathematical Association: 189–198, doi:10.2307/3620392, JSTOR 3620392, S2CID 118597450 Boutry, Pierre; Gries, Charly; Narboux, Julien; Schreck, Pascal (2019), "Parallel postulates and continuity axioms: a mechanized study in intuitionistic logic using Coq" (PDF), Journal of Automated Reasoning, 62: 1–68, doi:10.1007/s10817-017-9422-8, S2CID 25900234 Pambuccian, Victor; Schacht, Celia (2021), "The ubiquitous axiom", Results in Mathematics, 76 (3): 1–39, doi:10.1007/s00025-021-01424-3, S2CID 236236967 Pambuccian, Victor (2022), "On a splitting of the parallel postulate", Journal of Geometry, 113 (1): 1–13, doi:10.1007/s00022-022-00626-6, S2CID 246281748 Pambuccian, Victor (2025), "The parallel postulate", Annali dell'Università di Ferrara Sezioni VII - Scienze Matematiche, 71 (1): 1–26, doi:10.1007/s11565-024-00572-y, S2CID 274497557 == External links == On Gauss' Mountains Eder, Michelle (2000), Views of Euclid's Parallel Postulate in Ancient Greece and in Medieval Islam, Rutgers University, retrieved 2008-01-23
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Wikipedia:Parameshvara Nambudiri#0
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Vatasseri Parameshvara Nambudiri (c. 1380–1460) was a major Indian mathematician and astronomer of the Kerala school of astronomy and mathematics founded by Madhava of Sangamagrama. He was also an astrologer. Parameshvara was a proponent of observational astronomy in medieval India and he himself had made a series of eclipse observations to verify the accuracy of the computational methods then in use. Based on his eclipse observations, Parameshvara proposed several corrections to the astronomical parameters which had been in use since the times of Aryabhata. The computational scheme based on the revised set of parameters has come to be known as the Drgganita or Drig system. Parameshvara was also a prolific writer on matters relating to astronomy. At least 25 manuscripts have been identified as being authored by Parameshvara. == Biographical details == Parameshvara was a Hindu of Bhrgugotra following the Ashvalayanasutra of the Rigveda. Parameshvara's family name (Illam) was Vatasseri and his family resided in the village of Alathiyur (Sanskritised as Asvatthagrama) in Tirur, Kerala. Alathiyur is situated on the northern bank of the river Nila (river Bharathappuzha) at its mouth in Kerala. He was a grandson of a disciple of Govinda Bhattathiri (1237–1295 CE), a legendary figure in the astrological traditions of Kerala. Parameshvara studied under teachers Rudra and Narayana, and also under Madhava of Sangamagrama (c. 1350 – c. 1425) the founder of the Kerala school of astronomy and mathematics. Damodara, another prominent member of the Kerala school, was his son and also his pupil. Parameshvara was also a teacher of Nilakantha Somayaji (1444–1544) the author of the celebrated Tantrasamgraha. == Work == Parameshvara wrote commentaries on many mathematical and astronomical works such as those by Bhāskara I and Aryabhata. He made a series of eclipse observations over a 55-year period. Constantly attempted to compare these with the theoretically computed positions of the planets. He revised planetary parameters based on his observations. One of Parameshvara's more significant contributions was his mean value type formula for the inverse interpolation of the sine. He was the first mathematician to give a formula for the radius of the circle circumscribing a cyclic quadrilateral. The expression is sometimes attributed to Lhuilier [1782], 350 years later. With the sides of the cyclic quadrilateral being a, b, c, and d, the radius R of the circumscribed circle is: R = ( a b + c d ) ( a c + b d ) ( a d + b c ) ( − a + b + c + d ) ( a − b + c + d ) ( a + b − c + d ) ( a + b + c − d ) . {\displaystyle R={\sqrt {\frac {(ab+cd)(ac+bd)(ad+bc)}{(-a+b+c+d)(a-b+c+d)(a+b-c+d)(a+b+c-d)}}}.} == Works by Parameshvara == The following works of Parameshvara are well-known. A complete list of all manuscripts attributed to Parameshvara is available in Pingree. Bhatadipika – Commentary on Āryabhaṭīya of Āryabhaṭa I Karmadipika – Commentary on Mahabhaskariya of Bhaskara I Paramesvari – Commentary on Laghubhaskariya of Bhaskara I Sidhantadipika – Commentary on Mahabhaskariyabhashya of Govindasvāmi Vivarana – Commentary on Surya Siddhanta and Lilāvati Drgganita – Description of the Drig system (composed in 1431 CE) Goladipika – Spherical geometry and astronomy (composed in 1443 CE) Grahanamandana – Computation of eclipses (Its epoch is 15 July 1411 CE.) Grahanavyakhyadipika – On the rationale of the theory of eclipses Vakyakarana – Methods for the derivation of several astronomical tables == See also == List of astronomers and mathematicians of the Kerala school == References == == Further reading == David Pingree, Biography in Dictionary of Scientific Biography (New York 1970–1990). Bhaskara, Laghubhaskariyam : With Parameshvara's commentary (Poona, 1946). Bhaskara, Mahabhaskariyam: With Parameshvara's commentary called Karmadipika (Poona, 1945). Munjala, Laghumanasam : with commentary by Parameshvara (Poona, 1944). T.A. Sarasvati Amma (1979) Geometry in ancient and medieval India, (Delhi). K. Shankar Shukla (1957) The Surya-siddhanta with the commentary of Parameshvara (Lucknow). K. V. Sarma (2008), "Paramesvara", Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures (2nd edition) edited by Helaine Selin, Springer, ISBN 978-1-4020-4559-2. Kim Plofker (1996) "An example of the secant method of iterative approximation in a fifteenth-century Sanskrit text", Historia Mathematica 23 (3): 246–256. K. K. Raja (1963) "Astronomy and mathematics in Kerala", Brahmavidya 27; 136–143. K. Chandra Hari (2003). "Eclipse observations of Parameshvara, the 14th–15th-century astronomer of Kerala" (PDF). Indian Journal of History of Science. 38 (1): 43–57. Archived from the original (PDF) on 16 March 2012. Retrieved 28 January 2010. == External links == Achar, Narahari (2007). "Parameśvara of Vāṭaśśeri [Parmeśvara I]". In Thomas Hockey; et al. (eds.). The Biographical Encyclopedia of Astronomers. New York: Springer. p. 870. ISBN 978-0-387-31022-0. (PDF version) O'Connor, John J.; Robertson, Edmund F., "Parameshvara Nambudiri", MacTutor History of Mathematics Archive, University of St Andrews
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Wikipedia:Paraproduct#0
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In mathematics, a paraproduct is a non-commutative bilinear operator acting on functions that in some sense is like the product of the two functions it acts on. According to Svante Janson and Jaak Peetre, in an article from 1988, "the name 'paraproduct' denotes an idea rather than a unique definition; several versions exist and can be used for the same purposes." The concept emerged in J.-M. Bony’s theory of paradifferential operators. This said, for a given operator Λ {\displaystyle \Lambda } to be defined as a paraproduct, it is normally required to satisfy the following properties: It should "reconstruct the product" in the sense that for any pair of functions ( f , g ) {\displaystyle (f,g)} in its domain, f g = Λ ( f , g ) + Λ ( g , f ) . {\displaystyle fg=\Lambda (f,g)+\Lambda (g,f).} For any appropriate functions f {\displaystyle f} and h {\displaystyle h} with h ( 0 ) = 0 {\displaystyle h(0)=0} , it is the case that h ( f ) = Λ ( f , h ′ ( f ) ) {\displaystyle h(f)=\Lambda (f,h'(f))} . It should satisfy some form of the Leibniz rule. A paraproduct may also be required to satisfy some form of Hölder's inequality. == Notes == == Further references == Árpád Bényi, Diego Maldonado, and Virginia Naibo, "What is a Paraproduct?", Notices of the American Mathematical Society, Vol. 57, No. 7 (Aug., 2010), pp. 858–860.
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Wikipedia:Paratingent cone#0
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In mathematics, the paratingent cone and contingent cone were introduced by Bouligand (1932), and are closely related to tangent cones. == Definition == Let S {\displaystyle S} be a nonempty subset of a real normed vector space ( X , ‖ ⋅ ‖ ) {\displaystyle (X,\|\cdot \|)} . Let some x ¯ ∈ cl ( S ) {\displaystyle {\bar {x}}\in \operatorname {cl} (S)} be a point in the closure of S {\displaystyle S} . An element h ∈ X {\displaystyle h\in X} is called a tangent (or tangent vector) to S {\displaystyle S} at x ¯ {\displaystyle {\bar {x}}} , if there is a sequence ( x n ) n ∈ N {\displaystyle (x_{n})_{n\in \mathbb {N} }} of elements x n ∈ S {\displaystyle x_{n}\in S} and a sequence ( λ n ) n ∈ N {\displaystyle (\lambda _{n})_{n\in \mathbb {N} }} of positive real numbers λ n > 0 {\displaystyle \lambda _{n}>0} such that x ¯ = lim n → ∞ x n {\displaystyle {\bar {x}}=\lim _{n\to \infty }x_{n}} and h = lim n → ∞ λ n ( x n − x ¯ ) . {\displaystyle h=\lim _{n\to \infty }\lambda _{n}(x_{n}-{\bar {x}}).} The set T ( S , x ¯ ) {\displaystyle T(S,{\bar {x}})} of all tangents to S {\displaystyle S} at x ¯ {\displaystyle {\bar {x}}} is called the contingent cone (or the Bouligand tangent cone) to S {\displaystyle S} at x ¯ {\displaystyle {\bar {x}}} . An equivalent definition is given in terms of a distance function and the limit infimum. As before, let ( X , ‖ ⋅ ‖ ) {\displaystyle (X,\|\cdot \|)} be a normed vector space and take some nonempty set S ⊂ X {\displaystyle S\subset X} . For each x ∈ X {\displaystyle x\in X} , let the distance function to S {\displaystyle S} be d S ( x ) := inf { ‖ x − x ′ ‖ ∣ x ′ ∈ S } . {\displaystyle d_{S}(x):=\inf\{\|x-x'\|\mid x'\in S\}.} Then, the contingent cone to S ⊂ X {\displaystyle S\subset X} at x ∈ cl ( S ) {\displaystyle x\in \operatorname {cl} (S)} is defined by T S ( x ) := { v : lim inf h → 0 + d S ( x + h v ) h = 0 } . {\displaystyle T_{S}(x):=\left\{v:\liminf _{h\to 0^{+}}{\frac {d_{S}(x+hv)}{h}}=0\right\}.} == References ==
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Wikipedia:Parent function#0
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In mathematics education, a parent function is the core representation of a function type without manipulations such as translation and dilation. For example, for the family of quadratic functions having the general form y = a x 2 + b x + c , {\displaystyle y=ax^{2}+bx+c\,,} the simplest function is y = x 2 {\displaystyle y=x^{2}} , and every quadratic may be converted to that form by translations and dilations, which may be seen by completing the square. This is therefore the parent function of the family of quadratic equations. For linear and quadratic functions, the graph of any function can be obtained from the graph of the parent function by simple translations and stretches parallel to the axes. For example, the graph of y = x2 − 4x + 7 can be obtained from the graph of y = x2 by translating +2 units along the X axis and +3 units along Y axis. This is because the equation can also be written as y − 3 = (x − 2)2. For many trigonometric functions, the parent function is usually a basic sin(x), cos(x), or tan(x). For example, the graph of y = A sin(x) + B cos(x) can be obtained from the graph of y = sin(x) by translating it through an angle α along the positive X axis (where tan(α) = A⁄B), then stretching it parallel to the Y axis using a stretch factor R, where R2 = A2 + B2. This is because A sin(x) + B cos(x) can be written as R sin(x−α) (see List of trigonometric identities). Alternatively, the parent function may be interpreted as cos(x). The concept of parent function is less clear or inapplicable polynomials of higher degree because of the extra turning points, but for the family of n-degree polynomial functions for any given n, the parent function is sometimes taken as xn, or, to simplify further, x2 when n is even and x3 for odd n. Turning points may be established by differentiation to provide more detail of the graph. == See also == Curve sketching == References == == External links == Video explanation at VirtualNerd.com
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Wikipedia:Parity function#0
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In Boolean algebra, a parity function is a Boolean function whose value is one if and only if the input vector has an odd number of ones. The parity function of two inputs is also known as the XOR function. The parity function is notable for its role in theoretical investigation of circuit complexity of Boolean functions. The output of the parity function is the parity bit. == Definition == The n {\displaystyle n} -variable parity function is the Boolean function f : { 0 , 1 } n → { 0 , 1 } {\displaystyle f:\{0,1\}^{n}\to \{0,1\}} with the property that f ( x ) = 1 {\displaystyle f(x)=1} if and only if the number of ones in the vector x ∈ { 0 , 1 } n {\displaystyle x\in \{0,1\}^{n}} is odd. In other words, f {\displaystyle f} is defined as follows: f ( x ) = x 1 ⊕ x 2 ⊕ ⋯ ⊕ x n {\displaystyle f(x)=x_{1}\oplus x_{2}\oplus \dots \oplus x_{n}} where ⊕ {\displaystyle \oplus } denotes exclusive or. == Properties == Parity only depends on the number of ones and is therefore a symmetric Boolean function. The n-variable parity function and its negation are the only Boolean functions for which all disjunctive normal forms have the maximal number of 2 n − 1 monomials of length n and all conjunctive normal forms have the maximal number of 2 n − 1 clauses of length n. == Computational complexity == Some of the earliest work in computational complexity was 1961 bound of Bella Subbotovskaya showing the size of a Boolean formula computing parity must be at least Ω ( n 3 / 2 ) {\displaystyle \Omega (n^{3/2})} . This work uses the method of random restrictions. This exponent of 3 / 2 {\displaystyle 3/2} has been increased through careful analysis to 1.63 {\displaystyle 1.63} by Paterson and Zwick (1993) and then to 2 {\displaystyle 2} by Håstad (1998). In the early 1980s, Merrick Furst, James Saxe and Michael Sipser and independently Miklós Ajtai established super-polynomial lower bounds on the size of constant-depth Boolean circuits for the parity function, i.e., they showed that polynomial-size constant-depth circuits cannot compute the parity function. Similar results were also established for the majority, multiplication and transitive closure functions, by reduction from the parity function. Håstad (1987) established tight exponential lower bounds on the size of constant-depth Boolean circuits for the parity function. Håstad's Switching Lemma is the key technical tool used for these lower bounds and Johan Håstad was awarded the Gödel Prize for this work in 1994. The precise result is that depth-k circuits with AND, OR, and NOT gates require size exp ( Ω ( n 1 k − 1 ) ) {\displaystyle \exp(\Omega (n^{\frac {1}{k-1}}))} to compute the parity function. This is asymptotically almost optimal as there are depth-k circuits computing parity which have size exp ( O ( n 1 k − 1 ) t ) {\displaystyle \exp(O(n^{\frac {1}{k-1}})t)} . == Infinite version == An infinite parity function is a function f : { 0 , 1 } ω → { 0 , 1 } {\displaystyle f\colon \{0,1\}^{\omega }\to \{0,1\}} mapping every infinite binary string to 0 or 1, having the following property: if w {\displaystyle w} and v {\displaystyle v} are infinite binary strings differing only on finite number of coordinates then f ( w ) = f ( v ) {\displaystyle f(w)=f(v)} if and only if w {\displaystyle w} and v {\displaystyle v} differ on even number of coordinates. Assuming axiom of choice it can be proved that parity functions exist and there are 2 2 ℵ 0 {\displaystyle 2^{2^{\aleph _{0}}}} many of them; as many as the number of all functions from { 0 , 1 } ω {\displaystyle \{0,1\}^{\omega }} to { 0 , 1 } {\displaystyle \{0,1\}} . It is enough to take one representative per equivalence class of relation ≈ {\displaystyle \approx } defined as follows: w ≈ v {\displaystyle w\approx v} if w {\displaystyle w} and v {\displaystyle v} differ at finite number of coordinates. Having such representatives, we can map all of them to 0 {\displaystyle 0} ; the rest of f {\displaystyle f} values are deducted unambiguously. Another construction of an infinite parity function can be done using a non-principal ultrafilter U {\displaystyle U} on ω {\displaystyle \omega } . The existence of non-principal ultrafilters on ω {\displaystyle \omega } follows from – and is strictly weaker than – the axiom of choice. For any w : ω → { 0 , 1 } {\displaystyle w:\omega \to \{0,1\}} we consider the set A w = { n ∈ ω ∣ { k ≤ n ∣ w ( k ) = 0 } is even } {\displaystyle A_{w}=\{n\in \omega \mid \{k\leq n\mid w(k)=0\}{\text{ is even}}\}} . The infinite parity function f {\displaystyle f} is defined by mapping w {\displaystyle w} to 0 {\displaystyle 0} if and only if A w {\displaystyle A_{w}} is an element of the ultrafilter. It is necessary to assume at least some amount of choice to prove that infinite parity functions exist. If f {\displaystyle f} is an infinite parity function and we consider the inverse image f − 1 [ 0 ] {\displaystyle f^{-1}[0]} as a subset of the Cantor space { 0 , 1 } ω {\displaystyle \{0,1\}^{\omega }} , then f − 1 [ 0 ] {\displaystyle f^{-1}[0]} is a non-measurable set and does not have the property of Baire. Without the axiom of choice, it is consistent (relative to ZF) that all subsets of the Cantor space are measurable and have the property of Baire and thus that no infinite parity function exists; this holds in the Solovay model, for instance. == See also == Walsh function, a continuous equivalent Parity bit, the output of the function Piling-up lemma, a statistical property for independent inputs Multiway switching, a physical implementation often used to control lighting Related topics: Error Correction Error Detection == References ==
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Wikipedia:Parker–Sochacki method#0
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In mathematics, the Parker–Sochacki method is an algorithm for solving systems of ordinary differential equations (ODEs), developed by G. Edgar Parker and James Sochacki, of the James Madison University Mathematics Department. The method produces Maclaurin series solutions to systems of differential equations, with the coefficients in either algebraic or numerical form. == Summary == The Parker–Sochacki method rests on two simple observations: If a set of ODEs has a particular form, then the Picard method can be used to find their solution in the form of a power series. If the ODEs do not have the required form, it is nearly always possible to find an expanded set of equations that do have the required form, such that a subset of the solution is a solution of the original ODEs. Several coefficients of the power series are calculated in turn, a time step is chosen, the series is evaluated at that time, and the process repeats. The end result is a high order piecewise solution to the original ODE problem. The order of the solution desired is an adjustable variable in the program that can change between steps. The order of the solution is only limited by the floating point representation on the machine running the program. And in some cases can be either extended by using arbitrary precision floating point numbers, or for special cases by finding solution with only integer or rational coefficients. == Advantages == The method requires only addition, subtraction, and multiplication, making it very convenient for high-speed computation. (The only divisions are inverses of small integers, which can be precomputed.) Use of a high order—calculating many coefficients of the power series—is convenient. (Typically a higher order permits a longer time step without loss of accuracy, which improves efficiency.) The order and step size can be easily changed from one step to the next. It is possible to calculate a guaranteed error bound on the solution. Arbitrary precision floating point libraries allow this method to compute arbitrarily accurate solutions. With the Parker–Sochacki method, information between integration steps is developed at high order. As the Parker–Sochacki method integrates, the program can be designed to save the power series coefficients that provide a smooth solution between points in time. The coefficients can be saved and used so that polynomial evaluation provides the high order solution between steps. With most other classical integration methods, one would have to resort to interpolation to get information between integration steps, leading to an increase of error. There is an a priori error bound for a single step with the Parker–Sochacki method. This allows a Parker–Sochacki program to calculate the step size that guarantees that the error is below any non-zero given tolerance. Using this calculated step size with an error tolerance of less than half of the machine epsilon yields a symplectic integration. == Disadvantages == Most methods for numerically solving ODEs require only the evaluation of derivatives for chosen values of the variables, so systems like MATLAB include implementations of several methods all sharing the same calling sequence. Users can try different methods by simply changing the name of the function called. The Parker–Sochacki method requires more work to put the equations into the proper form, and cannot use the same calling sequence. == References == == External links == Polynomial ODEs – Examples, Solutions, Properties (PDF), retrieved August 27, 2017. A thorough explanation of the paradigm and application of the Parker–Sochacki method Joseph W. Rudmin (1998), "Application of the Parker–Sochacki Method to Celestial Mechanics", Journal of Computational Neuroscience, 27: 115–133, arXiv:1007.1677, doi:10.1007/s10827-008-0131-5. A demonstration of the theory and usage of the Parker–Sochacki method, including a solution for the classical Newtonian N-body problem with mutual gravitational attraction. The Modified Picard Method., retrieved November 11, 2013. A collection of papers and some Matlab code.
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Wikipedia:Parry–Sullivan invariant#0
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In mathematics, the Parry–Sullivan invariant (or Parry–Sullivan number) is a numerical quantity of interest in the study of incidence matrices in graph theory, and of certain one-dimensional dynamical systems. It provides a partial classification of non-trivial irreducible incidence matrices. It is named after the English mathematician Bill Parry and the American mathematician Dennis Sullivan, who introduced the invariant in a joint paper published in the journal Topology in 1975. == Definition == Let A be an n × n incidence matrix. Then the Parry–Sullivan number of A is defined to be P S ( A ) = det ( I − A ) , {\displaystyle \mathrm {PS} (A)=\det(I-A),} where I denotes the n × n identity matrix. == Properties == It can be shown that, for nontrivial irreducible incidence matrices, flow equivalence is completely determined by the Parry–Sullivan number and the Bowen–Franks group. == References ==
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Wikipedia:Part III of the Mathematical Tripos#0
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Part III of the Mathematical Tripos (officially Master of Mathematics/Master of Advanced Study) is a one-year master's-level taught course in mathematics offered at the Faculty of Mathematics, University of Cambridge. It is regarded as the most difficult and intensive mathematics course in the world. Roughly one third of the students take the course as a continuation at Cambridge after finishing the Parts IA, IB, and II of the Mathematical Tripos resulting in an integrated Master's (M.Math), whilst the remaining two thirds are external students who take the course as a one-year Master's (M.A.St). == History == The Smith's Prize Examination was founded by bequest of Robert Smith upon his death in 1768 to encourage the study of more advanced mathematics than that found in the undergraduate course. T. W. Körner notes Only a small handful of students took the Smith's prize examination in the nineteenth century. When Karl Pearson took the examination in 1879, the examiners were Stokes, Maxwell, Cayley, and Todhunter and the examinees went on each occasion to the examiner's house, did a morning paper, had lunch there, and continued their work on the paper in the afternoon. In 1883 this was replaced by an exam called Part III and the Smith's Prize awarded for an essay rather than examination. In 1886 this exam was renamed Part II, and later in 1909 Part II, Schedule B. In 1934 it was again renamed Part III. In the 1980s the Certificate of Advanced Study in Mathematics was introduced; for those students successfully completing Part III of the Mathematical Tripos in Easter Term 2011 CASM was replaced by two new degrees, the Master of Mathematics (M.Math.) and Master of Advanced Study (M.A.St.). All who have passed the course since 1962 are entitled to these new degrees. The first retrospective M.Math and M.A.St. degrees were conferred as part of a celebration of the university's 800th anniversary. The course is often still referred to as Part III. == Academics == === Course structure === The course lasts one year, divided into three eight-week terms. There is a wide variety of lectures on both pure and applied maths, mostly concentrated in the first two terms. The third term is primarily for examinations (and revision for said examinations) which, together with the option of writing a part III essay (introduced in the 1970s, a miniature thesis, often in the form of a literature review), determine one's final grade entirely. As of the 2024-2025 academic year, however, the part III essay is effectively mandatory for achieving a final mark of Distinction. === Degree awarded === Students who have completed their undergraduate degree at Cambridge will be awarded both a Bachelor of Arts (B.A.) and the Master of Mathematics (M.Math.) degree for four years of study, provided they have not previously graduated with a B.A. This allows Cambridge graduates to remain eligible for government funding for the course. Progression from Part II of the Mathematical Tripos to Part III requires either a first class degree in Part II or very good performances in Parts IB and Part II. Students who complete Part III of the Mathematical Tripos, but did not complete undergraduate studies at Cambridge (or have previously graduated with a B.A.) are awarded the Master of Advanced Study (M.A.St.) in Mathematics degree for the one-year course. Until 2011, the programme resulted in a Certificate of Advanced Study in Mathematics instead of a master's degree. === Grading === The grades available are Fail, Pass (Honours), Merit, and Distinction (the Merit grade was introduced in 2000). Cambridge recognises that in Part III of the mathematical tripos a merit is equivalent to a First Class in the other parts of the Tripos. The level of achievement required for a distinction is yet higher than a typical First Class degree. Traditionally, results are announced in the university's Senate House. Standing on the balcony, the examiner reads out the class results for each student, and printed copies of the results are then thrown to the audience below. The students' exact rankings are no longer announced, but the highest-ranked student is still identified, nowadays by the tipping of the examiner's academic hat when the relevant name is read out. === Prizes === In addition to the grades, there are six associated prizes. Five of these may be awarded at the discretion of the examiners: the Mayhew Prize for applied mathematics, the Tyson Medal for mathematics and astronomy, the Bartlett Prize for applied probability, the Wishart Prize for statistics and the Pure Mathematics Prize for pure mathematics. Several notable astronomers and astrophysicists have been awarded the Tyson Medal in the history of Part III maths, including Jayant Narlikar, Ray Lyttleton and Edmund Whittaker. In addition, the Thomas Bond Sprague Prize is awarded by the Rollo Davidson Trust for distinguished performance in actuarial science, finance, insurance, mathematics of operational research, probability, risk and statistics. == References ==
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Wikipedia:Partial algebra#0
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In abstract algebra, a partial algebra is a generalization of universal algebra to partial operations. == Example(s) == partial groupoid field — the multiplicative inversion is the only proper partial operation effect algebras == Structure == There is a "Meta Birkhoff Theorem" by Andreka, Nemeti and Sain (1982). == References == == Further reading == Peter Burmeister (2002) [1986]. A Model Theoretic Oriented Approach to Partial Algebras. CiteSeerX 10.1.1.92.6134. Horst Reichel (1984). Structural induction on partial algebras. Akademie-Verlag. Horst Reichel (1987). Initial computability, algebraic specifications, and partial algebras. Clarendon Press. ISBN 978-0-19-853806-6.
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Wikipedia:Partial derivative#0
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In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Partial derivatives are used in vector calculus and differential geometry. The partial derivative of a function f ( x , y , … ) {\displaystyle f(x,y,\dots )} with respect to the variable x {\displaystyle x} is variously denoted by It can be thought of as the rate of change of the function in the x {\displaystyle x} -direction. Sometimes, for z = f ( x , y , … ) {\displaystyle z=f(x,y,\ldots )} , the partial derivative of z {\displaystyle z} with respect to x {\displaystyle x} is denoted as ∂ z ∂ x . {\displaystyle {\tfrac {\partial z}{\partial x}}.} Since a partial derivative generally has the same arguments as the original function, its functional dependence is sometimes explicitly signified by the notation, such as in: f x ′ ( x , y , … ) , ∂ f ∂ x ( x , y , … ) . {\displaystyle f'_{x}(x,y,\ldots ),{\frac {\partial f}{\partial x}}(x,y,\ldots ).} The symbol used to denote partial derivatives is ∂. One of the first known uses of this symbol in mathematics is by Marquis de Condorcet from 1770, who used it for partial differences. The modern partial derivative notation was created by Adrien-Marie Legendre (1786), although he later abandoned it; Carl Gustav Jacob Jacobi reintroduced the symbol in 1841. == Definition == Like ordinary derivatives, the partial derivative is defined as a limit. Let U be an open subset of R n {\displaystyle \mathbb {R} ^{n}} and f : U → R {\displaystyle f:U\to \mathbb {R} } a function. The partial derivative of f at the point a = ( a 1 , … , a n ) ∈ U {\displaystyle \mathbf {a} =(a_{1},\ldots ,a_{n})\in U} with respect to the i-th variable xi is defined as ∂ ∂ x i f ( a ) = lim h → 0 f ( a 1 , … , a i − 1 , a i + h , a i + 1 … , a n ) − f ( a 1 , … , a i , … , a n ) h = lim h → 0 f ( a + h e i ) − f ( a ) h . {\displaystyle {\begin{aligned}{\frac {\partial }{\partial x_{i}}}f(\mathbf {a} )&=\lim _{h\to 0}{\frac {f(a_{1},\ldots ,a_{i-1},a_{i}+h,a_{i+1}\,\ldots ,a_{n})\ -f(a_{1},\ldots ,a_{i},\dots ,a_{n})}{h}}\\&=\lim _{h\to 0}{\frac {f(\mathbf {a} +h\mathbf {e_{i}} )-f(\mathbf {a} )}{h}}\,.\end{aligned}}} Where e i {\displaystyle \mathbf {e_{i}} } is the unit vector of i-th variable xi. Even if all partial derivatives ∂ f / ∂ x i ( a ) {\displaystyle \partial f/\partial x_{i}(a)} exist at a given point a, the function need not be continuous there. However, if all partial derivatives exist in a neighborhood of a and are continuous there, then f is totally differentiable in that neighborhood and the total derivative is continuous. In this case, it is said that f is a C1 function. This can be used to generalize for vector valued functions, f : U → R m {\displaystyle f:U\to \mathbb {R} ^{m}} , by carefully using a componentwise argument. The partial derivative ∂ f ∂ x {\textstyle {\frac {\partial f}{\partial x}}} can be seen as another function defined on U and can again be partially differentiated. If the direction of derivative is not repeated, it is called a mixed partial derivative. If all mixed second order partial derivatives are continuous at a point (or on a set), f is termed a C2 function at that point (or on that set); in this case, the partial derivatives can be exchanged by Clairaut's theorem: ∂ 2 f ∂ x i ∂ x j = ∂ 2 f ∂ x j ∂ x i . {\displaystyle {\frac {\partial ^{2}f}{\partial x_{i}\partial x_{j}}}={\frac {\partial ^{2}f}{\partial x_{j}\partial x_{i}}}.} == Notation == For the following examples, let f be a function in x, y, and z. First-order partial derivatives: ∂ f ∂ x = f x ′ = ∂ x f . {\displaystyle {\frac {\partial f}{\partial x}}=f'_{x}=\partial _{x}f.} Second-order partial derivatives: ∂ 2 f ∂ x 2 = f x x ″ = ∂ x x f = ∂ x 2 f . {\displaystyle {\frac {\partial ^{2}f}{\partial x^{2}}}=f''_{xx}=\partial _{xx}f=\partial _{x}^{2}f.} Second-order mixed derivatives: ∂ 2 f ∂ y ∂ x = ∂ ∂ y ( ∂ f ∂ x ) = ( f x ′ ) y ′ = f x y ″ = ∂ y x f = ∂ y ∂ x f . {\displaystyle {\frac {\partial ^{2}f}{\partial y\,\partial x}}={\frac {\partial }{\partial y}}\left({\frac {\partial f}{\partial x}}\right)=(f'_{x})'_{y}=f''_{xy}=\partial _{yx}f=\partial _{y}\partial _{x}f.} Higher-order partial and mixed derivatives: ∂ i + j + k f ∂ x i ∂ y j ∂ z k = f ( i , j , k ) = ∂ x i ∂ y j ∂ z k f . {\displaystyle {\frac {\partial ^{i+j+k}f}{\partial x^{i}\partial y^{j}\partial z^{k}}}=f^{(i,j,k)}=\partial _{x}^{i}\partial _{y}^{j}\partial _{z}^{k}f.} When dealing with functions of multiple variables, some of these variables may be related to each other, thus it may be necessary to specify explicitly which variables are being held constant to avoid ambiguity. In fields such as statistical mechanics, the partial derivative of f with respect to x, holding y and z constant, is often expressed as ( ∂ f ∂ x ) y , z . {\displaystyle \left({\frac {\partial f}{\partial x}}\right)_{y,z}.} Conventionally, for clarity and simplicity of notation, the partial derivative function and the value of the function at a specific point are conflated by including the function arguments when the partial derivative symbol (Leibniz notation) is used. Thus, an expression like ∂ f ( x , y , z ) ∂ x {\displaystyle {\frac {\partial f(x,y,z)}{\partial x}}} is used for the function, while ∂ f ( u , v , w ) ∂ u {\displaystyle {\frac {\partial f(u,v,w)}{\partial u}}} might be used for the value of the function at the point ( x , y , z ) = ( u , v , w ) {\displaystyle (x,y,z)=(u,v,w)} . However, this convention breaks down when we want to evaluate the partial derivative at a point like ( x , y , z ) = ( 17 , u + v , v 2 ) {\displaystyle (x,y,z)=(17,u+v,v^{2})} . In such a case, evaluation of the function must be expressed in an unwieldy manner as ∂ f ( x , y , z ) ∂ x ( 17 , u + v , v 2 ) {\displaystyle {\frac {\partial f(x,y,z)}{\partial x}}(17,u+v,v^{2})} or ∂ f ( x , y , z ) ∂ x | ( x , y , z ) = ( 17 , u + v , v 2 ) {\displaystyle \left.{\frac {\partial f(x,y,z)}{\partial x}}\right|_{(x,y,z)=(17,u+v,v^{2})}} in order to use the Leibniz notation. Thus, in these cases, it may be preferable to use the Euler differential operator notation with D i {\displaystyle D_{i}} as the partial derivative symbol with respect to the i-th variable. For instance, one would write D 1 f ( 17 , u + v , v 2 ) {\displaystyle D_{1}f(17,u+v,v^{2})} for the example described above, while the expression D 1 f {\displaystyle D_{1}f} represents the partial derivative function with respect to the first variable. For higher order partial derivatives, the partial derivative (function) of D i f {\displaystyle D_{i}f} with respect to the j-th variable is denoted D j ( D i f ) = D i , j f {\displaystyle D_{j}(D_{i}f)=D_{i,j}f} . That is, D j ∘ D i = D i , j {\displaystyle D_{j}\circ D_{i}=D_{i,j}} , so that the variables are listed in the order in which the derivatives are taken, and thus, in reverse order of how the composition of operators is usually notated. Of course, Clairaut's theorem implies that D i , j = D j , i {\displaystyle D_{i,j}=D_{j,i}} as long as comparatively mild regularity conditions on f are satisfied. == Gradient == An important example of a function of several variables is the case of a scalar-valued function f ( x 1 , … , x n ) {\displaystyle f(x_{1},\ldots ,x_{n})} on a domain in Euclidean space R n {\displaystyle \mathbb {R} ^{n}} (e.g., on R 2 {\displaystyle \mathbb {R} ^{2}} or R 3 {\displaystyle \mathbb {R} ^{3}} ). In this case f has a partial derivative ∂ f / ∂ x j {\displaystyle \partial f/\partial x_{j}} with respect to each variable xj. At the point a, these partial derivatives define the vector ∇ f ( a ) = ( ∂ f ∂ x 1 ( a ) , … , ∂ f ∂ x n ( a ) ) . {\displaystyle \nabla f(a)=\left({\frac {\partial f}{\partial x_{1}}}(a),\ldots ,{\frac {\partial f}{\partial x_{n}}}(a)\right).} This vector is called the gradient of f at a. If f is differentiable at every point in some domain, then the gradient is a vector-valued function ∇f which takes the point a to the vector ∇f(a). Consequently, the gradient produces a vector field. A common abuse of notation is to define the del operator (∇) as follows in three-dimensional Euclidean space R 3 {\displaystyle \mathbb {R} ^{3}} with unit vectors i ^ , j ^ , k ^ {\displaystyle {\hat {\mathbf {i} }},{\hat {\mathbf {j} }},{\hat {\mathbf {k} }}} : ∇ = [ ∂ ∂ x ] i ^ + [ ∂ ∂ y ] j ^ + [ ∂ ∂ z ] k ^ {\displaystyle \nabla =\left[{\frac {\partial }{\partial x}}\right]{\hat {\mathbf {i} }}+\left[{\frac {\partial }{\partial y}}\right]{\hat {\mathbf {j} }}+\left[{\frac {\partial }{\partial z}}\right]{\hat {\mathbf {k} }}} Or, more generally, for n-dimensional Euclidean space R n {\displaystyle \mathbb {R} ^{n}} with coordinates x 1 , … , x n {\displaystyle x_{1},\ldots ,x_{n}} and unit vectors e ^ 1 , … , e ^ n {\displaystyle {\hat {\mathbf {e} }}_{1},\ldots ,{\hat {\mathbf {e} }}_{n}} : ∇ = ∑ j = 1 n [ ∂ ∂ x j ] e ^ j = [ ∂ ∂ x 1 ] e ^ 1 + [ ∂ ∂ x 2 ] e ^ 2 + ⋯ + [ ∂ ∂ x n ] e ^ n {\displaystyle \nabla =\sum _{j=1}^{n}\left[{\frac {\partial }{\partial x_{j}}}\right]{\hat {\mathbf {e} }}_{j}=\left[{\frac {\partial }{\partial x_{1}}}\right]{\hat {\mathbf {e} }}_{1}+\left[{\frac {\partial }{\partial x_{2}}}\right]{\hat {\mathbf {e} }}_{2}+\dots +\left[{\frac {\partial }{\partial x_{n}}}\right]{\hat {\mathbf {e} }}_{n}} == Directional derivative == == Example == Suppose that f is a function of more than one variable. For instance, z = f ( x , y ) = x 2 + x y + y 2 . {\displaystyle z=f(x,y)=x^{2}+xy+y^{2}.} The graph of this function defines a surface in Euclidean space. To every point on this surface, there are an infinite number of tangent lines. Partial differentiation is the act of choosing one of these lines and finding its slope. Usually, the lines of most interest are those that are parallel to the xz-plane, and those that are parallel to the yz-plane (which result from holding either y or x constant, respectively). To find the slope of the line tangent to the function at P(1, 1) and parallel to the xz-plane, we treat y as a constant. The graph and this plane are shown on the right. Below, we see how the function looks on the plane y = 1. By finding the derivative of the equation while assuming that y is a constant, we find that the slope of f at the point (x, y) is: ∂ z ∂ x = 2 x + y . {\displaystyle {\frac {\partial z}{\partial x}}=2x+y.} So at (1, 1), by substitution, the slope is 3. Therefore, ∂ z ∂ x = 3 {\displaystyle {\frac {\partial z}{\partial x}}=3} at the point (1, 1). That is, the partial derivative of z with respect to x at (1, 1) is 3, as shown in the graph. The function f can be reinterpreted as a family of functions of one variable indexed by the other variables: f ( x , y ) = f y ( x ) = x 2 + x y + y 2 . {\displaystyle f(x,y)=f_{y}(x)=x^{2}+xy+y^{2}.} In other words, every value of y defines a function, denoted fy, which is a function of one variable x. That is, f y ( x ) = x 2 + x y + y 2 . {\displaystyle f_{y}(x)=x^{2}+xy+y^{2}.} In this section the subscript notation fy denotes a function contingent on a fixed value of y, and not a partial derivative. Once a value of y is chosen, say a, then f(x,y) determines a function fa which traces a curve x2 + ax + a2 on the xz-plane: f a ( x ) = x 2 + a x + a 2 . {\displaystyle f_{a}(x)=x^{2}+ax+a^{2}.} In this expression, a is a constant, not a variable, so fa is a function of only one real variable, that being x. Consequently, the definition of the derivative for a function of one variable applies: f a ′ ( x ) = 2 x + a . {\displaystyle f_{a}'(x)=2x+a.} The above procedure can be performed for any choice of a. Assembling the derivatives together into a function gives a function which describes the variation of f in the x direction: ∂ f ∂ x ( x , y ) = 2 x + y . {\displaystyle {\frac {\partial f}{\partial x}}(x,y)=2x+y.} This is the partial derivative of f with respect to x. Here '∂' is a rounded 'd' called the partial derivative symbol; to distinguish it from the letter 'd', '∂' is sometimes pronounced "partial". == Higher order partial derivatives == Second and higher order partial derivatives are defined analogously to the higher order derivatives of univariate functions. For the function f ( x , y , . . . ) {\displaystyle f(x,y,...)} the "own" second partial derivative with respect to x is simply the partial derivative of the partial derivative (both with respect to x):: 316–318 ∂ 2 f ∂ x 2 ≡ ∂ ∂ f / ∂ x ∂ x ≡ ∂ f x ∂ x ≡ f x x . {\displaystyle {\frac {\partial ^{2}f}{\partial x^{2}}}\equiv \partial {\frac {\partial f/\partial x}{\partial x}}\equiv {\frac {\partial f_{x}}{\partial x}}\equiv f_{xx}.} The cross partial derivative with respect to x and y is obtained by taking the partial derivative of f with respect to x, and then taking the partial derivative of the result with respect to y, to obtain ∂ 2 f ∂ y ∂ x ≡ ∂ ∂ f / ∂ x ∂ y ≡ ∂ f x ∂ y ≡ f x y . {\displaystyle {\frac {\partial ^{2}f}{\partial y\,\partial x}}\equiv \partial {\frac {\partial f/\partial x}{\partial y}}\equiv {\frac {\partial f_{x}}{\partial y}}\equiv f_{xy}.} Schwarz's theorem states that if the second derivatives are continuous, the expression for the cross partial derivative is unaffected by which variable the partial derivative is taken with respect to first and which is taken second. That is, ∂ 2 f ∂ x ∂ y = ∂ 2 f ∂ y ∂ x {\displaystyle {\frac {\partial ^{2}f}{\partial x\,\partial y}}={\frac {\partial ^{2}f}{\partial y\,\partial x}}} or equivalently f y x = f x y . {\displaystyle f_{yx}=f_{xy}.} Own and cross partial derivatives appear in the Hessian matrix which is used in the second order conditions in optimization problems. The higher order partial derivatives can be obtained by successive differentiation == Antiderivative analogue == There is a concept for partial derivatives that is analogous to antiderivatives for regular derivatives. Given a partial derivative, it allows for the partial recovery of the original function. Consider the example of ∂ z ∂ x = 2 x + y . {\displaystyle {\frac {\partial z}{\partial x}}=2x+y.} The so-called partial integral can be taken with respect to x (treating y as constant, in a similar manner to partial differentiation): z = ∫ ∂ z ∂ x d x = x 2 + x y + g ( y ) . {\displaystyle z=\int {\frac {\partial z}{\partial x}}\,dx=x^{2}+xy+g(y).} Here, the constant of integration is no longer a constant, but instead a function of all the variables of the original function except x. The reason for this is that all the other variables are treated as constant when taking the partial derivative, so any function which does not involve x will disappear when taking the partial derivative, and we have to account for this when we take the antiderivative. The most general way to represent this is to have the constant represent an unknown function of all the other variables. Thus the set of functions x 2 + x y + g ( y ) {\displaystyle x^{2}+xy+g(y)} , where g is any one-argument function, represents the entire set of functions in variables x, y that could have produced the x-partial derivative 2 x + y {\displaystyle 2x+y} . If all the partial derivatives of a function are known (for example, with the gradient), then the antiderivatives can be matched via the above process to reconstruct the original function up to a constant. Unlike in the single-variable case, however, not every set of functions can be the set of all (first) partial derivatives of a single function. In other words, not every vector field is conservative. == Applications == === Geometry === The volume V of a cone depends on the cone's height h and its radius r according to the formula V ( r , h ) = π r 2 h 3 . {\displaystyle V(r,h)={\frac {\pi r^{2}h}{3}}.} The partial derivative of V with respect to r is ∂ V ∂ r = 2 π r h 3 , {\displaystyle {\frac {\partial V}{\partial r}}={\frac {2\pi rh}{3}},} which represents the rate with which a cone's volume changes if its radius is varied and its height is kept constant. The partial derivative with respect to h equals 1 3 π r 2 {\textstyle {\frac {1}{3}}\pi r^{2}} , which represents the rate with which the volume changes if its height is varied and its radius is kept constant. By contrast, the total derivative of V with respect to r and h are respectively d V d r = 2 π r h 3 ⏞ ∂ V ∂ r + π r 2 3 ⏞ ∂ V ∂ h d h d r , d V d h = π r 2 3 ⏞ ∂ V ∂ h + 2 π r h 3 ⏞ ∂ V ∂ r d r d h . {\displaystyle {\begin{aligned}{\frac {dV}{dr}}&=\overbrace {\frac {2\pi rh}{3}} ^{\frac {\partial V}{\partial r}}+\overbrace {\frac {\pi r^{2}}{3}} ^{\frac {\partial V}{\partial h}}{\frac {dh}{dr}}\,,\\{\frac {dV}{dh}}&=\overbrace {\frac {\pi r^{2}}{3}} ^{\frac {\partial V}{\partial h}}+\overbrace {\frac {2\pi rh}{3}} ^{\frac {\partial V}{\partial r}}{\frac {dr}{dh}}\,.\end{aligned}}} The difference between the total and partial derivative is the elimination of indirect dependencies between variables in partial derivatives. If (for some arbitrary reason) the cone's proportions have to stay the same, and the height and radius are in a fixed ratio k, k = h r = d h d r . {\displaystyle k={\frac {h}{r}}={\frac {dh}{dr}}.} This gives the total derivative with respect to r, d V d r = 2 π r h 3 + π r 2 3 k , {\displaystyle {\frac {dV}{dr}}={\frac {2\pi rh}{3}}+{\frac {\pi r^{2}}{3}}k\,,} which simplifies to d V d r = k π r 2 , {\displaystyle {\frac {dV}{dr}}=k\pi r^{2},} Similarly, the total derivative with respect to h is d V d h = π r 2 . {\displaystyle {\frac {dV}{dh}}=\pi r^{2}.} The total derivative with respect to both r and h of the volume intended as scalar function of these two variables is given by the gradient vector ∇ V = ( ∂ V ∂ r , ∂ V ∂ h ) = ( 2 3 π r h , 1 3 π r 2 ) . {\displaystyle \nabla V=\left({\frac {\partial V}{\partial r}},{\frac {\partial V}{\partial h}}\right)=\left({\frac {2}{3}}\pi rh,{\frac {1}{3}}\pi r^{2}\right).} === Optimization === Partial derivatives appear in any calculus-based optimization problem with more than one choice variable. For example, in economics a firm may wish to maximize profit π(x, y) with respect to the choice of the quantities x and y of two different types of output. The first order conditions for this optimization are πx = 0 = πy. Since both partial derivatives πx and πy will generally themselves be functions of both arguments x and y, these two first order conditions form a system of two equations in two unknowns. === Thermodynamics, quantum mechanics and mathematical physics === Partial derivatives appear in thermodynamic equations like Gibbs-Duhem equation, in quantum mechanics as in Schrödinger wave equation, as well as in other equations from mathematical physics. The variables being held constant in partial derivatives here can be ratios of simple variables like mole fractions xi in the following example involving the Gibbs energies in a ternary mixture system: G 2 ¯ = G + ( 1 − x 2 ) ( ∂ G ∂ x 2 ) x 1 x 3 {\displaystyle {\bar {G_{2}}}=G+(1-x_{2})\left({\frac {\partial G}{\partial x_{2}}}\right)_{\frac {x_{1}}{x_{3}}}} Express mole fractions of a component as functions of other components' mole fraction and binary mole ratios: x 1 = 1 − x 2 1 + x 3 x 1 x 3 = 1 − x 2 1 + x 1 x 3 {\textstyle {\begin{aligned}x_{1}&={\frac {1-x_{2}}{1+{\frac {x_{3}}{x_{1}}}}}\\x_{3}&={\frac {1-x_{2}}{1+{\frac {x_{1}}{x_{3}}}}}\end{aligned}}} Differential quotients can be formed at constant ratios like those above: ( ∂ x 1 ∂ x 2 ) x 1 x 3 = − x 1 1 − x 2 ( ∂ x 3 ∂ x 2 ) x 1 x 3 = − x 3 1 − x 2 {\displaystyle {\begin{aligned}\left({\frac {\partial x_{1}}{\partial x_{2}}}\right)_{\frac {x_{1}}{x_{3}}}&=-{\frac {x_{1}}{1-x_{2}}}\\\left({\frac {\partial x_{3}}{\partial x_{2}}}\right)_{\frac {x_{1}}{x_{3}}}&=-{\frac {x_{3}}{1-x_{2}}}\end{aligned}}} Ratios X, Y, Z of mole fractions can be written for ternary and multicomponent systems: X = x 3 x 1 + x 3 Y = x 3 x 2 + x 3 Z = x 2 x 1 + x 2 {\displaystyle {\begin{aligned}X&={\frac {x_{3}}{x_{1}+x_{3}}}\\Y&={\frac {x_{3}}{x_{2}+x_{3}}}\\Z&={\frac {x_{2}}{x_{1}+x_{2}}}\end{aligned}}} which can be used for solving partial differential equations like: ( ∂ μ 2 ∂ n 1 ) n 2 , n 3 = ( ∂ μ 1 ∂ n 2 ) n 1 , n 3 {\displaystyle \left({\frac {\partial \mu _{2}}{\partial n_{1}}}\right)_{n_{2},n_{3}}=\left({\frac {\partial \mu _{1}}{\partial n_{2}}}\right)_{n_{1},n_{3}}} This equality can be rearranged to have differential quotient of mole fractions on one side. === Image resizing === Partial derivatives are key to target-aware image resizing algorithms. Widely known as seam carving, these algorithms require each pixel in an image to be assigned a numerical 'energy' to describe their dissimilarity against orthogonal adjacent pixels. The algorithm then progressively removes rows or columns with the lowest energy. The formula established to determine a pixel's energy (magnitude of gradient at a pixel) depends heavily on the constructs of partial derivatives. === Economics === Partial derivatives play a prominent role in economics, in which most functions describing economic behaviour posit that the behaviour depends on more than one variable. For example, a societal consumption function may describe the amount spent on consumer goods as depending on both income and wealth; the marginal propensity to consume is then the partial derivative of the consumption function with respect to income. == See also == == Notes == == External links == "Partial derivative", Encyclopedia of Mathematics, EMS Press, 2001 [1994] Partial Derivatives at MathWorld
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Wikipedia:Partial fraction decomposition#0
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In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator. The importance of the partial fraction decomposition lies in the fact that it provides algorithms for various computations with rational functions, including the explicit computation of antiderivatives, Taylor series expansions, inverse Z-transforms, and inverse Laplace transforms. The concept was discovered independently in 1702 by both Johann Bernoulli and Gottfried Leibniz. In symbols, the partial fraction decomposition of a rational fraction of the form f ( x ) g ( x ) , {\textstyle {\frac {f(x)}{g(x)}},} where f and g are polynomials, is the expression of the rational fraction as f ( x ) g ( x ) = p ( x ) + ∑ j f j ( x ) g j ( x ) {\displaystyle {\frac {f(x)}{g(x)}}=p(x)+\sum _{j}{\frac {f_{j}(x)}{g_{j}(x)}}} where p(x) is a polynomial, and, for each j, the denominator gj (x) is a power of an irreducible polynomial (i.e. not factorizable into polynomials of positive degrees), and the numerator fj (x) is a polynomial of a smaller degree than the degree of this irreducible polynomial. When explicit computation is involved, a coarser decomposition is often preferred, which consists of replacing "irreducible polynomial" by "square-free polynomial" in the description of the outcome. This allows replacing polynomial factorization by the much easier-to-compute square-free factorization. This is sufficient for most applications, and avoids introducing irrational coefficients when the coefficients of the input polynomials are integers or rational numbers. == Basic principles == Let R ( x ) = F G {\displaystyle R(x)={\frac {F}{G}}} be a rational fraction, where F and G are univariate polynomials in the indeterminate x over a field. The existence of the partial fraction can be proved by applying inductively the following reduction steps. === Polynomial part === There exist two polynomials E and F1 such that F G = E + F 1 G , {\displaystyle {\frac {F}{G}}=E+{\frac {F_{1}}{G}},} and deg F 1 < deg G , {\displaystyle \deg F_{1}<\deg G,} where deg P {\displaystyle \deg P} denotes the degree of the polynomial P. This results immediately from the Euclidean division of F by G, which asserts the existence of E and F1 such that F = E G + F 1 {\displaystyle F=EG+F_{1}} and deg F 1 < deg G . {\displaystyle \deg F_{1}<\deg G.} This allows supposing in the next steps that deg F < deg G . {\displaystyle \deg F<\deg G.} === Factors of the denominator === If deg F < deg G , {\displaystyle \deg F<\deg G,} and G = G 1 G 2 , {\displaystyle G=G_{1}G_{2},} where G1 and G2 are coprime polynomials, then there exist polynomials F 1 {\displaystyle F_{1}} and F 2 {\displaystyle F_{2}} such that F G = F 1 G 1 + F 2 G 2 , {\displaystyle {\frac {F}{G}}={\frac {F_{1}}{G_{1}}}+{\frac {F_{2}}{G_{2}}},} and deg F 1 < deg G 1 and deg F 2 < deg G 2 . {\displaystyle \deg F_{1}<\deg G_{1}\quad {\text{and}}\quad \deg F_{2}<\deg G_{2}.} This can be proved as follows. Bézout's identity asserts the existence of polynomials C and D such that C G 1 + D G 2 = 1 {\displaystyle CG_{1}+DG_{2}=1} (by hypothesis, 1 is a greatest common divisor of G1 and G2). Let D F = G 1 Q + F 1 {\displaystyle DF=G_{1}Q+F_{1}} with deg F 1 < deg G 1 {\displaystyle \deg F_{1}<\deg G_{1}} be the Euclidean division of DF by G 1 . {\displaystyle G_{1}.} Setting F 2 = C F + Q G 2 , {\displaystyle F_{2}=CF+QG_{2},} one gets F G = F ( C G 1 + D G 2 ) G 1 G 2 = D F G 1 + C F G 2 = F 1 + G 1 Q G 1 + F 2 − G 2 Q G 2 = F 1 G 1 + Q + F 2 G 2 − Q = F 1 G 1 + F 2 G 2 . {\displaystyle {\begin{aligned}{\frac {F}{G}}&={\frac {F(CG_{1}+DG_{2})}{G_{1}G_{2}}}={\frac {DF}{G_{1}}}+{\frac {CF}{G_{2}}}\\&={\frac {F_{1}+G_{1}Q}{G_{1}}}+{\frac {F_{2}-G_{2}Q}{G_{2}}}\\&={\frac {F_{1}}{G_{1}}}+Q+{\frac {F_{2}}{G_{2}}}-Q\\&={\frac {F_{1}}{G_{1}}}+{\frac {F_{2}}{G_{2}}}.\end{aligned}}} It remains to show that deg F 2 < deg G 2 . {\displaystyle \deg F_{2}<\deg G_{2}.} By reducing the last sum of fractions to a common denominator, one gets F = F 2 G 1 + F 1 G 2 , {\displaystyle F=F_{2}G_{1}+F_{1}G_{2},} and thus deg F 2 = deg ( F − F 1 G 2 ) − deg G 1 ≤ max ( deg F , deg ( F 1 G 2 ) ) − deg G 1 < max ( deg G , deg ( G 1 G 2 ) ) − deg G 1 = deg G 2 {\displaystyle {\begin{aligned}\deg F_{2}&=\deg(F-F_{1}G_{2})-\deg G_{1}\leq \max(\deg F,\deg(F_{1}G_{2}))-\deg G_{1}\\&<\max(\deg G,\deg(G_{1}G_{2}))-\deg G_{1}=\deg G_{2}\end{aligned}}} === Powers in the denominator === Using the preceding decomposition inductively one gets fractions of the form F G k , {\displaystyle {\frac {F}{G^{k}}},} with deg F < deg G k = k deg G , {\displaystyle \deg F<\deg G^{k}=k\deg G,} where G is an irreducible polynomial. If k > 1, one can decompose further, by using that an irreducible polynomial is a square-free polynomial, that is, 1 {\displaystyle 1} is a greatest common divisor of the polynomial and its derivative. If G ′ {\displaystyle G'} is the derivative of G, Bézout's identity provides polynomials C and D such that C G + D G ′ = 1 {\displaystyle CG+DG'=1} and thus F = F C G + F D G ′ . {\displaystyle F=FCG+FDG'.} Euclidean division of F D G ′ {\displaystyle FDG'} by G {\displaystyle G} gives polynomials H k {\displaystyle H_{k}} and Q {\displaystyle Q} such that F D G ′ = Q G + H k {\displaystyle FDG'=QG+H_{k}} and deg H k < deg G . {\displaystyle \deg H_{k}<\deg G.} Setting F k − 1 = F C + Q , {\displaystyle F_{k-1}=FC+Q,} one gets F G k = H k G k + F k − 1 G k − 1 , {\displaystyle {\frac {F}{G^{k}}}={\frac {H_{k}}{G^{k}}}+{\frac {F_{k-1}}{G^{k-1}}},} with deg H k < deg G . {\displaystyle \deg H_{k}<\deg G.} Iterating this process with F k − 1 G k − 1 {\displaystyle {\frac {F_{k-1}}{G^{k-1}}}} in place of F G k {\displaystyle {\frac {F}{G^{k}}}} leads eventually to the following theorem. === Statement === The uniqueness can be proved as follows. Let d = max(1 + deg f, deg g). All together, b and the aij have d coefficients. The shape of the decomposition defines a linear map from coefficient vectors to polynomials f of degree less than d. The existence proof means that this map is surjective. As the two vector spaces have the same dimension, the map is also injective, which means uniqueness of the decomposition. By the way, this proof induces an algorithm for computing the decomposition through linear algebra. If K is the field of complex numbers, the fundamental theorem of algebra implies that all pi have degree one, and all numerators a i j {\displaystyle a_{ij}} are constants. When K is the field of real numbers, some of the pi may be quadratic, so, in the partial fraction decomposition, quotients of linear polynomials by powers of quadratic polynomials may also occur. In the preceding theorem, one may replace "distinct irreducible polynomials" by "pairwise coprime polynomials that are coprime with their derivative". For example, the pi may be the factors of the square-free factorization of g. When K is the field of rational numbers, as it is typically the case in computer algebra, this allows to replace factorization by greatest common divisor computation for computing a partial fraction decomposition. == Application to symbolic integration == For the purpose of symbolic integration, the preceding result may be refined into This reduces the computation of the antiderivative of a rational function to the integration of the last sum, which is called the logarithmic part, because its antiderivative is a linear combination of logarithms. There are various methods to compute decomposition in the Theorem. One simple way is called Hermite's method. First, b is immediately computed by Euclidean division of f by g, reducing to the case where deg(f) < deg(g). Next, one knows deg(cij) < deg(pi), so one may write each cij as a polynomial with unknown coefficients. Reducing the sum of fractions in the Theorem to a common denominator, and equating the coefficients of each power of x in the two numerators, one gets a system of linear equations which can be solved to obtain the desired (unique) values for the unknown coefficients. == Procedure == Given two polynomials P ( x ) {\displaystyle P(x)} and Q ( x ) = ( x − α 1 ) ( x − α 2 ) ⋯ ( x − α n ) {\displaystyle Q(x)=(x-\alpha _{1})(x-\alpha _{2})\cdots (x-\alpha _{n})} , where the αn are distinct constants and deg P < n, explicit expressions for partial fractions can be obtained by supposing that P ( x ) Q ( x ) = c 1 x − α 1 + c 2 x − α 2 + ⋯ + c n x − α n {\displaystyle {\frac {P(x)}{Q(x)}}={\frac {c_{1}}{x-\alpha _{1}}}+{\frac {c_{2}}{x-\alpha _{2}}}+\cdots +{\frac {c_{n}}{x-\alpha _{n}}}} and solving for the ci constants, by substitution, by equating the coefficients of terms involving the powers of x, or otherwise. (This is a variant of the method of undetermined coefficients. After both sides of the equation are multiplied by Q(x), one side of the equation is a specific polynomial, and the other side is a polynomial with undetermined coefficients. The equality is possible only when the coefficients of like powers of x are equal. This yields n equations in n unknowns, the ck.) A more direct computation, which is strongly related to Lagrange interpolation, consists of writing P ( x ) Q ( x ) = ∑ i = 1 n P ( α i ) Q ′ ( α i ) 1 ( x − α i ) {\displaystyle {\frac {P(x)}{Q(x)}}=\sum _{i=1}^{n}{\frac {P(\alpha _{i})}{Q'(\alpha _{i})}}{\frac {1}{(x-\alpha _{i})}}} where Q ′ {\displaystyle Q'} is the derivative of the polynomial Q {\displaystyle Q} . The coefficients of 1 x − α j {\displaystyle {\tfrac {1}{x-\alpha _{j}}}} are called the residues of f/g. This approach does not account for several other cases, but can be modified accordingly: If deg P ≥ deg Q , {\displaystyle \deg P\geq \deg Q,} then it is necessary to perform the Euclidean division of P by Q, using polynomial long division, giving P(x) = E(x) Q(x) + R(x) with deg R < n. Dividing by Q(x) this gives P ( x ) Q ( x ) = E ( x ) + R ( x ) Q ( x ) , {\displaystyle {\frac {P(x)}{Q(x)}}=E(x)+{\frac {R(x)}{Q(x)}},} and then seek partial fractions for the remainder fraction (which by definition satisfies deg R < deg Q). If Q(x) contains factors which are irreducible over the given field, then the numerator N(x) of each partial fraction with such a factor F(x) in the denominator must be sought as a polynomial with deg N < deg F, rather than as a constant. For example, take the following decomposition over R: x 2 + 1 ( x + 2 ) ( x − 1 ) ( x 2 + x + 1 ) = a x + 2 + b x − 1 + c x + d x 2 + x + 1 . {\displaystyle {\frac {x^{2}+1}{(x+2)(x-1)\color {Blue}(x^{2}+x+1)}}={\frac {a}{x+2}}+{\frac {b}{x-1}}+{\frac {\color {OliveGreen}cx+d}{\color {Blue}x^{2}+x+1}}.} Suppose Q(x) = (x − α)r S(x) and S(α) ≠ 0, that is α is a root of Q(x) of multiplicity r. In the partial fraction decomposition, the r first powers of (x − α) will occur as denominators of the partial fractions (possibly with a zero numerator). For example, if S(x) = 1 the partial fraction decomposition has the form P ( x ) Q ( x ) = P ( x ) ( x − α ) r = c 1 x − α + c 2 ( x − α ) 2 + ⋯ + c r ( x − α ) r . {\displaystyle {\frac {P(x)}{Q(x)}}={\frac {P(x)}{(x-\alpha )^{r}}}={\frac {c_{1}}{x-\alpha }}+{\frac {c_{2}}{(x-\alpha )^{2}}}+\cdots +{\frac {c_{r}}{(x-\alpha )^{r}}}.} === Illustration === In an example application of this procedure, (3x + 5)/(1 − 2x)2 can be decomposed in the form 3 x + 5 ( 1 − 2 x ) 2 = A ( 1 − 2 x ) 2 + B ( 1 − 2 x ) . {\displaystyle {\frac {3x+5}{(1-2x)^{2}}}={\frac {A}{(1-2x)^{2}}}+{\frac {B}{(1-2x)}}.} Clearing denominators shows that 3x + 5 = A + B(1 − 2x). Expanding and equating the coefficients of powers of x gives Solving this system of linear equations for A and B yields A = 13/2 and B = −3/2. Hence, 3 x + 5 ( 1 − 2 x ) 2 = 13 / 2 ( 1 − 2 x ) 2 + − 3 / 2 ( 1 − 2 x ) . {\displaystyle {\frac {3x+5}{(1-2x)^{2}}}={\frac {13/2}{(1-2x)^{2}}}+{\frac {-3/2}{(1-2x)}}.} === Residue method === Over the complex numbers, suppose f(x) is a rational proper fraction, and can be decomposed into f ( x ) = ∑ i ( a i 1 x − x i + a i 2 ( x − x i ) 2 + ⋯ + a i k i ( x − x i ) k i ) . {\displaystyle f(x)=\sum _{i}\left({\frac {a_{i1}}{x-x_{i}}}+{\frac {a_{i2}}{(x-x_{i})^{2}}}+\cdots +{\frac {a_{ik_{i}}}{(x-x_{i})^{k_{i}}}}\right).} Let g i j ( x ) = ( x − x i ) j − 1 f ( x ) , {\displaystyle g_{ij}(x)=(x-x_{i})^{j-1}f(x),} then according to the uniqueness of Laurent series, aij is the coefficient of the term (x − xi)−1 in the Laurent expansion of gij(x) about the point xi, i.e., its residue a i j = Res ( g i j , x i ) . {\displaystyle a_{ij}=\operatorname {Res} (g_{ij},x_{i}).} This is given directly by the formula a i j = 1 ( k i − j ) ! lim x → x i d k i − j d x k i − j ( ( x − x i ) k i f ( x ) ) , {\displaystyle a_{ij}={\frac {1}{(k_{i}-j)!}}\lim _{x\to x_{i}}{\frac {d^{k_{i}-j}}{dx^{k_{i}-j}}}\left((x-x_{i})^{k_{i}}f(x)\right),} or in the special case when xi is a simple root, a i 1 = P ( x i ) Q ′ ( x i ) , {\displaystyle a_{i1}={\frac {P(x_{i})}{Q'(x_{i})}},} when f ( x ) = P ( x ) Q ( x ) . {\displaystyle f(x)={\frac {P(x)}{Q(x)}}.} == Over the reals == Partial fractions are used in real-variable integral calculus to find real-valued antiderivatives of rational functions. Partial fraction decomposition of real rational functions is also used to find their Inverse Laplace transforms. For applications of partial fraction decomposition over the reals, see Application to symbolic integration, above Partial fractions in Laplace transforms === General result === Let f ( x ) {\displaystyle f(x)} be any rational function over the real numbers. In other words, suppose there exist real polynomials functions p ( x ) {\displaystyle p(x)} and q ( x ) ≠ 0 {\displaystyle q(x)\neq 0} , such that f ( x ) = p ( x ) q ( x ) {\displaystyle f(x)={\frac {p(x)}{q(x)}}} By dividing both the numerator and the denominator by the leading coefficient of q ( x ) {\displaystyle q(x)} , we may assume without loss of generality that q ( x ) {\displaystyle q(x)} is monic. By the fundamental theorem of algebra, we can write q ( x ) = ( x − a 1 ) j 1 ⋯ ( x − a m ) j m ( x 2 + b 1 x + c 1 ) k 1 ⋯ ( x 2 + b n x + c n ) k n {\displaystyle q(x)=(x-a_{1})^{j_{1}}\cdots (x-a_{m})^{j_{m}}(x^{2}+b_{1}x+c_{1})^{k_{1}}\cdots (x^{2}+b_{n}x+c_{n})^{k_{n}}} where a 1 , … , a m {\displaystyle a_{1},\dots ,a_{m}} , b 1 , … , b n {\displaystyle b_{1},\dots ,b_{n}} , c 1 , … , c n {\displaystyle c_{1},\dots ,c_{n}} are real numbers with b i 2 − 4 c i < 0 {\displaystyle b_{i}^{2}-4c_{i}<0} , and j 1 , … , j m {\displaystyle j_{1},\dots ,j_{m}} , k 1 , … , k n {\displaystyle k_{1},\dots ,k_{n}} are positive integers. The terms ( x − a i ) {\displaystyle (x-a_{i})} are the linear factors of q ( x ) {\displaystyle q(x)} which correspond to real roots of q ( x ) {\displaystyle q(x)} , and the terms ( x i 2 + b i x + c i ) {\displaystyle (x_{i}^{2}+b_{i}x+c_{i})} are the irreducible quadratic factors of q ( x ) {\displaystyle q(x)} which correspond to pairs of complex conjugate roots of q ( x ) {\displaystyle q(x)} . Then the partial fraction decomposition of f ( x ) {\displaystyle f(x)} is the following: f ( x ) = p ( x ) q ( x ) = P ( x ) + ∑ i = 1 m ∑ r = 1 j i A i r ( x − a i ) r + ∑ i = 1 n ∑ r = 1 k i B i r x + C i r ( x 2 + b i x + c i ) r {\displaystyle f(x)={\frac {p(x)}{q(x)}}=P(x)+\sum _{i=1}^{m}\sum _{r=1}^{j_{i}}{\frac {A_{ir}}{(x-a_{i})^{r}}}+\sum _{i=1}^{n}\sum _{r=1}^{k_{i}}{\frac {B_{ir}x+C_{ir}}{(x^{2}+b_{i}x+c_{i})^{r}}}} Here, P(x) is a (possibly zero) polynomial, and the Air, Bir, and Cir are real constants. There are a number of ways the constants can be found. The most straightforward method is to multiply through by the common denominator q(x). We then obtain an equation of polynomials whose left-hand side is simply p(x) and whose right-hand side has coefficients which are linear expressions of the constants Air, Bir, and Cir. Since two polynomials are equal if and only if their corresponding coefficients are equal, we can equate the coefficients of like terms. In this way, a system of linear equations is obtained which always has a unique solution. This solution can be found using any of the standard methods of linear algebra. It can also be found with limits (see Example 5). == Examples == === Example 1 === f ( x ) = 1 x 2 + 2 x − 3 {\displaystyle f(x)={\frac {1}{x^{2}+2x-3}}} Here, the denominator splits into two distinct linear factors: q ( x ) = x 2 + 2 x − 3 = ( x + 3 ) ( x − 1 ) {\displaystyle q(x)=x^{2}+2x-3=(x+3)(x-1)} so we have the partial fraction decomposition f ( x ) = 1 x 2 + 2 x − 3 = A x + 3 + B x − 1 {\displaystyle f(x)={\frac {1}{x^{2}+2x-3}}={\frac {A}{x+3}}+{\frac {B}{x-1}}} Multiplying through by the denominator on the left-hand side gives us the polynomial identity 1 = A ( x − 1 ) + B ( x + 3 ) {\displaystyle 1=A(x-1)+B(x+3)} Substituting x = −3 into this equation gives A = −1/4, and substituting x = 1 gives B = 1/4, so that f ( x ) = 1 x 2 + 2 x − 3 = 1 4 ( − 1 x + 3 + 1 x − 1 ) {\displaystyle f(x)={\frac {1}{x^{2}+2x-3}}={\frac {1}{4}}\left({\frac {-1}{x+3}}+{\frac {1}{x-1}}\right)} === Example 2 === f ( x ) = x 3 + 16 x 3 − 4 x 2 + 8 x {\displaystyle f(x)={\frac {x^{3}+16}{x^{3}-4x^{2}+8x}}} After long division, we have f ( x ) = 1 + 4 x 2 − 8 x + 16 x 3 − 4 x 2 + 8 x = 1 + 4 x 2 − 8 x + 16 x ( x 2 − 4 x + 8 ) {\displaystyle f(x)=1+{\frac {4x^{2}-8x+16}{x^{3}-4x^{2}+8x}}=1+{\frac {4x^{2}-8x+16}{x(x^{2}-4x+8)}}} The factor x2 − 4x + 8 is irreducible over the reals, as its discriminant (−4)2 − 4×8 = −16 is negative. Thus the partial fraction decomposition over the reals has the shape 4 x 2 − 8 x + 16 x ( x 2 − 4 x + 8 ) = A x + B x + C x 2 − 4 x + 8 {\displaystyle {\frac {4x^{2}-8x+16}{x(x^{2}-4x+8)}}={\frac {A}{x}}+{\frac {Bx+C}{x^{2}-4x+8}}} Multiplying through by x3 − 4x2 + 8x, we have the polynomial identity 4 x 2 − 8 x + 16 = A ( x 2 − 4 x + 8 ) + ( B x + C ) x {\displaystyle 4x^{2}-8x+16=A\left(x^{2}-4x+8\right)+\left(Bx+C\right)x} Taking x = 0, we see that 16 = 8A, so A = 2. Comparing the x2 coefficients, we see that 4 = A + B = 2 + B, so B = 2. Comparing linear coefficients, we see that −8 = −4A + C = −8 + C, so C = 0. Altogether, f ( x ) = 1 + 2 ( 1 x + x x 2 − 4 x + 8 ) {\displaystyle f(x)=1+2\left({\frac {1}{x}}+{\frac {x}{x^{2}-4x+8}}\right)} The fraction can be completely decomposed using complex numbers. According to the fundamental theorem of algebra every complex polynomial of degree n has n (complex) roots (some of which can be repeated). The second fraction can be decomposed to: x x 2 − 4 x + 8 = D x − ( 2 + 2 i ) + E x − ( 2 − 2 i ) {\displaystyle {\frac {x}{x^{2}-4x+8}}={\frac {D}{x-(2+2i)}}+{\frac {E}{x-(2-2i)}}} Multiplying through by the denominator gives: x = D ( x − ( 2 − 2 i ) ) + E ( x − ( 2 + 2 i ) ) {\displaystyle x=D(x-(2-2i))+E(x-(2+2i))} Equating the coefficients of x and the constant (with respect to x) coefficients of both sides of this equation, one gets a system of two linear equations in D and E, whose solution is D = 1 + i 2 i = 1 − i 2 , E = 1 − i − 2 i = 1 + i 2 . {\displaystyle D={\frac {1+i}{2i}}={\frac {1-i}{2}},\qquad E={\frac {1-i}{-2i}}={\frac {1+i}{2}}.} Thus we have a complete decomposition: f ( x ) = x 3 + 16 x 3 − 4 x 2 + 8 x = 1 + 2 x + 1 − i x − ( 2 + 2 i ) + 1 + i x − ( 2 − 2 i ) {\displaystyle f(x)={\frac {x^{3}+16}{x^{3}-4x^{2}+8x}}=1+{\frac {2}{x}}+{\frac {1-i}{x-(2+2i)}}+{\frac {1+i}{x-(2-2i)}}} One may also compute directly A, D and E with the residue method (see also example 4 below). === Example 3 === This example illustrates almost all the "tricks" we might need to use, short of consulting a computer algebra system. f ( x ) = x 9 − 2 x 6 + 2 x 5 − 7 x 4 + 13 x 3 − 11 x 2 + 12 x − 4 x 7 − 3 x 6 + 5 x 5 − 7 x 4 + 7 x 3 − 5 x 2 + 3 x − 1 {\displaystyle f(x)={\frac {x^{9}-2x^{6}+2x^{5}-7x^{4}+13x^{3}-11x^{2}+12x-4}{x^{7}-3x^{6}+5x^{5}-7x^{4}+7x^{3}-5x^{2}+3x-1}}} After long division and factoring the denominator, we have f ( x ) = x 2 + 3 x + 4 + 2 x 6 − 4 x 5 + 5 x 4 − 3 x 3 + x 2 + 3 x ( x − 1 ) 3 ( x 2 + 1 ) 2 {\displaystyle f(x)=x^{2}+3x+4+{\frac {2x^{6}-4x^{5}+5x^{4}-3x^{3}+x^{2}+3x}{(x-1)^{3}(x^{2}+1)^{2}}}} The partial fraction decomposition takes the form 2 x 6 − 4 x 5 + 5 x 4 − 3 x 3 + x 2 + 3 x ( x − 1 ) 3 ( x 2 + 1 ) 2 = A x − 1 + B ( x − 1 ) 2 + C ( x − 1 ) 3 + D x + E x 2 + 1 + F x + G ( x 2 + 1 ) 2 . {\displaystyle {\frac {2x^{6}-4x^{5}+5x^{4}-3x^{3}+x^{2}+3x}{(x-1)^{3}(x^{2}+1)^{2}}}={\frac {A}{x-1}}+{\frac {B}{(x-1)^{2}}}+{\frac {C}{(x-1)^{3}}}+{\frac {Dx+E}{x^{2}+1}}+{\frac {Fx+G}{(x^{2}+1)^{2}}}.} Multiplying through by the denominator on the left-hand side we have the polynomial identity 2 x 6 − 4 x 5 + 5 x 4 − 3 x 3 + x 2 + 3 x = A ( x − 1 ) 2 ( x 2 + 1 ) 2 + B ( x − 1 ) ( x 2 + 1 ) 2 + C ( x 2 + 1 ) 2 + ( D x + E ) ( x − 1 ) 3 ( x 2 + 1 ) + ( F x + G ) ( x − 1 ) 3 {\displaystyle {\begin{aligned}&2x^{6}-4x^{5}+5x^{4}-3x^{3}+x^{2}+3x\\[4pt]={}&A\left(x-1\right)^{2}\left(x^{2}+1\right)^{2}+B\left(x-1\right)\left(x^{2}+1\right)^{2}+C\left(x^{2}+1\right)^{2}+\left(Dx+E\right)\left(x-1\right)^{3}\left(x^{2}+1\right)+\left(Fx+G\right)\left(x-1\right)^{3}\end{aligned}}} Now we use different values of x to compute the coefficients: { 4 = 4 C x = 1 2 + 2 i = ( F i + G ) ( 2 + 2 i ) x = i 0 = A − B + C − E − G x = 0 {\displaystyle {\begin{cases}4=4C&x=1\\2+2i=(Fi+G)(2+2i)&x=i\\0=A-B+C-E-G&x=0\end{cases}}} Solving this we have: { C = 1 F = 0 , G = 1 E = A − B {\displaystyle {\begin{cases}C=1\\F=0,G=1\\E=A-B\end{cases}}} Using these values we can write: 2 x 6 − 4 x 5 + 5 x 4 − 3 x 3 + x 2 + 3 x = A ( x − 1 ) 2 ( x 2 + 1 ) 2 + B ( x − 1 ) ( x 2 + 1 ) 2 + ( x 2 + 1 ) 2 + ( D x + ( A − B ) ) ( x − 1 ) 3 ( x 2 + 1 ) + ( x − 1 ) 3 = ( A + D ) x 6 + ( − A − 3 D ) x 5 + ( 2 B + 4 D + 1 ) x 4 + ( − 2 B − 4 D + 1 ) x 3 + ( − A + 2 B + 3 D − 1 ) x 2 + ( A − 2 B − D + 3 ) x {\displaystyle {\begin{aligned}&2x^{6}-4x^{5}+5x^{4}-3x^{3}+x^{2}+3x\\[4pt]={}&A\left(x-1\right)^{2}\left(x^{2}+1\right)^{2}+B\left(x-1\right)\left(x^{2}+1\right)^{2}+\left(x^{2}+1\right)^{2}+\left(Dx+\left(A-B\right)\right)\left(x-1\right)^{3}\left(x^{2}+1\right)+\left(x-1\right)^{3}\\[4pt]={}&\left(A+D\right)x^{6}+\left(-A-3D\right)x^{5}+\left(2B+4D+1\right)x^{4}+\left(-2B-4D+1\right)x^{3}+\left(-A+2B+3D-1\right)x^{2}+\left(A-2B-D+3\right)x\end{aligned}}} We compare the coefficients of x6 and x5 on both side and we have: { A + D = 2 − A − 3 D = − 4 ⇒ A = D = 1. {\displaystyle {\begin{cases}A+D=2\\-A-3D=-4\end{cases}}\quad \Rightarrow \quad A=D=1.} Therefore: 2 x 6 − 4 x 5 + 5 x 4 − 3 x 3 + x 2 + 3 x = 2 x 6 − 4 x 5 + ( 2 B + 5 ) x 4 + ( − 2 B − 3 ) x 3 + ( 2 B + 1 ) x 2 + ( − 2 B + 3 ) x {\displaystyle 2x^{6}-4x^{5}+5x^{4}-3x^{3}+x^{2}+3x=2x^{6}-4x^{5}+(2B+5)x^{4}+(-2B-3)x^{3}+(2B+1)x^{2}+(-2B+3)x} which gives us B = 0. Thus the partial fraction decomposition is given by: f ( x ) = x 2 + 3 x + 4 + 1 ( x − 1 ) + 1 ( x − 1 ) 3 + x + 1 x 2 + 1 + 1 ( x 2 + 1 ) 2 . {\displaystyle f(x)=x^{2}+3x+4+{\frac {1}{(x-1)}}+{\frac {1}{(x-1)^{3}}}+{\frac {x+1}{x^{2}+1}}+{\frac {1}{(x^{2}+1)^{2}}}.} Alternatively, instead of expanding, one can obtain other linear dependences on the coefficients computing some derivatives at x = 1 , ı {\displaystyle x=1,\imath } in the above polynomial identity. (To this end, recall that the derivative at x = a of (x − a)mp(x) vanishes if m > 1 and is just p(a) for m = 1.) For instance the first derivative at x = 1 gives 2 ⋅ 6 − 4 ⋅ 5 + 5 ⋅ 4 − 3 ⋅ 3 + 2 + 3 = A ⋅ ( 0 + 0 ) + B ⋅ ( 4 + 0 ) + 8 + D ⋅ 0 {\displaystyle 2\cdot 6-4\cdot 5+5\cdot 4-3\cdot 3+2+3=A\cdot (0+0)+B\cdot (4+0)+8+D\cdot 0} that is 8 = 4B + 8 so B = 0. === Example 4 (residue method) === f ( z ) = z 2 − 5 ( z 2 − 1 ) ( z 2 + 1 ) = z 2 − 5 ( z + 1 ) ( z − 1 ) ( z + i ) ( z − i ) {\displaystyle f(z)={\frac {z^{2}-5}{(z^{2}-1)(z^{2}+1)}}={\frac {z^{2}-5}{(z+1)(z-1)(z+i)(z-i)}}} Thus, f(z) can be decomposed into rational functions whose denominators are z+1, z−1, z+i, z−i. Since each term is of power one, −1, 1, −i and i are simple poles. Hence, the residues associated with each pole, given by P ( z i ) Q ′ ( z i ) = z i 2 − 5 4 z i 3 , {\displaystyle {\frac {P(z_{i})}{Q'(z_{i})}}={\frac {z_{i}^{2}-5}{4z_{i}^{3}}},} are 1 , − 1 , 3 i 2 , − 3 i 2 , {\displaystyle 1,-1,{\tfrac {3i}{2}},-{\tfrac {3i}{2}},} respectively, and f ( z ) = 1 z + 1 − 1 z − 1 + 3 i 2 1 z + i − 3 i 2 1 z − i . {\displaystyle f(z)={\frac {1}{z+1}}-{\frac {1}{z-1}}+{\frac {3i}{2}}{\frac {1}{z+i}}-{\frac {3i}{2}}{\frac {1}{z-i}}.} === Example 5 (limit method) === Limits can be used to find a partial fraction decomposition. Consider the following example: 1 x 3 − 1 {\displaystyle {\frac {1}{x^{3}-1}}} First, factor the denominator which determines the decomposition: 1 x 3 − 1 = 1 ( x − 1 ) ( x 2 + x + 1 ) = A x − 1 + B x + C x 2 + x + 1 . {\displaystyle {\frac {1}{x^{3}-1}}={\frac {1}{(x-1)(x^{2}+x+1)}}={\frac {A}{x-1}}+{\frac {Bx+C}{x^{2}+x+1}}.} Multiplying everything by x − 1 {\displaystyle x-1} , and taking the limit when x → 1 {\displaystyle x\to 1} , we get lim x → 1 ( ( x − 1 ) ( A x − 1 + B x + C x 2 + x + 1 ) ) = lim x → 1 A + lim x → 1 ( x − 1 ) ( B x + C ) x 2 + x + 1 = A . {\displaystyle \lim _{x\to 1}\left((x-1)\left({\frac {A}{x-1}}+{\frac {Bx+C}{x^{2}+x+1}}\right)\right)=\lim _{x\to 1}A+\lim _{x\to 1}{\frac {(x-1)(Bx+C)}{x^{2}+x+1}}=A.} On the other hand, lim x → 1 ( x − 1 ) ( x − 1 ) ( x 2 + x + 1 ) = lim x → 1 1 x 2 + x + 1 = 1 3 , {\displaystyle \lim _{x\to 1}{\frac {(x-1)}{(x-1)(x^{2}+x+1)}}=\lim _{x\to 1}{\frac {1}{x^{2}+x+1}}={\frac {1}{3}},} and thus: A = 1 3 . {\displaystyle A={\frac {1}{3}}.} Multiplying by x and taking the limit when x → ∞ {\displaystyle x\to \infty } , we have lim x → ∞ x ( A x − 1 + B x + C x 2 + x + 1 ) = lim x → ∞ A x x − 1 + lim x → ∞ B x 2 + C x x 2 + x + 1 = A + B , {\displaystyle \lim _{x\to \infty }x\left({\frac {A}{x-1}}+{\frac {Bx+C}{x^{2}+x+1}}\right)=\lim _{x\to \infty }{\frac {Ax}{x-1}}+\lim _{x\to \infty }{\frac {Bx^{2}+Cx}{x^{2}+x+1}}=A+B,} and lim x → ∞ x ( x − 1 ) ( x 2 + x + 1 ) = 0. {\displaystyle \lim _{x\to \infty }{\frac {x}{(x-1)(x^{2}+x+1)}}=0.} This implies A + B = 0 and so B = − 1 3 {\displaystyle B=-{\frac {1}{3}}} . For x = 0, we get − 1 = − A + C , {\displaystyle -1=-A+C,} and thus C = − 2 3 {\displaystyle C=-{\tfrac {2}{3}}} . Putting everything together, we get the decomposition 1 x 3 − 1 = 1 3 ( 1 x − 1 + − x − 2 x 2 + x + 1 ) . {\displaystyle {\frac {1}{x^{3}-1}}={\frac {1}{3}}\left({\frac {1}{x-1}}+{\frac {-x-2}{x^{2}+x+1}}\right).} === Example 6 (integral) === Suppose we have the indefinite integral: ∫ x 4 + x 3 + x 2 + 1 x 2 + x − 2 d x {\displaystyle \int {\frac {x^{4}+x^{3}+x^{2}+1}{x^{2}+x-2}}\,dx} Before performing decomposition, it is obvious we must perform polynomial long division and factor the denominator. Doing this would result in: ∫ ( x 2 + 3 + − 3 x + 7 ( x + 2 ) ( x − 1 ) ) d x {\displaystyle \int \left(x^{2}+3+{\frac {-3x+7}{(x+2)(x-1)}}\right)dx} Upon this, we may now perform partial fraction decomposition. ∫ ( x 2 + 3 + − 3 x + 7 ( x + 2 ) ( x − 1 ) ) d x = ∫ ( x 2 + 3 + A ( x + 2 ) + B ( x − 1 ) ) d x {\displaystyle \int \left(x^{2}+3+{\frac {-3x+7}{(x+2)(x-1)}}\right)dx=\int \left(x^{2}+3+{\frac {A}{(x+2)}}+{\frac {B}{(x-1)}}\right)dx} so: A ( x − 1 ) + B ( x + 2 ) = − 3 x + 7 {\displaystyle A(x-1)+B(x+2)=-3x+7} . Upon substituting our values, in this case, where x=1 to solve for B and x=-2 to solve for A, we will result in: A = − 13 3 , B = 4 3 {\displaystyle A={\frac {-13}{3}}\ ,B={\frac {4}{3}}} Plugging all of this back into our integral allows us to find the answer: ∫ ( x 2 + 3 + − 13 / 3 ( x + 2 ) + 4 / 3 ( x − 1 ) ) d x = x 3 3 + 3 x − 13 3 ln ( | x + 2 | ) + 4 3 ln ( | x − 1 | ) + C {\displaystyle \int \left(x^{2}+3+{\frac {-13/3}{(x+2)}}+{\frac {4/3}{(x-1)}}\right)\,dx={\frac {x^{3}}{3}}\ +3x-{\frac {13}{3}}\ln(|x+2|)+{\frac {4}{3}}\ln(|x-1|)+C} == The role of the Taylor polynomial == The partial fraction decomposition of a rational function can be related to Taylor's theorem as follows. Let P ( x ) , Q ( x ) , A 1 ( x ) , … , A r ( x ) {\displaystyle P(x),Q(x),A_{1}(x),\ldots ,A_{r}(x)} be real or complex polynomials assume that Q = ∏ j = 1 r ( x − λ j ) ν j , {\displaystyle Q=\prod _{j=1}^{r}(x-\lambda _{j})^{\nu _{j}},} satisfies deg A 1 < ν 1 , … , deg A r < ν r , and deg ( P ) < deg ( Q ) = ∑ j = 1 r ν j . {\displaystyle \deg A_{1}<\nu _{1},\ldots ,\deg A_{r}<\nu _{r},\quad {\text{and}}\quad \deg(P)<\deg(Q)=\sum _{j=1}^{r}\nu _{j}.} Also define Q i = ∏ j ≠ i ( x − λ j ) ν j = Q ( x − λ i ) ν i , 1 ⩽ i ⩽ r . {\displaystyle Q_{i}=\prod _{j\neq i}(x-\lambda _{j})^{\nu _{j}}={\frac {Q}{(x-\lambda _{i})^{\nu _{i}}}},\qquad 1\leqslant i\leqslant r.} Then we have P Q = ∑ j = 1 r A j ( x − λ j ) ν j {\displaystyle {\frac {P}{Q}}=\sum _{j=1}^{r}{\frac {A_{j}}{(x-\lambda _{j})^{\nu _{j}}}}} if, and only if, each polynomial A i ( x ) {\displaystyle A_{i}(x)} is the Taylor polynomial of P Q i {\displaystyle {\tfrac {P}{Q_{i}}}} of order ν i − 1 {\displaystyle \nu _{i}-1} at the point λ i {\displaystyle \lambda _{i}} : A i ( x ) := ∑ k = 0 ν i − 1 1 k ! ( P Q i ) ( k ) ( λ i ) ( x − λ i ) k . {\displaystyle A_{i}(x):=\sum _{k=0}^{\nu _{i}-1}{\frac {1}{k!}}\left({\frac {P}{Q_{i}}}\right)^{(k)}(\lambda _{i})\ (x-\lambda _{i})^{k}.} Taylor's theorem (in the real or complex case) then provides a proof of the existence and uniqueness of the partial fraction decomposition, and a characterization of the coefficients. === Sketch of the proof === The above partial fraction decomposition implies, for each 1 ≤ i ≤ r, a polynomial expansion P Q i = A i + O ( ( x − λ i ) ν i ) , for x → λ i , {\displaystyle {\frac {P}{Q_{i}}}=A_{i}+O((x-\lambda _{i})^{\nu _{i}}),\qquad {\text{for }}x\to \lambda _{i},} so A i {\displaystyle A_{i}} is the Taylor polynomial of P Q i {\displaystyle {\tfrac {P}{Q_{i}}}} , because of the unicity of the polynomial expansion of order ν i − 1 {\displaystyle \nu _{i}-1} , and by assumption deg A i < ν i {\displaystyle \deg A_{i}<\nu _{i}} . Conversely, if the A i {\displaystyle A_{i}} are the Taylor polynomials, the above expansions at each λ i {\displaystyle \lambda _{i}} hold, therefore we also have P − Q i A i = O ( ( x − λ i ) ν i ) , for x → λ i , {\displaystyle P-Q_{i}A_{i}=O((x-\lambda _{i})^{\nu _{i}}),\qquad {\text{for }}x\to \lambda _{i},} which implies that the polynomial P − Q i A i {\displaystyle P-Q_{i}A_{i}} is divisible by ( x − λ i ) ν i . {\displaystyle (x-\lambda _{i})^{\nu _{i}}.} For j ≠ i , Q j A j {\displaystyle j\neq i,Q_{j}A_{j}} is also divisible by ( x − λ i ) ν i {\displaystyle (x-\lambda _{i})^{\nu _{i}}} , so P − ∑ j = 1 r Q j A j {\displaystyle P-\sum _{j=1}^{r}Q_{j}A_{j}} is divisible by Q {\displaystyle Q} . Since deg ( P − ∑ j = 1 r Q j A j ) < deg ( Q ) {\displaystyle \deg \left(P-\sum _{j=1}^{r}Q_{j}A_{j}\right)<\deg(Q)} we then have P − ∑ j = 1 r Q j A j = 0 , {\displaystyle P-\sum _{j=1}^{r}Q_{j}A_{j}=0,} and we find the partial fraction decomposition dividing by Q {\displaystyle Q} . == Fractions of integers == The idea of partial fractions can be generalized to other integral domains, say the ring of integers where prime numbers take the role of irreducible denominators. For example: 1 18 = 1 2 − 1 3 − 1 3 2 . {\displaystyle {\frac {1}{18}}={\frac {1}{2}}-{\frac {1}{3}}-{\frac {1}{3^{2}}}.} == Notes == == References == Rao, K. R.; Ahmed, N. (1968). "Recursive techniques for obtaining the partial fraction expansion of a rational function". IEEE Trans. Educ. 11 (2): 152–154. Bibcode:1968ITEdu..11..152R. doi:10.1109/TE.1968.4320370. Henrici, Peter (1971). "An algorithm for the incomplete decomposition of a rational function into partial fractions". Z. Angew. Math. Phys. 22 (4): 751–755. Bibcode:1971ZaMP...22..751H. doi:10.1007/BF01587772. S2CID 120554693. Chang, Feng-Cheng (1973). "Recursive formulas for the partial fraction expansion of a rational function with multiple poles". Proc. IEEE. 61 (8): 1139–1140. doi:10.1109/PROC.1973.9216. Kung, H. T.; Tong, D. M. (1977). "Fast Algorithms for Partial Fraction Decomposition". SIAM Journal on Computing. 6 (3): 582. doi:10.1137/0206042. S2CID 5857432. Eustice, Dan; Klamkin, M. S. (1979). "On the coefficients of a partial fraction decomposition". American Mathematical Monthly. Vol. 86, no. 6. pp. 478–480. JSTOR 2320421. Mahoney, J. J.; Sivazlian, B. D. (1983). "Partial fractions expansion: a review of computational methodology and efficiency". J. Comput. Appl. Math. 9 (3): 247–269. doi:10.1016/0377-0427(83)90018-3. Miller, Charles D.; Lial, Margaret L.; Schneider, David I. (1990). Fundamentals of College Algebra (3rd ed.). Addison-Wesley Educational Publishers, Inc. pp. 364–370. ISBN 0-673-38638-4. Westreich, David (1991). "partial fraction expansion without derivative evaluation". IEEE Trans. Circ. Syst. 38 (6): 658–660. doi:10.1109/31.81863. Kudryavtsev, L. D. (2001) [1994], "Undetermined coefficients, method of", Encyclopedia of Mathematics, EMS Press Velleman, Daniel J. (2002). "Partial fractions, binomial coefficients and the integral of an odd power of sec theta". Amer. Math. Monthly. 109 (8): 746–749. doi:10.2307/3072399. JSTOR 3072399. Slota, Damian; Witula, Roman (2005). "Three brick method of the partial fraction decomposition of some type of rational expression". Computational Science – ICCS 2005. Lect. Not. Computer Sci. Vol. 33516. pp. 659–662. doi:10.1007/11428862_89. ISBN 978-3-540-26044-8. Kung, Sidney H. (2006). "Partial fraction decomposition by division". Coll. Math. J. 37 (2): 132–134. doi:10.2307/27646303. JSTOR 27646303. Witula, Roman; Slota, Damian (2008). "Partial fractions decompositions of some rational functions". Appl. Math. Comput. 197: 328–336. doi:10.1016/j.amc.2007.07.048. MR 2396331. == External links == Weisstein, Eric W. "Partial Fraction Decomposition". MathWorld. Blake, Sam. "Step-by-Step Partial Fractions". Make partial fraction decompositions with Scilab.
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Wikipedia:Partial function#0
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In mathematics, a partial function f from a set X to a set Y is a function from a subset S of X (possibly the whole X itself) to Y. The subset S, that is, the domain of f viewed as a function, is called the domain of definition or natural domain of f. If S equals X, that is, if f is defined on every element in X, then f is said to be a total function. In other words, a partial function is a binary relation over two sets that associates to every element of the first set at most one element of the second set; it is thus a univalent relation. This generalizes the concept of a (total) function by not requiring every element of the first set to be associated to an element of the second set. A partial function is often used when its exact domain of definition is not known, or is difficult to specify. However, even when the exact domain of definition is known, partial functions are often used for simplicity or brevity. This is the case in calculus, where, for example, the quotient of two functions is a partial function whose domain of definition cannot contain the zeros of the denominator; in this context, a partial function is generally simply called a function. In computability theory, a general recursive function is a partial function from the integers to the integers; no algorithm can exist for deciding whether an arbitrary such function is in fact total. When arrow notation is used for functions, a partial function f {\displaystyle f} from X {\displaystyle X} to Y {\displaystyle Y} is sometimes written as f : X ⇀ Y , {\displaystyle f:X\rightharpoonup Y,} f : X ↛ Y , {\displaystyle f:X\nrightarrow Y,} or f : X ↪ Y . {\displaystyle f:X\hookrightarrow Y.} However, there is no general convention, and the latter notation is more commonly used for inclusion maps or embeddings. Specifically, for a partial function f : X ⇀ Y , {\displaystyle f:X\rightharpoonup Y,} and any x ∈ X , {\displaystyle x\in X,} one has either: f ( x ) = y ∈ Y {\displaystyle f(x)=y\in Y} (it is a single element in Y), or f ( x ) {\displaystyle f(x)} is undefined. For example, if f {\displaystyle f} is the square root function restricted to the integers f : Z → N , {\displaystyle f:\mathbb {Z} \to \mathbb {N} ,} defined by: f ( n ) = m {\displaystyle f(n)=m} if, and only if, m 2 = n , {\displaystyle m^{2}=n,} m ∈ N , n ∈ Z , {\displaystyle m\in \mathbb {N} ,n\in \mathbb {Z} ,} then f ( n ) {\displaystyle f(n)} is only defined if n {\displaystyle n} is a perfect square (that is, 0 , 1 , 4 , 9 , 16 , … {\displaystyle 0,1,4,9,16,\ldots } ). So f ( 25 ) = 5 {\displaystyle f(25)=5} but f ( 26 ) {\displaystyle f(26)} is undefined. == Basic concepts == A partial function arises from the consideration of maps between two sets X and Y that may not be defined on the entire set X. A common example is the square root operation on the real numbers R {\displaystyle \mathbb {R} } : because negative real numbers do not have real square roots, the operation can be viewed as a partial function from R {\displaystyle \mathbb {R} } to R . {\displaystyle \mathbb {R} .} The domain of definition of a partial function is the subset S of X on which the partial function is defined; in this case, the partial function may also be viewed as a function from S to Y. In the example of the square root operation, the set S consists of the nonnegative real numbers [ 0 , + ∞ ) . {\displaystyle [0,+\infty ).} The notion of partial function is particularly convenient when the exact domain of definition is unknown or even unknowable. For a computer-science example of the latter, see Halting problem. In case the domain of definition S is equal to the whole set X, the partial function is said to be total. Thus, total partial functions from X to Y coincide with functions from X to Y. Many properties of functions can be extended in an appropriate sense of partial functions. A partial function is said to be injective, surjective, or bijective when the function given by the restriction of the partial function to its domain of definition is injective, surjective, bijective respectively. Because a function is trivially surjective when restricted to its image, the term partial bijection denotes a partial function which is injective. An injective partial function may be inverted to an injective partial function, and a partial function which is both injective and surjective has an injective function as inverse. Furthermore, a function which is injective may be inverted to a bijective partial function. The notion of transformation can be generalized to partial functions as well. A partial transformation is a function f : A ⇀ B , {\displaystyle f:A\rightharpoonup B,} where both A {\displaystyle A} and B {\displaystyle B} are subsets of some set X . {\displaystyle X.} == Function spaces == For convenience, denote the set of all partial functions f : X ⇀ Y {\displaystyle f:X\rightharpoonup Y} from a set X {\displaystyle X} to a set Y {\displaystyle Y} by [ X ⇀ Y ] . {\displaystyle [X\rightharpoonup Y].} This set is the union of the sets of functions defined on subsets of X {\displaystyle X} with same codomain Y {\displaystyle Y} : [ X ⇀ Y ] = ⋃ D ⊆ X [ D → Y ] , {\displaystyle [X\rightharpoonup Y]=\bigcup _{D\subseteq X}[D\to Y],} the latter also written as ⋃ D ⊆ X Y D . {\textstyle \bigcup _{D\subseteq {X}}Y^{D}.} In finite case, its cardinality is | [ X ⇀ Y ] | = ( | Y | + 1 ) | X | , {\displaystyle |[X\rightharpoonup Y]|=(|Y|+1)^{|X|},} because any partial function can be extended to a function by any fixed value c {\displaystyle c} not contained in Y , {\displaystyle Y,} so that the codomain is Y ∪ { c } , {\displaystyle Y\cup \{c\},} an operation which is injective (unique and invertible by restriction). == Discussion and examples == The first diagram at the top of the article represents a partial function that is not a function since the element 1 in the left-hand set is not associated with anything in the right-hand set. Whereas, the second diagram represents a function since every element on the left-hand set is associated with exactly one element in the right hand set. === Natural logarithm === Consider the natural logarithm function mapping the real numbers to themselves. The logarithm of a non-positive real is not a real number, so the natural logarithm function doesn't associate any real number in the codomain with any non-positive real number in the domain. Therefore, the natural logarithm function is not a function when viewed as a function from the reals to themselves, but it is a partial function. If the domain is restricted to only include the positive reals (that is, if the natural logarithm function is viewed as a function from the positive reals to the reals), then the natural logarithm is a function. === Subtraction of natural numbers === Subtraction of natural numbers (in which N {\displaystyle \mathbb {N} } is the non-negative integers) is a partial function: f : N × N ⇀ N {\displaystyle f:\mathbb {N} \times \mathbb {N} \rightharpoonup \mathbb {N} } f ( x , y ) = x − y . {\displaystyle f(x,y)=x-y.} It is defined only when x ≥ y . {\displaystyle x\geq y.} === Bottom element === In denotational semantics a partial function is considered as returning the bottom element when it is undefined. In computer science a partial function corresponds to a subroutine that raises an exception or loops forever. The IEEE floating point standard defines a not-a-number value which is returned when a floating point operation is undefined and exceptions are suppressed, e.g. when the square root of a negative number is requested. In a programming language where function parameters are statically typed, a function may be defined as a partial function because the language's type system cannot express the exact domain of the function, so the programmer instead gives it the smallest domain which is expressible as a type and contains the domain of definition of the function. === In category theory === In category theory, when considering the operation of morphism composition in concrete categories, the composition operation ∘ : hom ( C ) × hom ( C ) → hom ( C ) {\displaystyle \circ \;:\;\hom(C)\times \hom(C)\to \hom(C)} is a total function if and only if ob ( C ) {\displaystyle \operatorname {ob} (C)} has one element. The reason for this is that two morphisms f : X → Y {\displaystyle f:X\to Y} and g : U → V {\displaystyle g:U\to V} can only be composed as g ∘ f {\displaystyle g\circ f} if Y = U , {\displaystyle Y=U,} that is, the codomain of f {\displaystyle f} must equal the domain of g . {\displaystyle g.} The category of sets and partial functions is equivalent to but not isomorphic with the category of pointed sets and point-preserving maps. One textbook notes that "This formal completion of sets and partial maps by adding “improper,” “infinite” elements was reinvented many times, in particular, in topology (one-point compactification) and in theoretical computer science." The category of sets and partial bijections is equivalent to its dual. It is the prototypical inverse category. === In abstract algebra === Partial algebra generalizes the notion of universal algebra to partial operations. An example would be a field, in which the multiplicative inversion is the only proper partial operation (because division by zero is not defined). The set of all partial functions (partial transformations) on a given base set, X , {\displaystyle X,} forms a regular semigroup called the semigroup of all partial transformations (or the partial transformation semigroup on X {\displaystyle X} ), typically denoted by P T X . {\displaystyle {\mathcal {PT}}_{X}.} The set of all partial bijections on X {\displaystyle X} forms the symmetric inverse semigroup. === Charts and atlases for manifolds and fiber bundles === Charts in the atlases which specify the structure of manifolds and fiber bundles are partial functions. In the case of manifolds, the domain is the point set of the manifold. In the case of fiber bundles, the domain is the space of the fiber bundle. In these applications, the most important construction is the transition map, which is the composite of one chart with the inverse of another. The initial classification of manifolds and fiber bundles is largely expressed in terms of constraints on these transition maps. The reason for the use of partial functions instead of functions is to permit general global topologies to be represented by stitching together local patches to describe the global structure. The "patches" are the domains where the charts are defined. == See also == Analytic continuation – Extension of the domain of an analytic function (mathematics) Multivalued function – Generalized mathematical function Densely defined operator – Function that is defined almost everywhere (mathematics) == References == Martin Davis (1958), Computability and Unsolvability, McGraw–Hill Book Company, Inc, New York. Republished by Dover in 1982. ISBN 0-486-61471-9. Stephen Kleene (1952), Introduction to Meta-Mathematics, North-Holland Publishing Company, Amsterdam, Netherlands, 10th printing with corrections added on 7th printing (1974). ISBN 0-7204-2103-9. Harold S. Stone (1972), Introduction to Computer Organization and Data Structures, McGraw–Hill Book Company, New York. === Notes ===
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Wikipedia:Partial groupoid#0
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In abstract algebra, a partial groupoid (also called halfgroupoid, pargoid, or partial magma) is a set endowed with a partial binary operation. A partial groupoid is a partial algebra. == Partial semigroup == A partial groupoid ( G , ∘ ) {\displaystyle (G,\circ )} is called a partial semigroup if the following associative law holds: For all x , y , z ∈ G {\displaystyle x,y,z\in G} such that x ∘ y ∈ G {\displaystyle x\circ y\in G} and y ∘ z ∈ G {\displaystyle y\circ z\in G} , the following two statements hold: x ∘ ( y ∘ z ) ∈ G {\displaystyle x\circ (y\circ z)\in G} if and only if ( x ∘ y ) ∘ z ∈ G {\displaystyle (x\circ y)\circ z\in G} , and x ∘ ( y ∘ z ) = ( x ∘ y ) ∘ z {\displaystyle x\circ (y\circ z)=(x\circ y)\circ z} if x ∘ ( y ∘ z ) ∈ G {\displaystyle x\circ (y\circ z)\in G} (and, because of 1., also ( x ∘ y ) ∘ z ∈ G {\displaystyle (x\circ y)\circ z\in G} ). == References == == Further reading == E.S. Ljapin; A.E. Evseev (1997). The Theory of Partial Algebraic Operations. Springer Netherlands. ISBN 978-0-7923-4609-8.
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Wikipedia:Partial permutation#0
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In combinatorial mathematics, a partial permutation, or sequence without repetition, on a finite set S is a bijection between two specified subsets of S. That is, it is defined by two subsets U and V of equal size, and a one-to-one mapping from U to V. Equivalently, it is a partial function on S that can be extended to a permutation. == Representation == It is common to consider the case when the set S is simply the set {1, 2, ..., n} of the first n integers. In this case, a partial permutation may be represented by a string of n symbols, some of which are distinct numbers in the range from 1 to n {\displaystyle n} and the remaining ones of which are a special "hole" symbol ◊. In this formulation, the domain U of the partial permutation consists of the positions in the string that do not contain a hole, and each such position is mapped to the number in that position. For instance, the string "1 ◊ 2" would represent the partial permutation that maps 1 to itself and maps 3 to 2. The seven partial permutations on two items are ◊◊, ◊1, ◊2, 1◊, 2◊, 12, 21. == Combinatorial enumeration == The number of partial permutations on n items, for n = 0, 1, 2, ..., is given by the integer sequence 1, 2, 7, 34, 209, 1546, 13327, 130922, 1441729, 17572114, 234662231, ... (sequence A002720 in the OEIS) where the nth item in the sequence is given by the summation formula ∑ i = 0 n i ! ( n i ) 2 {\displaystyle \sum _{i=0}^{n}i!{\binom {n}{i}}^{2}} in which the ith term counts the number of partial permutations with support of size i, that is, the number of partial permutations with i non-hole entries. Alternatively, it can be computed by a recurrence relation P ( n ) = 2 n P ( n − 1 ) − ( n − 1 ) 2 P ( n − 2 ) . {\displaystyle P(n)=2nP(n-1)-(n-1)^{2}P(n-2).} This is determined as follows: P ( n − 1 ) {\displaystyle P(n-1)} partial permutations where the final elements of each set are omitted: P ( n − 1 ) {\displaystyle P(n-1)} partial permutations where the final elements of each set map to each other. ( n − 1 ) P ( n − 1 ) {\displaystyle (n-1)P(n-1)} partial permutations where the final element of the first set is included, but does not map to the final element of the second set ( n − 1 ) P ( n − 1 ) {\displaystyle (n-1)P(n-1)} partial permutations where the final element of the second set is included, but does not map to the final element of the first set − ( n − 1 ) 2 P ( n − 2 ) {\displaystyle -(n-1)^{2}P(n-2)} , the partial permutations included in both counts 3 and 4, those permutations where the final elements of both sets are included, but do not map to each other. == Restricted partial permutations == Some authors restrict partial permutations so that either the domain or the range of the bijection is forced to consist of the first k items in the set of n items being permuted, for some k. In the former case, a partial permutation of length k from an n-set is just a sequence of k terms from the n-set without repetition. (In elementary combinatorics, these objects are sometimes confusingly called "k-permutations" of the n-set.) == References ==
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Wikipedia:Partial trace#0
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In linear algebra and functional analysis, the partial trace is a generalization of the trace. Whereas the trace is a scalar-valued function on operators, the partial trace is an operator-valued function. The partial trace has applications in quantum information and decoherence which is relevant for quantum measurement and thereby to the decoherent approaches to interpretations of quantum mechanics, including consistent histories and the relative state interpretation. == Details == Suppose V {\displaystyle V} , W {\displaystyle W} are finite-dimensional vector spaces over a field, with dimensions m {\displaystyle m} and n {\displaystyle n} , respectively. For any space A {\displaystyle A} , let L ( A ) {\displaystyle L(A)} denote the space of linear operators on A {\displaystyle A} . The partial trace over W {\displaystyle W} is then written as Tr W : L ( V ⊗ W ) → L ( V ) {\displaystyle \operatorname {Tr} _{W}:\operatorname {L} (V\otimes W)\to \operatorname {L} (V)} , where ⊗ {\displaystyle \otimes } denotes the Kronecker product. It is defined as follows: For T ∈ L ( V ⊗ W ) {\displaystyle T\in \operatorname {L} (V\otimes W)} , let e 1 , … , e m {\displaystyle e_{1},\ldots ,e_{m}} , and f 1 , … , f n {\displaystyle f_{1},\ldots ,f_{n}} , be bases for V and W respectively; then T has a matrix representation { a k ℓ , i j } 1 ≤ k , i ≤ m , 1 ≤ ℓ , j ≤ n {\displaystyle \{a_{k\ell ,ij}\}\quad 1\leq k,i\leq m,\quad 1\leq \ell ,j\leq n} relative to the basis e k ⊗ f ℓ {\displaystyle e_{k}\otimes f_{\ell }} of V ⊗ W {\displaystyle V\otimes W} . Now for indices k, i in the range 1, ..., m, consider the sum b k , i = ∑ j = 1 n a k j , i j {\displaystyle b_{k,i}=\sum _{j=1}^{n}a_{kj,ij}} This gives a matrix bk,i. The associated linear operator on V is independent of the choice of bases and is by definition the partial trace. Among physicists, this is often called "tracing out" or "tracing over" W to leave only an operator on V in the context where W and V are Hilbert spaces associated with quantum systems (see below). === Invariant definition === The partial trace operator can be defined invariantly (that is, without reference to a basis) as follows: it is the unique linear map Tr W : L ( V ⊗ W ) → L ( V ) {\displaystyle \operatorname {Tr} _{W}:\operatorname {L} (V\otimes W)\rightarrow \operatorname {L} (V)} such that Tr W ( R ⊗ S ) = Tr ( S ) R ∀ R ∈ L ( V ) ∀ S ∈ L ( W ) . {\displaystyle \operatorname {Tr} _{W}(R\otimes S)=\operatorname {Tr} (S)\,R\quad \forall R\in \operatorname {L} (V)\quad \forall S\in \operatorname {L} (W).} To see that the conditions above determine the partial trace uniquely, let v 1 , … , v m {\displaystyle v_{1},\ldots ,v_{m}} form a basis for V {\displaystyle V} , let w 1 , … , w n {\displaystyle w_{1},\ldots ,w_{n}} form a basis for W {\displaystyle W} , let E i j : V → V {\displaystyle E_{ij}:V\to V} be the map that sends v i {\displaystyle v_{i}} to v j {\displaystyle v_{j}} (and all other basis elements to zero), and let F k l : W → W {\displaystyle F_{kl}\colon W\to W} be the map that sends w k {\displaystyle w_{k}} to w l {\displaystyle w_{l}} . Since the vectors v i ⊗ w k {\displaystyle v_{i}\otimes w_{k}} form a basis for V ⊗ W {\displaystyle V\otimes W} , the maps E i j ⊗ F k l {\displaystyle E_{ij}\otimes F_{kl}} form a basis for L ( V ⊗ W ) {\displaystyle \operatorname {L} (V\otimes W)} . From this abstract definition, the following properties follow: Tr W ( I V ⊗ W ) = dim W I V {\displaystyle \operatorname {Tr} _{W}(I_{V\otimes W})=\dim W\ I_{V}} Tr W ( T ( I V ⊗ S ) ) = Tr W ( ( I V ⊗ S ) T ) ∀ S ∈ L ( W ) ∀ T ∈ L ( V ⊗ W ) . {\displaystyle \operatorname {Tr} _{W}(T(I_{V}\otimes S))=\operatorname {Tr} _{W}((I_{V}\otimes S)T)\quad \forall S\in \operatorname {L} (W)\quad \forall T\in \operatorname {L} (V\otimes W).} === Category theoretic notion === It is the partial trace of linear transformations that is the subject of Joyal, Street, and Verity's notion of Traced monoidal category. A traced monoidal category is a monoidal category ( C , ⊗ , I ) {\displaystyle (C,\otimes ,I)} together with, for objects X, Y, U in the category, a function of Hom-sets, Tr X , Y U : Hom C ( X ⊗ U , Y ⊗ U ) → Hom C ( X , Y ) {\displaystyle \operatorname {Tr} _{X,Y}^{U}:\operatorname {Hom} _{C}(X\otimes U,Y\otimes U)\to \operatorname {Hom} _{C}(X,Y)} satisfying certain axioms. Another case of this abstract notion of partial trace takes place in the category of finite sets and bijections between them, in which the monoidal product is disjoint union. One can show that for any finite sets, X,Y,U and bijection X + U ≅ Y + U {\displaystyle X+U\cong Y+U} there exists a corresponding "partially traced" bijection X ≅ Y {\displaystyle X\cong Y} . == Partial trace for operators on Hilbert spaces == The partial trace generalizes to operators on infinite dimensional Hilbert spaces. Suppose V, W are Hilbert spaces, and let { f i } i ∈ I {\displaystyle \{f_{i}\}_{i\in I}} be an orthonormal basis for W. Now there is an isometric isomorphism ⨁ ℓ ∈ I ( V ⊗ C f ℓ ) → V ⊗ W {\displaystyle \bigoplus _{\ell \in I}(V\otimes \mathbb {C} f_{\ell })\rightarrow V\otimes W} Under this decomposition, any operator T ∈ L ( V ⊗ W ) {\displaystyle T\in \operatorname {L} (V\otimes W)} can be regarded as an infinite matrix of operators on V [ T 11 T 12 … T 1 j … T 21 T 22 … T 2 j … ⋮ ⋮ ⋮ T k 1 T k 2 … T k j … ⋮ ⋮ ⋮ ] , {\displaystyle {\begin{bmatrix}T_{11}&T_{12}&\ldots &T_{1j}&\ldots \\T_{21}&T_{22}&\ldots &T_{2j}&\ldots \\\vdots &\vdots &&\vdots \\T_{k1}&T_{k2}&\ldots &T_{kj}&\ldots \\\vdots &\vdots &&\vdots \end{bmatrix}},} where T k ℓ ∈ L ( V ) {\displaystyle T_{k\ell }\in \operatorname {L} (V)} . First suppose T is a non-negative operator. In this case, all the diagonal entries of the above matrix are non-negative operators on V. If the sum ∑ ℓ T ℓ ℓ {\displaystyle \sum _{\ell }T_{\ell \ell }} converges in the strong operator topology of L(V), it is independent of the chosen basis of W. The partial trace TrW(T) is defined to be this operator. The partial trace of a self-adjoint operator is defined if and only if the partial traces of the positive and negative parts are defined. === Computing the partial trace === Suppose W has an orthonormal basis, which we denote by ket vector notation as { | ℓ ⟩ } ℓ {\displaystyle \{\vert \ell \rangle \}_{\ell }} . Then Tr W ( ∑ k , ℓ T ( k ℓ ) ⊗ | k ⟩ ⟨ ℓ | ) = ∑ j T ( j j ) . {\displaystyle \operatorname {Tr} _{W}\left(\sum _{k,\ell }T^{(k\ell )}\,\otimes \,|k\rangle \langle \ell |\right)=\sum _{j}T^{(jj)}.} The superscripts in parentheses do not represent matrix components, but instead label the matrix itself. == Partial trace and invariant integration == In the case of finite dimensional Hilbert spaces, there is a useful way of looking at partial trace involving integration with respect to a suitably normalized Haar measure μ over the unitary group U(W) of W. Suitably normalized means that μ is taken to be a measure with total mass dim(W). Theorem. Suppose V, W are finite dimensional Hilbert spaces. Then ∫ U ( W ) ( I V ⊗ U ∗ ) T ( I V ⊗ U ) d μ ( U ) {\displaystyle \int _{\operatorname {U} (W)}(I_{V}\otimes U^{*})T(I_{V}\otimes U)\ d\mu (U)} commutes with all operators of the form I V ⊗ S {\displaystyle I_{V}\otimes S} and hence is uniquely of the form R ⊗ I W {\displaystyle R\otimes I_{W}} . The operator R is the partial trace of T. == Partial trace as a quantum operation == The partial trace can be viewed as a quantum operation. Consider a quantum mechanical system whose state space is the tensor product H A ⊗ H B {\displaystyle H_{A}\otimes H_{B}} of Hilbert spaces. A mixed state is described by a density matrix ρ, that is a non-negative trace-class operator of trace 1 on the tensor product H A ⊗ H B . {\displaystyle H_{A}\otimes H_{B}.} The partial trace of ρ with respect to the system B, denoted by ρ A {\displaystyle \rho ^{A}} , is called the reduced state of ρ on system A. In symbols, ρ A = Tr B ρ . {\displaystyle \rho ^{A}=\operatorname {Tr} _{B}\rho .} To show that this is indeed a sensible way to assign a state on the A subsystem to ρ, we offer the following justification. Let M be an observable on the subsystem A, then the corresponding observable on the composite system is M ⊗ I {\displaystyle M\otimes I} . However one chooses to define a reduced state ρ A {\displaystyle \rho ^{A}} , there should be consistency of measurement statistics. The expectation value of M after the subsystem A is prepared in ρ A {\displaystyle \rho ^{A}} and that of M ⊗ I {\displaystyle M\otimes I} when the composite system is prepared in ρ should be the same, i.e. the following equality should hold: Tr A ( M ⋅ ρ A ) = Tr ( M ⊗ I ⋅ ρ ) . {\displaystyle \operatorname {Tr} _{A}(M\cdot \rho ^{A})=\operatorname {Tr} (M\otimes I\cdot \rho ).} We see that this is satisfied if ρ A {\displaystyle \rho ^{A}} is as defined above via the partial trace. Furthermore, such operation is unique. Let T(H) be the Banach space of trace-class operators on the Hilbert space H. It can be easily checked that the partial trace, viewed as a map Tr B : T ( H A ⊗ H B ) → T ( H A ) {\displaystyle \operatorname {Tr} _{B}:T(H_{A}\otimes H_{B})\rightarrow T(H_{A})} is completely positive and trace-preserving. The density matrix ρ is Hermitian, positive semi-definite, and has a trace of 1. It has a spectral decomposition: ρ = ∑ m p m | Ψ m ⟩ ⟨ Ψ m | ; 0 ≤ p m ≤ 1 , ∑ m p m = 1 {\displaystyle \rho =\sum _{m}p_{m}|\Psi _{m}\rangle \langle \Psi _{m}|;\ 0\leq p_{m}\leq 1,\ \sum _{m}p_{m}=1} Its easy to see that the partial trace ρ A {\displaystyle \rho ^{A}} also satisfies these conditions. For example, for any pure state | ψ A ⟩ {\displaystyle |\psi _{A}\rangle } in H A {\displaystyle H_{A}} , we have ⟨ ψ A | ρ A | ψ A ⟩ = ∑ m p m Tr B [ ⟨ ψ A | Ψ m ⟩ ⟨ Ψ m | ψ A ⟩ ] ≥ 0 {\displaystyle \langle \psi _{A}|\rho ^{A}|\psi _{A}\rangle =\sum _{m}p_{m}\operatorname {Tr} _{B}[\langle \psi _{A}|\Psi _{m}\rangle \langle \Psi _{m}|\psi _{A}\rangle ]\geq 0} Note that the term Tr B [ ⟨ ψ A | Ψ m ⟩ ⟨ Ψ m | ψ A ⟩ ] {\displaystyle \operatorname {Tr} _{B}[\langle \psi _{A}|\Psi _{m}\rangle \langle \Psi _{m}|\psi _{A}\rangle ]} represents the probability of finding the state | ψ A ⟩ {\displaystyle |\psi _{A}\rangle } when the composite system is in the state | Ψ m ⟩ {\displaystyle |\Psi _{m}\rangle } . This proves the positive semi-definiteness of ρ A {\displaystyle \rho ^{A}} . The partial trace map as given above induces a dual map Tr B ∗ {\displaystyle \operatorname {Tr} _{B}^{*}} between the C*-algebras of bounded operators on H A {\displaystyle \;H_{A}} and H A ⊗ H B {\displaystyle H_{A}\otimes H_{B}} given by Tr B ∗ ( A ) = A ⊗ I . {\displaystyle \operatorname {Tr} _{B}^{*}(A)=A\otimes I.} Tr B ∗ {\displaystyle \operatorname {Tr} _{B}^{*}} maps observables to observables and is the Heisenberg picture representation of Tr B {\displaystyle \operatorname {Tr} _{B}} . === Comparison with classical case === Suppose instead of quantum mechanical systems, the two systems A and B are classical. The space of observables for each system are then abelian C*-algebras. These are of the form C(X) and C(Y) respectively for compact spaces X, Y. The state space of the composite system is simply C ( X ) ⊗ C ( Y ) = C ( X × Y ) . {\displaystyle C(X)\otimes C(Y)=C(X\times Y).} A state on the composite system is a positive element ρ of the dual of C(X × Y), which by the Riesz–Markov theorem corresponds to a regular Borel measure on X × Y. The corresponding reduced state is obtained by projecting the measure ρ to X. Thus the partial trace is the quantum mechanical equivalent of this operation. == References == Filipiak, Katarzyna; Klein, Daniel; Vojtková, Erika (2018). "The properties of partial trace and block trace operators of partitioned matrices". Electronic Journal of Linear Algebra. 33. doi:10.13001/1081-3810.3688. Johnston, Nathaniel (2021). Advanced Linear and Matrix Algebra. Springer. p. 367. doi:10.1007/978-3-030-52815-7. ISBN 978-3-030-52815-7.
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Wikipedia:Partition of an interval#0
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In mathematics, a partition of an interval [a, b] on the real line is a finite sequence x0, x1, x2, …, xn of real numbers such that a = x0 < x1 < x2 < … < xn = b. In other terms, a partition of a compact interval I is a strictly increasing sequence of numbers (belonging to the interval I itself) starting from the initial point of I and arriving at the final point of I. Every interval of the form [xi, xi + 1] is referred to as a subinterval of the partition x. == Refinement of a partition == Another partition Q of the given interval [a, b] is defined as a refinement of the partition P, if Q contains all the points of P and possibly some other points as well; the partition Q is said to be “finer” than P. Given two partitions, P and Q, one can always form their common refinement, denoted P ∨ Q, which consists of all the points of P and Q, in increasing order. == Norm of a partition == The norm (or mesh) of the partition x0 < x1 < x2 < … < xn is the length of the longest of these subintervals max{|xi − xi−1| : i = 1, … , n }. == Applications == Partitions are used in the theory of the Riemann integral, the Riemann–Stieltjes integral and the regulated integral. Specifically, as finer partitions of a given interval are considered, their mesh approaches zero and the Riemann sum based on a given partition approaches the Riemann integral. == Tagged partitions == A tagged partition or Perron Partition is a partition of a given interval together with a finite sequence of numbers t0, …, tn − 1 subject to the conditions that for each i, xi ≤ ti ≤ xi + 1. In other words, a tagged partition is a partition together with a distinguished point of every subinterval: its mesh is defined in the same way as for an ordinary partition. == See also == Regulated integral Riemann integral Riemann–Stieltjes integral Henstock–Kurzweil integral == References == == Further reading == Gordon, Russell A. (1994). The integrals of Lebesgue, Denjoy, Perron, and Henstock. Graduate Studies in Mathematics, 4. Providence, RI: American Mathematical Society. ISBN 0-8218-3805-9.
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Wikipedia:Pascale Garaud#0
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Pascale Garaud is a French astrophysicist and applied mathematician interested in fluid dynamics, magnetohydrodynamics, and their applications to astrophysics and geophysics. She is a professor of applied mathematics at the University of California, Santa Cruz, and currently serves as the department chair. Garaud was a student at Louis Pasteur University in Strasbourg, and came to Trinity College, Cambridge as a Knox Scholar to study for the Mathematical Tripos. Remaining at Cambridge, Garaud earned her Ph.D. in 2001. Her dissertation, The dynamics of the solar tachocline, was jointly supervised by Douglas Gough and Nigel Weiss. After postdoctoral research at Cambridge, she joined the faculty at the University of California, Santa Cruz in 2004. At Santa Cruz, Garaud is a Fellow of Oakes College. She founded the Kavli Summer Program in Astrophysics (formerly the International Summer Program for Modeling in Astrophysics), an annual meeting of graduate students and researchers, in 2010. In 2019, Garaud was elected a fellow of the American Physical Society. == References == == External links == Home page Pascale Garaud publications indexed by Google Scholar
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Wikipedia:Patrice Ossona de Mendez#0
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Patrice Ossona de Mendez is a French mathematician specializing in topological graph theory who works as a researcher at the Centre national de la recherche scientifique in Paris. He is editor-in-chief of the European Journal of Combinatorics, a position he has held since 2009. == Education and career == Ossona de Mendez was born on 13 December 1966 in Paris. He represented France in the International Mathematical Olympiad in 1985, earning a bronze medal there. He studied at the École Normale Supérieure from 1986 until 1990, and completed his Ph.D. in 1994 from the School for Advanced Studies in the Social Sciences. His dissertation, jointly supervised by Rosenstiehl and Hubert de Fraysseix, concerned bipolar orientations of graphs. He has worked at CNRS since 1995, and earned a habilitation in 2009 from the University of Bordeaux 1. == Book == With Jaroslav Nešetřil, he is the author of the book Sparsity: Graphs, Structures, and Algorithms (Algorithms and Combinatorics 28, Springer, 2012), concerning the properties and applications of different types of sparse graph. This book was included in ACM Computing Reviews list of Notable Books and Articles of 2012. == See also == Left-right planarity test Schnyder's theorem Bounded expansion == References == == External links == Home page Orcid profile
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Wikipedia:Patricia Fauring#0
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Ana María Patricia Fauring is an Argentine mathematician who won the Paul Erdős Award for being "the principal mathematician involved in training Argentine teams for the IMO and other international events, where they have done respectably". Fauring obtained her PhD in 1982 from the University of Buenos Aires under the supervision of Ángel Rafael Larotonda. Her thesis was titled Una noción de estabilidad para campos vectoriales complejos (A notion of stability for complex vector fields). == References ==
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Wikipedia:Patricia Gonçalves#0
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Ana Patrícia Carvalho Gonçalves is a Portuguese mathematician who works as a professor of mathematics at the Instituto Superior Técnico of the University of Lisbon. Her research concerns probability theory, and particularly the macroscopic properties of stochastic processes involving particle systems. == Early life, education, and career == Gonçalves is originally from Esposende, in Portugal. Her father owned a shoe company and her mother was a teacher. Her father died when she was young, of primary amyloidosis, leading her to aim for a career in medicine. However, this disruption in her life caused her schoolwork to suffer, and her scores were not high enough for medical school. Instead she went into mathematics at the University of Porto, inspired by one of her high school teachers. After taking a summer course in measure theory from Cláudio Landim at the Instituto Nacional de Matemática Pura e Aplicada (IMPA) in Rio de Janeiro, Brazil, she went to IMPA for her doctoral studies. She completed her PhD there in 2007. Her dissertation, supervised by Landim, was Central Limit Theorem for a Tagged Particle in Asymmetric Simple Exclusion. She became a postdoctoral researcher at the University of São Paulo in Brazil and the University of Minho in Portugal, and taught at the Pontifical Catholic University of Rio de Janeiro before returning to Portugal and taking her present position at the Instituto Superior Técnico. Gonçalves is the editor-in-chief of Electronic Communications in Probability for 2024–2026. == Recognition == She was an invited speaker at the 2022 (virtual) International Congress of Mathematicians, and is the 2024 Schramm Lecturer of the Institute of Mathematical Statistics and the Bernoulli Society at the 11th World Congress in Probability and Statistics. == References == == External links == Home page Patricia Gonçalves publications indexed by Google Scholar
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Wikipedia:Patrick Michael Grundy#0
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Patrick Michael Grundy (16 November 1917, Yarmouth, Isle of Wight – 4 November 1959) was an English mathematician and statistician. He was one of the eponymous co-discoverers of the Sprague–Grundy function and its application to the analysis of a wide class of combinatorial games. == Biography == Grundy received his secondary education from Malvern College, to which he had obtained a Major Scholarship in 1931, and from which he graduated in 1935. While there, he demonstrated his aptitude for mathematics by winning three prizes in that subject. After leaving school he entered Clare College, Cambridge, on a Foundation Scholarship, where he read for the Mathematical Tripos from 1936 to 1939, earning first class honours in Part II and a distinction in Part III. The work for which he is best known appeared in his first paper, Mathematics and Games, first published in the Cambridge University Mathematical Society's magazine, Eureka in 1939, and reprinted by the same magazine in 1964. The main results of this paper were discovered independently by Grundy and by Roland Sprague, and had already been published by the latter in 1935. The key idea is that of a function that assigns a non-negative integer to each position of a class of combinatorial games, now called impartial games, and which greatly assists in the identification of winning and losing positions, and of the winning moves from the former. The number assigned to a position by this function is called its Grundy value (or Grundy number), and the function itself is called the Sprague–Grundy function, in honour of its co-discoverers. The procedures developed by Sprague and Grundy for using their function to analyse impartial games are collectively called Sprague–Grundy theory, and at least two different theorems concerning these procedures have been called Sprague–Grundy theorems. The maximum number of colors used by a greedy coloring algorithm is called the Grundy number, also after this work on games, as its definition has some formal similarities with the Sprague–Grundy theory. In 1939 Grundy began research in algebraic geometry as a research student at the University of Cambridge, eventually specialising in the theory of ideals. In 1941 he won a Smith's Prize for an essay entitled On the theory of R-modules, and his first research paper in the area, A generalisation of additive ideal theory, was published in the following year. In 1943 he was appointed to an assistant lectureship at the University College of Hull, which he left in 1944. He was awarded a Ph.D. from the University of Cambridge in 1945. Shortly after the end of World War II, Grundy moved away from the field of algebra to take up work in statistics. In 1947 he began formal training in the latter discipline at the Rothamsted Experimental Station under a Ministry of Agriculture scholarship, graduating in 1949, when he then joined the permanent staff of the former organisation as an Experimental Officer. In 1951 he was promoted to Senior Experimental Officer. During his time at Rothamsted he performed most of his published statistical research, which included investigations of problems in the design and analysis of experiments, sampling, composition of animal populations, and fitting truncated distributions. From 1954 to 1958 Grundy worked as a statistician at the National Institute for Educational Research. During this period, he collaborated with Michael Healy and D.H. Rees to extend Frank Yates's work on cost–benefit analysis of experimentation. The results of this collaboration were reported in an influential paper, Economic choice of the amount of experimentation, published in series B of the Journal of the Royal Statistical Society in 1956. In 1958 Grundy moved to a position in the Biometry Unit at Oxford. However, he retired from this position after only one term, due to ill health. Early in 1959 Grundy married Hilary Taylor, a former colleague from the National Institute of Educational Research. Although his health then greatly improved throughout 1959, he was unfortunately killed in an accident in November of that year. == List of Grundy's papers == With the exception of the final item, this list is taken from Smith's obituary (1960). The first item is missing from Goddard's (1960) list, which is otherwise the same as Smith's. "Mathematics and games", Eureka, 2: 6–8, 1939 Grundy, P. M. (1942), "A generalisation of additive ideal theory", Mathematical Proceedings of the Cambridge Philosophical Society, 38 (3): 241–79, Bibcode:1942PCPS...38..241G, doi:10.1017/s0305004100021940, S2CID 120777795 Scorer, R. S.; Grundy, P. M.; Smith, C. A. B. (1944), "Some binary games", Mathematical Gazette, 28 (280): 96–103, doi:10.2307/3606393, JSTOR 3606393, S2CID 125099183 (with R.S. Scorer and C.A.B Smith) Grundy, P. M. (1947), "On integrally dependent Integral domains", Philosophical Transactions of the Royal Society of London, A, 240 (819): 295–326, Bibcode:1947RSPTA.240..295G, doi:10.1098/rsta.1947.0004 "Restricted randomization and quasi-latin squares", Journal of the Royal Statistical Society, Series B, 12: 286–91, 1950 (with M.J.R. Healy) Grundy, P. M. (1950), "The estimation of error in rectangular lattices", Biometrics, 6 (1): 25–33, doi:10.2307/3001421, JSTOR 3001421 "A general technique for the analysis of experiments with incorrectly treated plots", Journal of the Royal Statistical Society, Series B, 13: 272–83, 1951 Grundy, P. M. (1951), "The expected frequencies in a sample of an animal population in which the abundances of species are log-normally distributed (Part I)", Biometrika, 38 (3–4): 427–34, doi:10.1093/biomet/38.3-4.427 Grundy, P. M. (1952), "The fitting of grouped truncated and grouped censored distributions", Biometrika, 39 (3/4): 252–9, doi:10.2307/2334022, JSTOR 2334022 "Selection without replacement from within strata with probability proportional to size", Journal of the Royal Statistical Society, Series B, 15: 253–61, 1953 (with F. Yates) Leech, F. B.; Grundy, P. M. (1953), "A nomogram for assays in randomized blocks", British Journal of Pharmacology, 8 (3): 281–5, doi:10.1111/j.1476-5381.1953.tb00795.x, PMC 1509275, PMID 13093947 (with F. Leech) Grundy, P. M.; Rees, D. H.; Healy, M. J. R. (1954), "Decision between two alternatives—How many experiments?", Biometrics, 10 (3): 317–23, doi:10.2307/3001588, JSTOR 3001588 (with D.H. Rees and M.J.R. Healy) "A method of sampling with probability exactly proportional to size", Journal of the Royal Statistical Society, Series B, 16: 236–8, 1954 "Economic choice of the amount of experimentation", Journal of the Royal Statistical Society, Series B, 18: 32–49, 1956 (with D.H. Rees and M.J.R. Healy) "Fiducial distributions and prior distributions: an example in which the former cannot be associated with the latter", Journal of the Royal Statistical Society, Series B, 18: 217–21, 1956 Grundy, P. M.; Smith, C. A. B. (1956), "Disjunctive games with the last player losing", Mathematical Proceedings of the Cambridge Philosophical Society, 52 (3): 527–33, Bibcode:1956PCPS...52..527G, doi:10.1017/s0305004100031510, S2CID 122928717 (with C.A.B. Smith) "Mathematics and games", Eureka, 27: 9–11, 1964 [1939], archived from the original on 27 September 2007. Reprint of Grundy (1939). == Notes == == References == Berlekamp, Elwyn R.; Conway, John H.; Guy, Richard K. (1982), Winning Ways for your mathematical plays (2 volumes), London: Academic Press Conway, John Horton (2001), On Numbers and Games (2nd ed.), Wellesley, MA: A.K. Peters, ISBN 9781568811277 Goddard, L.S. (1960), "Patrick Michael Grundy", J. London Math. Soc., Series 1, Vol. 35 (3): 377–379, doi:10.1112/jlms/s1-35.3.377 Guy, Richard K., ed. (1991), Combinatorial Games, Proceedings of Symposia in Applied Mathematics, vol. 43, American Mathematical Society, ISBN 9780821867488 Siegel, Aaron N. (2013), Combinatorial Game Theory, Graduate Studies in Mathematics, vol. 146, American Mathematical Society, ISBN 9780821851906 Smith, Cedric A.B. (1960), "Patrick Michael Grundy, 1917–1959", Journal of the Royal Statistical Society, Series A, 123 (2): 221–22 Smith, Samuel Bruce (2015), Chance, Strategy, and Choice: An Introduction to the Mathematics of Games and Elections, Cambridge: Cambridge University Press, ISBN 9781316033708 Sprague, R.P. (1935), "Über mathematische Kampfspiele", Tohoku Mathematical Journal, 41: 438–444
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Wikipedia:Patrizia Gianni#0
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Patrizia M. Gianni (born 1952) is an Italian mathematician specializing in computer algebra. She is known for her early research on Gröbner bases including her discovery of the FGLM algorithm for changing monomial orderings in Gröbner bases, and for her development of the components of the Axiom computer algebra system concerning polynomials and rational functions. Gianni is a professor of algebra in the mathematics department of the University of Pisa. She earned a laurea from the University of Pisa, and has worked for IBM Research as well as for the University of Pisa. == References == == External links == Home page Patrizia Gianni publications indexed by Google Scholar
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Wikipedia:Paul Balmer#0
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Paul Balmer (born 1970) is a Swiss mathematician, working in algebra. He is a professor of mathematics at the University of California, Los Angeles. Balmer received his Ph.D. from the University of Lausanne in 1998, under the supervision of Manuel Ojanguren, with a thesis entitled Groupes de Witt dérivés des Schémas (in French). His research centers around triangulated categories. More specifically, he is a proponent of tensor-triangular geometry, an umbrella topic which covers geometric aspects of algebraic geometry, modular representation theory, stable homotopy theory, and other areas, by means of relevant tensor-triangulated categories. Balmer was an Invited Speaker at the International Congress of Mathematicians in Hyderabad in 2010, with a talk on Tensor Triangular Geometry. In 2012, he became a fellow of the American Mathematical Society. He was awarded the Humboldt Prize in 2015. == References ==
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Wikipedia:Paul Bernays#0
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Paul Isaac Bernays ( bur-NAYZ; Swiss Standard German: [bɛrˈnaɪs]; 17 October 1888 – 18 September 1977) was a Swiss mathematician who made significant contributions to mathematical logic, axiomatic set theory, and the philosophy of mathematics. He was an assistant and close collaborator of David Hilbert. == Biography == Bernays was born into a distinguished German-Jewish family of scholars and businessmen. His great-grandfather, Isaac ben Jacob Bernays, served as chief rabbi of Hamburg from 1821 to 1849. Bernays spent his childhood in Berlin, and attended the Köllnische Gymnasium, 1895–1907. At the University of Berlin, he studied mathematics under Issai Schur, Edmund Landau, Ferdinand Georg Frobenius, and Friedrich Schottky; philosophy under Alois Riehl, Carl Stumpf and Ernst Cassirer; and physics under Max Planck. At the University of Göttingen, he studied mathematics under David Hilbert, Edmund Landau, Hermann Weyl, and Felix Klein; physics under Voigt and Max Born; and philosophy under Leonard Nelson. In 1912, the University of Berlin awarded him a Ph.D. in mathematics for a thesis, supervised by Landau, on the analytic number theory of binary quadratic forms. That same year, the University of Zurich awarded him habilitation for a thesis on complex analysis and Picard's theorem. The examiner was Ernst Zermelo. Bernays was Privatdozent at the University of Zurich, 1912–1917, where he came to know George Pólya. His collected communications with Kurt Gödel span many decades. Starting in 1917, David Hilbert employed Bernays to assist him with his investigations of the foundation of arithmetic. Bernays also lectured on other areas of mathematics at the University of Göttingen. In 1918, that university awarded him a second habilitation for a thesis on the axiomatics of the propositional calculus of Principia Mathematica. In 1922, Göttingen appointed Bernays extraordinary professor without tenure. His most successful student there was Gerhard Gentzen. After Nazi Germany enacted the Law for the Restoration of the Professional Civil Service in 1933, the university fired Bernays because of his Jewish ancestry. After working privately for Hilbert for six months, Bernays and his family moved to Switzerland, whose nationality he had inherited from his father, and where the ETH Zurich employed him on occasion. He also visited the University of Pennsylvania and was a visiting scholar at the Institute for Advanced Study in 1935–36 and again in 1959–60. == Mathematical work == His habilitation thesis was written under the supervision of Hilbert himself, on the topic of the axiomatisation of propositional logic in Whitehead and Russell's Principia Mathematica. It contains the first known proof of semantic completeness of propositional logic, which was reproved independently also by Emil Post later on. Bernays's collaboration with Hilbert culminated in the two volume work, Grundlagen der Mathematik (English: Foundations of Mathematics) published in 1934 and 1939, which is discussed in Sieg and Ravaglia (2005). A proof in this work that a sufficiently strong consistent theory cannot contain its own reference functor is known as the Hilbert–Bernays paradox. In seven papers, published between 1937 and 1954 in the Journal of Symbolic Logic (republished in Müller 1976), Bernays set out an axiomatic set theory whose starting point was a related theory John von Neumann had set out in the 1920s. Von Neumann's theory took the notions of function and argument as primitive. Bernays recast von Neumann's theory so that classes and sets were primitive. Bernays's theory, with modifications by Kurt Gödel, is known as von Neumann–Bernays–Gödel set theory. == Publications == Hilbert, David; Bernays, Paul (1934), Grundlagen der Mathematik. I, Die Grundlehren der mathematischen Wissenschaften, vol. 40, Berlin, New York: Springer-Verlag, ISBN 978-3-540-04134-4, JFM 60.0017.02, MR 0237246, archived from the original on 2011-05-17 {{citation}}: ISBN / Date incompatibility (help) Hilbert, David; Bernays, Paul (1939), Grundlagen der Mathematik. II, Die Grundlehren der mathematischen Wissenschaften, vol. 50, Berlin, New York: Springer-Verlag, ISBN 978-3-540-05110-7, JFM 65.0021.02, MR 0272596, archived from the original on 2011-05-17 {{citation}}: ISBN / Date incompatibility (help) Bernays, Paul (1958), Axiomatic Set Theory, Studies in Logic and the Foundations of Mathematics, Amsterdam: North-Holland, ISBN 978-0-486-66637-2, MR 0106178 {{citation}}: ISBN / Date incompatibility (help) Bernays, Paul (1976), Abhandlungen zur Philosophie der Mathematik (in German), Darmstadt: Wissenschaftliche Buchgesellschaft, ISBN 978-3-534-06706-0, MR 0444417 Bernays, Paul; Schonfinkel, Moses (1928), "Zum Entscheidungsproblem der mathematischen Logik", Mathematische Annalen, 99 (99): 342–372, doi:10.1007/BF01459101, S2CID 122312654 == Notes == == References == Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15: 43–69, doi:10.2178/bsl/1231081769, S2CID 15567244. Kneebone, Geoffrey, 1963. Mathematical Logic and the Foundation of Mathematics. Van Nostrand. Dover reprint, 2001. A gentle introduction to some of the ideas in the Grundlagen der Mathematik. Lauener, Henri (1978), "Paul Bernays (1888--1977)", Zeitschrift für allgemeine Wissenschaftstheorie, 9 (1): 13–20, doi:10.1007/BF01801939, ISSN 0044-2216, MR 0546580, S2CID 147959212 Müller, Gert H., ed. (1976), Sets and classes. On the work by Paul Bernays, Studies in Logic and the Foundations of Mathematics, vol. 84, Amsterdam: North-Holland, ISBN 978-0-444-10907-1, MR 0414355 Sieg, Wilfried; Ravaglia, Mark (2005), "Chapter 77. David Hilbert and Paul Bernays, Grundlagen der Mathematik", in Grattan-Guinness, Ivor (ed.), Landmark writings in western mathematics 1640--1940, Elsevier B. V., Amsterdam, pp. 981–99, doi:10.1016/B978-044450871-3/50158-3, ISBN 978-0-444-50871-3, MR 2169816 == External links == Hilbert Bernays Project O'Connor, John J.; Robertson, Edmund F., "Paul Bernays", MacTutor History of Mathematics Archive, University of St Andrews Paul Bernays: A Short Biography (1976)
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Wikipedia:Paul Bressloff#0
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Paul C. Bressloff is a British applied mathematician, biophysicist and mathematical neuroscientist. As of 2022, Bressloff is currently a full professor in the Department of Mathematics at the University of Utah. == Education == Bressloff obtained an MA with First Class Honors from the University of Oxford in 1982, and obtained his Ph.D from the Department of Mathematics at King's College in 1988. His thesis was titled Quantum field theory of superstrings in the light-cone gauge. == Research == Bressloff has published extensively on a wide variety of applied and theoretical topics. As of 2022, he has an H-index of 54, and he has published over three-hundred and fifty articles, three textbooks, and has co-written a non-fiction popular science book. He has advised more than twenty PhD recipients. === Books === Paul is the author of three textbooks in computational biology, two of which deal with stochastic processes in cellular biology. Bursting: The Genesis of Rhythm in the Nervous System with Stephen Coombes (2003) Waves in Neural Media: From Single Neurons to Neural Fields (2013) Stochastic Processes in Cell Biology (2014) Stochastic Processes in Cell Biology: Volume II (2022) == References == == External links == Paul Bressloff publications indexed by Google Scholar
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Wikipedia:Paul Butzer#0
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Paul Leo Butzer (born 15 April 1928) is a German mathematician who specializes in Analysis (Approximation theory, Harmonic analysis). == Life and work == Butzer was born in Mülheim an der Ruhr on 15 April 1928. He is the son of an engineer, and his mother studied mathematics at RWTH Aachen University. As opponents of the National Socialists (Nazis), Butzer's parents left Germany with their children in 1937 and moved to England. During World War II, they relocated to Canada, where Butzer attended school in Montreal and studied mathematics at Loyola College (later Concordia University), completing his bachelor's degree in 1948. He then pursued further studies at the University of Toronto, including studying under Harold Scott MacDonald Coxeter and William Tutte, and obtained his Ph.D. in 1951 under the supervision of George G. Lorentz (On Bernstein polynomials). In 1952, he became a lecturer and then an assistant professor at McGill University. In 1955/56, he lived in Paris and then in Mainz. He decided to stay in Germany, where he completed his habilitation at the University of Freiburg, taught in Würzburg, and starting in 1958, at RWTH Aachen University. In 1962, he was appointed professor there. In 1963, he began organizing international conferences on Approximation theory at the Oberwolfach Research Institute for Mathematics, later together with Béla Szőkefalvi-Nagy. In addition to approximation theory and its connections to Fourier analysis and semigroups of operators in Banach spaces, Butzer also worked on probability theory (Central limit theorem and related Convergence tests issues), Sampling Theory, and Signal analysis. Paul Leo Butzer also delved into the history of mathematics, particularly in its connection with Aachen. He studied figures such as Peter Gustav Lejeune Dirichlet, Eduard Helly, Eugène Catalan, Pafnuty Chebyshev, Charles Jean de la Vallée Poussin, the history of splines, Otto Blumenthal, mathematics in the Carolingian era, and Elwin Bruno Christoffel (on whom he published a book). Butzer is a member of the Royal Society of Sciences in Liège and the Royal Belgian Academy of Sciences. He is an honorary member of the Mathematical Society in Hamburg. He has received honorary doctorates from three universities: Liège, York, and Timișoara. Paul Butzer is the brother of Karl W. Butzer. == Publications == With Hubert Berens: "Semi-groups of Operators and Approximation," Grundlehren der mathematischen Wissenschaften, Springer Verlag, 1967. With Hermann Schulte: "Ein Operatorenkalkül zur Lösung gewöhnlicher und partieller Differenzengleichungssysteme von Funktionen diskreter Veränderlicher und seine Anwendungen," Köln, Opladen, Westdeutscher Verlag, 1965. With Rolf Joachim Nessel: "Fourier Analysis and Approximation," Academic Press, Vol. 1 (One-dimensional Theory), 1971. With Walter Trebels: "Hilberttransformation, gebrochene Integration und Differentiation," Köln, Opladen, Westdeutscher Verlag, 1968. With Karl Scherer: "Approximationsprozesse und Intepolationsmethoden," BI Hochschultaschenbuch, Mannheim, 1968. With W. Oberdörster: "Darstellungssätze für beschränkte lineare Funktionale im Zusammenhang mit Hausdorff-, Stieltjes- und Hamburger-Momentenproblemen," Opladen, Westdeutscher Verlag, 1975. Edited with Dietrich Lohrmann: "Science in Western and Eastern Civilization in Carolingian Times," Birkhäuser, 1993. Edited with Walter Oberschelp and Max Kerner: "Karl der Grosse und sein Nachwirken: 1200 Jahre Kultur und Wissenschaft in Europa," 2 volumes, Turnhout, Brepols, 1997/98. "Mathematics in West and East from the fifth to tenth centuries: an overview," in P. L. Butzer, Dietrich Lohrmann (Eds.): "Science in Western and Eastern civilization in Carolingian times," Basel, 1993, pp. 443–481. With Karl W. Butzer: "Mathematics at Charlemagne's court and its transmission," in Catherine Cubitt (Ed.): "Court culture in the early middle ages," Turnhout, 2003, pp. 77–89. "The Mathematicians of the Aachen-Liège Region from the Carolingian to the Late Ottonian Period" (Die Mathematiker des Aachen-Lütticher Raumes von der karolingischen bis zur spätottonischen Epoche), in Annalen des Historischen Vereins für den Niederrhein, Volume 178, 1976, pp. 7–30. Edited with F. Féher: "E. B. Christoffel, the influence of his work on mathematics and the physical sciences," Birkhäuser, 1981. With Francois Jongmans: "P. L. Chebyshev (1821–1894): a guide to his life and work," Lehrstuhl für Mathematik A, RWTH Aachen, 1998. "Dirichlet and his role in the founding of mathematical physics," Lehrstuhl A für Mathematik, RWTH Aachen, 1983. == References == == External links == Homepage of the RWTH Aachen (German) O'Connor, John J.; Robertson, Edmund F., "Paul Butzer", MacTutor History of Mathematics Archive, University of St Andrews Mathematics Genealogy Project Author Profile in the zbMATH Databank
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Wikipedia:Paul C. Yang#0
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Paul C. Yang (Chinese: 杨建平; pinyin: Yáng Jiàn Píng; born 1947) is a Taiwanese-American mathematician specializing in differential geometry, partial differential equations and CR manifolds. He is best known for his work in Conformal geometry for his study of extremal metrics and his research on scalar curvature and Q-curvature. In CR Geometry he is known for his work on the CR embedding problem, the CR Paneitz operator and for introducing the Q' curvature in CR Geometry. == Career == Yang received a B.A. in mathematics from the University of California, Berkeley, in 1969. He then earned his doctorate from Berkeley in 1973 under the supervision of Hung-Hsi Wu (Chinese: 伍鴻熙). He held positions at Rice University, the University of Maryland, Indiana University and the University of Southern California before joining Princeton University in 2001. == Awards and honors == Yang was a Sloan Foundation Fellow in 1981. In 2012, he became a fellow of the American Mathematical Society. == Selected publications == Chang, Sun-Yung A.; Yang, Paul C. Conformal deformation of metrics on S 2 {\displaystyle S^{2}} . J. Differential Geom. 27 (1988), no. 2, 259–296. Chang, Sun-Yung A.; Yang, Paul C. Prescribing Gaussian curvature on S 2 {\displaystyle S^{2}} . Acta Math. 159 (1987), no. 3–4, 215–259. Chang, Sun-Yung A.; Yang, Paul C. Extremal metrics of zeta function determinants on 4-manifolds. Ann. of Math. (2) 142 (1995), no. 1, 171–212. Chang, Sun-Yung A.; Gursky, Matthew J.; Yang, Paul C. The scalar curvature equation on 2- and 3-spheres. Calc. Var. Partial Differential Equations 1 (1993), no. 2, 205–229. Chang, Sun-Yung A.; Gursky, Matthew J.; Yang, Paul C. An equation of Monge-Ampère type in conformal geometry, and four-manifolds of positive Ricci curvature. Ann. of Math. (2) 155 (2002), no. 3, 709–787. Yang, Paul C.; Yau, Shing-Tung Eigenvalues of the Laplacian of compact Riemann surfaces and minimal submanifolds. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 7 (1980), no. 1, 55–63. Chanillo, Sagun; Chiu, Hung-Lin; Yang, Paul C. Embeddability for Three Dimensional Cauchy-Riemann Manifolds and CR Yamabe Invariants, Duke Math. J.,161(15), (2012), 2909–2921. == References ==
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Wikipedia:Paul Cohn#0
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Paul Moritz Cohn FRS (8 January 1924 – 20 April 2006) was Astor Professor of Mathematics at University College London, 1986–1989, and author of many textbooks on algebra. His work was mostly in the area of algebra, especially non-commutative rings. == Early life == Cohn was the only child of Jewish parents, James (or Jakob) Cohn, owner of an import business, and Julia (née Cohen), a schoolteacher. Both of his parents were born in Hamburg, as were three of his grandparents. His ancestors came from various parts of Germany. His father fought in the German army in World War I; he was wounded several times and awarded the Iron Cross. A street in Hamburg is named in memory of his mother. When he was born, his parents were living with his mother's mother in Isestraße. After her death in October 1925, the family moved to a rented flat in a new building in Lattenkamp, in the Winterhude quarter. He attended a kindergarten then, in April 1930, moved to Alsterdorfer Straße School. After a while, he had a new teacher, a National Socialist, who picked on him and punished him without cause. Thus in 1931, he moved to the Meerweinstraße School where his mother taught. Following the rise of the Nazis in 1933, his father's business was confiscated and his mother dismissed. He moved to the Talmud-Tora-Schule, a Jewish school. In mid-1937, the family moved to Klosterallee. This was nearer the school, the synagogue and other pupils, being in the Jewish area. His German teacher was Dr. Ernst Loewenberg, the son of the poet Jakob Loewenberg. On the night of 9/10 November 1938 (Kristallnacht), his father was arrested and sent to Sachsenhausen concentration camp. He was released after four months but told to emigrate. Cohn went to Britain in May 1939 on the Kindertransport to work on a chicken farm, and never saw his parents again. He corresponded regularly with them until late 1941. At the end of the War, he learned that they were deported to Riga on 6 December 1941 and never returned. At the end of 1941, the farm closed. He trained as a precision engineer, acquired a work permit and worked in a factory for 4½ years. He passed the Cambridge Scholarship Examination, and won an exhibition to Trinity College, Cambridge. == Career == He received a B.A in mathematics from Cambridge University in 1948 and a Ph.D. (supervised by Philip Hall) in 1951. He then spent a year as a Chargé de Recherches at the University of Nancy. On his return, he became a lecturer in mathematics at Manchester University. He was a visiting professor at Yale University in 1961–1962, and for part of 1962 was at the University of California at Berkeley. On his return, he became Reader at Queen Mary College. He was a visiting professor at the University of Chicago in 1964 and at the State University of New York at Stony Brook in 1967. By then, he was regarded as one of the world's leading algebraists. Also in 1967, he became head of the Department of Mathematics at Bedford College, London. He held several visiting professorships, in America, Paris, Delhi, Canada, Haifa and Bielefeld. He was awarded the Lester R. Ford Award from the Mathematical Association of America in 1972 and the Senior Berwick Prize of the London Mathematical Society in 1974. In the early 1980s, funding cuts caused the closure of the small colleges of the University of London. Cohn moved to University College London in 1984, together with the two other experts at Bedford on ring theory, Bill Stephenson and Warren Dicks. He became Astor Professor of Mathematics there in 1986. He continued to be a visiting professor, for example to the University of Alberta in 1986 and to Bar Ilan University in 1987. He retired in 1989, but remained active as professor emeritus and honorary research fellow until his death. He was president of the London Mathematical Society, 1982–1984, having been its secretary, 1965–1967 and a council member in 1968–1971, 1972–1975 and 1979–1982. He was editor of the society's monographs in 1968–1977 and 1980–1993. He was elected a Fellow of the Royal Society in 1980 and was on its council, 1985–1987. He was a member of the Mathematical Committee of the Science Research Council, 1977–1980. He chaired the National Committee for Mathematics, 1988–1989. == Mathematical work == In all, Cohn wrote nearly 200 mathematical papers. He worked in many areas of algebra, mainly in non-commutative ring theory. His first papers, covering many topics, were published in 1952. He generalised a theorem due to Wilhelm Magnus, and worked on the structure of tensor spaces. In 1953 he published a joint paper with Kurt Mahler on pseudo-valuations and in 1954 he published a work on Lie algebras. Papers over the next few years covered areas such as group theory, field theory, Lie rings, semigroups, abelian groups and ring theory. After that, he moved into the areas of Jordan algebras, skew fields, and non-commutative unique factorisation domains. In 1957 Cohn published his first book, Lie Groups, on groups that are analytic manifolds: Lie groups. His second book, Linear Equations, appeared in 1958 and his third, Solid geometry, in 1961. Universal algebra appeared in 1965 (second edition 1981). After that, he concentrated on non-commutative ring theory and the theory of algebras. His monograph Free Rings and their Relations appeared in 1971. It covered the work of Cohn and others on free associative algebras and related classes of rings, especially free ideal rings. He included all of his own published results on the embedding of rings into skew fields. The second, enlarged edition appeared in 1985. Cohn also wrote undergraduate textbooks. Algebra I appeared in 1974 and Algebra II in 1977. The second edition, in three volumes, was published by Wiley between 1982 and 1991. These volumes were in line with the British (rather than American) curricula at the time and include both linear algebra and abstract algebra. Cohn wrote a subsequent revised iteration of the first volume as Classical Algebra (Wiley, 2000) as a more "user friendly" version for undergraduates (according to its preface); this book also includes a few selected topics from volumes II and III of Algebra. The final incarnation of Cohn's algebra textbooks appeared in 2003 as two Springer volumes Basic Algebra and Further Algebra and Applications. The material in Basic Algebra is (according to its preface) rather more concise and, while corresponding roughly with Algebra I, assumes knowledge of linear algebra. The material on basic theories (groups, rings, fields) is pursued in more depth in Basic Algebra compared to Algebra I. Further Algebra and Applications roughly corresponds to volumes II and III of Algebra, but reflects the shift of some material from these volumes to Basic Algebra. == Private life == His recreation was etymology and language in all its forms. He married Deirdre Sharon in 1958, and they had two daughters. == Publications == === Articles === Cohn, P. M. (1958). "Rings of zero-divisors". Proc. Amer. Math. Soc. 9 (6): 909–914. doi:10.1090/s0002-9939-1958-0103202-2. MR 0103202. Cohn, P. M. (1963). "Noncommutative unique factorization domains". Trans. Amer. Math. Soc. 109 (2): 313–331. doi:10.1090/s0002-9947-1963-0155851-x. MR 0155851. Cohn, P. M. (1963). "Rings with a weak algorithm". Trans. Amer. Math. Soc. 109 (2): 332–356. doi:10.1090/s0002-9947-1963-0153696-8. MR 0153696. 1995: "Skew Fields, Theory of General Division Rings", Encyclopedia of Mathematics and its Applications vol 57 Cohn, P. M. (2000). "From Hermite rings to Sylvester domains". Proc. Amer. Math. Soc. 128 (7): 1899–1904. doi:10.1090/s0002-9939-99-05189-8. MR 1646314. === Books === 1957: Lie Groups 1958: Linear Equations 1961: Solid Geometry 1965: Universal Algebra, second edition 1981 1971: Free Rings and Their Relations second edition 1985 1974: Algebra I, second edition 1982 1977: Algebra II, second edition 1989 1977: Skew Field Constructions 1990: Algebra III 1991: Algebraic Numbers and Algebraic Functions 1994: Elements of Linear Algebra 2000: Introduction to Ring Theory 2000: Classic Algebra 2002: Basic Algebra 2003: Further Algebra and Applications 2004: Oxford Dictionary of National Biography (contributor) 2006: Free Ideal Rings and Localization in General Rings == References == == Bibliography == Cohn, Paul; Struan Robertson (website owner) (February 2003). "Paul M. Cohn, (Childhood in Hamburg)" (Autobiography). A History of Jews in Hamburg: The Kindertransports. Retrieved 24 July 2008. Cooper, P. R. (ed.). "Paul M. Cohn". London Mathematical Society. Council for the Mathematical Sciences. Archived from the original (Newsletter) on 23 December 2012. Retrieved 4 August 2008. Schofield, Aidan (8 August 2006). "Professor Paul Cohn" (Newspaper obituary). The Independent. Retrieved 4 August 2008. The Times (29 June 2006). "Professor Paul Cohn: Mathematician who devoted himself to algebra" (Newspaper obituary). The Times. Retrieved 4 August 2008.
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Wikipedia:Paul Couderc#0
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Paul Couderc (15 July 1899 – 5 February 1981) was a French academic who held mathematics professorships at lycées in Chartres (1926–1929) and Paris (1930–1944). == Biography == Couderc completed his education at lycées in Nevers and Dijon, followed by a doctorate in mathematical sciences from the École Normale Supérieure in Paris. In 1926, he married Blanch Jurus. Throughout his career, Couderc authored approximately fifteen works in the field of astronomy. He provided an interpretation for the phenomena of light echoes around Nova Persei (1901), specifically their perceived superluminal expansion. This geometrical explanation later found application in the study of supernovae, quasars, and γ-ray bursts. == Awards and recognition == Kalinga Prize for the Popularization of Science (1966) == References ==
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Wikipedia:Paul Finsler#0
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Paul Finsler (born 11 April 1894, in Heilbronn, Germany, died 29 April 1970 in Zurich, Switzerland) was a German and Swiss mathematician. Finsler did his undergraduate studies at the Technische Hochschule Stuttgart, and his graduate studies at the University of Göttingen, where he received his Ph.D. in 1919 under the supervision of Constantin Carathéodory. He studied for his habilitation at the University of Cologne, receiving it in 1922. He joined the faculty of the University of Zurich in 1927, and was promoted to ordinary professor there in 1944. Finsler's thesis work concerned differential geometry, and Finsler spaces were named after him by Élie Cartan in 1934. The Hadwiger–Finsler inequality, a relation between the side lengths and area of a triangle in the Euclidean plane, is named after Finsler and his co-author Hugo Hadwiger, as is the Finsler–Hadwiger theorem on a square derived from two other squares that share a vertex. Finsler is also known for his work on the foundations of mathematics, developing a non-well-founded set theory with which he hoped to resolve the contradictions implied by Russell's paradox. == Publications == Finsler, Paul (1918), Über Kurven und Flächen in allgemeinen Räumen, Dissertation, Göttingen, JFM 46.1131.02 (Reprinted by Birkhäuser (1951)) Finsler, Paul (1926). "Gibt es Widersprüche in der Mathematik?". Jahresbericht der Deutschen Mathematiker-Vereinigung. 34: 143–154. Finsler, Paul (1926). "Formale Beweise und die Entscheidbarkeit". Mathematische Zeitschrift. 25: 676–682. doi:10.1007/bf01283861. S2CID 121054124. Finsler, Paul (1926). "Über die Grundlegung der Mengenlehre. Erster Teil". Mathematische Zeitschrift. 25: 683–713. doi:10.1007/bf01283862. Finsler, Paul (1963). "Über die Grundlegung der Mengenlehre. Zweiter Teil". Commentarii Mathematici Helvetici. 38 (1): 172–218. doi:10.1007/bf02566915. S2CID 124928448. Finsler, P. (1933). "Die Existenz der Zahlenreihe und des Kontinuums". Commentarii Mathematici Helvetici. 5: 88–94. doi:10.1007/BF01297507. S2CID 120768947. Finsler: Aufsätze zur Mengenlehre. (ed. G. Unger) 1975. Booth, David; Ziegler, Renatus, eds. (1996). Finsler Set Theory: Platonism and Circularity. "Translation of Paul Finsler's papers on set theory with introductory comments". Birkhäuser Basel. doi:10.1007/978-3-0348-9031-1. ISBN 978-3-0348-9876-8. == References == == Further reading == Burckhardt, J. J. (1980), Die Mathematik an der Universität Zurich 1916-1950 unter den Professoren R. Fueter, A. Speiser und P. Finsler, Basel{{citation}}: CS1 maint: location missing publisher (link).
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Wikipedia:Paul Glaister#0
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Paul Glaister is a British mathematician, the UK representative to the International Commission on Mathematical Instruction, the President of the 153 year old Mathematical Association and former Chair of the Joint Mathematical Council (JMC) of the United Kingdom, a body which set up the Advisory Committee on Mathematics Education along with the Royal Society, and comprises 31 bodies representing mathematics education in the UK; an External Expert for Ofqual and for the Standards and Testing Agency within the Department for Education; Honorary Secretary and a Council member of the Institute of Mathematics and its Applications; and works closely with the Education Development Trust in a number of areas in mathematics education. He is a professor of Mathematics and Mathematics Education in the Department of Mathematics and Statistics at the University of Reading. His contributions have been across the areas of computational fluid dynamics, numerical analysis, applied mathematics, mathematics education, and science education. Glaister was appointed Commander of the Order of the British Empire (CBE) in the 2023 New Year Honours for services to education. == References == == External links == Departmental webpage, University of Reading Personal webpage, University of Reading
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Wikipedia:Paul Gordan#0
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Paul Albert Gordan (27 April 1837 – 21 December 1912) was a German mathematician known for work in invariant theory and for the Clebsch–Gordan coefficients and Gordan's lemma. He was called "the king of invariant theory". His most famous result is that the ring of invariants of binary forms of fixed degree is finitely generated. Clebsch–Gordan coefficients are named after him and Alfred Clebsch. Gordan also served as the thesis advisor for Emmy Noether. == Life and Career == Gordan was born to Jewish parents in Breslau, Germany (now Wrocław, Poland), and died in Erlangen, Germany. He received his Dr. phil. at the University of Breslau with the thesis De Linea Geodetica, (On Geodesics of Spheroids) under Carl Jacobi in 1862. He moved to Erlangen in 1874 to become professor of mathematics at the University of Erlangen-Nuremberg. A famous quote attributed to Gordan about David Hilbert's proof of Hilbert's basis theorem, a result which vastly generalized his result on invariants, is "This is not mathematics; this is theology." The proof in question was the (non-constructive) existence of a finite basis for invariants. It is not clear if Gordan really said this since the earliest reference to it is 25 years after the events and after his death. Nor is it clear whether the quote was intended as criticism, or praise, or a subtle joke. Gordan himself encouraged Hilbert and used Hilbert's results and methods, and the widespread story that he opposed Hilbert's work on invariant theory is a myth (though he did correctly point out in a referee's report that some of the reasoning in Hilbert's paper was incomplete). He later said "I have convinced myself that even theology has its merits". He also published a simplified version of the proof. == Publications == Gordan, Paul (1885). Vorlesungen über Invariantentheorie. Vol. 1. Teubner. Retrieved 12 April 2014. Gordan, Paul (1887). Dr. Paul Gordan's Vorlesungen über Invariantentheorie. Vol. 2. B. G. Teubner. Retrieved 12 April 2014. Gordan, Paul (1987) [1885], Kerschensteiner, Georg (ed.), Vorlesungen über Invariantentheorie (2nd ed.), New York: Chelsea Publishing Co. or American Mathematical Society, ISBN 978-0-8284-0328-3, MR 0917266 == References == == See also == Dickson's lemma Invariant of a binary form Symbolic method == External links == Paul Gordan at the Mathematics Genealogy Project Gordan's publication catalog: "Katalog der Deutschen Nationalbibliothek". portal.dnb.de (in German). Retrieved 25 August 2024.
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Wikipedia:Paul Halmos#0
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Paul Richard Halmos (Hungarian: Halmos Pál; 3 March 3 1916 – 2 October 2006) was a Hungarian-born American mathematician and probabilist who made fundamental advances in the areas of mathematical logic, probability theory, operator theory, ergodic theory, and functional analysis (in particular, Hilbert spaces). He was also recognized as a great mathematical expositor. He has been described as one of The Martians. == Early life and education == Born in the Kingdom of Hungary into a Jewish family, Halmos immigrated to the United States at age 13. He obtained his B.A. from the University of Illinois, majoring in mathematics while also fulfilling the requirements for a degree in philosophy. He obtained the degree after only three years, and was 19 years old when he graduated. He then began a Ph.D. in philosophy, still at the Champaign–Urbana campus. However, after failing his masters' oral exams, he shifted to mathematics and graduated in 1938. Joseph L. Doob supervised his dissertation, titled Invariants of Certain Stochastic Transformations: The Mathematical Theory of Gambling Systems. == Career == Shortly after his graduation, Halmos left for the Institute for Advanced Study, lacking both job and grant money. Six months later, he was working under John von Neumann, which proved a decisive experience. While at the Institute, Halmos wrote his first book, Finite Dimensional Vector Spaces, which immediately established his reputation as a fine expositor of mathematics. From 1967 to 1968 he was the Donegall Lecturer in Mathematics at Trinity College Dublin. Halmos taught at Syracuse University, the University of Chicago (1946–60), the University of Michigan (~1961–67), the University of Hawaii (1967–68), Indiana University (1969–85), and the University of California at Santa Barbara (1976–78). From his 1985 retirement from Indiana until his death, he was affiliated with the Mathematics department at Santa Clara University (1985–2006). == Accomplishments == In a series of papers reprinted in his 1962 Algebraic Logic, Halmos devised polyadic algebras, an algebraic version of first-order logic differing from the better known cylindric algebras of Alfred Tarski and his students. An elementary version of polyadic algebra is described in monadic Boolean algebra. In addition to his original contributions to mathematics, Halmos was an unusually clear and engaging expositor of university mathematics. He won the Lester R. Ford Award in 1971 and again in 1977 (shared with W. P. Ziemer, W. H. Wheeler, S. H. Moolgavkar, J. H. Ewing and W. H. Gustafson). Halmos chaired the American Mathematical Society committee that wrote the AMS style guide for academic mathematics, published in 1973. In 1983, he received the AMS's Leroy P. Steele Prize for exposition. In the American Scientist 56(4): 375–389 (Winter 1968), Halmos argued that mathematics is a creative art, and that mathematicians should be seen as artists, not number crunchers. He discussed the division of the field into mathology and mathophysics, further arguing that mathematicians and painters think and work in related ways. Halmos's 1985 "automathography" I Want to Be a Mathematician is an account of what it was like to be an academic mathematician in 20th century America. He called the book "automathography" rather than "autobiography", because its focus is almost entirely on his life as a mathematician, not his personal life. The book contains the following quote on Halmos' view of what doing mathematics means: Don't just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis? In these memoirs, Halmos claims to have invented the "iff" notation for the words "if and only if" and to have been the first to use the "tombstone" notation to signify the end of a proof, and this is generally agreed to be the case. The tombstone symbol ∎ (Unicode U+220E) is sometimes called a halmos. In 1994, Halmos received the Deborah and Franklin Haimo Award for Distinguished College or University Teaching of Mathematics. In 2005, Halmos and his wife Virginia Halmos funded the Euler Book Prize, an annual award given by the Mathematical Association of America for a book that is likely to improve the view of mathematics among the public. The first prize was given in 2007, the 300th anniversary of Leonhard Euler's birth, to John Derbyshire for his book about Bernhard Riemann and the Riemann hypothesis: Prime Obsession. In 2009 George Csicsery featured Halmos in a documentary film also called I Want to Be a Mathematician. == Books == Books by Halmos have led to so many reviews that lists have been assembled. 1942. Finite-Dimensional Vector Spaces. Springer-Verlag. 1950. Measure Theory. Springer Verlag. 1951. Introduction to Hilbert Space and the Theory of Spectral Multiplicity. Chelsea. 1956. Lectures on Ergodic Theory. Chelsea. 1960. Naive Set Theory. Springer Verlag. 1962. Algebraic Logic. Chelsea. 1963. Lectures on Boolean Algebras. Van Nostrand. 1967. A Hilbert Space Problem Book. Springer-Verlag. 1973. (with Norman E. Steenrod, Menahem M. Schiffer, and Jean A. Dieudonne). How to Write Mathematics. American Mathematical Society. ISBN 978-0-8218-0055-3 1978. (with V. S. Sunder). Bounded Integral Operators on L² Spaces. Springer Verlag ISBN 978-3540088943 1985. I Want to Be a Mathematician. Springer-Verlag. ISBN 978-0387960784 1987. I Have a Photographic Memory. Mathematical Association of America. ISBN 978-0821819395 1991. Problems for Mathematicians, Young and Old, Dolciani Mathematical Expositions, Mathematical Association of America. ISBN 978-0883853207 1996. Linear Algebra Problem Book, Dolciani Mathematical Expositions, Mathematical Association of America. ISBN 978-0883853221 1998. (with Steven Givant). Logic as Algebra, Dolciani Mathematical Expositions No. 21, Mathematical Association of America.ISBN 978-1470451134 2009. (posthumous, with Steven Givant), Introduction to Boolean Algebras, Springer. == See also == Crinkled arc Commutator subspace Invariant subspace problem Naive set theory Criticism of non-standard analysis The Martians (scientists) == Notes == == References == J. H. Ewing; F. W. Gehring (1991). Paul Halmos: Celebrating 50 Years of Mathematics. Springer-Verlag. ISBN 0-387-97509-8. OCLC 22859036. Includes a bibliography of Halmos's writings through 1991. John Ewing (October 2007). "Paul Halmos: In His Own Words" (PDF). Notices of the American Mathematical Society. 54 (9): 1136–1144. Retrieved 15 January 2008. Paul Halmos (1985). I want to be a Mathematician: An Automathography. Springer-Verlag. ISBN 0-387-96470-3. OCLC 230812318. Paul R. Halmos (1970). "How to Write Mathematics" (PDF). L'Enseignement mathématique. 16 (2): 123–152. == External links == O'Connor, John J.; Robertson, Edmund F., "Paul Halmos", MacTutor History of Mathematics Archive, University of St Andrews "Paul Halmos: A Life in Mathematics", Mathematical Association of America (MAA) Finite-Dimensional Vector Spaces "Examples of Operators" a series of video lectures on operators in Hilbert Space given by Paul Halmos during his 2-week stay in Australia, Briscoe Center Digital Collections
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Wikipedia:Paul Painlevé#0
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Paul Painlevé (French: [pɔl pɛ̃ləve]; 5 December 1863 – 29 October 1933) was a French mathematician and statesman. He served twice as Prime Minister of the Third Republic: 12 September – 13 November 1917 and 17 April – 22 November 1925. His entry into politics came in 1906 after a professorship at the Sorbonne that began in 1892. His first term as prime minister lasted only nine weeks but dealt with weighty issues, such as the Russian Revolution, the American entry into the war, the failure of the Nivelle Offensive, quelling the French Army Mutinies and relations with the British. In the 1920s as Minister of War he was a key figure in building the Maginot Line. In his second term as prime minister he dealt with the outbreak of rebellion in Syria's Jabal Druze in July 1925 which had excited public and parliamentary anxiety over the general crisis of France's empire. == Biography == === Early life === Painlevé was born in Paris. Brought up within a family of skilled artisans (his father and grandfather were lithographic draughtsmen) Painlevé showed early promise across the range of elementary studies and was initially attracted by either an engineering or political career. However, he finally entered the École Normale Supérieure in 1883 to study mathematics, receiving his doctorate in 1887 following a period of study at Göttingen, Germany with Felix Klein and Hermann Amandus Schwarz. Intending an academic career he became professor at Université de Lille, returning to Paris in 1892 to teach at the Sorbonne, École Polytechnique and later at the Collège de France and the École Normale Supérieure. He was elected a member of the Académie des Sciences in 1900. He married Marguerite Petit de Villeneuve in 1901. She died during the birth of their son Jean Painlevé in the following year. Painlevé's mathematical work on differential equations led him to encounter their application to the theory of flight and, as ever, his broad interest in engineering topics fostered an enthusiasm for the emerging field of aviation. In 1908, he became Wilbur Wright's first airplane passenger in France and in 1909 created the first university course in aeronautics. === Mathematical work === Some differential equations can be solved using elementary algebraic operations that involve the trigonometric and exponential functions (sometimes called elementary functions). Many interesting special functions arise as solutions of linear second order ordinary differential equations. Around the turn of the century, Painlevé, É. Picard, and B. Gambier showed that of the class of nonlinear second order ordinary differential equations with polynomial coefficients, those that possess a certain desirable technical property, shared by the linear equations (nowadays commonly referred to as the 'Painlevé property') can always be transformed into one of fifty canonical forms. Of these fifty equations, just six require 'new' transcendental functions for their solution. These new transcendental functions, solving the remaining six equations, are called the Painlevé transcendents, and interest in them has revived recently due to their appearance in modern geometry, integrable systems and statistical mechanics. In 1895 he gave a series of lectures at Stockholm University on differential equations, at the end stating the Painlevé conjecture about singularities of the n-body problem. In the same year he published work on the Painlevé paradox, an apparent contradiction in simple models of friction. In the 1920s, Painlevé briefly turned his attention to the new theory of gravitation, general relativity, which had recently been introduced by Albert Einstein. In 1921, Painlevé proposed the Gullstrand–Painlevé coordinates for the Schwarzschild metric. The modification in the coordinate system was the first to reveal clearly that the Schwarzschild radius is a mere coordinate singularity (with however, profound global significance: it represents the event horizon of a black hole). This essential point was not generally appreciated by physicists until around 1963. In his diary, Harry Graf Kessler recorded that during a later visit to Berlin, Painlevé discussed pacifist international politics with Einstein, but there is no reference to discussions concerning the significance of the Schwarzschild radius. === Early political career === Between 1915 and 1917, Painlevé served as French Minister for Public Instruction and Inventions. In December 1915, he requested a scientific exchange agreement between France and Britain, resulting in Anglo-French collaboration that ultimately led to the parallel development by Paul Langevin in France and Robert Boyle in Britain of the first active sonar. He also established the Directorate of Inventions for National Defense, the predecessor of the French National Centre for Scientific Research. === First period as French Prime Minister === Painlevé took his aviation interests, along with those in naval and military matters, with him when he became, in 1906, Deputy for Paris's 5th arrondissement, the so-called Latin Quarter. By 1910, he had vacated his academic posts and World War I led to his active participation in military committees, joining Aristide Briand's cabinet in 1915 as Minister for Public Instruction and Inventions. On his appointment as War Minister in March 1917 he was immediately called upon to give his approval, albeit with some misgivings, to Robert Georges Nivelle's wildly optimistic plans for a breakthrough offensive in Champagne. Painlevé reacted to the disastrous public failure of the plan by dismissing Nivelle and controversially replacing him with Henri Philippe Pétain. He was also responsible for isolating the Russian Expeditionary Force in France in the La Courtine camp, located in a remote spot on the plateau of Millevaches. On 7 September 1917, Prime Minister Alexandre Ribot lost the support of the Socialists and Painlevé was called upon to form a new government. Painlevé was a leading voice at the Rapallo conference that led to the establishment of the Supreme Allied Council, a consultative body of Allied powers that anticipated the unified Allied command finally established in the following year. He appointed Ferdinand Foch as French representative knowing that he was the natural Allied commander. On Painlevé's return to Paris he was defeated and resigned on 13 November 1917 to be succeeded by Georges Clemenceau. Foch was finally named Allied generalissimo in March 1918, eventually becoming commander-in-chief of all Allied armies on the Western and Italian fronts. === Second period as French Prime Minister === Painlevé then played little active role in politics until the election of November 1919 when he emerged as a leftist critic of the right-wing Bloc National. By the time the next election approached in May 1924 his collaboration with Édouard Herriot, a fellow member of Briand's 1915 cabinet, had led to the formation of the Cartel des Gauches. Winning the election, Herriot became Prime Minister in June, while Painlevé became President of the Chamber of Deputies. Though Painlevé ran for President of France in 1924 he was defeated by Gaston Doumergue. Herriot's administration publicly recognised the Soviet Union, accepted the Dawes Plan and agreed to evacuate the Ruhr. However, a financial crisis arose from the ensuing devaluation of the franc and in April 1925, Herriot fell and Painlevé became Prime Minister for a second time on 17 April. Unfortunately, he was unable to offer convincing remedies for the financial problems and was forced to resign on 21 November. === Later political career === Following Painlevé's resignation, Briand formed a new government with Painlevé as Minister for War. Though Briand was defeated by Raymond Poincaré in 1926, Painlevé continued in office. Poincaré stabilised the franc with a return to the gold standard, but ultimately acceded power to Briand. During his tenure as Minister of War, Painlevé was instrumental in the creation of the Maginot Line. This line of military fortifications along France's Eastern border was largely designed by Painlevé, yet named for André Maginot, owing to Maginot's championing of public support and funding. Painlevé remained in office as Minister for War until July 1929. From 1925 to 1933, Painlevé represented France in the International Committee on Intellectual Cooperation of the League of Nations (he replaced Henri Bergson and was himself replaced by Édouard Herriot). Though he was proposed for President of France in 1932, Painlevé withdrew before the election. He became Minister of Air later that year, making proposals for an international treaty to ban the manufacture of bomber aircraft and to establish an international air force to enforce global peace. On the fall of the government in January 1933, his political career ended. Painlevé died in Paris in October of the same year. On 4 November, after a eulogy by Prime Minister Albert Sarraut, he was interred in the Panthéon. == Honours == Painlevé was elected an International Member of the American Philosophical Society in 1918. The aircraft carrier Painlevé was named in his honour. The asteroid 953 Painleva was named in his honour. The Laboratoire Paul Painlevé (fr), a French mathematics research lab, is named in his honour. Maurice Ravel dedicated the second of his Trois Chansons to him in 1915. == Composition of governments == === Painlevé's First Government, 12 September – 16 November 1917 === Paul Painlevé – President of the Council and Minister of War Alexandre Ribot – Minister of Foreign Affairs Louis Loucheur – Minister of Armaments and War Manufacturing Théodore Steeg – Minister of the Interior Louis Lucien Klotz – Minister of Finance André Renard – Minister of Labour and Social Security Provisions Raoul Péret – Minister of Justice Charles Chaumet – Minister of Marine Charles Daniel-Vincent – Minister of Public Instruction and Fine Arts Fernand David – Minister of Agriculture Maurice Long – Minister of General Supply René Besnard – Minister of Colonies Albert Claveille – Minister of Public Works and Transport Étienne Clémentel – Minister of Commerce, Industry, Posts, and Telegraphs Louis Barthou – Minister of State Léon Bourgeois – Minister of State Paul Doumer – Minister of State Jean Dupuy – Minister of State Changes 27 September 1917 – Henry Franklin-Bouillon entered the ministry as Minister of State. 23 October 1917 – Louis Barthou succeeded Ribot as Minister of Foreign Affairs === Painlevé's Second Ministry, 17 April – 29 October 1925 === Paul Painlevé – President of the Council and Minister of War Aristide Briand – Minister of Foreign Affairs Abraham Schrameck – Minister of the Interior Joseph Caillaux – Minister of Finance Antoine Durafour – Minister of Labour, Hygiene, Welfare Work, and Social Security Provisions Théodore Steeg – Minister of Justice Émile Borel – Minister of Marine Anatole de Monzie – Minister of Public Instruction and Fine Arts. Louis Antériou – Minister of Pensions Jean Durand – Minister of Agriculture Orly André-Hesse – Minister of Colonies Pierre Laval – Minister of Public Works Charles Chaumet – Minister of Commerce and Industry Changes 11 October 1925 – Anatole de Monzie succeeded Steeg as Minister of Justice. Yvon Delbos succeeded Monzie as Minister of Public Instruction and Fine Arts. === Painlevé's Third Ministry, 29 October – 28 November 1925 === Paul Painlevé – President of the Council and Minister of Finance Aristide Briand – Minister of Foreign Affairs Édouard Daladier – Minister of War Abraham Schrameck – Minister of the Interior Georges Bonnet – Minister of Budget Antoine Durafour – Minister of Labour, Hygiene, Welfare Work, and Social Security Provisions Camille Chautemps – Minister of Justice Émile Borel – Minister of Marine Yvon Delbos – Minister of Public Instruction and Fine Arts Louis Antériou – Minister of Pensions Jean Durand – Minister of Agriculture Léon Perrier – Minister of Colonies Anatole de Monazie – Minister of Public Works Charles Daniel-Vincent – Minister of Commerce and Industry == Works == Sur les lignes singulières des fonctions analytiques - 1887/On singular lines of analytic functions. Mémoire sur les équations différentielles du premier ordre - 1892/Memory on first order differential equations. Leçons sur la théorie analytique des équations différentielles, A. Hermann (Paris), 1897/A course on analytic theory of differential equations. Leçons sur les fonctions de variables réelles et les développements en séries de polynômes - 1905/A course on real variable functions and polynomial development series. Cours de mécanique et machines (Paris), 1907/A course on mechanics and machines. Cours de mécanique et machines 2 (Paris), 1908/A course on mechanics and machines 2. Leçons sur les fonctions définies par les équations différentielles du premier ordre, Gauthier-Villars (Paris), 1908/A course on functions defined by first order differential equations. L'aéroplane, Lille, 1909/Aeroplane. Cours de mécanique et machines (Paris), 1909/A course on mechanics and machines. L'aviation, Paris, Felix Alcan, 1910/Aviation. Les axiomes de la mécanique, examen critique; Note sur la propagation de la lumière - 1922/Mechanics axioms, a critical study; Notes on light spread. Leçons sur la théorie analytique des équations différentielles, Hermann, Paris, 1897/A course on analytical theory of differential equations. Trois mémoires de Painlevé sur la relativité (1921-1922)/Painlevé's three memories on relativity. == See also == List of people on the cover of Time Magazine: 1920s == References == == Further reading == Dutton, David (1981). "Paul Painlevé and the end of the sacred union in Wartime France". Journal of Strategic Studies. 4 (1): 46–59. doi:10.1080/01402398108437065. Greenhalgh, Elizabeth (2011). "Paul Painlevé and Franco-British Relations in 1917". Contemporary British History. 25 (1): 5–27. doi:10.1080/13619462.2011.546094. S2CID 144569843. == External links == Paul Painlevé at the Mathematics Genealogy Project Biography (French) Newspaper clippings about Paul Painlevé in the 20th Century Press Archives of the ZBW
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Wikipedia:Paul Sophus Epstein#0
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Paul Sophus Epstein (Russian: Павел Зигмундович Эпштейн, romanized: Pavel Zigmundovich Epshteyn; March 20, 1883 – February 8, 1966) was a Russian-American mathematical physicist. He was known for his contributions to fluid dynamics and to the development of quantum mechanics. == Early life and studies == Paul Epstein's parents, Siegmund Simon Epstein and Sarah Sophia (Lurie) Epstein were of a middle class Jewish family. He said that his mother recognized his potential at the age of four years and predicted that he would be a mathematician. He went to the Hochschule in Minsk, and from 1901 to 1905 studied mathematics and physics at the Imperial University of Moscow under Pyotr Nikolaevich Lebedev. In 1909 he graduated, and became a Privatdozent at the University of Moscow. In 1910 he went to Munich, Germany, to do research under Arnold Sommerfeld, who was his advisor, and Epstein was granted a Ph.D. on a problem in the theory of diffraction of electromagnetic waves. from the Technische Universität München, in 1914. == Career == At the outbreak of World War I he was in Munich, and considered an enemy alien. Thanks to Sommerfeld's intervention he was allowed to stay in Munich as a private citizen, and could continue with his research. By that time Epstein became interested in the quantum theory of atomic structure. In 1916, he published a seminal paper explaining the Stark effect using the Bohr–Sommerfeld quantization from old quantum theory. After the war he went to Zurich, where he remained for two years. He moved to Leiden in 1921, to become Hendrik Lorentz' and Paul Ehrenfest's assistant. Shortly afterwards he was recruited by Robert Millikan to come to the California Institute of Technology (Caltech), where he remained for the rest of his career. In 1930, he was elected a member of the National Academy of Sciences. After World War II, concerned that some young American intellectuals were becoming attracted to communism, he joined the American Committee for Cultural Freedom and in 1951 he served as one of the three US delegates to the seminar the Congress held in Strasbourg. He retired as emeritus professor at Caltech in 1953 and continued to serve as a consultant for a number of industrial companies. The Epstein Memorial Fund was established through donations from more than fifty of his former students and a bronze bust of Epstein stands in the Physics and Mathematics section of the Millikan library at Caltech. == Research == === Quantum mechanics === In 1922 he published 3 papers, on the explanations of the Stark effect, Zeeman effect and magnetic dipoles using Bohr's quantum theory. === Air and fluid drag === As well as quantum theory, Epstein also published papers in other fields. He worked on were the settling of gases in the atmosphere, the theory of vibrations of shells and plates and the absorption of sound in fogs and suspensions. Epstein calculated in 1924 the drag on a sphere moving in a gas in the rarefied (i.e., high, or at least not small, Knudsen number) flow regime. This calculation was an important ingredient in improved measurements of the charge of the electron in the Millikan's oil drop experiment, as cited and described as "masterly" by Millikan himself in his review paper of 1930. Epstein's important result is now known as Epstein drag in honor of his work. === Continental drift === Epstein calculated the effect of Polflucht in 1920 as a possible causing force of Alfred Wegener's continental drift. His value was about 10−6 of the gravitational force. Some years later other geophysicists could show that the Polflucht force is far too weak to cause plate tectonics. The toughness of the sublayers of the Earth's crust is much more stronger than assumed by Epstein. == Other interests == Epstein had other interests as well. He was very interested in Freudian psychology and was one of the founding members of a Psychoanalytic Study Group (together with Thomas Libbin) that later merged into the Los Angeles Institute for Psychoanalysis. In the 1930s he published two articles in the monthly literary and scientific magazine Reflex - "The Frontiers of Science" and "Uses and Abuses of Nationalism". Although he was not an active Zionist, Epstein served as a member of the American National Society of Friends of the Hebrew University and was very friendly with the prominent Israeli mathematician Abraham Fraenkel. == Personal life == In 1930, Epstein married Alice Emelie Ryckman, and the couple had a daughter Sari (Mittelbach). == See also == Stark effect == Notes == == External links == Jesse W.M. DuMond, Paul Sophus Epstein, Biographical Memoirs (1974), National Academy of Sciences (NAS) Biographical Memoirs National Academy of Sciences zu Epstein von Jesse DuMond, 1974 Caltech Archives Oral Histories Online Interview with Paul S. Epstein Oral history interview transcript with Paul Sophus Epstein on 25 May 1962, American Institute of Physics, Niels Bohr Library & Archives - Session I Oral history interview transcript with Paul Sophus Epstein on 26 May 1962, American Institute of Physics, Niels Bohr Library & Archives - Session II Oral history interview transcript with Paul Sophus Epstein on 2 June 1962, American Institute of Physics, Niels Bohr Library & Archives - Session III
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Wikipedia:Paul Tseng#0
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Paul Tseng (Chinese: 曾匀) was a Taiwanese-born American-Canadian applied mathematician and a professor at the Department of Mathematics at the University of Washington, in Seattle, Washington. Tseng was recognized by his peers to be one of the leading optimization researchers of his generation. On August 13, 2009, Paul Tseng went missing while kayaking in the Jinsha River in the Yunnan province of China and is presumed dead. == Biography == Tseng was born September 21, 1959, in Hsinchu, Taiwan. In December 1970, Tseng's family moved to Vancouver, British Columbia. Tseng received his B.Sc. from Queen's University in 1981 and his Ph.D. from Massachusetts Institute of Technology in 1986. In 1990 Tseng moved to the University of Washington's Department of Mathematics. Tseng has conducted research primarily in continuous optimization and secondarily in discrete optimization and distributed computation. == Research == Tseng made many contributions to mathematical optimization, publishing many articles and helping to develop quality software that has been widely used. He published over 120 papers in optimization and had close collaborations with several colleagues, including Dimitri Bertsekas and Zhi-Quan Tom Luo. Tseng's research subjects include: Efficient algorithms for structured convex programs and network flow problems, Complexity analysis of interior point methods for linear programming, Parallel and distributed computing, Error bounds and convergence analysis of iterative algorithms for optimization problems and variational inequalities, Interior point methods and semidefinite relaxations for hard quadratic and matrix optimization problems, and Applications of large scale optimization techniques in signal processing and machine learning. In his research, Tseng gave a new proof for the sharpest complexity result for path-following interior-point methods for linear programming. Furthermore, together with Tom Luo, he resolved a long-standing open question on the convergence of matrix splitting algorithms for linear complementarity problems and affine variational inequalities. Tseng was the first to establish the convergence of the affine scaling algorithm for linear programming in the presence of degeneracy. Tseng has coauthored (with his Ph.D. advisor, Dimitri Bertsekas) a publicly available network optimization program, called RELAX, which has been widely used in industry and academia for research purposes. This software has been used by statisticians like Paul R. Rosenbaum and Donald Rubin in their work on propensity score matching. Tseng's software for matching has similarly been used in nonparametric statistics to implement exact tests. Tseng has also developed a program called ERELAXG, for network optimization problems with gains. In 2010 conferences in his honor were held at the University of Washington and at Fudan University in Shanghai. Tseng's personal web page can be accessed in the exact state it was at the time of his disappearance, and contains many of his writings. == Travels and disappearance == Paul Tseng was an ardent bicyclist, kayaker and backpacker. He took many adventurous trips, including kayak tours along the Mekong, the Danube, the Nile and the Amazon. On August 13, 2009, Paul Tseng went missing while kayaking in the Yantze river near Lijiang, in Yunnan province of China and is now presumed dead. == See also == == Notes == == External links == Math Programming Society Publications from DBLP. Publications from Google Scholar.
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Wikipedia:Paul Ver Eecke#0
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Paul-Louis Ver Eecke (23 February 1867 – 14 October 1959) was a Belgian mining engineer and historian of Greek mathematics. He produced influential French translations of the mathematical works of ancient Greece, including those of Archimedes, Pappus, and Theodosius. Ver Eecke was born in Menen where he received an early education in Greek and Latin. He completed his secondary studies at the Royal Athenaeum in Bruges before going to study at the mining school at Liege (1888-1891). He then worked in the mining industry. While serving as an engineer for the Fortis Powder Company Ltd at Herentals, Antwerp, he was nearly killed in an explosion. His family forced him to move out of such dangerous work and he then joined the Labor Administration in 1894, then a newly created department. He became a principal inspector but during World War I, he was forced to take leave and he studied Greek mathematical works. This would later become his most influential work and included translations into French, incorporating modern mathematical notation, of the works of Apollonius of Perga (1924), Diophantus (1926), Theodosius (1927), Serenus of Antinoe (1929), Pappus of Alexandria (1933), Euclid (1938); and the works of Didymus, Diophanes, Anthemius, and [the palimpsests of] Bobbio (1940). He became an inspector general of labor in 1922 and retired in 1923. Although worked largely in isolation, he collaborated with Johan Ludvig Heiberg. For his contribution, Eecke was made Commander of the Order of Leopold and of the Order of the Crown, and later Grand Officer of the Order of Leopold II, Officer of the Order of Orange-Nassau by the Belgian government. He was also decorated by the Greek and French governments. == References == == External links == Biography in French (with portrait and list of publications)
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Wikipedia:Paul Vincensini#0
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Paul Félix Vincensini (30 April 1896, in Bastia – 9 August 1978, in La Ciotat) was a French mathematician. In 1927, he wrote his dissertation Sur trois types de congruences rectilignes at the University of Toulouse. In 1945, working as a Professor at the University of Besançon, he was awarded the Charles Dupin Prize of the French Academy of Sciences, for his work in higher geometry. In 1949, he got the Prix de la Pensée Française. In the same year, he went to Marseille University. He retired in 1967, but still accepted presidency of a symposium of the Florence Institute for Pure and Applied Mathematical Sciences in 1978. == Selected works == Paul Félix Vincensini (Sep 1950). "Sur certains réseaux tracés sur une surface et leur rôle en géométrie différentielle". In Lawrence M. Graves and Paul A. Smith and Einar Hille and Oscar Zariski (ed.). Proc. International Congress of Mathematicians (PDF). American Mathematical Society. p. 509. Paul Félix Vincensini (1957). "Vue d'ensemble sur l'œuvre géométrique de Luigi Bianchi". Rendiconti del Seminario Matematico. 16: 115–157. Paul Félix Vincensini (Nov 1958). "Sur une représentation dans E4 des congruences W à nappes focales réglées de E3". Archiv der Mathematik 1. XI. 9 (4): 360–365. Vincensini, P. (1972). "La géométrie différentielle au XIXe siècle". Scientia. 101: 617–696. == References ==
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Wikipedia:Paul Zimmermann (mathematician)#0
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Paul Zimmermann (born 13 November 1964) is a French computational mathematician, working at INRIA. == Education == After engineering studies at École Polytechnique 1984 to 1987, he got a master's degree in computer science in 1988 from University Paris VII and a magister from École Normale Supérieure in mathematics and computer science. His doctoral degree from École Polytechnique in 1991 was entitled Séries génératrices et analyse automatique d’algorithmes, and advised by Philippe Flajolet. == Research == His interests include asymptotically fast arithmetic. He has developed some of the fastest available code for manipulating polynomials over GF(2), and for calculating hypergeometric constants to billions of decimal places. He is associated with the CARAMEL project to develop efficient arithmetic, in a general context and in particular in the context of algebraic curves of small genus; arithmetic on polynomials of very large degree turns out to be useful in algorithms for point-counting on such curves. He is also interested in computational number theory. In particular, he has contributed to some of the record computations in integer factorisation and discrete logarithm. Zimmermann co-authored the book Computational Mathematics, published in 2018 on SageMath used by Mathematical students worldwide. In 2010, he co-authored a book on algorithms for computer arithmetic with Richard Brent. He has been an active developer of the GMP-ECM implementation of the elliptic curve method for integer factorisation and of MPFR, an arbitrary precision floating point library with correct rounding. He is also a coauthor of the CADO-NFS software tool, which was used to factor RSA-240 in record time. In a 2014 blog post, Zimmermann said that he would refuse invitations to review papers submitted to gold (author-pays) open access and hybrid open access journals, because he disagrees with the publication mechanism. == References == Flajolet, Philippe; Zimmerman, Paul; Van Cutsem, Bernard (1994). "A calculus for the random generation of labelled combinatorial structures". Theoretical Computer Science. 132 (1): 1–35. doi:10.1016/0304-3975(94)90226-7. MR 1290534. == External links == http://www.loria.fr/~zimmerma/
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Wikipedia:Paul de Casteljau#0
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Paul de Casteljau (19 November 1930 – 24 March 2022) was a French physicist and mathematician. In 1959, while working at Citroën, he developed an algorithm for evaluating calculations on a certain family of curves, which would later be formalized and popularized by engineer Pierre Bézier, leading to the curves widely known as Bézier curves. He studied at École Normale Supérieure, and worked at Citroën from 1958 until his retirement in 1992. When he arrived there, "Specialists admitted that all electrical, electronic and mechanical problems had more or less been solved. All—except for one single formality which made up for 5%, but certainly not for 20% of the problem; in other words, how to express component parts by equations." A short autobiographic sketch goes back to the early 1990s, a longer autobiography talks about his education and life at Citroën until his retirement. He continued publishing in retirement, which led to three monographs and ten academic papers, most of his publications written in French. == De Casteljau curves == De Casteljau's algorithm is widely used, with some modifications, as it is the most robust and numerically stable method for evaluating polynomials. Other methods, such as Horner's method and forward differencing, are faster for calculating single points but are less robust. De Casteljau's algorithm is still very fast for subdividing a De Casteljau curve or Bézier curve into two curve segments at an arbitrary parametric location. == Further contributions == Noteworthy are his contributions beyond geometric modeling, which only became known internationally posthumously a generalization of the Euclidean algorithm to several variables, with numerous applications in number theory a generalization of the golden ratio for use with regular polygons and thus for solving polynomial equations a generalization of the 9-point circle to the 14-point strophoid a representation of the relativistic Lorentz transformation as a quaternion a view of geometric optics that complements the Abbe sine condition == Awards == Paul de Casteljau received the 1987 Seymour Cray Prize from the French National Center for Scientific Research, the 1993 John Gregory Memorial Award, and the 2012 Bézier Award from the Solid Modeling Association (SMA). The SMA's announcement highlights de Casteljau's eponymous algorithm: Paul de Castlejau's contributions are less widely known than should be the case because he was not able to publish them until equivalent ideas had been reinvented independently by others, sometimes in a rather different form but now recognisably related. Because he was not permitted to publish his early work, we now call polynomials with a Bernstein basis "Bézier polynomials", although Bézier himself did not use control points but their first difference vectors as the coefficients. We also call the multilinear polynomials "blossoming", following Lyle Ramshaw who in turn credited de Casteljau with the underlying "polar approach" to the mathematical theory of splines. We do call the algorithm for the stable evaluation of the Bernstein-Bézier form for polynomials "de Casteljau algorithm" although it is Carl de Boor's more general result applying it to B-splines which is now widely used in CAD/CAM systems. The SMA also quotes Pierre Bézier on de Casteljau's contributions: There is no doubt that Citroën was the first company in France that paid attention to CAD, as early as 1958. Paul de Casteljau, a highly gifted mathematician, devised a system based on the use of Bernstein polynomials. ... the system devised by de Casteljau was oriented towards translating already existing shapes into patches, defined in terms of numerical data. ... Due to Citroën's policy, the results obtained by de Casteljau were not published until 1974, and this excellent mathematician was deprived of part of the well deserved fame that his discoveries and inventions should have earned him. == Publications == (in French) Paul De Casteljau, Outillage Méthodes Calcul, INPI Enveloppe Soleau No. 40.040, 1959, Citroen Internal Document P2108 (in French) Paul De Casteljau, Courbes et Surfaces à Pôles, 1963, Citroen Internal Document P_4147 (in French) Mathématiques et CAO. Vol. 2 : Formes à pôles, Hermes, 1986 Shape Mathematics and CAD, KoganPage, London 1986 (in French) Les quaternions: Hermès, 1987, ISBN 978-2866011031 (in French) Le Lissage: Hermès, 1990 POLynomials, POLar Forms, and InterPOLation, September 1992, In Lychee / Schumaker: Mathematical methods in computer aided geometric design II, Addison-Wesley 1992, pp.57-68 Polar Forms as Curve and Surface Modeling as used by Citroën, In: Piegl (ed.) Fundamental Developments of Computer-Aided Geometric Modeling, Academic Press, 1993 (in French) Splines Focales, In Laurent / Le Méhauté / Schumaker: Curves and Surfaces in Geometric Design, AK Peters 1994, pp.91-103 (in French) Courbes et Profils Esthétiques contre Fonctions Orthogonales (Histoire Vécue), In: Dæhlen, Lyche, Schumaker (eds.) Mathematical Methods for Curves and Surfaces, S. 73-82,1995 (in French) La Tolérance d'Usinage chez Citroën dans les Années (19)60, In: Le Méhauté, Rabut, Schumaker (eds.), Curves and Surfaces with Applications in CAGD, S. 69-76, 1997 De Faget De Casteljau, Paul (1998). "Intersection Methods of Convergence". Computing [Suppl]. 13: 77–80. doi:10.1007/978-3-7091-6444-0_7. (in French) Intersections et Convergence, In: Laurent, Sablonnière, Schumaker (eds.), Curve and Surface Design: Saint-Malo 1999 (in French) In mémoriam Henri de Faget de Casteljau: Son autre passe-temps, la géométrie à travers l'hexagone de Pascal, Procès-verbaux et Mémoires de l'Académie des Sciences, Belles Lettres et Arts de Besançon et de Franche-Comté, Band 193 (1998-1999), S. 91-114, 1999 De Faget De Casteljau, Paul (August 1999). "De Casteljau's autobiography: My time at Citroën". Computer Aided Geometric Design. 16 (7): 583–586. doi:10.1016/S0167-8396(99)00024-2. (in French) Au dela du Nombre d'Or, Revue Internationale de CFAO et d'Informatique Graphique, S. 19-31, 2001 (in French) Fantastique strophoïde rectangle, Revue Internationale de CFAO et d'Informatique Graphique, S. 357-370, 2001 == References ==
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Wikipedia:Paula Tretkoff#0
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Paula Tretkoff (née Paula Beazley Cohen) is an Australian-American mathematician who studies number theory, noncommutative geometry, and hypergeometric functions. She is a professor of mathematics at Texas A&M University, and a director of research at the Centre national de la recherche scientifique (CNRS) associated with the University of Lille. == Education and career == Tretkoff was born in Sydney, Australia, but is a US citizen. She studied mathematics at the University of Sydney, earning first class honours in applied mathematics in 1978 and in pure mathematics in 1979. She completed her Ph.D. in 1985 at the University of Nottingham, in England. Her dissertation, Height Problems and Modular Forms, was supervised by David Masser. She completed a habilitation in 1995 at Pierre and Marie Curie University. Tretkoff joined CNRS as a researcher in 1983, associated with Pierre and Marie Curie University. She moved to Lille, and became a director of research, in 1995. In 2002 she took up her present position as a professor at Texas A&M, while retaining her position at CNRS. == Books == Tretkoff is the author of two books: Complex Ball Quotients and Line Arrangements in the Projective Plane, Mathematical Notes, vol. 51, Princeton University Press, 2016, ISBN 978-0-691-14477-1 Periods and Special Functions in Transcendence, Advanced Textbooks in Mathematics, World Scientific Publishing, 2017, ISBN 978-1-78634-294-2 == References == == External links == Home page
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Wikipedia:Paulette Libermann#0
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Paulette Libermann (14 November 1919 – 10 July 2007) was a French mathematician, specializing in differential geometry. == Early life and education == Libermann was one of three sisters born to a family of Russian-Ukrainian Yiddish-speaking Jewish immigrants to Paris. After attending the Lycée Lamartine, she began her university studies in 1938 at the École normale supérieure de jeunes filles, a college in Sèvres for training women to become school teachers. Due to the reforms of the new director Eugénie Cotton, who wanted her school to be at the same level of École Normale Supérieure, Libermann was taught by leading mathematicians including Élie Cartan, Jacqueline Ferrand and André Lichnerowicz. Two years later, upon completion of her studies, she was prevented from taking the agrégation and becoming a teacher because of the anti-Jewish laws instituted by the German occupation. However, thanks to a scholarship provided by Cotton, she began doing research under Cartan's supervision. In 1942, she and her family escaped Paris for Lyon, where they hid from the persecutions by Klaus Barbie for two years. After the liberation of Paris in 1944, she returned to Sèvres and completed her studies, obtaining the agrégation. == Career == Libermann taught briefly in a school at Douai, and then got a scholarship to study at Oxford University between 1945 and 1947, where she obtained a bachelor's degree under the supervision of J. H. C. Whitehead. From 1947 to 1951 she hold a teaching position at a school for girls in Strasbourg. However, at the encouragement of Élie Cartan, during this period she also continued her research at Université Louis Pasteur. In 1951 she left teaching for a research position at the Centre national de la recherche scientifique, and in 1953 she completed her doctoral thesis, entitled Sur le problème d’équivalence de certaines structures infinitésimales [On the equivalence problem of certain infinitesimal structures], under the supervision of Charles Ehresmann. After her PhD, Libermann was appointed assistant professor at the University of Rennes in 1954 and full professor at the same university in 1958. In 1966 she moved to the University of Paris, and when the university split in 1968, she joined Paris Diderot University, from which she retired in 1986. == Research == Libermann's research involved many different aspects of differential geometry and global analysis. In particular, she worked on G-structures and Cartan's equivalence method, Lie groupoids and Lie pseudogroups, higher-order connections, and contact geometry. In 1987 she wrote together with Charles-Michel Marle one of the first textbooks on symplectic geometry and analytical mechanics. == Selected publications == == References ==
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Wikipedia:Pauline van den Driessche#0
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Pauline van den Driessche (born 1941) is a British and Canadian applied mathematician who is a professor emerita in the department of mathematics and statistics at the University of Victoria, where she has also held an affiliation in the department of computer science. Her research interests include mathematical biology, matrix analysis, and stability theory. == Education and career == Van den Driessche earned bachelor's and master's degrees in 1961 and 1963 respectively from Imperial College London. She completed her doctorate in 1964 from the University College of Wales; her dissertation concerned fluid mechanics. She stayed on for a year in Wales as an assistant lecturer; she was hired as an assistant professor at the University of Victoria in 1965, and retired in 2006. == Contributions == In mathematical biology, van den Driessche's contributions include important work on delay differential equations and on Hopf bifurcations, and the effects of changing population size and immigration on epidemics. She has also done more fundamental research in linear algebra, motivated by applications in mathematical biology. Her work in this area includes pioneering contributions to the theory of combinatorial matrix theory in which she proved connections between the sign pattern of a matrix and its stability, as well as results on matrix decomposition. == Awards and honors == In 2005, the journal Linear Algebra and its Applications published a special issue in her honor. She was the 2007 winner of the Krieger–Nelson Prize of the Canadian Mathematical Society, and in the same year became the inaugural Olga Taussky-Todd Lecturer, an award given every four years at the International Congress on Industrial and Applied Mathematics by the International Council for Industrial and Applied Mathematics and Association for Women in Mathematics. In 2013 she became a fellow of the Society for Industrial and Applied Mathematics "for contributions to linear algebra and mathematical biology". She received the CAIMS Research Prize from the Canadian Applied and Industrial Mathematics Society in 2019, and the 2022 Hans Schneider Prize in Linear Algebra. == References ==
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Wikipedia:Paulius Slavėnas#0
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Paulius Slavėnas (21 July 1901 – 24 February 1991) was a Lithuanian astronomer, mathematician, and science historian who headed the Vilnius University Astronomical Observatory twice, from 1944 to 1952, and from 1956 to 1969. == Biography == Paulius Slavėnas was born on 21 July 1901 in Moscow, the Russian Empire. His father was Vincas Slavėnas (1874–1939), and his mother, Liubovė Kvalkova-Burštein was an accoucher. In 1918, he graduated from the Grigory Shelaputin gymnasium and entered the faculty of physics and mathematics of the Moscow State University. From 1921 to 1923, Slavėnas served in the Red Army. His father returned to Lithuania in 1922 and settled in Marijampolė. After returning to Lithuania himself, Slavėnas continued his studies at the University of Lithuania, from which he graduated in 1925. After receiving a stipend from the Rockefeller Foundation, Slavėnas traveled to the United States. He graduated from Yale University with a doctoral degree in 1928. His doctoral advisor was Ernest William Brown, and his dissertation was on the three-body problem. During his study years in the country, Slavėnas visited (and sometimes worked at) many observatories, read science popularization lectures, communicated with Lithuanian-Americans, and published scientific press. He received attention from various specialists, and Slavėnas became a member of the International Astronomical Union, the American Astronomical Society, the Royal Astronomical Society, the Société astronomique de France, and the Astronomische Gesellschaft. From 1929 to 1930, Slavėnas served in the Lithuanian Army. From 1930 to 1931 he was a lecturer and professor at Vytautas Magnus University. During the German occupation of Lithuania, he was fired by the authorities for his participation in establishing the Union of Cultural Cooperation between Greater Lithuania and Lithuania Minor in 1933–1936. Additionally, he was the union's chairman from 1936 to 1939. From 1944 Slavėnas became a professor of Vilnius University. In 1944–1952, and later in 1956–1969 Slavėnas headed the Vilnius University Astronomical Observatory and the university's astronomy department. Slavėnas also continued to lecture at Vytautas Magnus University from 1944 to 1952. In 1949 he was elected as a corresponding member of the Lithuanian Academy of Sciences of the Lithuanian SSR, and was made a real member in 1968. From 1949 to 1953 Slavėnas was the scientific secretary of the academy's presidium. From 1950 to 1987, Slavėnas headed the Society of Lithuanian Science Historians and Philosophers. From 1950 to 1987 Slavėnas was chairman of the Commission for the History of Natural Sciences and Technology at the presidium of the academy. From 1954 to 1971 Slavėnas was a member of the Council of Astronomers of the Academy of Sciences of the Soviet Union. In 1959 he founded a group of astronomers at the academy's institute of physics and mathematics. He headed the work of Lithuanian historians of science. From 1960 to 1968, he was the editor of the Bulletin of the Vilnius Astronomical Observatory. From 1966, Slavėnas was a member correspondent of the International Academy of the History of Science. From 1959 to 1971, Slavėnas was deputy chief of Soviet encyclopedia of Lithuania Minor. From 1960 onwards, under his care, the Commission for the History of Nature and Technology began to publish "From the History of Sciences in Lithuania". The most important areas of Slavėnas's scientific research were the structure and evolution of the universe and stars, astrophotometric research, the basics of the theory of relativity, history and systematics of science, and the scientific worldview. From 1926 to 1990, Slavėnas published 15 books. He published more than 870 scientific and science popularization articles. For his active scientific work he was awarded the title of merited scientific figure of the Lithuanian SSR. Furthermore, Slavėnas was awarded several Soviet orders and medals letters of honor. Slavėnas died on 24 February 1991 in Vilnius. A memorial plaque was uncovered at his former home in 1996. == References ==
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Wikipedia:Paulo Pinheiro#0
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Paulo Pinheiro is a Brazilian American computer scientist working in the areas of provenance and semantic web in support of sciences. Pinheiro has been a research scientist at the Rensselaer Polytechnic Institute's Tetherless World Constellation since 2013. Between 2011 and 2013, he was a staff scientist at the U.S. Department of Energy's Pacific Northwest National Laboratory. Between 2006 and 2012, he was an associate professor of computer science at the University of Texas at El Paso. Pinheiro is from a long line of scientists and engineers: his father is a retired professor of material sciences at the Universidade Federal de Minas Gerais; his paternal grandfather was the founding Mine Superintendent of Vale S.A., the second largest mining company in the world and Israel Pinheiro da Silva, a great-granduncle, was the chief engineer responsible for the construction of Brasília, the capital of Brazil. == Education == Pinheiro received a licenciatura in mathematics and a master's degree in computer science from the Universidade Federal de Minas Gerais, Brazil, and a Ph.D. in computer science from University of Manchester, United Kingdom, in 2002. Between 2002 and 2005, he was a postdoctoral fellow in the Knowledge Systems Laboratory at Stanford University, United States, and a Sistemas de Informação's faculty member at Faculdades INESC/Unai, Brazil. == Research == Pinheiro's research focuses on innovative ways of using semantically enable resources such as ontologies, abstract process specifications, and distributed provenance in support of trust and uncertainty management for sciences. Pinheiro is the author of the Unified Modeling Language for Interactive Systems (UMLi) developed as part of his PhD work at the Information Management Group at the University of Manchester. Pinheiro is a co-author of the Provenance Markup Language (PML) originally developed at Stanford's Knowledge Systems Laboratory. == References == == External links == Personal web page Publications
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Wikipedia:Paulus Gerdes#0
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Paulus Pierre Joseph Gerdes (November 11, 1952 – November 10, 2014) was a Dutch mathematician and university professor, who was one of the pioneers in the field of ethnomathematics research, particularly in Africa. == Education and career == Gerdes was a student of mathematics and physics as an undergraduate at Radboud University Nijmegen. After visiting Vietnam he returned to Radboud for a second bachelor's degree, in cultural anthropology in 1974, and a master's degree in mathematics in 1975. He moved to Mozambique in 1976, started working at Eduardo Mondlane University in 1977, and remained in the country for the rest of his life, with positions in the Ministry of Education and Culture and as dean and later rector at the Instituto Superior Pedagógico (now Pedagogical University of Maputo). In 1986, he completed a doctorate (Dr. rer. nat.) through the Pädagogische Hochschule „Karl Friedrich Wilhelm Wander“ Dresden in Germany, with the dissertation Zum erwachenden geometrischen Denken: Tätigkeit und die mögliche Herkunft einiger früher geometrischer Begriffe und Relationen, unter besonderer Berücksichtigung der Mathematik der Entwicklungsländer [On the awakening of geometric thinking: activity and the possible origin of some early geometric concepts and relations, with special reference to the mathematics of the developing countries] supervised by Heiner Meyer. He also headed the founding commission of Lúrio University, and chaired the Commission on the History of Mathematics in Africa of the African Mathematical Union. He served as the secretary of the Southern African Mathematical Sciences Association, was vice president of the African Academy of Sciences, and was president of the International Association for Science and Cultural Diversity and of the International Group of Ethno-mathematical Studies. == Recognition == Gerdes was elected to the African Academy of Sciences in 2001, and as a corresponding member of the International Academy of the History of Science in 2005. A special issue of the Journal of Mathematics and Culture was published in his memory in 2021. == Books == Gerdes's books include: Ethnogeometrie. Kulturanthropologische Beiträge zur Genese und Didaktik der Geometrie [Ethnogeometry. Contributions to the genesis and didactics of geometry from the point of view of cultural anthropology] (in German; Franzbecker, 1990). Revised and translated into English as Awakening of Geometrical Thought in Early Culture (MEP Publications, 2003) L'ethnomathématique comme nouveau domaine de recherche en. Quelques réflexions et expériences du Mozambique [Ethnomathematics as a new field of research in Africa. Some reflections about and experiences in Mozambique] (in French; Inst. Sup. Ped. Mozambique, 1993) African Pythagoras: A study in culture and mathematics education (originally in Portuguese, 1992; Inst. Sup. Ped. Mozambique, 1994; color ed., Mozambican Ethnomathematics Research Centre, 2011) Sipatsi: Technology, Art and Geometry in Inhambane (with Gildo Bufalo, Inst. Sup. Ped. Mozambique 1994) Sona Geometry: Reflections on the tradition of sand drawings in Africa south of the Equator (Inst. Sup. Ped. Mozambique 1994) Ethnomathematics and Education in Africa (Univ. Pedagógica, Gabinet do Reitor, Stockholm, 1995) Women and Geometry in Southern Africa: Some suggestions for further research (Univ. Ped. Maputo 1995); translated as Femmes et géométrie en Afrique Australe (L'Harmattan, 1996) Une tradition géométrique en Afrique—les dessins sur le sable [A geometric tradition in Africa—sand drawings], Vols. I–III (in French; L’Harmattan, 1995) Ethnomathematik — dargestellt am Beispiel der Sona Geometrie [Ethnomathematics — described by the example of Sona geometry] (in German, Spektrum, 1997) Lusona: récréations géométriques d'Afrique [Lusona: geometrical recreations of Africa] (1997) Geometry from Africa: Mathematical and Educational Explorations (Mathematical Association of America, 1999) Women, Art and Geometry in Southern Africa (Africa World Press, 1999) Mathematics in African History and Cultures: An annotated bibliography (with Ahmed Djebbar, African Mathematical Union, 2004; 2nd ed., 2007) Sona Geometry from Angola: Mathematics of an African tradition (Polimetrica International Scientific Publisher, 2006) African Doctorates in Mathematics: A Catalogue (African Mathematical Union, 2007) Otthava: Fazer Cestos E Geometria Na Cultura Makhuwa Do Nordeste de Moçambique (self-published, 2007; translated as Otthava: making baskets and doing geometry in the Makhuwa culture in the Northeast of Mozambique, 2011) Drawings from Angola: Living Mathematics (self-published, 2007) Tinlhèlò. Interweaving art and mathematics. Colourful basket trays from the south of Mozambique (self-published, 2010) From Ethnomathematics to Art: Design Matrices and Cyclic Matrices (UNESP, 2010) == References ==
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Wikipedia:Pavel Winternitz#0
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Pavel Winternitz (July 25, 1936 – February 13, 2021) was a Czech-born Canadian mathematical physicist. He completed undergraduate studies at Prague University and received a doctorate from Leningrad University (Ph.D. 1962) under the supervision of J. A. Smorodinsky. His research is on integrable systems and symmetries. == Life and career == Winternitz was a member of the Mathematical Physics group at the Centre de recherches mathématiques (CRM), a national research centre in mathematics at the Université de Montréal and Professor in the Department of Mathematics and Statistics at Université de Montréal. His work has had a strong impact in several domains of mathematical physics, and his publications are very widely cited. In 2001, he was recipient of the CAP-CRM Prize in Theoretical and Mathematical Physics . In 2018, he was recipient of the Wigner medal. Winternitz died in Montreal on February 13, 2021, at the age of 84. == References == == External links == Centre de recherches mathématiques Pavel Winternitz at the Mathematics Genealogy Project P Winternitz publications in Google Scholar
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Wikipedia:Pavle Papić#0
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Pavle Papić (1919, Antofagasta – 2005, Zagreb) was a Croatian mathematician. Papić graduated mathematics from the University of Zagreb where in 1953 he received his doctorate degree in mathematics under the supervision of Đuro Kurepa. From 1966 until 1968 he was a dean of Faculty of Natural Sciences and Mathematics at the University of Zagreb, from 1968 until 1974 a director of Institute of Mathematics in Zagreb. Since 1977 he was a corresponding member of the Yugoslav Academy of Arts and Sciences and since 1994 a corresponding member of the Croatian Academy of Arts and Sciences. His interests were in set theory and general topology. He found necessary and sufficient conditions for metrizability and orderability of pseudometric and ultrametric spaces. == References ==
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Wikipedia:Pavol Hell#0
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Pavol Hell is a Canadian mathematician and computer scientist, born in Czechoslovakia. He is a professor of computing science at Simon Fraser University. Hell started his mathematical studies at Charles University in Prague, and moved to Canada in August 1968 after the Warsaw Pact invasion of Czechoslovakia. He obtained his MSc from McMaster University in Hamilton, under the joint supervision of Gert Sabidussi and Alex Rosa, and his PhD at the Universite de Montreal, with Gert Sabidussi. In his PhD research he pioneered, on the suggestion of Gert Sabidussi, the study of graph retracts. He describes his area of interest as "computational combinatorics", including algorithmic graph theory and complexity of graph problems. His current focus is on nicely structured graph classes, and on the complexity of various versions of graph homomorphism problems. Hell has written the book Graph and Homomorphisms with his long-term collaborator Jaroslav Nešetřil, and many highly cited papers, including "On the complexity of H-coloring" also with Nešetřil, "On the history of the minimum spanning tree problem", with Ron Graham, "On the completeness of a generalized matching problem" with David Kirkpatrick, and "List homomorphisms and circular arc graphs" with Tomas Feder and Jing Huang. He is a managing editor of the Journal of Graph Theory, and was named a fellow of the Society for Industrial and Applied Mathematics (SIAM) in 2012. == References == == External links == Pavol Hell at DBLP Bibliography Server Journal of Graph Theory
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Wikipedia:Pawel Bartoszek#0
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Paweł Bartoszek (born 25 September 1980 Poland) is a Polish-born Icelandic politician. In 2010 he was elected in the Constitutional Assembly election. In the 2016 Icelandic parliamentary election, he was elected as a Member of the Althing, representing Viðreisn. In the 2018 Icelandic municipal elections, he was elected to the Reykjavík City Council. In the 2024 Icelandic parliamentary election, he was once again elected, representing Viðreisn. He serves as chairman of the Foreign Affairs Committee since 2025. Bartoszek holds a master's degree in mathematics from the University of Iceland. == References ==
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Wikipedia:Pañcabodha#0
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Pañcabodha is the title of several different Sanskrit treatises on astronomy and mathematics composed by members of the Kerala school of astronomy and mathematics. All these works are karaṅa texts, that is, books which explain the various computations in astronomy especially with regard to those related to the preparation of Panchangam-s (calendar). They are essentially manuals of computations. The authorship of only three Pañcabodha-s have been identified. The Pañcabodha-s generally contain five sections dealing with five topics. But this is not a strict rule. There are Pañcabodha-s which treat less than five topics and which treat more than five topics. The topics treated are not always the same. However many of them treat the five advanced topics in traditional Indian astronomy, namely, Vyātīpāta (computation of the instants at which the sum of the (true) longitudes of the Sun and the Moon amounts to half a circle), Grahaṇa (computation of eclipses), Chāyā (shadow computations), Śṛṅgonnati (the computation of the elevation of the lunar horn, or the angle between the line of cusps and the horizontal plane) and Mauḍhya (computation of the instants when a planet becomes invisible due to its direction/longitude being close to that of the Sun). == The works titled Pañcabodha and their commentaries == The various Pañcabodha-s are listed below. Following K. V. Sarma, the various texts are identified by assigning them numbers written in Roman numerals. Pañcabodha I (also called Prakīrṇasaṅgraha) by anonymous author: Has a commentary in Malayalam also of anonymous authorship. The work has been published with a modern commentary in Malayalam by Kanippayyoor Sankaran Namputirippad, by Panchangam Press, Kunnamkulam, Kerala. Pañcabodha II by anonymous author: Has a Malayalam commentary, called pañcabodhakriyākrma by Mahishamangalam Sankaran Namputiri. Pañcabodha III by Puthumana Somayaji: Divided into five section dealing with Vyātīpāta, Grahaṇa, Chāyā, Śṛṅgonnati and Mauḍhya. Has an anonymous Malayalam commentary. Pañcabodha IV by anonymous author: Has five sections dealing with Vyātīpāta, Grahaṇa, Chāyā, Śṛṅgonnati and Mauḍhya in 105 cerses. This has several commentaries. Laghuvivṛtī by Nārāyaṇa (1529 CE) Pañcabodhārpadarpaṅaṃ by Mahishamangalam Sankaran Namputiri Bālasaṅkaraṃ by Mahishamangalam Sankaran Namputiri Pañcabodhakriyākrama by anonymous author Pañcabodha V by Purușotttama Pañcabodha VI by anonymous author. Has commentary in Malayalam called Pañcabodhaṃ Bhāșā of anonymous authorship. Pañcabodha VII by anonymous author. treats the topics Vyātīpāta, Grahaṇa, Chāyā, Śṛṅgonnati and Mauḍhya. Has a commentary in Malayalam. Pañcabodha VIII by anonymous author. The work is in a mixture of Sanskrit and Malayalam languages. Pañcabodha IX by anonymous author. Pañcabodha X by also calledPañcabodhagaṇitaṃ Bhāṣā. the text is in Malayalam. Pañcabodha XI by anonymous author. Has a commentary in Malayalam by Vāsuṇṇi Mūssatu of Veḷḷānaśśeri == See also == List of astronomers and mathematicians of the Kerala school == References ==
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Wikipedia:Peano existence theorem#0
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In mathematics, specifically in the study of ordinary differential equations, the Peano existence theorem, Peano theorem or Cauchy–Peano theorem, named after Giuseppe Peano and Augustin-Louis Cauchy, is a fundamental theorem which guarantees the existence of solutions to certain initial value problems. == History == Peano first published the theorem in 1886 with an incorrect proof. In 1890 he published a new correct proof using successive approximations. == Theorem == Let D {\displaystyle D} be an open subset of R × R {\displaystyle \mathbb {R} \times \mathbb {R} } with f : D → R {\displaystyle f\colon D\to \mathbb {R} } a continuous function and y ′ ( t ) = f ( t , y ( t ) ) {\displaystyle y'(t)=f\left(t,y(t)\right)} a continuous, explicit first-order differential equation defined on D, then every initial value problem y ( t 0 ) = y 0 {\displaystyle y\left(t_{0}\right)=y_{0}} for f with ( t 0 , y 0 ) ∈ D {\displaystyle (t_{0},y_{0})\in D} has a local solution z : I → R {\displaystyle z\colon I\to \mathbb {R} } where I {\displaystyle I} is a neighbourhood of t 0 {\displaystyle t_{0}} in R {\displaystyle \mathbb {R} } , such that z ′ ( t ) = f ( t , z ( t ) ) {\displaystyle z'(t)=f\left(t,z(t)\right)} for all t ∈ I {\displaystyle t\in I} . The solution need not be unique: one and the same initial value ( t 0 , y 0 ) {\displaystyle (t_{0},y_{0})} may give rise to many different solutions z {\displaystyle z} . == Proof == By replacing y {\displaystyle y} with y − y 0 {\displaystyle y-y_{0}} , t {\displaystyle t} with t − t 0 {\displaystyle t-t_{0}} , we may assume t 0 = y 0 = 0 {\displaystyle t_{0}=y_{0}=0} . As D {\displaystyle D} is open there is a rectangle R = [ − t 1 , t 1 ] × [ − y 1 , y 1 ] ⊂ D {\displaystyle R=[-t_{1},t_{1}]\times [-y_{1},y_{1}]\subset D} . Because R {\displaystyle R} is compact and f {\displaystyle f} is continuous, we have sup R | f | ≤ C < ∞ {\displaystyle \textstyle \sup _{R}|f|\leq C<\infty } and by the Stone–Weierstrass theorem there exists a sequence of Lipschitz functions f k : R → R {\displaystyle f_{k}:R\to \mathbb {R} } converging uniformly to f {\displaystyle f} in R {\displaystyle R} . Without loss of generality, we assume sup R | f k | ≤ 2 C {\displaystyle \textstyle \sup _{R}|f_{k}|\leq 2C} for all k {\displaystyle k} . We define Picard iterations y k , n : I = [ − t 2 , t 2 ] → R {\displaystyle y_{k,n}:I=[-t_{2},t_{2}]\to \mathbb {R} } as follows, where t 2 = min { t 1 , y 1 / ( 2 C ) } {\displaystyle t_{2}=\min\{t_{1},y_{1}/(2C)\}} . y k , 0 ( t ) ≡ 0 {\displaystyle y_{k,0}(t)\equiv 0} , and y k , n + 1 ( t ) = ∫ 0 t f k ( t ′ , y k , n ( t ′ ) ) d t ′ {\displaystyle \textstyle y_{k,n+1}(t)=\int _{0}^{t}f_{k}(t',y_{k,n}(t'))\,\mathrm {d} t'} . They are well-defined by induction: as | y k , n + 1 ( t ) | ≤ | ∫ 0 t | f k ( t ′ , y k , n ( t ′ ) ) | d t ′ | ≤ | t | sup R | f k | ≤ t 2 ⋅ 2 C ≤ y 1 , {\displaystyle {\begin{aligned}|y_{k,n+1}(t)|&\leq \textstyle \left|\int _{0}^{t}|f_{k}(t',y_{k,n}(t'))|\,\mathrm {d} t'\right|\\&\leq \textstyle |t|\sup _{R}|f_{k}|\\&\leq t_{2}\cdot 2C\leq y_{1},\end{aligned}}} ( t ′ , y k , n + 1 ( t ′ ) ) {\displaystyle (t',y_{k,n+1}(t'))} is within the domain of f k {\displaystyle f_{k}} . We have | y k , n + 1 ( t ) − y k , n ( t ) | ≤ | ∫ 0 t | f k ( t ′ , y k , n ( t ′ ) ) − f k ( t ′ , y k , n − 1 ( t ′ ) ) | d t ′ | ≤ L k | ∫ 0 t | y k , n ( t ′ ) − y k , n − 1 ( t ′ ) | d t ′ | , {\displaystyle {\begin{aligned}|y_{k,n+1}(t)-y_{k,n}(t)|&\leq \textstyle \left|\int _{0}^{t}|f_{k}(t',y_{k,n}(t'))-f_{k}(t',y_{k,n-1}(t'))|\,\mathrm {d} t'\right|\\&\leq \textstyle L_{k}\left|\int _{0}^{t}|y_{k,n}(t')-y_{k,n-1}(t')|\,\mathrm {d} t'\right|,\end{aligned}}} where L k {\displaystyle L_{k}} is the Lipschitz constant of f k {\displaystyle f_{k}} . Thus for maximal difference M k , n ( t ) = sup t ′ ∈ [ 0 , t ] | y k , n + 1 ( t ′ ) − y k , n ( t ′ ) | {\displaystyle \textstyle M_{k,n}(t)=\sup _{t'\in [0,t]}|y_{k,n+1}(t')-y_{k,n}(t')|} , we have a bound M k , n ( t ) ≤ L k | ∫ 0 t M k , n − 1 ( t ′ ) d t ′ | {\displaystyle \textstyle M_{k,n}(t)\leq L_{k}\left|\int _{0}^{t}M_{k,n-1}(t')\,\mathrm {d} t'\right|} , and M k , 0 ( t ) ≤ | ∫ 0 t | f k ( t ′ , 0 ) | d t ′ | ≤ | t | sup R | f k | ≤ 2 C | t | . {\displaystyle {\begin{aligned}M_{k,0}(t)&\leq \textstyle \left|\int _{0}^{t}|f_{k}(t',0)|\,\mathrm {d} t'\right|\\&\leq |t|\textstyle \sup _{R}|f_{k}|\leq 2C|t|.\end{aligned}}} By induction, this implies the bound M k , n ( t ) ≤ 2 C L k n | t | n + 1 / ( n + 1 ) ! {\displaystyle M_{k,n}(t)\leq 2CL_{k}^{n}|t|^{n+1}/(n+1)!} which tends to zero as n → ∞ {\displaystyle n\to \infty } for all t ∈ I {\displaystyle t\in I} . The functions y k , n {\displaystyle y_{k,n}} are equicontinuous as for − t 2 ≤ t < t ′ ≤ t 2 {\displaystyle -t_{2}\leq t<t'\leq t_{2}} we have | y k , n + 1 ( t ′ ) − y k , n + 1 ( t ) | ≤ ∫ x t ′ | f k ( t ″ , y k , n ( t ″ ) ) | d t ″ ≤ | t ′ − t | sup R | f k | ≤ 2 C | t ′ − t | , {\displaystyle {\begin{aligned}|y_{k,n+1}(t')-y_{k,n+1}(t)|&\leq \textstyle \int _{x}^{t'}|f_{k}(t'',y_{k,n}(t''))|\,\mathrm {d} t''\\&\textstyle \leq |t'-t|\sup _{R}|f_{k}|\leq 2C|t'-t|,\end{aligned}}} so by the Arzelà–Ascoli theorem they are relatively compact. In particular, for each k {\displaystyle k} there is a subsequence ( y k , φ k ( n ) ) n ∈ N {\displaystyle (y_{k,\varphi _{k}(n)})_{n\in \mathbb {N} }} converging uniformly to a continuous function y k : I → R {\displaystyle y_{k}:I\to \mathbb {R} } . Taking limit n → ∞ {\displaystyle n\to \infty } in | y k , φ k ( n ) ( t ) − ∫ 0 x f k ( t ′ , y k , φ k ( n ) ( t ′ ) ) d t ′ | = | y k , φ k ( n ) ( t ) − y k , φ k ( n ) + 1 ( t ) | ≤ M k , φ k ( n ) ( t 2 ) {\displaystyle {\begin{aligned}\textstyle \left|y_{k,\varphi _{k}(n)}(t)-\int _{0}^{x}f_{k}(t',y_{k,\varphi _{k}(n)}(t'))\,\mathrm {d} t'\right|&=|y_{k,\varphi _{k}(n)}(t)-y_{k,\varphi _{k}(n)+1}(t)|\\&\leq M_{k,\varphi _{k}(n)}(t_{2})\end{aligned}}} we conclude that y k ( t ) = ∫ 0 t f k ( t ′ , y k ( t ′ ) ) d t ′ {\displaystyle \textstyle y_{k}(t)=\int _{0}^{t}f_{k}(t',y_{k}(t'))\,\mathrm {d} t'} . The functions y k {\displaystyle y_{k}} are in the closure of a relatively compact set, so they are themselves relatively compact. Thus there is a subsequence y ψ ( k ) {\displaystyle y_{\psi (k)}} converging uniformly to a continuous function z : I → R {\displaystyle z:I\to \mathbb {R} } . Taking limit k → ∞ {\displaystyle k\to \infty } in y ψ ( k ) ( t ) = ∫ 0 t f ψ ( k ) ( t ′ , y ψ ( k ) ( t ′ ) ) d t ′ {\displaystyle \textstyle y_{\psi (k)}(t)=\int _{0}^{t}f_{\psi (k)}(t',y_{\psi (k)}(t'))\,\mathrm {d} t'} we conclude that z ( t ) = ∫ 0 t f ( t ′ , z ( t ′ ) ) d t ′ {\displaystyle \textstyle z(t)=\int _{0}^{t}f(t',z(t'))\,\mathrm {d} t'} , using the fact that f ψ ( k ) {\displaystyle f_{\psi (k)}} are equicontinuous by the Arzelà–Ascoli theorem. By the fundamental theorem of calculus, z ′ ( t ) = f ( t , z ( t ) ) {\displaystyle z'(t)=f(t,z(t))} in I {\displaystyle I} . == Related theorems == The Peano theorem can be compared with another existence result in the same context, the Picard–Lindelöf theorem. The Picard–Lindelöf theorem both assumes more and concludes more. It requires Lipschitz continuity, while the Peano theorem requires only continuity; but it proves both existence and uniqueness where the Peano theorem proves only the existence of solutions. To illustrate, consider the ordinary differential equation y ′ = | y | 1 2 {\displaystyle y'=\left\vert y\right\vert ^{\frac {1}{2}}} on the domain [ 0 , 1 ] . {\displaystyle \left[0,1\right].} According to the Peano theorem, this equation has solutions, but the Picard–Lindelöf theorem does not apply since the right hand side is not Lipschitz continuous in any neighbourhood containing 0. Thus we can conclude existence but not uniqueness. It turns out that this ordinary differential equation has two kinds of solutions when starting at y ( 0 ) = 0 {\displaystyle y(0)=0} , either y ( t ) = 0 {\displaystyle y(t)=0} or y ( t ) = t 2 / 4 {\displaystyle y(t)=t^{2}/4} . The transition between y = 0 {\displaystyle y=0} and y = ( t − C ) 2 / 4 {\displaystyle y=(t-C)^{2}/4} can happen at any C {\displaystyle C} . The Carathéodory existence theorem is a generalization of the Peano existence theorem with weaker conditions than continuity. The Peano existence theorem cannot be straightforwardly extended to a general Hilbert space H {\displaystyle {\mathcal {H}}} : for an open subset D {\displaystyle D} of R × H {\displaystyle \mathbb {R} \times {\mathcal {H}}} , the continuity of f : D → R {\displaystyle f\colon D\to \mathbb {R} } alone is insufficient for guaranteeing the existence of solutions for the associated initial value problem. == Notes == == References == Osgood, W. F. (1898). "Beweis der Existenz einer Lösung der Differentialgleichung dy/dx = f(x, y) ohne Hinzunahme der Cauchy-Lipschitzchen Bedingung". Monatshefte für Mathematik. 9: 331–345. doi:10.1007/BF01707876. S2CID 122312261. Coddington, Earl A.; Levinson, Norman (1955). Theory of Ordinary Differential Equations. New York: McGraw-Hill. Murray, Francis J.; Miller, Kenneth S. (1976) [1954]. Existence Theorems for Ordinary Differential Equations (Reprint ed.). New York: Krieger. Teschl, Gerald (2012). Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society. ISBN 978-0-8218-8328-0.
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Wikipedia:Peano kernel theorem#0
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In numerical analysis, the Peano kernel theorem is a general result on error bounds for a wide class of numerical approximations (such as numerical quadratures), defined in terms of linear functionals. It is attributed to Giuseppe Peano. == Statement == Let V [ a , b ] {\displaystyle {\mathcal {V}}[a,b]} be the space of all functions f {\displaystyle f} that are differentiable on ( a , b ) {\displaystyle (a,b)} that are of bounded variation on [ a , b ] {\displaystyle [a,b]} , and let L {\displaystyle L} be a linear functional on V [ a , b ] {\displaystyle {\mathcal {V}}[a,b]} . Assume that that L {\displaystyle L} annihilates all polynomials of degree ≤ ν {\displaystyle \leq \nu } , i.e. L p = 0 , ∀ p ∈ P ν [ x ] . {\displaystyle Lp=0,\qquad \forall p\in \mathbb {P} _{\nu }[x].} Suppose further that for any bivariate function g ( x , θ ) {\displaystyle g(x,\theta )} with g ( x , ⋅ ) , g ( ⋅ , θ ) ∈ C ν + 1 [ a , b ] {\displaystyle g(x,\cdot ),\,g(\cdot ,\theta )\in C^{\nu +1}[a,b]} , the following is valid: L ∫ a b g ( x , θ ) d θ = ∫ a b L g ( x , θ ) d θ , {\displaystyle L\int _{a}^{b}g(x,\theta )\,d\theta =\int _{a}^{b}Lg(x,\theta )\,d\theta ,} and define the Peano kernel of L {\displaystyle L} as k ( θ ) = L [ ( x − θ ) + ν ] , θ ∈ [ a , b ] , {\displaystyle k(\theta )=L[(x-\theta )_{+}^{\nu }],\qquad \theta \in [a,b],} using the notation ( x − θ ) + ν = { ( x − θ ) ν , x ≥ θ , 0 , x ≤ θ . {\displaystyle (x-\theta )_{+}^{\nu }={\begin{cases}(x-\theta )^{\nu },&x\geq \theta ,\\0,&x\leq \theta .\end{cases}}} The Peano kernel theorem states that, if k ∈ V [ a , b ] {\displaystyle k\in {\mathcal {V}}[a,b]} , then for every function f {\displaystyle f} that is ν + 1 {\textstyle \nu +1} times continuously differentiable, we have L f = 1 ν ! ∫ a b k ( θ ) f ( ν + 1 ) ( θ ) d θ . {\displaystyle Lf={\frac {1}{\nu !}}\int _{a}^{b}k(\theta )f^{(\nu +1)}(\theta )\,d\theta .} === Bounds === Several bounds on the value of L f {\displaystyle Lf} follow from this result: | L f | ≤ 1 ν ! ‖ k ‖ 1 ‖ f ( ν + 1 ) ‖ ∞ | L f | ≤ 1 ν ! ‖ k ‖ ∞ ‖ f ( ν + 1 ) ‖ 1 | L f | ≤ 1 ν ! ‖ k ‖ 2 ‖ f ( ν + 1 ) ‖ 2 {\displaystyle {\begin{aligned}|Lf|&\leq {\frac {1}{\nu !}}\|k\|_{1}\|f^{(\nu +1)}\|_{\infty }\\[5pt]|Lf|&\leq {\frac {1}{\nu !}}\|k\|_{\infty }\|f^{(\nu +1)}\|_{1}\\[5pt]|Lf|&\leq {\frac {1}{\nu !}}\|k\|_{2}\|f^{(\nu +1)}\|_{2}\end{aligned}}} where ‖ ⋅ ‖ 1 {\displaystyle \|\cdot \|_{1}} , ‖ ⋅ ‖ 2 {\displaystyle \|\cdot \|_{2}} and ‖ ⋅ ‖ ∞ {\displaystyle \|\cdot \|_{\infty }} are the taxicab, Euclidean and maximum norms respectively. == Application == In practice, the main application of the Peano kernel theorem is to bound the error of an approximation that is exact for all f ∈ P ν {\displaystyle f\in \mathbb {P} _{\nu }} . The theorem above follows from the Taylor polynomial for f {\displaystyle f} with integral remainder: f ( x ) = f ( a ) + ( x − a ) f ′ ( a ) + ( x − a ) 2 2 f ″ ( a ) + ⋯ ⋯ + ( x − a ) ν ν ! f ( ν ) ( a ) + 1 ν ! ∫ a x ( x − θ ) ν f ( ν + 1 ) ( θ ) d θ , {\displaystyle {\begin{aligned}f(x)=f(a)+{}&(x-a)f'(a)+{\frac {(x-a)^{2}}{2}}f''(a)+\cdots \\[6pt]&\cdots +{\frac {(x-a)^{\nu }}{\nu !}}f^{(\nu )}(a)+{\frac {1}{\nu !}}\int _{a}^{x}(x-\theta )^{\nu }f^{(\nu +1)}(\theta )\,d\theta ,\end{aligned}}} defining L ( f ) {\displaystyle L(f)} as the error of the approximation, using the linearity of L {\displaystyle L} together with exactness for f ∈ P ν {\displaystyle f\in \mathbb {P} _{\nu }} to annihilate all but the final term on the right-hand side, and using the ( ⋅ ) + {\displaystyle (\cdot )_{+}} notation to remove the x {\displaystyle x} -dependence from the integral limits. == See also == Divided differences == References ==
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Wikipedia:Pedro Luis García Pérez#0
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Pedro Luis García Pérez (23 March 1938 – 2 January 2025) was a Spanish mathematician, physicist, and academic. He was president of the Royal Spanish Mathematical Society from 1982 to 1988. García Pérez died on 2 January 2025, at the age of 86. == References ==
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Wikipedia:Pedro Ontaneda#0
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Pedro Ontaneda Portal is a Peruvian-American mathematician specializing in topology and differential geometry. He is a distinguished professor at Binghamton University, a unit of the State University of New York. == Education and career == Ontaneda received his Ph.D. in 1994 from Stony Brook University (another unit of SUNY), advised by Lowell Jones. Subsequently he taught at the Federal University of Pernambuco in Brazil. He moved to Binghamton University in 2005. == Mathematical contributions == Ontaneda's work deals with the geometry and topology of aspherical spaces, with particular attention to the relationship between exotic structures and negative or non-positive curvature on manifolds. Classical examples of Riemannian manifolds of negative curvature are given by real hyperbolic manifolds, or more generally by locally symmetric spaces of rank 1. One of Ontaneda's most celebrated contributions is the construction of manifolds that admit negatively curved Riemannian metrics but do not admit locally symmetric ones. More precisely, he showed that for any n ≥ 4 {\displaystyle n\geq 4} and for any ε > 0 {\displaystyle \varepsilon >0} there exists a closed Riemannian n {\displaystyle n} -manifold N {\displaystyle N} satisfying the following two properties: All the sectional curvatures of N {\displaystyle N} are in [ − 1 − ε , − 1 ] {\displaystyle [-1-\varepsilon ,-1]} . N {\displaystyle N} is not homeomorphic to a locally symmetric space. In particular, the fundamental group of N {\displaystyle N} is Gromov hyperbolic but not isomorphic to a uniform lattice in a Lie group of rank 1. These manifolds are obtained via the Riemannian hyperbolization procedure developed by Ontaneda in a series of papers, which is a smooth version of the strict hyperbolization procedure introduced by Ruth Charney and Michael W. Davis. The obstruction to being locally symmetric comes from the fact that Ontaneda's manifolds have nontrivial rational Pontryagin classes. The restriction to dimension n ≥ 4 {\displaystyle n\geq 4} is necessary. Indeed, if a surface admits a negatively curved metric, then it admits one that is locally isometric to the real hyperbolic plane, as a consequence of the uniformization theorem. A similar statement holds for 3 {\displaystyle 3} -manifolds thanks to the hyperbolization theorem. Ontaneda also made a "remarkable" contribution to the classification of dynamical systems by constructing partially hyperbolic diffeomorphisms (a generalization of Anosov diffeomorphisms) on some simply connected manifolds of high dimension; see his 2015 paper. == Selected publications == F. T. Farrell, L. E. Jones, and P. Ontaneda (2007), "Negative curvature and exotic topology." In Surveys in Differential Geometry, Vol. XI, pp. 329–347, International Press, Somerville, MA. F. Thomas Farrell and Pedro Ontaneda (2010), "On the topology of the space of negatively curved metrics." Journal of Differential Geometry 86, no. 2, pp. 273–301. Andrey Gogolev, Pedro Ontaneda, and Federico Rodriguez Hertz (2015), "New partially hyperbolic dynamical systems I." Acta Mathematica 215, no. 2, pp. 363–393. Pedro Ontaneda (2020), "Riemannian hyperbolization." Publ. Math. Inst. Hautes Études Sci. 131, pp. 1–72. == References == == External links == Pedro Ontaneda's Author Profile on MathSciNet
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Wikipedia:Pedro Peralta y Barnuevo#0
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Pedro Peralta y Barnuevo (Lima, 26 November 1663 – 30 April 1743) was an Enlightenment-era Peruvian mathematician, cosmographer, historian, scholar, poet, and astronomer, and was considered a polymath. He was rector of University of San Marcos in Lima. Peralta's parents were Spaniard Francisco Peralta Barnuevo and Magdalena E. Rocha Benavides from Lima. He was the brother of José de Peralta Barnuevo, Bishop of Buenos Aires. He studied Roman and canonical art and law at the University of San Marcos, from which he obtained the degree of doctor in canons and laws (1680-1686). Subsequently, he obtained the title of lawyer before the Royal Court (1686). He mastered Latin, Greek, French, Portuguese, Italian, English and Quechua, and had in his library works that reveal an all-embracing curiosity: grammar, poliorcetics, astronomy and metallurgy, among others. Upon the death of his father, he inherited from him the position of royal accountant of the Court of Audit. He also received income from his wife's landed estates. He became rector of the University of San Marcos in very difficult circumstances for the university in 1715 and 1716. He was a member of the Académie des sciences of Paris, because of his decision to collaborate in a very important Franco-Spanish geodesic expedition, and the head of the expedition, begun in 1735, was the French naturalist and geographer Charles Marie de la Condamine. It was sought (and was done after long and very careful work), determine the length of the meridian arc, and numerous observations of the nature of that area were also carried out. Spaniards Antonio de Ulloa and Jorge Juan participated as principals. == Works == Oración que dixo el rector de esta Real Universidad de San Marcos a su ilustre claustro, el 30 de junio de 1716. Lima, 1716. Jubileos de Lima y fiestas reales. Lima, 1723. Historia de España Vindicada. Lima, 1730. Lima fundada o Conquista del Peru: poema heroico. Lima, 1732. Passion y triumpho de Christo: dividida en diez oraciones. Lima, 1738. == References == == Further reading == Brading, D.A. The First America: The Spanish Monarchy, Creole Patriots, and the Liberal State, 1492=1867. New York: Cambridge University Press 1991, pp. 391–399, Williams, Jerry M. Peralta Barnuevo and the Art of Propaganda. Newark, Del., 2001.
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Wikipedia:Peeter Lorents#0
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Peeter Lorents (born 25 September 1951 Pärnu) is an Estonian mathematician and politician. He was a member of VII Riigikogu. == References ==
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Wikipedia:Peetre theorem#0
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In mathematics, the (linear) Peetre theorem, named after Jaak Peetre, is a result of functional analysis that gives a characterisation of differential operators in terms of their effect on generalized function spaces, and without mentioning differentiation in explicit terms. The Peetre theorem is an example of a finite order theorem in which a function or a functor, defined in a very general way, can in fact be shown to be a polynomial because of some extraneous condition or symmetry imposed upon it. This article treats two forms of the Peetre theorem. The first is the original version which, although quite useful in its own right, is actually too general for most applications. == The original Peetre theorem == Let M be a smooth manifold and let E and F be two vector bundles on M. Let Γ ∞ ( E ) , and Γ ∞ ( F ) {\displaystyle \Gamma ^{\infty }(E),\ {\hbox{and}}\ \Gamma ^{\infty }(F)} be the spaces of smooth sections of E and F. An operator D : Γ ∞ ( E ) → Γ ∞ ( F ) {\displaystyle D:\Gamma ^{\infty }(E)\rightarrow \Gamma ^{\infty }(F)} is a morphism of sheaves which is linear on sections such that the support of D is non-increasing: supp Ds ⊆ supp s for every smooth section s of E. The original Peetre theorem asserts that, for every point p in M, there is a neighborhood U of p and an integer k (depending on U) such that D is a differential operator of order k over U. This means that D factors through a linear mapping iD from the k-jet of sections of E into the space of smooth sections of F: D = i D ∘ j k {\displaystyle D=i_{D}\circ j^{k}} where j k : Γ ∞ E → J k E {\displaystyle j^{k}:\Gamma ^{\infty }E\rightarrow J^{k}E} is the k-jet operator and i D : J k E → F {\displaystyle i_{D}:J^{k}E\rightarrow F} is a linear mapping of vector bundles. === Proof === The problem is invariant under local diffeomorphism, so it is sufficient to prove it when M is an open set in Rn and E and F are trivial bundles. At this point, it relies primarily on two lemmas: Lemma 1. If the hypotheses of the theorem are satisfied, then for every x∈M and C > 0, there exists a neighborhood V of x and a positive integer k such that for any y∈V\{x} and for any section s of E whose k-jet vanishes at y (jks(y)=0), we have |Ds(y)|<C. Lemma 2. The first lemma is sufficient to prove the theorem. We begin with the proof of Lemma 1. Suppose the lemma is false. Then there is a sequence xk tending to x, and a sequence of very disjoint balls Bk around the xk (meaning that the geodesic distance between any two such balls is non-zero), and sections sk of E over each Bk such that jksk(xk)=0 but |Dsk(xk)|≥C>0. Let ρ(x) denote a standard bump function for the unit ball at the origin: a smooth real-valued function which is equal to 1 on B1/2(0), which vanishes to infinite order on the boundary of the unit ball. Consider every other section s2k. At x2k, these satisfy j2ks2k(x2k)=0. Suppose that 2k is given. Then, since these functions are smooth and each satisfy j2k(s2k)(x2k)=0, it is possible to specify a smaller ball B′δ(x2k) such that the higher order derivatives obey the following estimate: ∑ | α | ≤ k sup y ∈ B δ ′ ( x 2 k ) | ∇ α s k ( y ) | ≤ 1 M k ( δ 2 ) k {\displaystyle \sum _{|\alpha |\leq k}\ \sup _{y\in B'_{\delta }(x_{2k})}|\nabla ^{\alpha }s_{k}(y)|\leq {\frac {1}{M_{k}}}\left({\frac {\delta }{2}}\right)^{k}} where M k = ∑ | α | ≤ k sup | ∇ α ρ | . {\displaystyle M_{k}=\sum _{|\alpha |\leq k}\sup |\nabla ^{\alpha }\rho |.} Now ρ 2 k ( y ) := ρ ( y − x 2 k δ ) {\displaystyle \rho _{2k}(y):=\rho \left({\frac {y-x_{2k}}{\delta }}\right)} is a standard bump function supported in B′δ(x2k), and the derivative of the product s2kρ2k is bounded in such a way that max | α | ≤ k sup y ∈ B δ ′ ( x 2 k ) | ∇ α ( ρ 2 k s 2 k ) | ≤ 2 − k . {\displaystyle \max _{|\alpha |\leq k}\ \sup _{y\in B'_{\delta }(x_{2k})}|\nabla ^{\alpha }(\rho _{2k}s_{2k})|\leq 2^{-k}.} As a result, because the following series and all of the partial sums of its derivatives converge uniformly q ( y ) = ∑ k = 1 ∞ ρ 2 k ( y ) s 2 k ( y ) , {\displaystyle q(y)=\sum _{k=1}^{\infty }\rho _{2k}(y)s_{2k}(y),} q(y) is a smooth function on all of V. We now observe that since s2k and ρ {\displaystyle \rho } 2ks2k are equal in a neighborhood of x2k, lim k → ∞ | D q ( x 2 k ) | ≥ C {\displaystyle \lim _{k\rightarrow \infty }|Dq(x_{2k})|\geq C} So by continuity |Dq(x)|≥ C>0. On the other hand, lim k → ∞ D q ( x 2 k + 1 ) = 0 {\displaystyle \lim _{k\rightarrow \infty }Dq(x_{2k+1})=0} since Dq(x2k+1)=0 because q is identically zero in B2k+1 and D is support non-increasing. So Dq(x)=0. This is a contradiction. We now prove Lemma 2. First, let us dispense with the constant C from the first lemma. We show that, under the same hypotheses as Lemma 1, |Ds(y)|=0. Pick a y in V\{x} so that jks(y)=0 but |Ds(y)|=g>0. Rescale s by a factor of 2C/g. Then if g is non-zero, by the linearity of D, |Ds(y)|=2C>C, which is impossible by Lemma 1. This proves the theorem in the punctured neighborhood V\{x}. Now, we must continue the differential operator to the central point x in the punctured neighborhood. D is a linear differential operator with smooth coefficients. Furthermore, it sends germs of smooth functions to germs of smooth functions at x as well. Thus the coefficients of D are also smooth at x. == A specialized application == Let M be a compact smooth manifold (possibly with boundary), and E and F be finite dimensional vector bundles on M. Let Γ ∞ ( E ) {\displaystyle \Gamma ^{\infty }(E)} be the collection of smooth sections of E. An operator D : Γ ∞ ( E ) → Γ ∞ ( F ) {\displaystyle D:\Gamma ^{\infty }(E)\rightarrow \Gamma ^{\infty }(F)} is a smooth function (of Fréchet manifolds) which is linear on the fibres and respects the base point on M: π ∘ D p = p . {\displaystyle \pi \circ D_{p}=p.} The Peetre theorem asserts that for each operator D, there exists an integer k such that D is a differential operator of order k. Specifically, we can decompose D = i D ∘ j k {\displaystyle D=i_{D}\circ j^{k}} where i D {\displaystyle i_{D}} is a mapping from the jets of sections of E to the bundle F. See also intrinsic differential operators. == Example: Laplacian == Consider the following operator: ( L f ) ( x 0 ) = lim r → 0 2 d r 2 1 | S r | ∫ S r ( f ( x ) − f ( x 0 ) ) d x {\displaystyle (Lf)(x_{0})=\lim _{r\to 0}{\frac {2d}{r^{2}}}{\frac {1}{|S_{r}|}}\int _{S_{r}}(f(x)-f(x_{0}))dx} where f ∈ C ∞ ( R d ) {\displaystyle f\in C^{\infty }(\mathbb {R} ^{d})} and S r {\displaystyle S_{r}} is the sphere centered at x 0 {\displaystyle x_{0}} with radius r {\displaystyle r} . This is in fact the Laplacian, as can be seen using Taylor's theorem. We show will show L {\displaystyle L} is a differential operator by Peetre's theorem. The main idea is that since L f ( x 0 ) {\displaystyle Lf(x_{0})} is defined only in terms of f {\displaystyle f} 's behavior near x 0 {\displaystyle x_{0}} , it is local in nature; in particular, if f {\displaystyle f} is locally zero, so is L f {\displaystyle Lf} , and hence the support cannot grow. The technical proof goes as follows. Let M = R d {\displaystyle M=\mathbb {R} ^{d}} and E {\displaystyle E} and F {\displaystyle F} be the rank 1 {\displaystyle 1} trivial bundles. Then Γ ∞ ( E ) {\displaystyle \Gamma ^{\infty }(E)} and Γ ∞ ( F ) {\displaystyle \Gamma ^{\infty }(F)} are simply the space C ∞ ( R d ) {\displaystyle C^{\infty }(\mathbb {R} ^{d})} of smooth functions on R d {\displaystyle \mathbb {R} ^{d}} . As a sheaf, F ( U ) {\displaystyle {\mathcal {F}}(U)} is the set of smooth functions on the open set U {\displaystyle U} and restriction is function restriction. To see L {\displaystyle L} is indeed a morphism, we need to check ( L u ) | V = L ( u | V ) {\displaystyle (Lu)|V=L(u|V)} for open sets U {\displaystyle U} and V {\displaystyle V} such that V ⊆ U {\displaystyle V\subseteq U} and u ∈ C ∞ ( U ) {\displaystyle u\in C^{\infty }(U)} . This is clear because for x ∈ V {\displaystyle x\in V} , both [ ( L u ) | V ] ( x ) {\displaystyle [(Lu)|V](x)} and [ L ( u | V ) ] ( x ) {\displaystyle [L(u|V)](x)} are simply lim r → 0 2 d r 2 1 | S r | ∫ S r ( u ( y ) − u ( x ) ) d y {\displaystyle \lim _{r\to 0}{\frac {2d}{r^{2}}}{\frac {1}{|S_{r}|}}\int _{S_{r}}(u(y)-u(x))dy} , as the S r {\displaystyle S_{r}} eventually sits inside both U {\displaystyle U} and V {\displaystyle V} anyway. It is easy to check that L {\displaystyle L} is linear: L ( f + g ) = L ( f ) + L ( g ) {\displaystyle L(f+g)=L(f)+L(g)} and L ( a f ) = a L ( f ) {\displaystyle L(af)=aL(f)} Finally, we check that L {\displaystyle L} is local in the sense that s u p p L f ⊆ s u p p f {\displaystyle suppLf\subseteq suppf} . If x 0 ∉ s u p p ( f ) {\displaystyle x_{0}\notin supp(f)} , then ∃ r > 0 {\displaystyle \exists r>0} such that f = 0 {\displaystyle f=0} in the ball of radius r {\displaystyle r} centered at x 0 {\displaystyle x_{0}} . Thus, for x ∈ B ( x 0 , r ) {\displaystyle x\in B(x_{0},r)} , ∫ S r ′ ( f ( y ) − f ( x ) ) d y = 0 {\displaystyle \int _{S_{r'}}(f(y)-f(x))dy=0} for r ′ < r − | x − x 0 | {\displaystyle r'<r-|x-x_{0}|} , and hence ( L f ) ( x ) = 0 {\displaystyle (Lf)(x)=0} . Therefore, x 0 ∉ s u p p L f {\displaystyle x_{0}\notin suppLf} . So by Peetre's theorem, L {\displaystyle L} is a differential operator. == References == Peetre, J., Une caractérisation abstraite des opérateurs différentiels, Math. Scand. 7 (1959), 211-218. Peetre, J., Rectification à l'article Une caractérisation abstraite des opérateurs différentiels, Math. Scand. 8 (1960), 116-120. Terng, C.L., Natural vector bundles and natural differential operators, Am. J. Math. 100 (1978), 775-828.
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Wikipedia:Peetre's inequality#0
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In mathematics, Peetre's inequality, named after Jaak Peetre, says that for any real number t {\displaystyle t} and any vectors x {\displaystyle x} and y {\displaystyle y} in R n , {\displaystyle \mathbb {R} ^{n},} the following inequality holds: ( 1 + | x | 2 1 + | y | 2 ) t ≤ 2 | t | ( 1 + | x − y | 2 ) | t | . {\displaystyle \left({\frac {1+|x|^{2}}{1+|y|^{2}}}\right)^{t}~\leq ~2^{|t|}(1+|x-y|^{2})^{|t|}.} The inequality was proved by J. Peetre in 1959 and has founds applications in functional analysis and Sobolev spaces. == See also == List of inequalities == References == Chazarain, J.; Piriou, A. (2011), Introduction to the Theory of Linear Partial Differential Equations, Studies in Mathematics and its Applications, Elsevier, p. 90, ISBN 9780080875354. Ruzhansky, Michael; Turunen, Ville (2009), Pseudo-Differential Operators and Symmetries: Background Analysis and Advanced Topics, Pseudo-Differential Operators, Theory and Applications, vol. 2, Springer, p. 321, ISBN 9783764385132. Saint Raymond, Xavier (1991), Elementary Introduction to the Theory of Pseudodifferential Operators, Studies in Advanced Mathematics, vol. 3, CRC Press, p. 21, ISBN 9780849371585. This article incorporates material from Peetre's inequality on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License. == External links == Planetmath.org: Peetre's inequality
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Wikipedia:Pekka Myrberg#0
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Pekka Juhana Myrberg (30 December 1892, Viipuri – 8 November 1976, Helsinki) was a Finnish mathematician known for developing the concept of period-doubling bifurcation in a paper published in the 1950s. The concept was further developed by Mitchell Feigenbaum during the 1970s. Myrberg received his PhD in 1916 at the University of Helsinki under Ernst Lindelöf with thesis Zur Theorie der Konvergenz der Poincaré´schen Reihen ('On the theory of the convergence of Poincaré's series'). He began his career by teaching at a gymnasium, and then became professor extraordinarius at the University of Helsinki in 1921 and professor ordinarius in 1926. In 1952 he became the rector and then served as the chancellor of the University of Helsinki from 1952 to 1962. In 1962 he retired as professor emeritus but continued publishing mathematical papers into the 1970s. In the 1950s, Myberg published several fundamental papers on the iteration of rational functions (especially quadratic functions). His research revived interest in the results of Gaston Julia and Pierre Fatou published during the beginning of the 20th century. Myberg was a member of the Finnish Academy of Sciences. In 1954 he was an invited speaker (Über die Integration der Poissonschen Gleichung auf Riemannschen Flächen) at the International Mathematical Congress in Amsterdam. == References == == External links == Pekka Myrberg at Find a Grave
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Wikipedia:Pekka Tukia#0
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Pekka Pertti Tukia (born 3 November 1945 in Pihtipudas) is a Finnish mathematician who does research on Kleinian groups and their geometric properties (such as limit sets). Tukia received his PhD in 1972 with thesis advisor Kaarlo Virtanen in Helsinki. Tukia is a professor at the University of Helsinki. He made substantial contributions to the collective work of about a dozen mathematicians who proved the Seifert fiber space conjecture. In 1992 he was an invited speaker with talk Generalizations of Fuchsian and Kleinian groups at the European Congress of European Mathematicians in Paris. In 1994 he was an invited speaker with talk A survey of Möbius groups at the International Congress of Mathematicians in Zürich. == Selected publications == Tukia, Pekka (1980). "On two dimensional quasiconformal groups". Annales Academiæ Scientiarum Fennicæ. Series A. 5: 73–78. CiteSeerX 10.1.1.732.909. doi:10.5186/aasfm.1980.0530. Tukia, Pakka (1981). "A quasiconformal group not isomorphic to the Möbius group". Annales Academiæ Scientiarum Fennicæ. Series A. 6: 149–160. doi:10.5186/aasfm.1981.0625. Tukia, Pekka (1984). "The Hausdorff dimension of the limit set of a geometrically finite Kleinian groups". Acta Mathematica. 152: 127–140. doi:10.1007/BF02392194. Tukia, Pekka (1985). "On limit sets of geometrically finite Kleinian groups". Mathematica Scandinavica. 57: 29–43. doi:10.7146/math.scand.a-12104. Tukia, Pekka (1985). "Differentiability and rigidity of Möbius groups". Inventiones Mathematicae. 82 (3): 557–578. Bibcode:1985InMat..82..557T. doi:10.1007/BF01388870. S2CID 123260399. Tukia, Pekka (1985). "Quasiconformal extension of quasisymmetric mappings compatible with a Möbius group". Acta Mathematica. 154 (3–4): 153–193. doi:10.1007/BF02392471. Tukia, Pekka (1985). "On isomorphisms of geometrically finite Möbius groups" (PDF). Publications Mathématiques de l'IHÉS. 61: 171–214. doi:10.1007/BF02698805. Tukia, Pekka (1986). "On quasiconformal groups". Journal d'Analyse Mathématique. 46: 318–346. doi:10.1007/BF02796595. S2CID 120093947. Tukia, P. (1989). "A rigidity theorem for Möbius groups". Inventiones Mathematicae. 97 (2): 405–431. Bibcode:1989InMat..97..405T. doi:10.1007/BF01389048. S2CID 121594611. Tukia, Pekka (1989). "Hausdorff dimension of quasisymmetric mappings". Mathematica Scandinavica. 65 (1): 152–160. doi:10.7146/math.scand.a-12274. JSTOR 24492095. Tukia, Pekka (1991). "Mostow-rigidity and non-compact hyperbolic manifolds". Quarterly Journal of Mathematics. 42 (1): 219–226. doi:10.1093/qmath/42.1.221. "Convergence groups and Gromov's metric hyperbolic spaces" (PDF). New Zealand Journal of Mathematics. 23: 157–187. 1994. Tukia, Pekka (2006). "Teichmüller sequences on trajectories invariant under a Kleinian group". Journal d'Analyse Mathématique. 99: 35–87. doi:10.1007/BF02789442. S2CID 121882279. Tukia, Pekka (2015). "Limits of Teichmüller maps". Journal d'Analyse Mathématique. 125: 71–111. doi:10.1007/s11854-015-0003-7. S2CID 255398787. with James W. Anderson and Kurt Falk: Anderson, James W.; Falk, Kurt; Tukia, Pekka (2007). "Conformal measures associated to ends of hyperbolic n-manifolds". Quarterly Journal of Mathematics. 58: 1–15. arXiv:math/0409582. doi:10.1093/qmath/hal019. == References ==
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Wikipedia:Peng Tsu Ann#0
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Peng Tsu Ann (born 1936) is a Singaporean mathematician, and the first University of Singapore (now the National University of Singapore, Abbreviation: NUS) graduate to obtain a PhD in mathematics. Peng was the Head of the Department of Mathematics at NUS from 1982 to 1996 and oversaw its rapid growth during the period. In mathematics, Peng's research interests are in group theory. He was a visiting member at the Institute for Advanced Study (IAS) in the spring of 1989. The Peng Tsu Ann Assistant Professorship at the Department of Mathematics in NUS is named after him. == Biography == Peng obtained his BSc from the University of Singapore in 1962 and PhD from the University of London in 1965, under the direction of Karl W. Gruenberg. He received a British Commonwealth Scholarship in 1962 and a Fellowship in 1972 under the Commonwealth Scholarship and Fellowship Plan. Peng served as president of the Singapore Mathematical Society from 1980 to 1982, and in 1987. Peng played a major role in organizing the Singapore Group Theory Conference in 1987, where the invited speakers included Walter Feit, Graham Higman, Jean-Pierre Serre, Michio Suzuki, and John G. Thompson. Peng retired from the Department of Mathematics at NUS in 1996. == References ==
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Wikipedia:Peng Yee Lee#0
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Peng Yee Lee (born 1938) is a Singaporean mathematician and mathematics educator. Lee is an associate professor of mathematics at the National Institute of Education in Singapore. He is a former president of the Southeast Asian Mathematical Society, a former vice president of the International Commission on Mathematical Instruction, and a former president of the Association of Mathematics Educators of Singapore. In mathematics, Lee's work is in analysis focusing on integration theory. == Early life and education == Lee received his PhD from Queens University of Belfast (UK) in 1965, under the direction of Ralph Henstock. == Career == Lee has taught at the University of Malawi (1965–67), the University of Auckland (1967–71), Nanyang University (1971–81), the National University of Singapore (1981–94), and the National Institute of Education (NIE) in Nanyang Technological University (1994–2013). Lee was a president of the Southeast Asian Mathematical Society in 1981 and 1982, and vice president of the International Commission on Mathematical Instruction (ICMI) from 1987 to 1990 and from 1991 to 1994. He was President of the Association of Mathematics Educators in Singapore in 2000 and 2001. In 2012, he became one of the inaugural Fellows of the American Mathematical Society. == Contributions == Lee worked for more than 40 years to promote the development of mathematics and mathematics education in Southeast Asia and in China. He trained a significant number of mathematicians from these regions, including those from Singapore, Indonesia and the Philippines. Lee has published about 100 research papers and written several books on analysis and mathematics education. Lee contributed to the development of Singapore math, a teaching method and mathematics curriculum that focus on problem solving and heuristic model drawing, which originated in Singapore in the 1980s. == Selected works == Peng Yee Lee (1989) Lanzhou Lectures on Henstock Integration, World Scientific. ISBN 9971-50-891-5. Peng Yee Lee and Redolf Výborný (2000) Integral: an easy approach after Kurzweil and Henstock, Cambridge University Press. ISBN 0-521-77968-5. Peng Yee Lee (2006) Mathematics for Teaching or Mathematics for Teachers? Guest Editorial, The Mathematics Educator, Vol. 16, No. 2, 2–3. Peng Yee Lee, Jan de Lange, and William Schmidt (2006) What are PISA and TIMSS? What do they tell us? International Congress of Mathematicians. Vol. III, 1663–1672, European Mathematical Society, Zürich. Bin Xiong and Peng Yee Lee (2007) Mathematical Olympiad in China: Problems and Solutions, East China Normal University Press. Lee, P. Y, “Sixty years of mathematics syllabi and textbooks in Singapore (1945-2005)”. Paper presented for The First International Conference on Mathematics Curriculum at the University of Chicago, 2005. == References == == External links == Mathematics Genealogy Project – Peng Yee Lee "Google Scholar report
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Wikipedia:Penny Haxell#0
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Penelope Evelyn Haxell is a Canadian mathematician who works as a professor in the department of combinatorics and optimization at the University of Waterloo. Her research interests include extremal combinatorics and graph theory. == Education and career == Haxell earned a bachelor's degree in 1988 from the University of Waterloo, and completed a doctorate in 1993 from the University of Cambridge under the supervision of Béla Bollobás. Since then, she has worked at the University of Waterloo, where she was promoted to full professor in 2004. == Research == Haxell's research accomplishments include results on the Szemerédi regularity lemma, hypergraph generalizations of Hall's marriage theorem (see Haxell's matching theorem), fractional graph packing problems, and strong coloring of graphs. == Recognition == Haxell was the 2006 winner of the Krieger–Nelson Prize of the Canadian Mathematical Society. == References ==
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Wikipedia:Pepijn van Erp#0
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Pepijn van Erp (born 1972) is a Dutch mathematician and skeptical activist. Van Erp studied mathematics at the Radboud University Nijmegen, graduating in 1999. After graduating, Van Erp worked as a statistics consultant at the PTT. Between 2002 and 2005, he lived in Tanzania. Until 2012, Van Erp was secretary at the Stichting Nijmeegs Universitair Fonds (SNUF). In 2011, Van Erp became involved with Stichting Skepsis, where he has served as board member since March 2012. Over a number of years, he has occupied himself with all sorts of dubious claims to determine whether they are scientific or pseudoscientific in nature. In February 2013, Van Erp was one of several commentators who accused the editorial staff of talkshow De Wereld Draait Door to uncritically present the story of the heavily handicapped Niek Zervaas, who was alleged to have been able to communicate using facilitated communication, a practice characterised as pseudoscience. The editorial staff later admitted that they had been 'blinded' by the wondrous story of Niek's parents, and should have been more skeptical. After van Erp criticised the ideas of American-Italian nuclear physicist Ruggero Santilli, the latter sued him, his webhost, and the chairman of Skepsis Foundation in 2016. The suit against the foundation was dismissed in August 2018 and shortly thereafter the suit against van Erp was settled. In October 2016, the electronics company Philips planned to hold a workshop called "Energy Medicine meets technology" with speakers from companies developing an electroacupuncture device and an app for mapping meridians; after van Erp posted a blog describing his efforts to get information from the companies and criticizing these devices as pseudoscience and quackery, Philips cancelled the workshop. He is also a chess player. == References == == External links == Personal website
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Wikipedia:Per Enflo#0
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Per H. Enflo (Swedish: [ˈpæːr ˈěːnfluː]; born 20 May 1944) is a Swedish mathematician working primarily in functional analysis, a field in which he solved problems that had been considered fundamental. Three of these problems had been open for more than forty years: The basis problem and the approximation problem and later the invariant subspace problem for Banach spaces. In solving these problems, Enflo developed new techniques which were then used by other researchers in functional analysis and operator theory for years. Some of Enflo's research has been important also in other mathematical fields, such as number theory, and in computer science, especially computer algebra and approximation algorithms. Enflo works at Kent State University, where he holds the title of University Professor. Enflo has earlier held positions at the Miller Institute for Basic Research in Science at the University of California, Berkeley, Stanford University, École Polytechnique, (Paris) and The Royal Institute of Technology, Stockholm. Enflo is also a concert pianist. == Enflo's contributions to functional analysis and operator theory == In mathematics, functional analysis is concerned with the study of vector spaces and operators acting upon them. It has its historical roots in the study of functional spaces, in particular transformations of functions, such as the Fourier transform, as well as in the study of differential and integral equations. In functional analysis, an important class of vector spaces consists of the complete normed vector spaces over the real or complex numbers, which are called Banach spaces. An important example of a Banach space is a Hilbert space, where the norm arises from an inner product. Hilbert spaces are of fundamental importance in many areas, including the mathematical formulation of quantum mechanics, stochastic processes, and time-series analysis. Besides studying spaces of functions, functional analysis also studies the continuous linear operators on spaces of functions. === Hilbert's fifth problem and embeddings === At Stockholm University, Hans Rådström suggested that Enflo consider Hilbert's fifth problem in the spirit of functional analysis. In two years, 1969–1970, Enflo published five papers on Hilbert's fifth problem; these papers are collected in Enflo (1970), along with a short summary. Some of the results of these papers are described in Enflo (1976) and in the last chapter of Benyamini and Lindenstrauss. ==== Applications in computer science ==== Enflo's techniques have found application in computer science. Algorithm theorists derive approximation algorithms that embed finite metric spaces into low-dimensional Euclidean spaces with low "distortion" (in Gromov's terminology for the Lipschitz category; cf. Banach–Mazur distance). Low-dimensional problems have lower computational complexity, of course. More importantly, if the problems embed well in either the Euclidean plane or the three-dimensional Euclidean space, then geometric algorithms become exceptionally fast. However, such embedding techniques have limitations, as shown by Enflo's (1969) theorem: For every m ≥ 2 {\displaystyle m\geq 2} , the Hamming cube C m {\displaystyle C_{m}} cannot be embedded with "distortion D {\displaystyle D} " (or less) into 2 m {\displaystyle 2^{m}} -dimensional Euclidean space if D < m {\displaystyle D<{\sqrt {m}}} . Consequently, the optimal embedding is the natural embedding, which realizes { 0 , 1 } m {\displaystyle \{0,1\}^{m}} as a subspace of m {\displaystyle m} -dimensional Euclidean space. This theorem, "found by Enflo [1969], is probably the first result showing an unbounded distortion for embeddings into Euclidean spaces. Enflo considered the problem of uniform embeddability among Banach spaces, and the distortion was an auxiliary device in his proof." === Geometry of Banach spaces === A uniformly convex space is a Banach space so that, for every ϵ > 0 {\displaystyle \epsilon >0} there is some δ > 0 {\displaystyle \delta >0} so that for any two vectors with ‖ x ‖ ≤ 1 {\displaystyle \|x\|\leq 1} and ‖ y ‖ ≤ 1 , {\displaystyle \|y\|\leq 1,} ‖ x + y ‖ > 2 − δ {\displaystyle \|x+y\|>2-\delta } implies that ‖ x − y ‖ < ϵ . {\displaystyle \|x-y\|<\epsilon .} Intuitively, the center of a line segment inside the unit ball must lie deep inside the unit ball unless the segment is short. In 1972 Enflo proved that "every super-reflexive Banach space admits an equivalent uniformly convex norm". === The basis problem and Mazur's goose === With one paper, which was published in 1973, Per Enflo solved three problems that had stumped functional analysts for decades: The basis problem of Stefan Banach, the "Goose problem" of Stanisław Mazur, and the approximation problem of Alexander Grothendieck. Grothendieck had shown that his approximation problem was the central problem in the theory of Banach spaces and continuous linear operators. ==== Basis problem of Banach ==== The basis problem was posed by Stefan Banach in his book, Theory of Linear Operators. Banach asked whether every separable Banach space has a Schauder basis. A Schauder basis or countable basis is similar to the usual (Hamel) basis of a vector space; the difference is that for Hamel bases we use linear combinations that are finite sums, while for Schauder bases they may be infinite sums. This makes Schauder bases more suitable for the analysis of infinite-dimensional topological vector spaces including Banach spaces. Schauder bases were described by Juliusz Schauder in 1927. Let V denote a Banach space over the field F. A Schauder basis is a sequence (bn) of elements of V such that for every element v ∈ V there exists a unique sequence (αn) of elements in F so that v = ∑ n ∈ N α n b n {\displaystyle v=\sum _{n\in \mathbb {N} }\alpha _{n}b_{n}\,} where the convergence is understood with respect to the norm topology. Schauder bases can also be defined analogously in a general topological vector space. ==== Problem 153 in the Scottish Book: Mazur's goose ==== Banach and other Polish mathematicians would work on mathematical problems at the Scottish Café. When a problem was especially interesting and when its solution seemed difficult, the problem would be written down in the book of problems, which soon became known as the Scottish Book. For problems that seemed especially important or difficult or both, the problem's proposer would often pledge to award a prize for its solution. On 6 November 1936, Stanisław Mazur posed a problem on representing continuous functions. Formally writing down problem 153 in the Scottish Book, Mazur promised as the reward a "live goose", an especially rich price during the Great Depression and on the eve of World War II. Fairly soon afterwards, it was realized that Mazur's problem was closely related to Banach's problem on the existence of Schauder bases in separable Banach spaces. Most of the other problems in the Scottish Book were solved regularly. However, there was little progress on Mazur's problem and a few other problems, which became famous open problems to mathematicians around the world. ==== Grothendieck's formulation of the approximation problem ==== Grothendieck's work on the theory of Banach spaces and continuous linear operators introduced the approximation property. A Banach space is said to have the approximation property, if every compact operator is a limit of finite-rank operators. The converse is always true. In a long monograph, Grothendieck proved that if every Banach space had the approximation property, then every Banach space would have a Schauder basis. Grothendieck thus focused the attention of functional analysts on deciding whether every Banach space have the approximation property. ==== Enflo's solution ==== In 1972, Per Enflo constructed a separable Banach space that lacks the approximation property and a Schauder basis. In 1972, Mazur awarded a live goose to Enflo in a ceremony at the Stefan Banach Center in Warsaw; the "goose reward" ceremony was broadcast throughout Poland. === Invariant subspace problem and polynomials === In functional analysis, one of the most prominent problems was the invariant subspace problem, which required the evaluation of the truth of the following proposition: Given a complex Banach space H of dimension > 1 and a bounded linear operator T : H → H, then H has a non-trivial closed T-invariant subspace, i.e. there exists a closed linear subspace W of H which is different from {0} and H such that T(W) ⊆ W. For Banach spaces, the first example of an operator without an invariant subspace was constructed by Enflo. (For Hilbert spaces, the invariant subspace problem remains open.) Enflo proposed a solution to the invariant subspace problem in 1975, publishing an outline in 1976. Enflo submitted the full article in 1981 and the article's complexity and length delayed its publication to 1987 Enflo's long "manuscript had a world-wide circulation among mathematicians" and some of its ideas were described in publications besides Enflo (1976). Enflo's works inspired a similar construction of an operator without an invariant subspace for example by Beauzamy, who acknowledged Enflo's ideas. In the 1990s, Enflo developed a "constructive" approach to the invariant subspace problem on Hilbert spaces. ==== Multiplicative inequalities for homogeneous polynomials ==== An essential idea in Enflo's construction was "concentration of polynomials at low degrees": For all positive integers m {\displaystyle m} and n {\displaystyle n} , there exists C ( m , n ) > 0 {\displaystyle C(m,n)>0} such that for all homogeneous polynomials P {\displaystyle P} and Q {\displaystyle Q} of degrees m {\displaystyle m} and n {\displaystyle n} (in k {\displaystyle k} variables), then | P Q | ≥ C ( m , n ) | P | | Q | , {\displaystyle |PQ|\geq C(m,n)|P|\,|Q|,} where | P | {\displaystyle |P|} denotes the sum of the absolute values of the coefficients of P {\displaystyle P} . Enflo proved that C ( m , n ) {\displaystyle C(m,n)} does not depend on the number of variables k {\displaystyle k} . Enflo's original proof was simplified by Montgomery. This result was generalized to other norms on the vector space of homogeneous polynomials. Of these norms, the most used has been the Bombieri norm. ===== Bombieri norm ===== The Bombieri norm is defined in terms of the following scalar product: For all α , β ∈ N N {\displaystyle \alpha ,\beta \in \mathbb {N} ^{N}} we have ⟨ X α | X β ⟩ = 0 {\displaystyle \langle X^{\alpha }|X^{\beta }\rangle =0} if α ≠ β {\displaystyle \alpha \neq \beta } For every α ∈ N N {\displaystyle \alpha \in \mathbb {N} ^{N}} we define | | X α | | 2 = | α | ! α ! , {\displaystyle ||X^{\alpha }||^{2}={\frac {|\alpha |!}{\alpha !}},} where we use the following notation: if α = ( α 1 , … , α N ) ∈ N N {\displaystyle \alpha =(\alpha _{1},\dots ,\alpha _{N})\in \mathbb {N} ^{N}} , we write | α | = Σ i = 1 N α i {\displaystyle |\alpha |=\Sigma _{i=1}^{N}\alpha _{i}} and α ! = Π i = 1 N ( α i ! ) {\displaystyle \alpha !=\Pi _{i=1}^{N}(\alpha _{i}!)} and X α = Π i = 1 N X i α i . {\displaystyle X^{\alpha }=\Pi _{i=1}^{N}X_{i}^{\alpha _{i}}.} The most remarkable property of this norm is the Bombieri inequality: Let P , Q {\displaystyle P,Q} be two homogeneous polynomials respectively of degree d ∘ ( P ) {\displaystyle d^{\circ }(P)} and d ∘ ( Q ) {\displaystyle d^{\circ }(Q)} with N {\displaystyle N} variables, then, the following inequality holds: d ∘ ( P ) ! d ∘ ( Q ) ! ( d ∘ ( P ) + d ∘ ( Q ) ) ! | | P | | 2 | | Q | | 2 ≤ | | P ⋅ Q | | 2 ≤ | | P | | 2 | | Q | | 2 . {\displaystyle {\frac {d^{\circ }(P)!d^{\circ }(Q)!}{(d^{\circ }(P)+d^{\circ }(Q))!}}||P||^{2}\,||Q||^{2}\leq ||P\cdot Q||^{2}\leq ||P||^{2}\,||Q||^{2}.} In the above statement, the Bombieri inequality is the left-hand side inequality; the right-hand side inequality means that the Bombieri norm is a norm of the algebra of polynomials under multiplication. The Bombieri inequality implies that the product of two polynomials cannot be arbitrarily small, and this lower-bound is fundamental in applications like polynomial factorization (or in Enflo's construction of an operator without an invariant subspace). ==== Applications ==== Enflo's idea of "concentration of polynomials at low degrees" has led to important publications in number theory algebraic and Diophantine geometry, and polynomial factorization. == Mathematical biology: Population dynamics == In applied mathematics, Per Enflo has published several papers in mathematical biology, specifically in population dynamics. === Human evolution === Enflo has also published in population genetics and paleoanthropology. Today, all humans belong to one population of Homo sapiens sapiens, which is individed by species barrier. However, according to the "Out of Africa" model this is not the first species of hominids: the first species of genus Homo, Homo habilis, evolved in East Africa at least 2 Ma, and members of this species populated different parts of Africa in a relatively short time. Homo erectus evolved more than 1.8 Ma, and by 1.5 Ma had spread throughout the Old World. Anthropologists have been divided as to whether current human population evolved as one interconnected population (as postulated by the Multiregional Evolution hypothesis), or evolved only in East Africa, speciated, and then migrating out of Africa and replaced human populations in Eurasia (called the "Out of Africa" Model or the "Complete Replacement" Model). Neanderthals and modern humans coexisted in Europe for several thousand years, but the duration of this period is uncertain. Modern humans may have first migrated to Europe 40–43,000 years ago. Neanderthals may have lived as recently as 24,000 years ago in refugia on the south coast of the Iberian peninsula such as Gorham's Cave. Inter-stratification of Neanderthal and modern human remains has been suggested, but is disputed. With Hawks and Wolpoff, Enflo published an explanation of fossil evidence on the DNA of Neanderthal and modern humans. This article tries to resolve a debate in the evolution of modern humans between theories suggesting either multiregional and single African origins. In particular, the extinction of Neanderthals could have happened due to waves of modern humans entered Europe – in technical terms, due to "the continuous influx of modern human DNA into the Neandertal gene pool." Enflo has also written about the population dynamics of zebra mussels in Lake Erie. == Piano == Per Enflo is also a concert pianist. A child prodigy in both music and mathematics, Enflo won the Swedish competition for young pianists at age 11 in 1956, and he won the same competition in 1961. At age 12, Enflo appeared as a soloist with the Royal Opera Orchestra of Sweden. He debuted in the Stockholm Concert Hall in 1963. Enflo's teachers included Bruno Seidlhofer, Géza Anda, and Gottfried Boon (who himself was a student of Arthur Schnabel). In 1999 Enflo competed in the first annual Van Cliburn Foundation's International Piano Competition for Outstanding Amateurs Archived 2009-04-19 at the Wayback Machine. Enflo performs regularly around Kent and in a Mozart series in Columbus, Ohio (with the Triune Festival Orchestra). His solo piano recitals have appeared on the Classics Network of the radio station WOSU, which is sponsored by Ohio State University. == References == === Notes === "Recipients of 2005 Distinguished Scholar Award at Kent State University Announced", eInside, 2005-4-11. Retrieved on February 4, 2007. === Bibliography === Enflo, Per. (1970) Investigations on Hilbert's fifth problem for non locally compact groups (Stockholm University). Enflo's thesis contains reprints of exactly five papers: Enflo, Per (1969a). "Topological groups in which multiplication on one side is differentiable or linear". Math. Scand. 24: 195–197. doi:10.7146/math.scand.a-10930. Per Enflo (1969). "On the nonexistence of uniform homeomorphisms between Lp spaces". Ark. Mat. 8 (2): 103–5. Bibcode:1970ArM.....8..103E. doi:10.1007/BF02589549. Enflo, Per (1969b). "On a problem of Smirnov". Ark. Mat. 8 (2): 107–109. Bibcode:1970ArM.....8..107E. doi:10.1007/bf02589550. Enflo, Per (1970a). "Uniform structures and square roots in topological groups I". Israel Journal of Mathematics. 8 (3): 230–252. doi:10.1007/BF02771560. S2CID 189773170. Enflo, Per (1970b). "Uniform structures and square roots in topological groups II". Israel Journal of Mathematics. 8 (3): 253–272. doi:10.1007/BF02771561. S2CID 121193430. Enflo, Per. 1976. Uniform homeomorphisms between Banach spaces. Séminaire Maurey-Schwartz (1975—1976), Espaces, L p {\displaystyle L^{p}} , applications radonifiantes et géométrie des espaces de Banach, Exp. No. 18, 7 pp. Centre Math., École Polytech., Palaiseau. MR0477709 (57 #17222) [Highlights of papers on Hilbert's fifth problem and on independent results of Martin Ribe, another student of Hans Rådström] Enflo, Per (1972). "Banach spaces which can be given an equivalent uniformly convex norm". Israel Journal of Mathematics. 13 (3–4): 281–288. doi:10.1007/BF02762802. MR 0336297. S2CID 120895135. Enflo, Per (1973). "A counterexample to the approximation problem in Banach spaces". Acta Mathematica. 130: 309–317. doi:10.1007/BF02392270. MR 0402468. Enflo, Per (1976). "On the invariant subspace problem in Banach spaces" (PDF). Séminaire Maurey--Schwartz (1975--1976) Espaces Lp, applications radonifiantes et géométrie des espaces de Banach, Exp. Nos. 14–15. Centre Math., École Polytech., Palaiseau. pp. 1–7. MR 0473871. Enflo, Per (1987). "On the invariant subspace problem for Banach spaces". Acta Mathematica. 158 (3): 213–313. doi:10.1007/BF02392260. ISSN 0001-5962. MR 0892591. Beauzamy, Bernard; Bombieri, Enrico; Enflo, Per; Montgomery, Hugh L. (1990). "Products of polynomials in many variables". Journal of Number Theory. 36 (2): 219–245. doi:10.1016/0022-314X(90)90075-3. hdl:2027.42/28840. MR 1072467. Beauzamy, Bernard; Enflo, Per; Wang, Paul (October 1994). "Quantitative Estimates for Polynomials in One or Several Variables: From Analysis and Number Theory to Symbolic and Massively Parallel Computation". Mathematics Magazine. 67 (4): 243–257. JSTOR 2690843. (accessible to readers with undergraduate mathematics) P. Enflo, John D. Hawks, M. Wolpoff. "A simple reason why Neanderthal ancestry can be consistent with current DNA information". American Journal Physical Anthropology, 2001 Enflo, Per; Lomonosov, Victor (2001). "Some aspects of the invariant subspace problem". Handbook of the geometry of Banach spaces. Vol. I. Amsterdam: North-Holland. pp. 533–559. Bartle, R. G. (1977). "Review of Per Enflo's "A counterexample to the approximation problem in Banach spaces" Acta Mathematica 130 (1973), 309–317". Mathematical Reviews. 130: 309–317. doi:10.1007/BF02392270. MR 0402468. Beauzamy, Bernard (1985) [1982]. Introduction to Banach Spaces and their Geometry (Second revised ed.). North-Holland. ISBN 0-444-86416-4. MR 0889253. Beauzamy, Bernard (1988). Introduction to Operator Theory and Invariant Subspaces. North Holland. ISBN 0-444-70521-X. MR 0967989. Enrico Bombieri and Walter Gubler (2006). Heights in Diophantine Geometry. Cambridge U. P. ISBN 0-521-84615-3. Roger B. Eggleton (1984). "Review of Mauldin's The Scottish Book: Mathematics from the Scottish Café". Mathematical Reviews. MR 0666400. Grothendieck, A.: Produits tensoriels topologiques et espaces nucleaires. Memo. Amer. Math. Soc. 16 (1955). Halmos, Paul R. (1978). "Schauder bases". American Mathematical Monthly. 85 (4): 256–257. doi:10.2307/2321165. JSTOR 2321165. MR 0488901. Paul R. Halmos, "Has progress in mathematics slowed down?" Amer. Math. Monthly 97 (1990), no. 7, 561–588. MR1066321 William B. Johnson "Complementably universal separable Banach spaces" in Robert G. Bartle (ed.), 1980 Studies in functional analysis, Mathematical Association of America. Kałuża, Roman (1996). Ann Kostant and Wojbor Woyczyński (ed.). Through a Reporter's Eyes: The Life of Stefan Banach. Birkhäuser. ISBN 0-8176-3772-9. MR 1392949. Knuth, Donald E (1997). "4.6.2 Factorization of Polynomials". Seminumerical Algorithms. The Art of Computer Programming. Vol. 2 (Third ed.). Reading, Massachusetts: Addison-Wesley. pp. 439–461, 678–691. ISBN 0-201-89684-2. Kwapień, S. "On Enflo's example of a Banach space without the approximation property". Séminaire Goulaouic-Schwartz 1972—1973: Équations aux dérivées partielles et analyse fonctionnelle, Exp. No. 8, 9 pp. Centre de Math., École Polytech., Paris, 1973. MR407569 Lindenstrauss, Joram and Benyamini, Yoav. Geometric nonlinear functional analysis Colloquium publications, 48. American Mathematical Society. Lindenstrauss, J.; Tzafriri, L.: Classical Banach Spaces I, Sequence spaces, 1977. Springer-Verlag. Matoušek, Jiří (2002). Lectures on Discrete Geometry. Graduate Texts in Mathematics. Springer-Verlag. ISBN 978-0-387-95373-1.. R. Daniel Mauldin, ed. (1981). The Scottish Book: Mathematics from the Scottish Café (Including selected papers presented at the Scottish Book Conference held at North Texas State University, Denton, Tex., May 1979). Boston, Mass.: Birkhäuser. pp. xiii+268 pp. (2 plates). ISBN 3-7643-3045-7. MR 0666400. Nedevski, P.; Trojanski, S. (1973). "P. Enflo solved in the negative Banach's problem on the existence of a basis for every separable Banach space". Fiz.-Mat. Spis. Bulgar. Akad. Nauk. 16 (49): 134–138. MR 0458132. Pietsch, Albrecht (2007). History of Banach spaces and linear operators]. Boston, MA: Birkhäuser Boston, Inc. pp. xxiv+855 pp. ISBN 978-0-8176-4367-6. MR 2300779. Pisier, Gilles (1975). "Martingales with values in uniformly convex spaces". Israel Journal of Mathematics. 20 (3–4): 326–350. doi:10.1007/BF02760337. MR 0394135. S2CID 120947324. Heydar Radjavi & Peter Rosenthal (March 1982). "The invariant subspace problem". The Mathematical Intelligencer. 4 (1): 33–37. doi:10.1007/BF03022994. S2CID 122811130. Karen Saxe, Beginning Functional Analysis, Undergraduate Texts in Mathematics, 2002 Springer-Verlag, New York. (Pages 122–123 sketch a biography of Per Enflo.) Schmidt, Wolfgang M. (1980 [1996 with minor corrections]) Diophantine approximation. Lecture Notes in Mathematics 785. Springer. Singer, Ivan. Bases in Banach spaces. II. Editura Academiei Republicii Socialiste România, Bucharest; Springer-Verlag, Berlin-New York, 1981. viii+880 pp. ISBN 3-540-10394-5. MR610799 Yadav, B. S. (2005). "The present state and heritages of the invariant subspace problem". Milan Journal of Mathematics. 73: 289–316. doi:10.1007/s00032-005-0048-7. ISSN 1424-9286. MR 2175046. S2CID 121068326. == External sources == Biography of Per Enflo at Canisius College Homepage of Per Enflo at Kent State University Enflo, Per (25 April 2011). "Personal notes, in my own words". perenflo.com. Archived from the original on 26 April 2012. Retrieved 13 December 2011. Per Enflo at the Mathematics Genealogy Project
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Wikipedia:Per Lindström#0
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Per "Pelle" Lindström (9 April 1936 – 21 August 2009, Gothenburg) was a Swedish logician, after whom Lindström's theorem and the Lindström quantifier are named. (He also independently discovered Ehrenfeucht–Fraïssé games.) He was one of the key followers of Lars Svenonius. Lindström was awarded a PhD from the University of Gothenburg in 1966. His thesis was titled Some Results in the Theory of Models of First Order Languages. A festschrift for Lindström was published in 1986. == Selected publications == Per Lindström, First Order Predicate Logic with Generalized Quantifiers, Theoria 32, 1966, 186–195. Per Lindström, On Extensions of Elementary Logic, Theoria 35, 1969, 1–11. Per Lindström (1997). Aspects of incompleteness. Springer-Verlag. ISBN 978-3-540-63213-9.; 2nd ed. published by ASL in 2003, ISBN 978-1-56881-173-4 == References == == Further reading == Väänänen, J.; Westerståhl, D. (2010). "In Memoriam: Per Lindström" (PDF). Theoria. 76 (2): 100–107. doi:10.1111/j.1755-2567.2010.01069.x. == External links == Per Lindström at the Mathematics Genealogy Project
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Wikipedia:Percy Deift#0
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Percy Alec Deift (born September 10, 1945) is a mathematician known for his work on spectral theory, integrable systems, random matrix theory and Riemann–Hilbert problems. == Life == Deift was born in Durban, South Africa, where he obtained degrees in chemical engineering, physics, and mathematics, and received a Ph.D. in mathematical physics from Princeton University in 1977. He is a Silver Professor at the Courant Institute of Mathematical Sciences, New York University. == Honors and awards == Deift is a fellow of the American Mathematical Society (elected 2012), a member of the American Academy of Arts and Sciences (elected 2003), and of the U.S. National Academy of Sciences (elected 2009). He is a co-winner of the 1998 Pólya Prize, and was named a Guggenheim Fellow in 1999. He gave an invited address at the International Congress of Mathematicians in Berlin in 1998 and plenary addresses in 2006 at the International Congress of Mathematicians in Madrid and at the International Congress on Mathematical Physics in Rio de Janeiro. Deift gave the Gibbs Lecture at the Joint Meeting of the American Mathematical Society in 2009. Along with Michael Aizenman and Giovanni Gallavotti, he won the Henri Poincare Prize in 2018. == Selected works == with Eugene Trubowitz: Inverse scattering on the line, Communications on pure and applied Mathematics, vol. 32, 1979, pp. 121–251 doi:10.1002/cpa.3160320202 with Fernando Lund, E. Trubowitz: Deift, P; Lund, F; Trubowitz, E (February 1980). "Nonlinear wave equations and constrained harmonic motion" (PDF). Proc Natl Acad Sci U S A. 77 (2): 716–719. Bibcode:1980PNAS...77..716D. doi:10.1073/pnas.77.2.716. PMC 348351. PMID 16592777. with Richard Beals, Carlos Tomei: Direct and inverse scattering on the line, AMS, 1988 with Luen-Chau Li, C. Tomei: Loop groups, discrete versions of some classical integrable systems, and rank 2 extensions, AMS, 1992 with K. T-R McLaughlin: A continuum limit of the Toda lattice, AMS, 1998 Orthogonal polynomials and random matrices: a Riemann-Hilbert approach, AMS (American Mathematical Society), 2000 (and Courant Institute, 1999) with Dmitri Gioev: Random matrix theory: invariant embeddings and universality, AMS, 2009 with Jinho Baik and Toufic Suidan Combinatorics and Random Matrix Theory. American Mathematical Society. 22 June 2016. ISBN 978-0-8218-4841-8. == See also == Riemann–Hilbert problems random matrix theory integrable systems == References == == External links == Percy Deift personal webpage, Courant Institute, New York University Percy Alec Deift, Mathematics Genealogy Project
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Wikipedia:Percy Nunn#0
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Sir Thomas Percy Nunn (28 December 1870 – 12 December 1944) was a British educationalist, Professor of Education, 1913–36 at Institute of Education, University of London. He was knighted in 1930. == Early life == Nunn was born in Bristol in 1870. His grandfather and father were schoolmasters. He was interested in making of mathematical instruments and writing plays. He got his education at Bristol University College. He received his B.A in 1895. == Career == His career started as a secondary school teacher at grammar school in London in 1891. From 1891 till 1901 he developed methods of teaching which revolutionised the teaching of mathematics in the UK. In 1903 he became a member of the staff in the London Day Training college. He worked as a part-time lecturer. In 1915 he attended the third Conference of the New Ideals in Education in Stratford where a group including Belle Rennie, William Mather and Nunn agreed that a new teacher training facility was required. This would lead to the Gipsy Hill College in South London which in time became a key part of Kingston University. Nunn became a professor of education at the University of London. In 1922 was appointed Principal. He was the president of the Aristotelian Society from 1923-1924. == Selected publications == The teaching of algebra (including trigonometry). 1914. Education: its data and first principles. 1920. Relativity and gravitation: an elementary treatise upon Einstein's theory. 1923. == References == Richard Aldrich, 'Nunn, Sir (Thomas) Percy (1870–1944)’, Oxford Dictionary of National Biography, Oxford University Press, 2004 accessed Sir (Thomas) Percy Nunn (1870–1944): doi:10.1093/ref:odnb/35268 * Tibble, J. W. (1961). "Sir Percy Nunn: 1870-1944". British Journal of Educational Studies. 10 (1): 58–75. doi:10.2307/3118702. JSTOR 3118702. 'NUNN, Sir Percy’, Who Was Who, A & C Black, 1920–2008; online edn, Oxford University Press, Dec 2007, retrieved 26 January 2012 http://www.aim25.ac.uk/cgi-bin/vcdf/detail?coll_id=2542&inst_id=5
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Wikipedia:Perdita Stevens#0
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Perdita Emma Stevens (born 1966) is a British mathematician, theoretical computer scientist, and software engineer who holds a personal chair in the mathematics of software engineering as part of the School of Informatics at the University of Edinburgh. Her research includes work on model-driven engineering, including model transformation, model checking, and the Unified Modeling Language. == Education and career == Stevens read mathematics at the University of Cambridge, earning a bachelor's degree in 1987. She went to the University of Warwick for graduate study in abstract algebra, earning a master's degree in 1988 and completing a PhD in 1992. Her doctoral dissertation, Integral Forms for Weyl Modules of G L ( 2 , Q ) {\displaystyle \mathrm {GL} (2,\mathrm {Q} )} , was supervised by Sandy Green. After working in industry as a software engineer, Stevens joined the Department of Computer Science at the University of Edinburgh in 1984. She became a reader there in 2003 and in 2014 was given a personal chair as Professor of Mathematics of Software Engineering. == Books == Stevens is the author of books including: Using UML: Software Engineering with Objects and Components (with Rob Pooley, Addison-Wesley, 1999; 2nd ed., 2006) How to Write Good Programs: A Guide for Students (Cambridge University Press, 2020) == References == == External links == Perdita Stevens publications indexed by Google Scholar Home page
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Wikipedia:Perfect complex#0
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In algebra, a perfect complex of modules over a commutative ring A is an object in the derived category of A-modules that is quasi-isomorphic to a bounded complex of finite projective A-modules. A perfect module is a module that is perfect when it is viewed as a complex concentrated at degree zero. For example, if A is Noetherian, a module over A is perfect if and only if it is finitely generated and of finite projective dimension. == Other characterizations == Perfect complexes are precisely the compact objects in the unbounded derived category D ( A ) {\displaystyle D(A)} of A-modules. They are also precisely the dualizable objects in this category. A compact object in the ∞-category of (say right) module spectra over a ring spectrum is often called perfect; see also module spectrum. == Pseudo-coherent sheaf == When the structure sheaf O X {\displaystyle {\mathcal {O}}_{X}} is not coherent, working with coherent sheaves has awkwardness (namely the kernel of a finite presentation can fail to be coherent). Because of this, SGA 6 Expo I introduces the notion of a pseudo-coherent sheaf. By definition, given a ringed space ( X , O X ) {\displaystyle (X,{\mathcal {O}}_{X})} , an O X {\displaystyle {\mathcal {O}}_{X}} -module is called pseudo-coherent if for every integer n ≥ 0 {\displaystyle n\geq 0} , locally, there is a free presentation of finite type of length n; i.e., L n → L n − 1 → ⋯ → L 0 → F → 0 {\displaystyle L_{n}\to L_{n-1}\to \cdots \to L_{0}\to F\to 0} . A complex F of O X {\displaystyle {\mathcal {O}}_{X}} -modules is called pseudo-coherent if, for every integer n, there is locally a quasi-isomorphism L → F {\displaystyle L\to F} where L has degree bounded above and consists of finite free modules in degree ≥ n {\displaystyle \geq n} . If the complex consists only of the zero-th degree term, then it is pseudo-coherent if and only if it is so as a module. Roughly speaking, a pseudo-coherent complex may be thought of as a limit of perfect complexes. == See also == Hilbert–Burch theorem elliptic complex (related notion; discussed at SGA 6 Exposé II, Appendix II.) == References == Ben-Zvi, David; Francis, John; Nadler, David (2010), "Integral transforms and Drinfeld centers in derived algebraic geometry", Journal of the American Mathematical Society, 23 (4): 909–966, arXiv:0805.0157, doi:10.1090/S0894-0347-10-00669-7, MR 2669705, S2CID 2202294 == Bibliography == Berthelot, Pierre; Alexandre Grothendieck; Luc Illusie, eds. (1971). Séminaire de Géométrie Algébrique du Bois Marie - 1966-67 - Théorie des intersections et théorème de Riemann-Roch - (SGA 6) (Lecture notes in mathematics 225). Lecture Notes in Mathematics (in French). Vol. 225. Berlin; New York: Springer-Verlag. xii+700. doi:10.1007/BFb0066283. ISBN 978-3-540-05647-8. MR 0354655. Kerz, Moritz; Strunk, Florian; Tamme, Georg (2018). "Algebraic K-theory and descent for blow-ups". Inventiones Mathematicae. 211 (2): 523–577. arXiv:1611.08466. Bibcode:2018InMat.211..523K. doi:10.1007/s00222-017-0752-2. Lurie, Jacob (2014). "Algebraic K-Theory and Manifold Topology (Math 281), Lecture 19: K-Theory of Ring Spectra" (PDF). == External links == "Determinantal identities for perfect complexes". MathOverflow. "An alternative definition of pseudo-coherent complex". MathOverflow. "15.74 Perfect complexes". The Stacks project. "perfect module". ncatlab.org.
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Wikipedia:Periodic continued fraction#0
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In mathematics, an infinite periodic continued fraction is a simple continued fraction that can be placed in the form x = a 0 + 1 a 1 + 1 a 2 + 1 ⋱ a k + 1 a k + 1 + ⋱ ⋱ a k + m − 1 + 1 a k + m + 1 a k + 1 + 1 a k + 2 + ⋱ {\displaystyle x=a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{\quad \ddots \quad a_{k}+{\cfrac {1}{a_{k+1}+{\cfrac {\ddots }{\quad \ddots \quad a_{k+m-1}+{\cfrac {1}{a_{k+m}+{\cfrac {1}{a_{k+1}+{\cfrac {1}{a_{k+2}+{\ddots }}}}}}}}}}}}}}}}}} where the initial block [ a 0 ; a 1 , … , a k ] {\displaystyle [a_{0};a_{1},\dots ,a_{k}]} of k+1 partial denominators is followed by a block [ a k + 1 , a k + 2 , … , a k + m ] {\displaystyle [a_{k+1},a_{k+2},\dots ,a_{k+m}]} of m partial denominators that repeats ad infinitum. For example, 2 {\displaystyle {\sqrt {2}}} can be expanded to the periodic continued fraction [ 1 ; 2 , 2 , 2 , . . . ] {\displaystyle [1;2,2,2,...]} . This article considers only the case of periodic regular continued fractions. In other words, the remainder of this article assumes that all the partial denominators ai (i ≥ 1) are positive integers. The general case, where the partial denominators ai are arbitrary real or complex numbers, is treated in the article convergence problem. == Purely periodic and periodic fractions == Since all the partial numerators in a regular continued fraction are equal to unity we can adopt a shorthand notation in which the continued fraction shown above is written as x = [ a 0 ; a 1 , a 2 , … , a k , a k + 1 , a k + 2 , … , a k + m , a k + 1 , a k + 2 , … , a k + m , … ] = [ a 0 ; a 1 , a 2 , … , a k , a k + 1 , a k + 2 , … , a k + m ¯ ] {\displaystyle {\begin{aligned}x&=[a_{0};a_{1},a_{2},\dots ,a_{k},a_{k+1},a_{k+2},\dots ,a_{k+m},a_{k+1},a_{k+2},\dots ,a_{k+m},\dots ]\\&=[a_{0};a_{1},a_{2},\dots ,a_{k},{\overline {a_{k+1},a_{k+2},\dots ,a_{k+m}}}]\end{aligned}}} where, in the second line, a vinculum marks the repeating block. Some textbooks use the notation x = [ a 0 ; a 1 , a 2 , … , a k , a ˙ k + 1 , a k + 2 , … , a ˙ k + m ] {\displaystyle {\begin{aligned}x&=[a_{0};a_{1},a_{2},\dots ,a_{k},{\dot {a}}_{k+1},a_{k+2},\dots ,{\dot {a}}_{k+m}]\end{aligned}}} where the repeating block is indicated by dots over its first and last terms. If the initial non-repeating block is not present – that is, if k = -1, a0 = am and x = [ a 0 ; a 1 , a 2 , … , a m − 1 ¯ ] , {\displaystyle x=[{\overline {a_{0};a_{1},a_{2},\dots ,a_{m-1}}}],} the regular continued fraction x is said to be purely periodic. For example, the regular continued fraction [ 1 ; 1 , 1 , 1 , … ] {\displaystyle [1;1,1,1,\dots ]} of the golden ratio φ is purely periodic, while the regular continued fraction [ 1 ; 2 , 2 , 2 , … ] {\displaystyle [1;2,2,2,\dots ]} of 2 {\displaystyle {\sqrt {2}}} is periodic, but not purely periodic. However, the regular continued fraction [ 2 ; 2 , 2 , 2 , … ] {\displaystyle [2;2,2,2,\dots ]} of the silver ratio σ = 2 + 1 {\displaystyle \sigma ={\sqrt {2}}+1} is purely periodic. == As unimodular matrices == Periodic continued fractions are in one-to-one correspondence with the real quadratic irrationals. The correspondence is explicitly provided by Minkowski's question-mark function. That article also reviews tools that make it easy to work with such continued fractions. Consider first the purely periodic part x = [ 0 ; a 1 , a 2 , … , a m ¯ ] , {\displaystyle x=[0;{\overline {a_{1},a_{2},\dots ,a_{m}}}],} This can, in fact, be written as x = α x + β γ x + δ {\displaystyle x={\frac {\alpha x+\beta }{\gamma x+\delta }}} with the α , β , γ , δ {\displaystyle \alpha ,\beta ,\gamma ,\delta } being integers, and satisfying α δ − β γ = 1. {\displaystyle \alpha \delta -\beta \gamma =1.} Explicit values can be obtained by writing S = ( 1 0 1 1 ) {\displaystyle S={\begin{pmatrix}1&0\\1&1\end{pmatrix}}} which is termed a "shift", so that S n = ( 1 0 n 1 ) {\displaystyle S^{n}={\begin{pmatrix}1&0\\n&1\end{pmatrix}}} and similarly a reflection, given by T ↦ ( − 1 1 0 1 ) {\displaystyle T\mapsto {\begin{pmatrix}-1&1\\0&1\end{pmatrix}}} so that T 2 = I {\displaystyle T^{2}=I} . Both of these matrices are unimodular, arbitrary products remain unimodular. Then, given x {\displaystyle x} as above, the corresponding matrix is of the form S a 1 T S a 2 T ⋯ T S a m = ( α β γ δ ) {\displaystyle S^{a_{1}}TS^{a_{2}}T\cdots TS^{a_{m}}={\begin{pmatrix}\alpha &\beta \\\gamma &\delta \end{pmatrix}}} and one has x = [ 0 ; a 1 , a 2 , … , a m ¯ ] = α x + β γ x + δ {\displaystyle x=[0;{\overline {a_{1},a_{2},\dots ,a_{m}}}]={\frac {\alpha x+\beta }{\gamma x+\delta }}} as the explicit form. As all of the matrix entries are integers, this matrix belongs to the modular group S L ( 2 , Z ) . {\displaystyle SL(2,\mathbb {Z} ).} == Relation to quadratic irrationals == A quadratic irrational number is an irrational real root of the quadratic equation a x 2 + b x + c = 0 {\displaystyle ax^{2}+bx+c=0} where the coefficients a, b, and c are integers, and the discriminant, b 2 − 4 a c {\displaystyle b^{2}-4ac} , is greater than zero. By the quadratic formula, every quadratic irrational can be written in the form ζ = P + D Q {\displaystyle \zeta ={\frac {P+{\sqrt {D}}}{Q}}} where P, D, and Q are integers, D > 0 is not a perfect square (but not necessarily square-free), and Q divides the quantity P 2 − D {\displaystyle P^{2}-D} (for example ( 6 + 8 ) / 4 {\displaystyle (6+{\sqrt {8}})/4} ). Such a quadratic irrational may also be written in another form with a square-root of a square-free number (for example ( 3 + 2 ) / 2 {\displaystyle (3+{\sqrt {2}})/2} ) as explained for quadratic irrationals. By considering the complete quotients of periodic continued fractions, Euler was able to prove that if x is a regular periodic continued fraction, then x is a quadratic irrational number. The proof is straightforward. From the fraction itself, one can construct the quadratic equation with integral coefficients that x must satisfy. Lagrange proved the converse of Euler's theorem: if x is a quadratic irrational, then the regular continued fraction expansion of x is periodic. Given a quadratic irrational x one can construct m different quadratic equations, each with the same discriminant, that relate the successive complete quotients of the regular continued fraction expansion of x to one another. Since there are only finitely many of these equations (the coefficients are bounded), the complete quotients (and also the partial denominators) in the regular continued fraction that represents x must eventually repeat. == Reduced surds == The quadratic surd ζ = P + D Q {\displaystyle \zeta ={\frac {P+{\sqrt {D}}}{Q}}} is said to be reduced if ζ > 1 {\displaystyle \zeta >1} and its conjugate η = P − D Q {\displaystyle \eta ={\frac {P-{\sqrt {D}}}{Q}}} satisfies the inequalities − 1 < η < 0 {\displaystyle -1<\eta <0} . For instance, the golden ratio ϕ = ( 1 + 5 ) / 2 = 1.618033... {\displaystyle \phi =(1+{\sqrt {5}})/2=1.618033...} is a reduced surd because it is greater than one and its conjugate ( 1 − 5 ) / 2 = − 0.618033... {\displaystyle (1-{\sqrt {5}})/2=-0.618033...} is greater than −1 and less than zero. On the other hand, the square root of two 2 = ( 0 + 8 ) / 2 {\displaystyle {\sqrt {2}}=(0+{\sqrt {8}})/2} is greater than one but is not a reduced surd because its conjugate − 2 = ( 0 − 8 ) / 2 {\displaystyle -{\sqrt {2}}=(0-{\sqrt {8}})/2} is less than −1. Galois proved that the regular continued fraction which represents a quadratic surd ζ is purely periodic if and only if ζ is a reduced surd. In fact, Galois showed more than this. He also proved that if ζ is a reduced quadratic surd and η is its conjugate, then the continued fractions for ζ and for (−1/η) are both purely periodic, and the repeating block in one of those continued fractions is the mirror image of the repeating block in the other. In symbols we have ζ = [ a 0 ; a 1 , a 2 , … , a m − 1 ¯ ] − 1 η = [ a m − 1 ; a m − 2 , a m − 3 , … , a 0 ¯ ] {\displaystyle {\begin{aligned}\zeta &=[{\overline {a_{0};a_{1},a_{2},\dots ,a_{m-1}}}]\\[3pt]{\frac {-1}{\eta }}&=[{\overline {a_{m-1};a_{m-2},a_{m-3},\dots ,a_{0}}}]\,\end{aligned}}} where ζ is any reduced quadratic surd, and η is its conjugate. From these two theorems of Galois a result already known to Lagrange can be deduced. If r > 1 is a rational number that is not a perfect square, then r = [ a 0 ; a 1 , a 2 , … , a 2 , a 1 , 2 a 0 ¯ ] . {\displaystyle {\sqrt {r}}=[a_{0};{\overline {a_{1},a_{2},\dots ,a_{2},a_{1},2a_{0}}}].} In particular, if n is any non-square positive integer, the regular continued fraction expansion of √n contains a repeating block of length m, in which the first m − 1 partial denominators form a palindromic string. == Length of the repeating block == By analyzing the sequence of combinations P n + D Q n {\displaystyle {\frac {P_{n}+{\sqrt {D}}}{Q_{n}}}} that can possibly arise when ζ = P + D Q {\displaystyle \zeta ={\frac {P+{\sqrt {D}}}{Q}}} is expanded as a regular continued fraction, Lagrange showed that the largest partial denominator ai in the expansion is less than 2 D {\displaystyle 2{\sqrt {D}}} , and that the length of the repeating block is less than 2D. More recently, sharper arguments based on the divisor function have shown that the length of the repeating block for a quadratic surd of discriminant D is on the order of O ( D ln D ) . {\displaystyle {\mathcal {O}}({\sqrt {D}}\ln {D}).} === Canonical form and repetend === The following iterative algorithm can be used to obtain the continued fraction expansion in canonical form (S is any natural number that is not a perfect square): m 0 = 0 {\displaystyle m_{0}=0\,\!} d 0 = 1 {\displaystyle d_{0}=1\,\!} a 0 = ⌊ S ⌋ {\displaystyle a_{0}=\left\lfloor {\sqrt {S}}\right\rfloor \,\!} m n + 1 = d n a n − m n {\displaystyle m_{n+1}=d_{n}a_{n}-m_{n}\,\!} d n + 1 = S − m n + 1 2 d n {\displaystyle d_{n+1}={\frac {S-m_{n+1}^{2}}{d_{n}}}\,\!} a n + 1 = ⌊ S + m n + 1 d n + 1 ⌋ = ⌊ a 0 + m n + 1 d n + 1 ⌋ . {\displaystyle a_{n+1}=\left\lfloor {\frac {{\sqrt {S}}+m_{n+1}}{d_{n+1}}}\right\rfloor =\left\lfloor {\frac {a_{0}+m_{n+1}}{d_{n+1}}}\right\rfloor \!.} Notice that mn, dn, and an are always integers. The algorithm terminates when this triplet is the same as one encountered before. The algorithm can also terminate on ai when ai = 2 a0, which is easier to implement. The expansion will repeat from then on. The sequence [ a 0 ; a 1 , a 2 , a 3 , … ] {\displaystyle [a_{0};a_{1},a_{2},a_{3},\dots ]} is the continued fraction expansion: S = a 0 + 1 a 1 + 1 a 2 + 1 a 3 + ⋱ {\displaystyle {\sqrt {S}}=a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{a_{3}+\,\ddots }}}}}}} ==== Example ==== To obtain √114 as a continued fraction, begin with m0 = 0; d0 = 1; and a0 = 10 (102 = 100 and 112 = 121 > 114 so 10 chosen). 114 = 114 + 0 1 = 10 + 114 − 10 1 = 10 + ( 114 − 10 ) ( 114 + 10 ) 114 + 10 = 10 + 114 − 100 114 + 10 = 10 + 1 114 + 10 14 . {\displaystyle {\begin{aligned}{\sqrt {114}}&={\frac {{\sqrt {114}}+0}{1}}=10+{\frac {{\sqrt {114}}-10}{1}}=10+{\frac {({\sqrt {114}}-10)({\sqrt {114}}+10)}{{\sqrt {114}}+10}}\\&=10+{\frac {114-100}{{\sqrt {114}}+10}}=10+{\frac {1}{\frac {{\sqrt {114}}+10}{14}}}.\end{aligned}}} m 1 = d 0 ⋅ a 0 − m 0 = 1 ⋅ 10 − 0 = 10 . {\displaystyle m_{1}=d_{0}\cdot a_{0}-m_{0}=1\cdot 10-0=10\,.} d 1 = S − m 1 2 d 0 = 114 − 10 2 1 = 14 . {\displaystyle d_{1}={\frac {S-m_{1}^{2}}{d_{0}}}={\frac {114-10^{2}}{1}}=14\,.} a 1 = ⌊ a 0 + m 1 d 1 ⌋ = ⌊ 10 + 10 14 ⌋ = ⌊ 20 14 ⌋ = 1 . {\displaystyle a_{1}=\left\lfloor {\frac {a_{0}+m_{1}}{d_{1}}}\right\rfloor =\left\lfloor {\frac {10+10}{14}}\right\rfloor =\left\lfloor {\frac {20}{14}}\right\rfloor =1\,.} So, m1 = 10; d1 = 14; and a1 = 1. 114 + 10 14 = 1 + 114 − 4 14 = 1 + 114 − 16 14 ( 114 + 4 ) = 1 + 1 114 + 4 7 . {\displaystyle {\frac {{\sqrt {114}}+10}{14}}=1+{\frac {{\sqrt {114}}-4}{14}}=1+{\frac {114-16}{14({\sqrt {114}}+4)}}=1+{\frac {1}{\frac {{\sqrt {114}}+4}{7}}}.} Next, m2 = 4; d2 = 7; and a2 = 2. 114 + 4 7 = 2 + 114 − 10 7 = 2 + 14 7 ( 114 + 10 ) = 2 + 1 114 + 10 2 . {\displaystyle {\frac {{\sqrt {114}}+4}{7}}=2+{\frac {{\sqrt {114}}-10}{7}}=2+{\frac {14}{7({\sqrt {114}}+10)}}=2+{\frac {1}{\frac {{\sqrt {114}}+10}{2}}}.} 114 + 10 2 = 10 + 114 − 10 2 = 10 + 14 2 ( 114 + 10 ) = 10 + 1 114 + 10 7 . {\displaystyle {\frac {{\sqrt {114}}+10}{2}}=10+{\frac {{\sqrt {114}}-10}{2}}=10+{\frac {14}{2({\sqrt {114}}+10)}}=10+{\frac {1}{\frac {{\sqrt {114}}+10}{7}}}.} 114 + 10 7 = 2 + 114 − 4 7 = 2 + 98 7 ( 114 + 4 ) = 2 + 1 114 + 4 14 . {\displaystyle {\frac {{\sqrt {114}}+10}{7}}=2+{\frac {{\sqrt {114}}-4}{7}}=2+{\frac {98}{7({\sqrt {114}}+4)}}=2+{\frac {1}{\frac {{\sqrt {114}}+4}{14}}}.} 114 + 4 14 = 1 + 114 − 10 14 = 1 + 14 14 ( 114 + 10 ) = 1 + 1 114 + 10 1 . {\displaystyle {\frac {{\sqrt {114}}+4}{14}}=1+{\frac {{\sqrt {114}}-10}{14}}=1+{\frac {14}{14({\sqrt {114}}+10)}}=1+{\frac {1}{\frac {{\sqrt {114}}+10}{1}}}.} 114 + 10 1 = 20 + 114 − 10 1 = 20 + 14 114 + 10 = 20 + 1 114 + 10 14 . {\displaystyle {\frac {{\sqrt {114}}+10}{1}}=20+{\frac {{\sqrt {114}}-10}{1}}=20+{\frac {14}{{\sqrt {114}}+10}}=20+{\frac {1}{\frac {{\sqrt {114}}+10}{14}}}.} Now, loop back to the second equation above. Consequently, the simple continued fraction for the square root of 114 is 114 = [ 10 ; 1 , 2 , 10 , 2 , 1 , 20 ¯ ] . {\displaystyle {\sqrt {114}}=[10;{\overline {1,2,10,2,1,20}}].\,} (sequence A010179 in the OEIS) √114 is approximately 10.67707 82520. After one expansion of the repetend, the continued fraction yields the rational fraction 21194 1985 {\displaystyle {\frac {21194}{1985}}} whose decimal value is approx. 10.67707 80856, a relative error of 0.0000016% or 1.6 parts in 100,000,000. ==== Generalized continued fraction ==== A more rapid method is to evaluate its generalized continued fraction. From the formula derived there: z = x 2 + y = x + y 2 x + y 2 x + y 2 x + ⋱ = x + 2 x ⋅ y 2 ( 2 z − y ) − y − y 2 2 ( 2 z − y ) − y 2 2 ( 2 z − y ) − ⋱ {\displaystyle {\begin{aligned}{\sqrt {z}}={\sqrt {x^{2}+y}}&=x+{\cfrac {y}{2x+{\cfrac {y}{2x+{\cfrac {y}{2x+\ddots }}}}}}\\&=x+{\cfrac {2x\cdot y}{2(2z-y)-y-{\cfrac {y^{2}}{2(2z-y)-{\cfrac {y^{2}}{2(2z-y)-\ddots }}}}}}\end{aligned}}} and the fact that 114 is 2/3 of the way between 102=100 and 112=121 results in 114 = 1026 3 = 32 2 + 2 3 = 32 3 + 2 / 3 64 + 2 64 + 2 64 + 2 64 + ⋱ = 32 3 + 2 192 + 18 192 + 18 192 + ⋱ , {\displaystyle {\begin{aligned}{\sqrt {114}}={\cfrac {\sqrt {1026}}{3}}={\cfrac {\sqrt {32^{2}+2}}{3}}&={\cfrac {32}{3}}+{\cfrac {2/3}{64+{\cfrac {2}{64+{\cfrac {2}{64+{\cfrac {2}{64+\ddots }}}}}}}}\\&={\cfrac {32}{3}}+{\cfrac {2}{192+{\cfrac {18}{192+{\cfrac {18}{192+\ddots }}}}}},\end{aligned}}} which is simply the aforementioned [ 10 ; 1 , 2 , 10 , 2 , 1 , 20 , 1 , 2 ] {\displaystyle [10;1,2,\,10,2,1,\,20,1,2]} evaluated at every third term. Combining pairs of fractions produces 114 = 32 2 + 2 3 = 32 3 + 64 / 3 2050 − 1 − 1 2050 − 1 2050 − ⋱ = 32 3 + 64 6150 − 3 − 9 6150 − 9 6150 − ⋱ , {\displaystyle {\begin{aligned}{\sqrt {114}}={\cfrac {\sqrt {32^{2}+2}}{3}}&={\cfrac {32}{3}}+{\cfrac {64/3}{2050-1-{\cfrac {1}{2050-{\cfrac {1}{2050-\ddots }}}}}}\\&={\cfrac {32}{3}}+{\cfrac {64}{6150-3-{\cfrac {9}{6150-{\cfrac {9}{6150-\ddots }}}}}},\end{aligned}}} which is now [ 10 ; 1 , 2 , 10 , 2 , 1 , 20 , 1 , 2 ¯ ] {\displaystyle [10;1,2,{\overline {10,2,1,20,1,2}}]} evaluated at the third term and every six terms thereafter. == See also == Continued fraction – Mathematical expression Generalized continued fraction – Mathematical expressionPages displaying short descriptions of redirect targets Hermite's problem Continued fraction method of computing square roots – Algorithms for calculating square rootsPages displaying short descriptions of redirect targets Restricted partial quotients – Analytic series Continued fraction factorization – an integer factorization algorithmPages displaying wikidata descriptions as a fallback == Notes == == References ==
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Wikipedia:Periodic points of complex quadratic mappings#0
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This article describes periodic points of some complex quadratic maps. A map is a formula for computing a value of a variable based on its own previous value or values; a quadratic map is one that involves the previous value raised to the powers one and two; and a complex map is one in which the variable and the parameters are complex numbers. A periodic point of a map is a value of the variable that occurs repeatedly after intervals of a fixed length. These periodic points play a role in the theories of Fatou and Julia sets. == Definitions == Let f c ( z ) = z 2 + c {\displaystyle f_{c}(z)=z^{2}+c\,} be the complex quadric mapping, where z {\displaystyle z} and c {\displaystyle c} are complex numbers. Notationally, f c ( k ) ( z ) {\displaystyle f_{c}^{(k)}(z)} is the k {\displaystyle k} -fold composition of f c {\displaystyle f_{c}} with itself (not to be confused with the k {\displaystyle k} th derivative of f c {\displaystyle f_{c}} )—that is, the value after the k-th iteration of the function f c . {\displaystyle f_{c}.} Thus f c ( k ) ( z ) = f c ( f c ( k − 1 ) ( z ) ) . {\displaystyle f_{c}^{(k)}(z)=f_{c}(f_{c}^{(k-1)}(z)).} Periodic points of a complex quadratic mapping of period p {\displaystyle p} are points z {\displaystyle z} of the dynamical plane such that f c ( p ) ( z ) = z , {\displaystyle f_{c}^{(p)}(z)=z,} where p {\displaystyle p} is the smallest positive integer for which the equation holds at that z. We can introduce a new function: F p ( z , f ) = f c ( p ) ( z ) − z , {\displaystyle F_{p}(z,f)=f_{c}^{(p)}(z)-z,} so periodic points are zeros of function F p ( z , f ) {\displaystyle F_{p}(z,f)} : points z satisfying F p ( z , f ) = 0 , {\displaystyle F_{p}(z,f)=0,} which is a polynomial of degree 2 p . {\displaystyle 2^{p}.} == Number of periodic points == The degree of the polynomial F p ( z , f ) {\displaystyle F_{p}(z,f)} describing periodic points is d = 2 p {\displaystyle d=2^{p}} so it has exactly d = 2 p {\displaystyle d=2^{p}} complex roots (= periodic points), counted with multiplicity. == Stability of periodic points (orbit) - multiplier == The multiplier (or eigenvalue, derivative) m ( f p , z 0 ) = λ {\displaystyle m(f^{p},z_{0})=\lambda } of a rational map f {\displaystyle f} iterated p {\displaystyle p} times at cyclic point z 0 {\displaystyle z_{0}} is defined as: m ( f p , z 0 ) = λ = { f p ′ ( z 0 ) , if z 0 ≠ ∞ 1 f p ′ ( z 0 ) , if z 0 = ∞ {\displaystyle m(f^{p},z_{0})=\lambda ={\begin{cases}f^{p\prime }(z_{0}),&{\mbox{if }}z_{0}\neq \infty \\{\frac {1}{f^{p\prime }(z_{0})}},&{\mbox{if }}z_{0}=\infty \end{cases}}} where f p ′ ( z 0 ) {\displaystyle f^{p\prime }(z_{0})} is the first derivative of f p {\displaystyle f^{p}} with respect to z {\displaystyle z} at z 0 {\displaystyle z_{0}} . Because the multiplier is the same at all periodic points on a given orbit, it is called a multiplier of the periodic orbit. The multiplier is: a complex number; invariant under conjugation of any rational map at its fixed point; used to check stability of periodic (also fixed) points with stability index a b s ( λ ) . {\displaystyle abs(\lambda ).\,} A periodic point is attracting when a b s ( λ ) < 1 ; {\displaystyle abs(\lambda )<1;} super-attracting when a b s ( λ ) = 0 ; {\displaystyle abs(\lambda )=0;} attracting but not super-attracting when 0 < a b s ( λ ) < 1 ; {\displaystyle 0<abs(\lambda )<1;} indifferent when a b s ( λ ) = 1 ; {\displaystyle abs(\lambda )=1;} rationally indifferent or parabolic if λ {\displaystyle \lambda } is a root of unity; irrationally indifferent if a b s ( λ ) = 1 {\displaystyle abs(\lambda )=1} but multiplier is not a root of unity; repelling when a b s ( λ ) > 1. {\displaystyle abs(\lambda )>1.} Periodic points that are attracting are always in the Fatou set; that are repelling are in the Julia set; that are indifferent fixed points may be in one or the other. A parabolic periodic point is in the Julia set. == Period-1 points (fixed points) == === Finite fixed points === Let us begin by finding all finite points left unchanged by one application of f {\displaystyle f} . These are the points that satisfy f c ( z ) = z {\displaystyle f_{c}(z)=z} . That is, we wish to solve z 2 + c = z , {\displaystyle z^{2}+c=z,\,} which can be rewritten as z 2 − z + c = 0. {\displaystyle \ z^{2}-z+c=0.} Since this is an ordinary quadratic equation in one unknown, we can apply the standard quadratic solution formula: α 1 = 1 − 1 − 4 c 2 {\displaystyle \alpha _{1}={\frac {1-{\sqrt {1-4c}}}{2}}} and α 2 = 1 + 1 − 4 c 2 . {\displaystyle \alpha _{2}={\frac {1+{\sqrt {1-4c}}}{2}}.} So for c ∈ C ∖ { 1 / 4 } {\displaystyle c\in \mathbb {C} \setminus \{1/4\}} we have two finite fixed points α 1 {\displaystyle \alpha _{1}} and α 2 {\displaystyle \alpha _{2}} . Since α 1 = 1 2 − m {\displaystyle \alpha _{1}={\frac {1}{2}}-m} and α 2 = 1 2 + m {\displaystyle \alpha _{2}={\frac {1}{2}}+m} where m = 1 − 4 c 2 , {\displaystyle m={\frac {\sqrt {1-4c}}{2}},} we have α 1 + α 2 = 1 {\displaystyle \alpha _{1}+\alpha _{2}=1} . Thus fixed points are symmetrical about z = 1 / 2 {\displaystyle z=1/2} . ==== Complex dynamics ==== Here different notation is commonly used: α c = 1 − 1 − 4 c 2 {\displaystyle \alpha _{c}={\frac {1-{\sqrt {1-4c}}}{2}}} with multiplier λ α c = 1 − 1 − 4 c {\displaystyle \lambda _{\alpha _{c}}=1-{\sqrt {1-4c}}} and β c = 1 + 1 − 4 c 2 {\displaystyle \beta _{c}={\frac {1+{\sqrt {1-4c}}}{2}}} with multiplier λ β c = 1 + 1 − 4 c . {\displaystyle \lambda _{\beta _{c}}=1+{\sqrt {1-4c}}.} Again we have α c + β c = 1. {\displaystyle \alpha _{c}+\beta _{c}=1.} Since the derivative with respect to z is P c ′ ( z ) = d d z P c ( z ) = 2 z , {\displaystyle P_{c}'(z)={\frac {d}{dz}}P_{c}(z)=2z,} we have P c ′ ( α c ) + P c ′ ( β c ) = 2 α c + 2 β c = 2 ( α c + β c ) = 2. {\displaystyle P_{c}'(\alpha _{c})+P_{c}'(\beta _{c})=2\alpha _{c}+2\beta _{c}=2(\alpha _{c}+\beta _{c})=2.} This implies that P c {\displaystyle P_{c}} can have at most one attractive fixed point. These points are distinguished by the facts that: β c {\displaystyle \beta _{c}} is: the landing point of the external ray for angle=0 for c ∈ M ∖ { 1 / 4 } {\displaystyle c\in M\setminus \left\{1/4\right\}} the most repelling fixed point of the Julia set the one on the right (whenever fixed point are not symmetrical around the real axis), it is the extreme right point for connected Julia sets (except for cauliflower). α c {\displaystyle \alpha _{c}} is: the landing point of several rays attracting when c {\displaystyle c} is in the main cardioid of the Mandelbrot set, in which case it is in the interior of a filled-in Julia set, and therefore belongs to the Fatou set (strictly to the basin of attraction of finite fixed point) parabolic at the root point of the limb of the Mandelbrot set repelling for other values of c {\displaystyle c} ==== Special cases ==== An important case of the quadratic mapping is c = 0 {\displaystyle c=0} . In this case, we get α 1 = 0 {\displaystyle \alpha _{1}=0} and α 2 = 1 {\displaystyle \alpha _{2}=1} . In this case, 0 is a superattractive fixed point, and 1 belongs to the Julia set. ==== Only one fixed point ==== We have α 1 = α 2 {\displaystyle \alpha _{1}=\alpha _{2}} exactly when 1 − 4 c = 0. {\displaystyle 1-4c=0.} This equation has one solution, c = 1 / 4 , {\displaystyle c=1/4,} in which case α 1 = α 2 = 1 / 2 {\displaystyle \alpha _{1}=\alpha _{2}=1/2} . In fact c = 1 / 4 {\displaystyle c=1/4} is the largest positive, purely real value for which a finite attractor exists. === Infinite fixed point === We can extend the complex plane C {\displaystyle \mathbb {C} } to the Riemann sphere (extended complex plane) C ^ {\displaystyle \mathbb {\hat {C}} } by adding infinity: C ^ = C ∪ { ∞ } {\displaystyle \mathbb {\hat {C}} =\mathbb {C} \cup \{\infty \}} and extend f c {\displaystyle f_{c}} such that f c ( ∞ ) = ∞ . {\displaystyle f_{c}(\infty )=\infty .} Then infinity is: superattracting a fixed point of f c {\displaystyle f_{c}} : f c ( ∞ ) = ∞ = f c − 1 ( ∞ ) . {\displaystyle f_{c}(\infty )=\infty =f_{c}^{-1}(\infty ).} == Period-2 cycles == Period-2 cycles are two distinct points β 1 {\displaystyle \beta _{1}} and β 2 {\displaystyle \beta _{2}} such that f c ( β 1 ) = β 2 {\displaystyle f_{c}(\beta _{1})=\beta _{2}} and f c ( β 2 ) = β 1 {\displaystyle f_{c}(\beta _{2})=\beta _{1}} , and hence f c ( f c ( β n ) ) = β n {\displaystyle f_{c}(f_{c}(\beta _{n}))=\beta _{n}} for n ∈ { 1 , 2 } {\displaystyle n\in \{1,2\}} : f c ( f c ( z ) ) = ( z 2 + c ) 2 + c = z 4 + 2 c z 2 + c 2 + c . {\displaystyle f_{c}(f_{c}(z))=(z^{2}+c)^{2}+c=z^{4}+2cz^{2}+c^{2}+c.} Equating this to z, we obtain z 4 + 2 c z 2 − z + c 2 + c = 0. {\displaystyle z^{4}+2cz^{2}-z+c^{2}+c=0.} This equation is a polynomial of degree 4, and so has four (possibly non-distinct) solutions. However, we already know two of the solutions. They are α 1 {\displaystyle \alpha _{1}} and α 2 {\displaystyle \alpha _{2}} , computed above, since if these points are left unchanged by one application of f {\displaystyle f} , then clearly they will be unchanged by more than one application of f {\displaystyle f} . Our 4th-order polynomial can therefore be factored in 2 ways: === First method of factorization === ( z − α 1 ) ( z − α 2 ) ( z − β 1 ) ( z − β 2 ) = 0. {\displaystyle (z-\alpha _{1})(z-\alpha _{2})(z-\beta _{1})(z-\beta _{2})=0.\,} This expands directly as x 4 − A x 3 + B x 2 − C x + D = 0 {\displaystyle x^{4}-Ax^{3}+Bx^{2}-Cx+D=0} (note the alternating signs), where D = α 1 α 2 β 1 β 2 , {\displaystyle D=\alpha _{1}\alpha _{2}\beta _{1}\beta _{2},\,} C = α 1 α 2 β 1 + α 1 α 2 β 2 + α 1 β 1 β 2 + α 2 β 1 β 2 , {\displaystyle C=\alpha _{1}\alpha _{2}\beta _{1}+\alpha _{1}\alpha _{2}\beta _{2}+\alpha _{1}\beta _{1}\beta _{2}+\alpha _{2}\beta _{1}\beta _{2},\,} B = α 1 α 2 + α 1 β 1 + α 1 β 2 + α 2 β 1 + α 2 β 2 + β 1 β 2 , {\displaystyle B=\alpha _{1}\alpha _{2}+\alpha _{1}\beta _{1}+\alpha _{1}\beta _{2}+\alpha _{2}\beta _{1}+\alpha _{2}\beta _{2}+\beta _{1}\beta _{2},\,} A = α 1 + α 2 + β 1 + β 2 . {\displaystyle A=\alpha _{1}+\alpha _{2}+\beta _{1}+\beta _{2}.\,} We already have two solutions, and only need the other two. Hence the problem is equivalent to solving a quadratic polynomial. In particular, note that α 1 + α 2 = 1 − 1 − 4 c 2 + 1 + 1 − 4 c 2 = 1 + 1 2 = 1 {\displaystyle \alpha _{1}+\alpha _{2}={\frac {1-{\sqrt {1-4c}}}{2}}+{\frac {1+{\sqrt {1-4c}}}{2}}={\frac {1+1}{2}}=1} and α 1 α 2 = ( 1 − 1 − 4 c ) ( 1 + 1 − 4 c ) 4 = 1 2 − ( 1 − 4 c ) 2 4 = 1 − 1 + 4 c 4 = 4 c 4 = c . {\displaystyle \alpha _{1}\alpha _{2}={\frac {(1-{\sqrt {1-4c}})(1+{\sqrt {1-4c}})}{4}}={\frac {1^{2}-({\sqrt {1-4c}})^{2}}{4}}={\frac {1-1+4c}{4}}={\frac {4c}{4}}=c.} Adding these to the above, we get D = c β 1 β 2 {\displaystyle D=c\beta _{1}\beta _{2}} and A = 1 + β 1 + β 2 {\displaystyle A=1+\beta _{1}+\beta _{2}} . Matching these against the coefficients from expanding f {\displaystyle f} , we get D = c β 1 β 2 = c 2 + c {\displaystyle D=c\beta _{1}\beta _{2}=c^{2}+c} and A = 1 + β 1 + β 2 = 0. {\displaystyle A=1+\beta _{1}+\beta _{2}=0.} From this, we easily get β 1 β 2 = c + 1 {\displaystyle \beta _{1}\beta _{2}=c+1} and β 1 + β 2 = − 1 {\displaystyle \beta _{1}+\beta _{2}=-1} . From here, we construct a quadratic equation with A ′ = 1 , B = 1 , C = c + 1 {\displaystyle A'=1,B=1,C=c+1} and apply the standard solution formula to get β 1 = − 1 − − 3 − 4 c 2 {\displaystyle \beta _{1}={\frac {-1-{\sqrt {-3-4c}}}{2}}} and β 2 = − 1 + − 3 − 4 c 2 . {\displaystyle \beta _{2}={\frac {-1+{\sqrt {-3-4c}}}{2}}.} Closer examination shows that: f c ( β 1 ) = β 2 {\displaystyle f_{c}(\beta _{1})=\beta _{2}} and f c ( β 2 ) = β 1 , {\displaystyle f_{c}(\beta _{2})=\beta _{1},} meaning these two points are the two points on a single period-2 cycle. === Second method of factorization === We can factor the quartic by using polynomial long division to divide out the factors ( z − α 1 ) {\displaystyle (z-\alpha _{1})} and ( z − α 2 ) , {\displaystyle (z-\alpha _{2}),} which account for the two fixed points α 1 {\displaystyle \alpha _{1}} and α 2 {\displaystyle \alpha _{2}} (whose values were given earlier and which still remain at the fixed point after two iterations): ( z 2 + c ) 2 + c − z = ( z 2 + c − z ) ( z 2 + z + c + 1 ) . {\displaystyle (z^{2}+c)^{2}+c-z=(z^{2}+c-z)(z^{2}+z+c+1).\,} The roots of the first factor are the two fixed points. They are repelling outside the main cardioid. The second factor has the two roots − 1 ± − 3 − 4 c 2 . {\displaystyle {\frac {-1\pm {\sqrt {-3-4c}}}{2}}.\,} These two roots, which are the same as those found by the first method, form the period-2 orbit. ==== Special cases ==== Again, let us look at c = 0 {\displaystyle c=0} . Then β 1 = − 1 − i 3 2 {\displaystyle \beta _{1}={\frac {-1-i{\sqrt {3}}}{2}}} and β 2 = − 1 + i 3 2 , {\displaystyle \beta _{2}={\frac {-1+i{\sqrt {3}}}{2}},} both of which are complex numbers. We have | β 1 | = | β 2 | = 1 {\displaystyle |\beta _{1}|=|\beta _{2}|=1} . Thus, both these points are "hiding" in the Julia set. Another special case is c = − 1 {\displaystyle c=-1} , which gives β 1 = 0 {\displaystyle \beta _{1}=0} and β 2 = − 1 {\displaystyle \beta _{2}=-1} . This gives the well-known superattractive cycle found in the largest period-2 lobe of the quadratic Mandelbrot set. == Cycles for period greater than 2 == The degree of the equation f ( n ) ( z ) = z {\displaystyle f^{(n)}(z)=z} is 2n; thus for example, to find the points on a 3-cycle we would need to solve an equation of degree 8. After factoring out the factors giving the two fixed points, we would have a sixth degree equation. There is no general solution in radicals to polynomial equations of degree five or higher, so the points on a cycle of period greater than 2 must in general be computed using numerical methods. However, in the specific case of period 4 the cyclical points have lengthy expressions in radicals. In the case c = –2, trigonometric solutions exist for the periodic points of all periods. The case z n + 1 = z n 2 − 2 {\displaystyle z_{n+1}=z_{n}^{2}-2} is equivalent to the logistic map case r = 4: x n + 1 = 4 x n ( 1 − x n ) . {\displaystyle x_{n+1}=4x_{n}(1-x_{n}).} Here the equivalence is given by z = 2 − 4 x . {\displaystyle z=2-4x.} One of the k-cycles of the logistic variable x (all of which cycles are repelling) is sin 2 ( 2 π 2 k − 1 ) , sin 2 ( 2 ⋅ 2 π 2 k − 1 ) , sin 2 ( 2 2 ⋅ 2 π 2 k − 1 ) , sin 2 ( 2 3 ⋅ 2 π 2 k − 1 ) , … , sin 2 ( 2 k − 1 2 π 2 k − 1 ) . {\displaystyle \sin ^{2}\left({\frac {2\pi }{2^{k}-1}}\right),\,\sin ^{2}\left(2\cdot {\frac {2\pi }{2^{k}-1}}\right),\,\sin ^{2}\left(2^{2}\cdot {\frac {2\pi }{2^{k}-1}}\right),\,\sin ^{2}\left(2^{3}\cdot {\frac {2\pi }{2^{k}-1}}\right),\dots ,\sin ^{2}\left(2^{k-1}{\frac {2\pi }{2^{k}-1}}\right).} == References == == Further reading == Geometrical properties of polynomial roots Alan F. Beardon, Iteration of Rational Functions, Springer 1991, ISBN 0-387-95151-2 Michael F. Barnsley (Author), Stephen G. Demko (Editor), Chaotic Dynamics and Fractals (Notes and Reports in Mathematics in Science and Engineering Series) Academic Pr (April 1986), ISBN 0-12-079060-2 Wolf Jung : Homeomorphisms on Edges of the Mandelbrot Set. Ph.D. thesis of 2002 The permutations of periodic points in quadratic polynominials by J Leahy == External links == Algebraic solution of Mandelbrot orbital boundaries by Donald D. Cross Brown Method by Robert P. Munafo arXiv:hep-th/0501235v2 V.Dolotin, A.Morozov: Algebraic Geometry of Discrete Dynamics. The case of one variable.
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Wikipedia:Periodic summation#0
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In mathematics, any integrable function s ( t ) {\displaystyle s(t)} can be made into a periodic function s P ( t ) {\displaystyle s_{P}(t)} with period P by summing the translations of the function s ( t ) {\displaystyle s(t)} by integer multiples of P. This is called periodic summation: s P ( t ) = ∑ n = − ∞ ∞ s ( t + n P ) {\displaystyle s_{P}(t)=\sum _{n=-\infty }^{\infty }s(t+nP)} When s P ( t ) {\displaystyle s_{P}(t)} is alternatively represented as a Fourier series, the Fourier coefficients are equal to the values of the continuous Fourier transform, S ( f ) ≜ F { s ( t ) } , {\displaystyle S(f)\triangleq {\mathcal {F}}\{s(t)\},} at intervals of 1 P {\displaystyle {\tfrac {1}{P}}} . That identity is a form of the Poisson summation formula. Similarly, a Fourier series whose coefficients are samples of s ( t ) {\displaystyle s(t)} at constant intervals (T) is equivalent to a periodic summation of S ( f ) , {\displaystyle S(f),} which is known as a discrete-time Fourier transform. The periodic summation of a Dirac delta function is the Dirac comb. Likewise, the periodic summation of an integrable function is its convolution with the Dirac comb. == Quotient space as domain == If a periodic function is instead represented using the quotient space domain R / ( P Z ) {\displaystyle \mathbb {R} /(P\mathbb {Z} )} then one can write: φ P : R / ( P Z ) → R {\displaystyle \varphi _{P}:\mathbb {R} /(P\mathbb {Z} )\to \mathbb {R} } φ P ( x ) = ∑ τ ∈ x s ( τ ) . {\displaystyle \varphi _{P}(x)=\sum _{\tau \in x}s(\tau )~.} The arguments of φ P {\displaystyle \varphi _{P}} are equivalence classes of real numbers that share the same fractional part when divided by P {\displaystyle P} . == Citations == == See also == Dirac comb Circular convolution Discrete-time Fourier transform
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Wikipedia:Perkins Professorship of Astronomy and Mathematics#0
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The Perkins Professorship of Astronomy and Mathematics is an endowed professorship established at Harvard College in 1842 by James Perkins, Jr., (1761–1822). == History of the Perkins Chair == James Perkins, Jr., was a Boston philanthropist, benefactor of the Boston Athenæum, and co-founder with his younger brother Thomas Handasyd Perkins of the Perkins School for the Blind. In his will, Perkins left $20,000 to Harvard College to establish a chair in "whatever field the President and Fellows should find the most useful".: 406 The funds were transferred to Harvard on February 20, 1842, upon the death of Perkins' wife. At that time the Harvard Corporation voted ...that a Professorship of Astronomy and Mathematics be established in the College to be denominated the Perkins Professorship of Astronomy and Mathematics.: 406 The Perkins chair was the second chair in mathematics, the first and most famous being the Hollis Chair in Mathematics and Natural Philosophy endowed by Thomas Hollis in 1727. The Hollis Chair in Mathematics was in turn the second professorship endowed at Harvard, the first being the Hollis Chair of Divinity endowed in 1721. Hollis Chair of Divinity is in its turn the oldest endowed chair in the United States. In 1906, noting that chairs in astronomy had been endowed in 1858 (Phillips) and in 1887 (Paine), the Corporation voted ...that the title of the Perkins Professorship of Astronomy and Mathematics ... be amended so that it shall read Perkins Professorship in Mathematics.: 406 Starting with the most recent appointment in 1991 the name of the chair informally became the Perkins Professor of Applied Mathematics when the chair was moved to Harvard's School of Engineering and Applied Sciences. The formal name of the chair remains unchanged. == Holders of the Perkins Professorship of Astronomy and Mathematics == The holders of the Perkins Professorship of Astronomy and Mathematics have been: Benjamin Peirce (1842–1880) James Mills Peirce (1885–1906) William Elwood Byerly (1906–1913) William Fogg Osgood (1914–1933) George David Birkhoff (1933–1944) Joseph Leonard Walsh (1946–1966) Richard Dagobert Brauer (1966–1971) John Torrence Tate, Jr (1971–1991) David Kazhdan (1991–2003) Mark Kisin (2018–) == References ==
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Wikipedia:Permanent (mathematics)#0
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In linear algebra, the permanent of a square matrix is a function of the matrix similar to the determinant. The permanent, as well as the determinant, is a polynomial in the entries of the matrix. Both are special cases of a more general function of a matrix called the immanant. == Definition == The permanent of an n×n matrix A = (ai,j) is defined as perm ( A ) = ∑ σ ∈ S n ∏ i = 1 n a i , σ ( i ) . {\displaystyle \operatorname {perm} (A)=\sum _{\sigma \in S_{n}}\prod _{i=1}^{n}a_{i,\sigma (i)}.} The sum here extends over all elements σ of the symmetric group Sn; i.e. over all permutations of the numbers 1, 2, ..., n. For example, perm ( a b c d ) = a d + b c , {\displaystyle \operatorname {perm} {\begin{pmatrix}a&b\\c&d\end{pmatrix}}=ad+bc,} and perm ( a b c d e f g h i ) = a e i + b f g + c d h + c e g + b d i + a f h . {\displaystyle \operatorname {perm} {\begin{pmatrix}a&b&c\\d&e&f\\g&h&i\end{pmatrix}}=aei+bfg+cdh+ceg+bdi+afh.} The definition of the permanent of A differs from that of the determinant of A in that the signatures of the permutations are not taken into account. The permanent of a matrix A is denoted per A, perm A, or Per A, sometimes with parentheses around the argument. Minc uses Per(A) for the permanent of rectangular matrices, and per(A) when A is a square matrix. Muir and Metzler use the notation | + | + {\displaystyle {\overset {+}{|}}\quad {\overset {+}{|}}} . The word, permanent, originated with Cauchy in 1812 as “fonctions symétriques permanentes” for a related type of function, and was used by Muir and Metzler in the modern, more specific, sense. == Properties == If one views the permanent as a map that takes n vectors as arguments, then it is a multilinear map and it is symmetric (meaning that any order of the vectors results in the same permanent). Furthermore, given a square matrix A = ( a i j ) {\displaystyle A=\left(a_{ij}\right)} of order n: perm(A) is invariant under arbitrary permutations of the rows and/or columns of A. This property may be written symbolically as perm(A) = perm(PAQ) for any appropriately sized permutation matrices P and Q, multiplying any single row or column of A by a scalar s changes perm(A) to s⋅perm(A), perm(A) is invariant under transposition, that is, perm(A) = perm(AT). If A = ( a i j ) {\displaystyle A=\left(a_{ij}\right)} and B = ( b i j ) {\displaystyle B=\left(b_{ij}\right)} are square matrices of order n then, perm ( A + B ) = ∑ s , t perm ( a i j ) i ∈ s , j ∈ t perm ( b i j ) i ∈ s ¯ , j ∈ t ¯ , {\displaystyle \operatorname {perm} \left(A+B\right)=\sum _{s,t}\operatorname {perm} \left(a_{ij}\right)_{i\in s,j\in t}\operatorname {perm} \left(b_{ij}\right)_{i\in {\bar {s}},j\in {\bar {t}}},} where s and t are subsets of the same size of {1,2,...,n} and s ¯ , t ¯ {\displaystyle {\bar {s}},{\bar {t}}} are their respective complements in that set. If A {\displaystyle A} is a triangular matrix, i.e. a i j = 0 {\displaystyle a_{ij}=0} , whenever i > j {\displaystyle i>j} or, alternatively, whenever i < j {\displaystyle i<j} , then its permanent (and determinant as well) equals the product of the diagonal entries: perm ( A ) = a 11 a 22 ⋯ a n n = ∏ i = 1 n a i i . {\displaystyle \operatorname {perm} \left(A\right)=a_{11}a_{22}\cdots a_{nn}=\prod _{i=1}^{n}a_{ii}.} == Relation to determinants == Laplace's expansion by minors for computing the determinant along a row, column or diagonal extends to the permanent by ignoring all signs. For every i {\textstyle i} , p e r m ( B ) = ∑ j = 1 n B i , j M i , j , {\displaystyle \mathbb {perm} (B)=\sum _{j=1}^{n}B_{i,j}M_{i,j},} where B i , j {\displaystyle B_{i,j}} is the entry of the ith row and the jth column of B, and M i , j {\textstyle M_{i,j}} is the permanent of the submatrix obtained by removing the ith row and the jth column of B. For example, expanding along the first column, perm ( 1 1 1 1 2 1 0 0 3 0 1 0 4 0 0 1 ) = 1 ⋅ perm ( 1 0 0 0 1 0 0 0 1 ) + 2 ⋅ perm ( 1 1 1 0 1 0 0 0 1 ) + 3 ⋅ perm ( 1 1 1 1 0 0 0 0 1 ) + 4 ⋅ perm ( 1 1 1 1 0 0 0 1 0 ) = 1 ( 1 ) + 2 ( 1 ) + 3 ( 1 ) + 4 ( 1 ) = 10 , {\displaystyle {\begin{aligned}\operatorname {perm} \left({\begin{matrix}1&1&1&1\\2&1&0&0\\3&0&1&0\\4&0&0&1\end{matrix}}\right)={}&1\cdot \operatorname {perm} \left({\begin{matrix}1&0&0\\0&1&0\\0&0&1\end{matrix}}\right)+2\cdot \operatorname {perm} \left({\begin{matrix}1&1&1\\0&1&0\\0&0&1\end{matrix}}\right)\\&{}+\ 3\cdot \operatorname {perm} \left({\begin{matrix}1&1&1\\1&0&0\\0&0&1\end{matrix}}\right)+4\cdot \operatorname {perm} \left({\begin{matrix}1&1&1\\1&0&0\\0&1&0\end{matrix}}\right)\\={}&1(1)+2(1)+3(1)+4(1)=10,\end{aligned}}} while expanding along the last row gives, perm ( 1 1 1 1 2 1 0 0 3 0 1 0 4 0 0 1 ) = 4 ⋅ perm ( 1 1 1 1 0 0 0 1 0 ) + 0 ⋅ perm ( 1 1 1 2 0 0 3 1 0 ) + 0 ⋅ perm ( 1 1 1 2 1 0 3 0 0 ) + 1 ⋅ perm ( 1 1 1 2 1 0 3 0 1 ) = 4 ( 1 ) + 0 + 0 + 1 ( 6 ) = 10. {\displaystyle {\begin{aligned}\operatorname {perm} \left({\begin{matrix}1&1&1&1\\2&1&0&0\\3&0&1&0\\4&0&0&1\end{matrix}}\right)={}&4\cdot \operatorname {perm} \left({\begin{matrix}1&1&1\\1&0&0\\0&1&0\end{matrix}}\right)+0\cdot \operatorname {perm} \left({\begin{matrix}1&1&1\\2&0&0\\3&1&0\end{matrix}}\right)\\&{}+\ 0\cdot \operatorname {perm} \left({\begin{matrix}1&1&1\\2&1&0\\3&0&0\end{matrix}}\right)+1\cdot \operatorname {perm} \left({\begin{matrix}1&1&1\\2&1&0\\3&0&1\end{matrix}}\right)\\={}&4(1)+0+0+1(6)=10.\end{aligned}}} On the other hand, the basic multiplicative property of determinants is not valid for permanents. A simple example shows that this is so. 4 = perm ( 1 1 1 1 ) perm ( 1 1 1 1 ) ≠ perm ( ( 1 1 1 1 ) ( 1 1 1 1 ) ) = perm ( 2 2 2 2 ) = 8. {\displaystyle {\begin{aligned}4&=\operatorname {perm} \left({\begin{matrix}1&1\\1&1\end{matrix}}\right)\operatorname {perm} \left({\begin{matrix}1&1\\1&1\end{matrix}}\right)\\&\neq \operatorname {perm} \left(\left({\begin{matrix}1&1\\1&1\end{matrix}}\right)\left({\begin{matrix}1&1\\1&1\end{matrix}}\right)\right)=\operatorname {perm} \left({\begin{matrix}2&2\\2&2\end{matrix}}\right)=8.\end{aligned}}} Unlike the determinant, the permanent has no easy geometrical interpretation; it is mainly used in combinatorics, in treating boson Green's functions in quantum field theory, and in determining state probabilities of boson sampling systems. However, it has two graph-theoretic interpretations: as the sum of weights of cycle covers of a directed graph, and as the sum of weights of perfect matchings in a bipartite graph. == Applications == === Symmetric tensors === The permanent arises naturally in the study of the symmetric tensor power of Hilbert spaces. In particular, for a Hilbert space H {\displaystyle H} , let ∨ k H {\displaystyle \vee ^{k}H} denote the k {\displaystyle k} th symmetric tensor power of H {\displaystyle H} , which is the space of symmetric tensors. Note in particular that ∨ k H {\displaystyle \vee ^{k}H} is spanned by the symmetric products of elements in H {\displaystyle H} . For x 1 , x 2 , … , x k ∈ H {\displaystyle x_{1},x_{2},\dots ,x_{k}\in H} , we define the symmetric product of these elements by x 1 ∨ x 2 ∨ ⋯ ∨ x k = ( k ! ) − 1 / 2 ∑ σ ∈ S k x σ ( 1 ) ⊗ x σ ( 2 ) ⊗ ⋯ ⊗ x σ ( k ) {\displaystyle x_{1}\vee x_{2}\vee \cdots \vee x_{k}=(k!)^{-1/2}\sum _{\sigma \in S_{k}}x_{\sigma (1)}\otimes x_{\sigma (2)}\otimes \cdots \otimes x_{\sigma (k)}} If we consider ∨ k H {\displaystyle \vee ^{k}H} (as a subspace of ⊗ k H {\displaystyle \otimes ^{k}H} , the kth tensor power of H {\displaystyle H} ) and define the inner product on ∨ k H {\displaystyle \vee ^{k}H} accordingly, we find that for x j , y j ∈ H {\displaystyle x_{j},y_{j}\in H} ⟨ x 1 ∨ x 2 ∨ ⋯ ∨ x k , y 1 ∨ y 2 ∨ ⋯ ∨ y k ⟩ = perm [ ⟨ x i , y j ⟩ ] i , j = 1 k {\displaystyle \langle x_{1}\vee x_{2}\vee \cdots \vee x_{k},y_{1}\vee y_{2}\vee \cdots \vee y_{k}\rangle =\operatorname {perm} \left[\langle x_{i},y_{j}\rangle \right]_{i,j=1}^{k}} Applying the Cauchy–Schwarz inequality, we find that perm [ ⟨ x i , x j ⟩ ] i , j = 1 k ≥ 0 {\displaystyle \operatorname {perm} \left[\langle x_{i},x_{j}\rangle \right]_{i,j=1}^{k}\geq 0} , and that | perm [ ⟨ x i , y j ⟩ ] i , j = 1 k | 2 ≤ perm [ ⟨ x i , x j ⟩ ] i , j = 1 k ⋅ perm [ ⟨ y i , y j ⟩ ] i , j = 1 k {\displaystyle \left|\operatorname {perm} \left[\langle x_{i},y_{j}\rangle \right]_{i,j=1}^{k}\right|^{2}\leq \operatorname {perm} \left[\langle x_{i},x_{j}\rangle \right]_{i,j=1}^{k}\cdot \operatorname {perm} \left[\langle y_{i},y_{j}\rangle \right]_{i,j=1}^{k}} === Cycle covers === Any square matrix A = ( a i j ) i , j = 1 n {\displaystyle A=(a_{ij})_{i,j=1}^{n}} can be viewed as the adjacency matrix of a weighted directed graph on vertex set V = { 1 , 2 , … , n } {\displaystyle V=\{1,2,\dots ,n\}} , with a i j {\displaystyle a_{ij}} representing the weight of the arc from vertex i to vertex j. A cycle cover of a weighted directed graph is a collection of vertex-disjoint directed cycles in the digraph that covers all vertices in the graph. Thus, each vertex i in the digraph has a unique "successor" σ ( i ) {\displaystyle \sigma (i)} in the cycle cover, and so σ {\displaystyle \sigma } represents a permutation on V. Conversely, any permutation σ {\displaystyle \sigma } on V corresponds to a cycle cover with arcs from each vertex i to vertex σ ( i ) {\displaystyle \sigma (i)} . If the weight of a cycle-cover is defined to be the product of the weights of the arcs in each cycle, then weight ( σ ) = ∏ i = 1 n a i , σ ( i ) , {\displaystyle \operatorname {weight} (\sigma )=\prod _{i=1}^{n}a_{i,\sigma (i)},} implying that perm ( A ) = ∑ σ weight ( σ ) . {\displaystyle \operatorname {perm} (A)=\sum _{\sigma }\operatorname {weight} (\sigma ).} Thus the permanent of A is equal to the sum of the weights of all cycle-covers of the digraph. === Perfect matchings === A square matrix A = ( a i j ) {\displaystyle A=(a_{ij})} can also be viewed as the adjacency matrix of a bipartite graph which has vertices x 1 , x 2 , … , x n {\displaystyle x_{1},x_{2},\dots ,x_{n}} on one side and y 1 , y 2 , … , y n {\displaystyle y_{1},y_{2},\dots ,y_{n}} on the other side, with a i j {\displaystyle a_{ij}} representing the weight of the edge from vertex x i {\displaystyle x_{i}} to vertex y j {\displaystyle y_{j}} . If the weight of a perfect matching σ {\displaystyle \sigma } that matches x i {\displaystyle x_{i}} to y σ ( i ) {\displaystyle y_{\sigma (i)}} is defined to be the product of the weights of the edges in the matching, then weight ( σ ) = ∏ i = 1 n a i , σ ( i ) . {\displaystyle \operatorname {weight} (\sigma )=\prod _{i=1}^{n}a_{i,\sigma (i)}.} Thus the permanent of A is equal to the sum of the weights of all perfect matchings of the graph. == Permanents of (0, 1) matrices == === Enumeration === The answers to many counting questions can be computed as permanents of matrices that only have 0 and 1 as entries. Let Ω(n,k) be the class of all (0, 1)-matrices of order n with each row and column sum equal to k. Every matrix A in this class has perm(A) > 0. The incidence matrices of projective planes are in the class Ω(n2 + n + 1, n + 1) for n an integer > 1. The permanents corresponding to the smallest projective planes have been calculated. For n = 2, 3, and 4 the values are 24, 3852 and 18,534,400 respectively. Let Z be the incidence matrix of the projective plane with n = 2, the Fano plane. Remarkably, perm(Z) = 24 = |det (Z)|, the absolute value of the determinant of Z. This is a consequence of Z being a circulant matrix and the theorem: If A is a circulant matrix in the class Ω(n,k) then if k > 3, perm(A) > |det (A)| and if k = 3, perm(A) = |det (A)|. Furthermore, when k = 3, by permuting rows and columns, A can be put into the form of a direct sum of e copies of the matrix Z and consequently, n = 7e and perm(A) = 24e. Permanents can also be used to calculate the number of permutations with restricted (prohibited) positions. For the standard n-set {1, 2, ..., n}, let A = ( a i j ) {\displaystyle A=(a_{ij})} be the (0, 1)-matrix where aij = 1 if i → j is allowed in a permutation and aij = 0 otherwise. Then perm(A) is equal to the number of permutations of the n-set that satisfy all the restrictions. Two well known special cases of this are the solution of the derangement problem and the ménage problem: the number of permutations of an n-set with no fixed points (derangements) is given by perm ( J − I ) = perm ( 0 1 1 … 1 1 0 1 … 1 1 1 0 … 1 ⋮ ⋮ ⋮ ⋱ ⋮ 1 1 1 … 0 ) = n ! ∑ i = 0 n ( − 1 ) i i ! , {\displaystyle \operatorname {perm} (J-I)=\operatorname {perm} \left({\begin{matrix}0&1&1&\dots &1\\1&0&1&\dots &1\\1&1&0&\dots &1\\\vdots &\vdots &\vdots &\ddots &\vdots \\1&1&1&\dots &0\end{matrix}}\right)=n!\sum _{i=0}^{n}{\frac {(-1)^{i}}{i!}},} where J is the n×n all 1's matrix and I is the identity matrix, and the ménage numbers are given by perm ( J − I − I ′ ) = perm ( 0 0 1 … 1 1 0 0 … 1 1 1 0 … 1 ⋮ ⋮ ⋮ ⋱ ⋮ 0 1 1 … 0 ) = ∑ k = 0 n ( − 1 ) k 2 n 2 n − k ( 2 n − k k ) ( n − k ) ! , {\displaystyle {\begin{aligned}\operatorname {perm} (J-I-I')&=\operatorname {perm} \left({\begin{matrix}0&0&1&\dots &1\\1&0&0&\dots &1\\1&1&0&\dots &1\\\vdots &\vdots &\vdots &\ddots &\vdots \\0&1&1&\dots &0\end{matrix}}\right)\\&=\sum _{k=0}^{n}(-1)^{k}{\frac {2n}{2n-k}}{2n-k \choose k}(n-k)!,\end{aligned}}} where I' is the (0, 1)-matrix with nonzero entries in positions (i, i + 1) and (n, 1). Permanent of n×n all 1's matrix is a number of possible arrangements of n mutually non-attacking rooks in the positions of the board of size n×n. === Bounds === The Bregman–Minc inequality, conjectured by H. Minc in 1963 and proved by L. M. Brégman in 1973, gives an upper bound for the permanent of an n × n (0, 1)-matrix. If A has ri ones in row i for each 1 ≤ i ≤ n, the inequality states that perm A ≤ ∏ i = 1 n ( r i ) ! 1 / r i . {\displaystyle \operatorname {perm} A\leq \prod _{i=1}^{n}(r_{i})!^{1/r_{i}}.} == Van der Waerden's conjecture == In 1926, Van der Waerden conjectured that the minimum permanent among all n × n doubly stochastic matrices is n!/nn, achieved by the matrix for which all entries are equal to 1/n. Proofs of this conjecture were published in 1980 by B. Gyires and in 1981 by G. P. Egorychev and D. I. Falikman; Egorychev's proof is an application of the Alexandrov–Fenchel inequality. For this work, Egorychev and Falikman won the Fulkerson Prize in 1982. == Computation == The naïve approach, using the definition, of computing permanents is computationally infeasible even for relatively small matrices. One of the fastest known algorithms is due to H. J. Ryser. Ryser's method is based on an inclusion–exclusion formula that can be given as follows: Let A k {\displaystyle A_{k}} be obtained from A by deleting k columns, let P ( A k ) {\displaystyle P(A_{k})} be the product of the row-sums of A k {\displaystyle A_{k}} , and let Σ k {\displaystyle \Sigma _{k}} be the sum of the values of P ( A k ) {\displaystyle P(A_{k})} over all possible A k {\displaystyle A_{k}} . Then perm ( A ) = ∑ k = 0 n − 1 ( − 1 ) k Σ k . {\displaystyle \operatorname {perm} (A)=\sum _{k=0}^{n-1}(-1)^{k}\Sigma _{k}.} It may be rewritten in terms of the matrix entries as follows: perm ( A ) = ( − 1 ) n ∑ S ⊆ { 1 , … , n } ( − 1 ) | S | ∏ i = 1 n ∑ j ∈ S a i j . {\displaystyle \operatorname {perm} (A)=(-1)^{n}\sum _{S\subseteq \{1,\dots ,n\}}(-1)^{|S|}\prod _{i=1}^{n}\sum _{j\in S}a_{ij}.} The permanent is believed to be more difficult to compute than the determinant. While the determinant can be computed in polynomial time by Gaussian elimination, Gaussian elimination cannot be used to compute the permanent. Moreover, computing the permanent of a (0,1)-matrix is #P-complete. Thus, if the permanent can be computed in polynomial time by any method, then FP = #P, which is an even stronger statement than P = NP. When the entries of A are nonnegative, however, the permanent can be computed approximately in probabilistic polynomial time, up to an error of ε M {\displaystyle \varepsilon M} , where M {\displaystyle M} is the value of the permanent and ε > 0 {\displaystyle \varepsilon >0} is arbitrary. The permanent of a certain set of positive semidefinite matrices is NP-hard to approximate within any subexponential factor. If further conditions on the spectrum are imposed, the permanent can be approximated in probabilistic polynomial time: the best achievable error of this approximation is ε M {\displaystyle \varepsilon {\sqrt {M}}} ( M {\displaystyle M} is again the value of the permanent). The hardness in these instances is closely linked with difficulty of simulating boson sampling experiments. == MacMahon's master theorem == Another way to view permanents is via multivariate generating functions. Let A = ( a i j ) {\displaystyle A=(a_{ij})} be a square matrix of order n. Consider the multivariate generating function: F ( x 1 , x 2 , … , x n ) = ∏ i = 1 n ( ∑ j = 1 n a i j x j ) = ( ∑ j = 1 n a 1 j x j ) ( ∑ j = 1 n a 2 j x j ) ⋯ ( ∑ j = 1 n a n j x j ) . {\displaystyle {\begin{aligned}F(x_{1},x_{2},\dots ,x_{n})&=\prod _{i=1}^{n}\left(\sum _{j=1}^{n}a_{ij}x_{j}\right)\\&=\left(\sum _{j=1}^{n}a_{1j}x_{j}\right)\left(\sum _{j=1}^{n}a_{2j}x_{j}\right)\cdots \left(\sum _{j=1}^{n}a_{nj}x_{j}\right).\end{aligned}}} The coefficient of x 1 x 2 … x n {\displaystyle x_{1}x_{2}\dots x_{n}} in F ( x 1 , x 2 , … , x n ) {\displaystyle F(x_{1},x_{2},\dots ,x_{n})} is perm(A). As a generalization, for any sequence of n non-negative integers, s 1 , s 2 , … , s n {\displaystyle s_{1},s_{2},\dots ,s_{n}} define: perm ( s 1 , s 2 , … , s n ) ( A ) {\displaystyle \operatorname {perm} ^{(s_{1},s_{2},\dots ,s_{n})}(A)} as the coefficient of x 1 s 1 x 2 s 2 ⋯ x n s n {\displaystyle x_{1}^{s_{1}}x_{2}^{s_{2}}\cdots x_{n}^{s_{n}}} in ( ∑ j = 1 n a 1 j x j ) s 1 ( ∑ j = 1 n a 2 j x j ) s 2 ⋯ ( ∑ j = 1 n a n j x j ) s n . {\displaystyle \left(\sum _{j=1}^{n}a_{1j}x_{j}\right)^{s_{1}}\left(\sum _{j=1}^{n}a_{2j}x_{j}\right)^{s_{2}}\cdots \left(\sum _{j=1}^{n}a_{nj}x_{j}\right)^{s_{n}}.} MacMahon's master theorem relating permanents and determinants is: perm ( s 1 , s 2 , … , s n ) ( A ) = coefficient of x 1 s 1 x 2 s 2 ⋯ x n s n in 1 det ( I − X A ) , {\displaystyle \operatorname {perm} ^{(s_{1},s_{2},\dots ,s_{n})}(A)={\text{ coefficient of }}x_{1}^{s_{1}}x_{2}^{s_{2}}\cdots x_{n}^{s_{n}}{\text{ in }}{\frac {1}{\det(I-XA)}},} where I is the order n identity matrix and X is the diagonal matrix with diagonal [ x 1 , x 2 , … , x n ] . {\displaystyle [x_{1},x_{2},\dots ,x_{n}].} == Rectangular matrices == The permanent function can be generalized to apply to non-square matrices. Indeed, several authors make this the definition of a permanent and consider the restriction to square matrices a special case. Specifically, for an m × n matrix A = ( a i j ) {\displaystyle A=(a_{ij})} with m ≤ n, define perm ( A ) = ∑ σ ∈ P ( n , m ) a 1 σ ( 1 ) a 2 σ ( 2 ) … a m σ ( m ) {\displaystyle \operatorname {perm} (A)=\sum _{\sigma \in \operatorname {P} (n,m)}a_{1\sigma (1)}a_{2\sigma (2)}\ldots a_{m\sigma (m)}} where P(n,m) is the set of all m-permutations of the n-set {1,2,...,n}. Ryser's computational result for permanents also generalizes. If A is an m × n matrix with m ≤ n, let A k {\displaystyle A_{k}} be obtained from A by deleting k columns, let P ( A k ) {\displaystyle P(A_{k})} be the product of the row-sums of A k {\displaystyle A_{k}} , and let σ k {\displaystyle \sigma _{k}} be the sum of the values of P ( A k ) {\displaystyle P(A_{k})} over all possible A k {\displaystyle A_{k}} . Then perm ( A ) = ∑ k = 0 m − 1 ( − 1 ) k ( n − m + k k ) σ n − m + k . {\displaystyle \operatorname {perm} (A)=\sum _{k=0}^{m-1}(-1)^{k}{\binom {n-m+k}{k}}\sigma _{n-m+k}.} === Systems of distinct representatives === The generalization of the definition of a permanent to non-square matrices allows the concept to be used in a more natural way in some applications. For instance: Let S1, S2, ..., Sm be subsets (not necessarily distinct) of an n-set with m ≤ n. The incidence matrix of this collection of subsets is an m × n (0,1)-matrix A. The number of systems of distinct representatives (SDR's) of this collection is perm(A). == See also == Computing the permanent Bapat–Beg theorem, an application of permanents in order statistics Slater determinant, an application of permanents in quantum mechanics Hafnian == Notes == == References == Brualdi, Richard A. (2006). Combinatorial matrix classes. Encyclopedia of Mathematics and Its Applications. Vol. 108. Cambridge: Cambridge University Press. ISBN 978-0-521-86565-4. Zbl 1106.05001. Minc, Henryk (1978). Permanents. Encyclopedia of Mathematics and its Applications. Vol. 6. With a foreword by Marvin Marcus. Reading, MA: Addison–Wesley. ISSN 0953-4806. OCLC 3980645. Zbl 0401.15005. Muir, Thomas; Metzler, William H. (1960) [1882]. A Treatise on the Theory of Determinants. New York: Dover. OCLC 535903. Percus, J.K. (1971), Combinatorial Methods, Applied Mathematical Sciences #4, New York: Springer-Verlag, ISBN 978-0-387-90027-8 Ryser, Herbert John (1963), Combinatorial Mathematics, The Carus Mathematical Monographs #14, The Mathematical Association of America van Lint, J.H.; Wilson, R.M. (2001), A Course in Combinatorics, Cambridge University Press, ISBN 978-0521422604 == Further reading == Hall Jr., Marshall (1986), Combinatorial Theory (2nd ed.), New York: John Wiley & Sons, pp. 56–72, ISBN 978-0-471-09138-7 Contains a proof of the Van der Waerden conjecture. Marcus, M.; Minc, H. (1965), "Permanents", The American Mathematical Monthly, 72 (6): 577–591, doi:10.2307/2313846, JSTOR 2313846 == External links == Permanent at PlanetMath. Van der Waerden's permanent conjecture at PlanetMath.
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Wikipedia:Perron's irreducibility criterion#0
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Perron's irreducibility criterion is a sufficient condition for a polynomial to be irreducible in Z [ x ] {\displaystyle \mathbb {Z} [x]} —that is, for it to be unfactorable into the product of lower-degree polynomials with integer coefficients. This criterion is applicable only to monic polynomials. However, unlike other commonly used criteria, Perron's criterion does not require any knowledge of prime decomposition of the polynomial's coefficients. == Criterion == Suppose we have the following polynomial with integer coefficients f ( x ) = x n + a n − 1 x n − 1 + ⋯ + a 1 x + a 0 , {\displaystyle f(x)=x^{n}+a_{n-1}x^{n-1}+\cdots +a_{1}x+a_{0},} where a 0 ≠ 0 {\displaystyle a_{0}\neq 0} . If either of the following two conditions applies: | a n − 1 | > 1 + | a n − 2 | + ⋯ + | a 0 | {\displaystyle |a_{n-1}|>1+|a_{n-2}|+\cdots +|a_{0}|} | a n − 1 | = 1 + | a n − 2 | + ⋯ + | a 0 | , f ( ± 1 ) ≠ 0 {\displaystyle |a_{n-1}|=1+|a_{n-2}|+\cdots +|a_{0}|,\quad f(\pm 1)\neq 0} then f {\displaystyle f} is irreducible over the integers (and by Gauss's lemma also over the rational numbers). == History == The criterion was first published by Oskar Perron in 1907 in Journal für die reine und angewandte Mathematik. == Proof == A short proof can be given based on the following lemma due to Panaitopol: Lemma. Let f ( x ) = x n + a n − 1 x n − 1 + ⋯ + a 1 x + a 0 {\displaystyle f(x)=x^{n}+a_{n-1}x^{n-1}+\cdots +a_{1}x+a_{0}} be a polynomial with | a n − 1 | > 1 + | a n − 2 | + ⋯ + | a 1 | + | a 0 | {\displaystyle |a_{n-1}|>1+|a_{n-2}|+\cdots +|a_{1}|+|a_{0}|} . Then exactly one zero z {\displaystyle z} of f {\displaystyle f} satisfies | z | > 1 {\displaystyle |z|>1} , and the other n − 1 {\displaystyle n-1} zeroes of f {\displaystyle f} satisfy | z | < 1 {\displaystyle |z|<1} . Suppose that f ( x ) = g ( x ) h ( x ) {\displaystyle f(x)=g(x)h(x)} where g {\displaystyle g} and h {\displaystyle h} are integer polynomials. Since, by the above lemma, f {\displaystyle f} has only one zero with modulus not less than 1 {\displaystyle 1} , one of the polynomials g , h {\displaystyle g,h} has all its zeroes strictly inside the unit circle. Suppose that z 1 , … , z k {\displaystyle z_{1},\dots ,z_{k}} are the zeroes of g {\displaystyle g} , and | z 1 | , … , | z k | < 1 {\displaystyle |z_{1}|,\dots ,|z_{k}|<1} . Note that g ( 0 ) {\displaystyle g(0)} is a nonzero integer, and | g ( 0 ) | = | z 1 ⋯ z k | < 1 {\displaystyle |g(0)|=|z_{1}\cdots z_{k}|<1} , contradiction. Therefore, f {\displaystyle f} is irreducible. == Generalizations == In his publication Perron provided variants of the criterion for multivariate polynomials over arbitrary fields. In 2010, Bonciocat published novel proofs of these criteria. == See also == Eisenstein's criterion Cohn's irreducibility criterion == References ==
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Wikipedia:Perspective (graphical)#0
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Linear or point-projection perspective (from Latin perspicere 'to see through') is one of two types of graphical projection perspective in the graphic arts; the other is parallel projection. Linear perspective is an approximate representation, generally on a flat surface, of an image as it is seen by the eye. Perspective drawing is useful for representing a three-dimensional scene in a two-dimensional medium, like paper. It is based on the optical fact that for a person an object looks N times (linearly) smaller if it has been moved N times further from the eye than the original distance was. The most characteristic features of linear perspective are that objects appear smaller as their distance from the observer increases, and that they are subject to foreshortening, meaning that an object's dimensions parallel to the line of sight appear shorter than its dimensions perpendicular to the line of sight. All objects will recede to points in the distance, usually along the horizon line, but also above and below the horizon line depending on the view used. Italian Renaissance painters and architects including Filippo Brunelleschi, Leon Battista Alberti, Masaccio, Paolo Uccello, Piero della Francesca and Luca Pacioli studied linear perspective, wrote treatises on it, and incorporated it into their artworks. == Overview == Linear or point-projection perspective works by putting an imaginary flat plane that is close to an object under observation and directly facing an observer's eyes (i.e., the observer is on a normal, or perpendicular line to the plane). Then draw straight lines from every point in the object to the observer. The area on the plane where those lines pass through the plane is a point-projection prospective image resembling what is seen by the observer. === Examples of one-point perspective === === Examples of two-point perspective === === Examples of three-point perspective === === Examples of curvilinear perspective === Additionally, a central vanishing point can be used (just as with one-point perspective) to indicate frontal (foreshortened) depth. == History == === Early history === The earliest art paintings and drawings typically sized many objects and characters hierarchically according to their spiritual or thematic importance, not their distance from the viewer, and did not use foreshortening. The most important figures are often shown as the highest in a composition, also from hieratic motives, leading to the so-called "vertical perspective", common in the art of Ancient Egypt, where a group of "nearer" figures are shown below the larger figure or figures; simple overlapping was also employed to relate distance. Additionally, oblique foreshortening of round elements like shields and wheels is evident in Ancient Greek red-figure pottery. Systematic attempts to evolve a system of perspective are usually considered to have begun around the fifth century BC in the art of ancient Greece, as part of a developing interest in illusionism allied to theatrical scenery. This was detailed within Aristotle's Poetics as skenographia: using flat panels on a stage to give the illusion of depth. The philosophers Anaxagoras and Democritus worked out geometric theories of perspective for use with skenographia. Alcibiades had paintings in his house designed using skenographia, so this art was not confined merely to the stage. Euclid in his Optics (c. 300 BC) argues correctly that the perceived size of an object is not related to its distance from the eye by a simple proportion. In the first-century BC frescoes of the Villa of P. Fannius Synistor, multiple vanishing points are used in a systematic but not fully consistent manner. Chinese artists made use of oblique projection from the first or second century until the 18th century. It is not certain how they came to use the technique; Dubery and Willats (1983) speculate that the Chinese acquired the technique from India, which acquired it from Ancient Rome, while others credit it as an indigenous invention of Ancient China. Oblique projection is also seen in Japanese art, such as in the Ukiyo-e paintings of Torii Kiyonaga (1752–1815). By the later periods of antiquity, artists, especially those in less popular traditions, were well aware that distant objects could be shown smaller than those close at hand for increased realism, but whether this convention was actually used in a work depended on many factors. Some of the paintings found in the ruins of Pompeii show a remarkable realism and perspective for their time. It has been claimed that comprehensive systems of perspective were evolved in antiquity, but most scholars do not accept this. Hardly any of the many works where such a system would have been used have survived. A passage in Philostratus suggests that classical artists and theorists thought in terms of "circles" at equal distance from the viewer, like a classical semi-circular theatre seen from the stage. The roof beams in rooms in the Vatican Virgil, from about 400 AD, are shown converging, more or less, on a common vanishing point, but this is not systematically related to the rest of the composition. Medieval artists in Europe, like those in the Islamic world and China, were aware of the general principle of varying the relative size of elements according to distance, but even more than classical art were perfectly ready to override it for other reasons. Buildings were often shown obliquely according to a particular convention. The use and sophistication of attempts to convey distance increased steadily during the period, but without a basis in a systematic theory. Byzantine art was also aware of these principles, but also used the reverse perspective convention for the setting of principal figures. Ambrogio Lorenzetti painted a floor with convergent lines in his Presentation at the Temple (1342), though the rest of the painting lacks perspective elements. === Renaissance === It is generally accepted that Filippo Brunelleschi conducted a series of experiments between 1415 and 1420, which included making drawings of various Florentine buildings in correct perspective. According to Vasari and Antonio Manetti, in about 1420, Brunelleschi demonstrated his discovery of perspective by having people look through a hole on his painting from the backside. Through it, they would see a building such as the Florence Baptistery for which the painting was made. When Brunelleschi lifted a mirror between the building and the painting, the mirror reflected the painting to an observer looking through the hole, so that the observer can compare how similar the building and the painting of it are. (The vanishing point is centered from the perspective of an experiment participant.) Brunelleschi applied this new system of perspective to his paintings around 1425. This scenario is indicative, but faces several problems that are still debated. First of all, nothing can be said for certain about the correctness of his perspective construction of the Baptistery of San Giovanni because Brunelleschi's panel is lost. Second, no other perspective painting or drawing by Brunelleschi is known. (In fact, Brunelleschi was not known to have painted at all.) Third, in the account written by Antonio Manetti in his Vita di Ser Brunellesco at the end of the 15th century on Brunelleschi's panel, there is not a single occurrence of the word "experiment". Fourth, the conditions listed by Manetti are contradictory with each other. For example, the description of the eyepiece sets a visual field of 15°, much narrower than the visual field resulting from the urban landscape described. Soon after Brunelleschi's demonstrations, nearly every interested artist in Florence and in Italy used geometrical perspective in their paintings and sculpture, notably Donatello, Masaccio,Lorenzo Ghiberti, Masolino da Panicale, Paolo Uccello, and Filippo Lippi. Not only was perspective a way of showing depth, it was also a new method of creating a composition. Visual art could now depict a single, unified scene rather than a combination of several. Early examples include Masolino's St. Peter Healing a Cripple and the Raising of Tabitha (c. 1423), Donatello's The Feast of Herod (c. 1427), as well as Ghiberti's Jacob and Esau and other panels from the east doors of the Florence Baptistery. Masaccio (d. 1428) achieved an illusionistic effect by placing the vanishing point at the viewer's eye level in his Holy Trinity (c. 1427), and in The Tribute Money, it is placed behind the face of Jesus. In the late 15th century, Melozzo da Forlì first applied the technique of foreshortening (in Rome, Loreto, Forlì and others). This overall story is based on qualitative judgments, and would need to be faced against the material evaluations that have been conducted on Renaissance perspective paintings. Apart from the paintings of Piero della Francesca, which are a model of the genre, the majority of 15th century works show serious errors in their geometric construction. This is true of Masaccio's Trinity fresco and of many works, including those by renowned artists like Leonardo da Vinci. As shown by the quick proliferation of accurate perspective paintings in Florence, Brunelleschi likely understood (with help from his friend the mathematician Toscanelli), but did not publish the mathematics behind perspective. Decades later, his friend Leon Battista Alberti wrote De pictura (c. 1435), a treatise on proper methods of showing distance in painting. Alberti's primary breakthrough was not to show the mathematics in terms of conical projections, as it actually appears to the eye. Instead, he formulated the theory based on planar projections, or how the rays of light, passing from the viewer's eye to the landscape, would strike the picture plane (the painting). He was then able to calculate the apparent height of a distant object using two similar triangles. The mathematics behind similar triangles is relatively simple, having been long ago formulated by Euclid. Alberti was also trained in the science of optics through the school of Padua and under the influence of Biagio Pelacani da Parma who studied Alhazen's Book of Optics. This book, translated around 1200 into Latin, had laid the mathematical foundation for perspective in Europe. Piero della Francesca elaborated on De pictura in his De Prospectiva pingendi in the 1470s, making many references to Euclid. Alberti had limited himself to figures on the ground plane and giving an overall basis for perspective. Della Francesca fleshed it out, explicitly covering solids in any area of the picture plane. Della Francesca also started the now common practice of using illustrated figures to explain the mathematical concepts, making his treatise easier to understand than Alberti's. Della Francesca was also the first to accurately draw the Platonic solids as they would appear in perspective. Luca Pacioli's 1509 Divina proportione (Divine Proportion), illustrated by Leonardo da Vinci, summarizes the use of perspective in painting, including much of Della Francesca's treatise. Leonardo applied one-point perspective as well as shallow focus to some of his works. Two-point perspective was demonstrated as early as 1525 by Albrecht Dürer, who studied perspective by reading Piero and Pacioli's works, in his Unterweisung der Messung ("Instruction of the Measurement"). == Limitations == Perspective images are created with reference to a particular center of vision for the picture plane. In order for the resulting image to appear identical to the original scene, a viewer must view the image from the exact vantage point used in the calculations relative to the image. When viewed from a different point, this cancels out what would appear to be distortions in the image. For example, a sphere drawn in perspective will be stretched into an ellipse. These apparent distortions are more pronounced away from the center of the image as the angle between a projected ray (from the scene to the eye) becomes more acute relative to the picture plane. Artists may choose to "correct" perspective distortions, for example by drawing all spheres as perfect circles, or by drawing figures as if centered on the direction of view. In practice, unless the viewer observes the image from an extreme angle, like standing far to the side of a painting, the perspective normally looks more or less correct. This is referred to as "Zeeman's Paradox". == See also == Anamorphosis Camera angle Cutaway drawing Perspective control Perspective (geometry) Trompe-l'œil Uki-e Zograscope == Notes == == References == === Sources === Edgerton, Samuel Y. (2009). The Mirror, the Window & the Telescope: How Renaissance Linear Perspective Changed Our Vision of the Universe. Ithaca, NY: Cornell University Press. ISBN 978-0-8014-4758-7. == Further reading == Andersen, Kirsti (2007). The Geometry of an Art: The History of the Mathematical Theory of Perspective from Alberti to Monge. Springer. Damisch, Hubert (1994). The Origin of Perspective, Translated by John Goodman. Cambridge, Massachusetts: MIT Press. Gill, Robert W (1974). Perspective From Basic to Creative. Australia: Thames & Hudson. Hyman, Isabelle, comp (1974). Brunelleschi in Perspective. Englewood Cliffs, New Jersey: Prentice-Hall.{{cite book}}: CS1 maint: multiple names: authors list (link) Kemp, Martin (1992). The Science of Art: Optical Themes in Western Art from Brunelleschi to Seurat. Yale University Press. Pérez-Gómez, Alberto; Pelletier, Louise (1997). Architectural Representation and the Perspective Hinge. Cambridge, Massachusetts: MIT Press. Raynaud, Dominique (2003). "Linear perspective in Masaccio's Trinity fresco: Demonstration or self-persuasion?". Nuncius. 18 (1): 331–344. doi:10.1163/182539103X00684. Raynaud, Dominique (2014). Optics and the Rise of Perspective. A Study in Network Knowledge Diffusion. Oxford: Bardwell Press. Raynaud, Dominique (2016). Studies on Binocular Vision. Archimedes. Vol. 47. Bibcode:2016sbvo.book.....R. doi:10.1007/978-3-319-42721-8. ISBN 978-3-319-42720-1. S2CID 151589160. Vasari, Giorgio (1568). The Lives of the Artists. Florence, Italy. == External links == Teaching Perspective in Art and Mathematics through Leonardo da Vinci's Work at Mathematical Association of America Metaphysical Perspective in Ancient Roman-Wall Painting How to Draw a Two Point Perspective Grid at Creating Comics
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Wikipedia:Perspectivity#0
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In geometry and in its applications to drawing, a perspectivity is the formation of an image in a picture plane of a scene viewed from a fixed point. == Graphics == The science of graphical perspective uses perspectivities to make realistic images in proper proportion. According to Kirsti Andersen, the first author to describe perspectivity was Leon Alberti in his De Pictura (1435). In English, Brook Taylor presented his Linear Perspective in 1715, where he explained "Perspective is the Art of drawing on a Plane the Appearances of any Figures, by the Rules of Geometry". In a second book, New Principles of Linear Perspective (1719), Taylor wrote When Lines drawn according to a certain Law from the several Parts of any Figure, cut a Plane, and by that Cutting or Intersection describe a figure on that Plane, that Figure so described is called the Projection of the other Figure. The Lines producing that Projection, taken all together, are called the System of Rays. And when those Rays all pass thro’ one and same Point, they are called the Cone of Rays. And when that Point is consider’d as the Eye of a Spectator, that System of Rays is called the Optic Cone == Projective geometry == In projective geometry the points of a line are called a projective range, and the set of lines in a plane on a point is called a pencil. Given two lines ℓ {\displaystyle \ell } and m {\displaystyle m} in a projective plane and a point P of that plane on neither line, the bijective mapping between the points of the range of ℓ {\displaystyle \ell } and the range of m {\displaystyle m} determined by the lines of the pencil on P is called a perspectivity (or more precisely, a central perspectivity with center P). A special symbol has been used to show that points X and Y are related by a perspectivity; X ⩞ Y . {\displaystyle X\doublebarwedge Y.} In this notation, to show that the center of perspectivity is P, write X ⩞ P Y . {\displaystyle X\ {\overset {P}{\doublebarwedge }}\ Y.} The existence of a perspectivity means that corresponding points are in perspective. The dual concept, axial perspectivity, is the correspondence between the lines of two pencils determined by a projective range. === Projectivity === The composition of two perspectivities is, in general, not a perspectivity. A perspectivity or a composition of two or more perspectivities is called a projectivity (projective transformation, projective collineation and homography are synonyms). There are several results concerning projectivities and perspectivities which hold in any pappian projective plane: Theorem: Any projectivity between two distinct projective ranges can be written as the composition of no more than two perspectivities. Theorem: Any projectivity from a projective range to itself can be written as the composition of three perspectivities. Theorem: A projectivity between two distinct projective ranges which fixes a point is a perspectivity. === Higher-dimensional perspectivities === The bijective correspondence between points on two lines in a plane determined by a point of that plane not on either line has higher-dimensional analogues which will also be called perspectivities. Let Sm and Tm be two distinct m-dimensional projective spaces contained in an n-dimensional projective space Rn. Let Pn−m−1 be an (n − m − 1)-dimensional subspace of Rn with no points in common with either Sm or Tm. For each point X of Sm, the space L spanned by X and Pn-m-1 meets Tm in a point Y = fP(X). This correspondence fP is also called a perspectivity. The central perspectivity described above is the case with n = 2 and m = 1. === Perspective collineations === Let S2 and T2 be two distinct projective planes in a projective 3-space R3. With O and O* being points of R3 in neither plane, use the construction of the last section to project S2 onto T2 by the perspectivity with center O followed by the projection of T2 back onto S2 with the perspectivity with center O*. This composition is a bijective map of the points of S2 onto itself which preserves collinear points and is called a perspective collineation (central collineation in more modern terminology). Let φ be a perspective collineation of S2. Each point of the line of intersection of S2 and T2 will be fixed by φ and this line is called the axis of φ. Let point P be the intersection of line OO* with the plane S2. P is also fixed by φ and every line of S2 that passes through P is stabilized by φ (fixed, but not necessarily pointwise fixed). P is called the center of φ. The restriction of φ to any line of S2 not passing through P is the central perspectivity in S2 with center P between that line and the line which is its image under φ. == See also == Perspective projection Desargues's theorem == Notes == == References == Andersen, Kirsti (1992), Brook Taylor's Work on Linear Perspective, Springer, ISBN 0-387-97486-5 Coxeter, Harold Scott MacDonald (1969), Introduction to Geometry (2nd ed.), New York: John Wiley & Sons, ISBN 978-0-471-50458-0, MR 0123930 Fishback, W.T. (1969), Projective and Euclidean Geometry, John Wiley & Sons Pedoe, Dan (1988), Geometry/A Comprehensive Course, Dover, ISBN 0-486-65812-0 Young, John Wesley (1930), Projective Geometry, The Carus Mathematical Monographs (#4), Mathematical Association of America == External links == Christopher Cooper Perspectivities and Projectivities. James C. Morehead Jr. (1911) Perspective and Projective Geometries: A Comparison from Rice University. John Taylor Projective Geometry from University of Brighton.
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Wikipedia:Perturbation problem beyond all orders#0
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In mathematics, perturbation theory works typically by expanding unknown quantity in a power series in a small parameter. However, in a perturbation problem beyond all orders, all coefficients of the perturbation expansion vanish and the difference between the function and the constant function 0 cannot be detected by a power series. A simple example is understood by an attempt at trying to expand e − 1 / ϵ {\displaystyle e^{-1/\epsilon }} in a Taylor series in ϵ > 0 {\displaystyle \epsilon >0} about 0. All terms in a naïve Taylor expansion are identically zero. This is because the function e − 1 / z {\displaystyle e^{-1/z}} possesses an essential singularity at z = 0 {\displaystyle z=0} in the complex z {\displaystyle z} -plane, and therefore the function is most appropriately modeled by a Laurent series -- a Taylor series has a zero radius of convergence. Thus, if a physical problem possesses a solution of this nature, possibly in addition to an analytic part that may be modeled by a power series, the perturbative analysis fails to recover the singular part. Terms of nature similar to e − 1 / ϵ {\displaystyle e^{-1/\epsilon }} are considered to be "beyond all orders" of the standard perturbative power series. == See also == Asymptotic expansion == References == J P Boyd, "The Devil's Invention: Asymptotic, Superasymptotic and Hyperasymptotic Series", https://link.springer.com/article/10.1023/A:1006145903624 C. M. Bender and S. A. Orszag, "Advanced Mathematical Methods for Scientists and Engineers", https://link.springer.com/book/10.1007%2F978-1-4757-3069-2 C. M. Bender, Lectures on Mathematical Physics, https://www.perimeterinstitute.ca/video-library/collection/11/12-psi-mathematical-physics Archived 2017-01-09 at the Wayback Machine
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Wikipedia:Perturbation theory#0
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In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle step that breaks the problem into "solvable" and "perturbative" parts. In regular perturbation theory, the solution is expressed as a power series in a small parameter ε {\displaystyle \varepsilon } . The first term is the known solution to the solvable problem. Successive terms in the series at higher powers of ε {\displaystyle \varepsilon } usually become smaller. An approximate 'perturbation solution' is obtained by truncating the series, often keeping only the first two terms, the solution to the known problem and the 'first order' perturbation correction. Perturbation theory is used in a wide range of fields and reaches its most sophisticated and advanced forms in quantum field theory. Perturbation theory (quantum mechanics) describes the use of this method in quantum mechanics. The field in general remains actively and heavily researched across multiple disciplines. == Description == Perturbation theory develops an expression for the desired solution in terms of a formal power series known as a perturbation series in some "small" parameter, that quantifies the deviation from the exactly solvable problem. The leading term in this power series is the solution of the exactly solvable problem, while further terms describe the deviation in the solution, due to the deviation from the initial problem. Formally, we have for the approximation to the full solution A , {\displaystyle \ A\ ,} a series in the small parameter (here called ε), like the following: A ≡ A 0 + ε 1 A 1 + ε 2 A 2 + ε 3 A 3 + ⋯ {\displaystyle A\equiv A_{0}+\varepsilon ^{1}A_{1}+\varepsilon ^{2}A_{2}+\varepsilon ^{3}A_{3}+\cdots } In this example, A 0 {\displaystyle \ A_{0}\ } would be the known solution to the exactly solvable initial problem, and the terms A 1 , A 2 , A 3 , … {\displaystyle \ A_{1},A_{2},A_{3},\ldots \ } represent the first-order, second-order, third-order, and higher-order terms, which may be found iteratively by a mechanistic but increasingly difficult procedure. For small ε {\displaystyle \ \varepsilon \ } these higher-order terms in the series generally (but not always) become successively smaller. An approximate "perturbative solution" is obtained by truncating the series, often by keeping only the first two terms, expressing the final solution as a sum of the initial (exact) solution and the "first-order" perturbative correction A → A 0 + ε A 1 f o r ε → 0 {\displaystyle A\to A_{0}+\varepsilon A_{1}\qquad {\mathsf {for}}\qquad \varepsilon \to 0} Some authors use big O notation to indicate the order of the error in the approximate solution: A = A 0 + ε A 1 + O ( ε 2 ) . {\displaystyle \;A=A_{0}+\varepsilon A_{1}+{\mathcal {O}}{\bigl (}\ \varepsilon ^{2}\ {\bigr )}~.} If the power series in ε {\displaystyle \ \varepsilon \ } converges with a nonzero radius of convergence, the perturbation problem is called a regular perturbation problem. In regular perturbation problems, the asymptotic solution smoothly approaches the exact solution. However, the perturbation series can also diverge, and the truncated series can still be a good approximation to the true solution if it is truncated at a point at which its elements are minimum. This is called an asymptotic series. If the perturbation series is divergent or not a power series (for example, if the asymptotic expansion must include non-integer powers ε ( 1 / 2 ) {\displaystyle \ \varepsilon ^{\left(1/2\right)}\ } or negative powers ε − 2 {\displaystyle \ \varepsilon ^{-2}\ } ) then the perturbation problem is called a singular perturbation problem. Many special techniques in perturbation theory have been developed to analyze singular perturbation problems. == Prototypical example == The earliest use of what would now be called perturbation theory was to deal with the otherwise unsolvable mathematical problems of celestial mechanics: for example the orbit of the Moon, which moves noticeably differently from a simple Keplerian ellipse because of the competing gravitation of the Earth and the Sun. Perturbation methods start with a simplified form of the original problem, which is simple enough to be solved exactly. In celestial mechanics, this is usually a Keplerian ellipse. Under Newtonian gravity, an ellipse is exactly correct when there are only two gravitating bodies (say, the Earth and the Moon) but not quite correct when there are three or more objects (say, the Earth, Moon, Sun, and the rest of the Solar System) and not quite correct when the gravitational interaction is stated using formulations from general relativity. == Perturbative expansion == Keeping the above example in mind, one follows a general recipe to obtain the perturbation series. The perturbative expansion is created by adding successive corrections to the simplified problem. The corrections are obtained by forcing consistency between the unperturbed solution, and the equations describing the system in full. Write D {\displaystyle \ D\ } for this collection of equations; that is, let the symbol D {\displaystyle \ D\ } stand in for the problem to be solved. Quite often, these are differential equations, thus, the letter "D". The process is generally mechanical, if laborious. One begins by writing the equations D {\displaystyle \ D\ } so that they split into two parts: some collection of equations D 0 {\displaystyle \ D_{0}\ } which can be solved exactly, and some additional remaining part ε D 1 {\displaystyle \ \varepsilon D_{1}\ } for some small ε ≪ 1 . {\displaystyle \ \varepsilon \ll 1~.} The solution A 0 {\displaystyle \ A_{0}\ } (to D 0 {\displaystyle \ D_{0}\ } ) is known, and one seeks the general solution A {\displaystyle \ A\ } to D = D 0 + ε D 1 . {\displaystyle \ D=D_{0}+\varepsilon D_{1}~.} Next the approximation A ≈ A 0 + ε A 1 {\displaystyle \ A\approx A_{0}+\varepsilon A_{1}\ } is inserted into ε D 1 {\displaystyle \ \varepsilon D_{1}} . This results in an equation for A 1 , {\displaystyle \ A_{1}\ ,} which, in the general case, can be written in closed form as a sum over integrals over A 0 . {\displaystyle \ A_{0}~.} Thus, one has obtained the first-order correction A 1 {\displaystyle \ A_{1}\ } and thus A ≈ A 0 + ε A 1 {\displaystyle \ A\approx A_{0}+\varepsilon A_{1}\ } is a good approximation to A . {\displaystyle \ A~.} It is a good approximation, precisely because the parts that were ignored were of size ε 2 . {\displaystyle \ \varepsilon ^{2}~.} The process can then be repeated, to obtain corrections A 2 , {\displaystyle \ A_{2}\ ,} and so on. In practice, this process rapidly explodes into a profusion of terms, which become extremely hard to manage by hand. Isaac Newton is reported to have said, regarding the problem of the Moon's orbit, that "It causeth my head to ache." This unmanageability has forced perturbation theory to develop into a high art of managing and writing out these higher order terms. One of the fundamental breakthroughs in quantum mechanics for controlling the expansion are the Feynman diagrams, which allow quantum mechanical perturbation series to be represented by a sketch. == Examples == Perturbation theory has been used in a large number of different settings in physics and applied mathematics. Examples of the "collection of equations" D {\displaystyle D} include algebraic equations, differential equations (e.g., the equations of motion and commonly wave equations), thermodynamic free energy in statistical mechanics, radiative transfer, and Hamiltonian operators in quantum mechanics. Examples of the kinds of solutions that are found perturbatively include the solution of the equation of motion (e.g., the trajectory of a particle), the statistical average of some physical quantity (e.g., average magnetization), and the ground state energy of a quantum mechanical problem. Examples of exactly solvable problems that can be used as starting points include linear equations, including linear equations of motion (harmonic oscillator, linear wave equation), statistical or quantum-mechanical systems of non-interacting particles (or in general, Hamiltonians or free energies containing only terms quadratic in all degrees of freedom). Examples of systems that can be solved with perturbations include systems with nonlinear contributions to the equations of motion, interactions between particles, terms of higher powers in the Hamiltonian/free energy. For physical problems involving interactions between particles, the terms of the perturbation series may be displayed (and manipulated) using Feynman diagrams. == History == Perturbation theory was first devised to solve otherwise intractable problems in the calculation of the motions of planets in the solar system. For instance, Newton's law of universal gravitation explained the gravitation between two astronomical bodies, but when a third body is added, the problem was, "How does each body pull on each?" Kepler's orbital equations only solve Newton's gravitational equations when the latter are limited to just two bodies interacting. The gradually increasing accuracy of astronomical observations led to incremental demands in the accuracy of solutions to Newton's gravitational equations, which led many eminent 18th and 19th century mathematicians, notably Joseph-Louis Lagrange and Pierre-Simon Laplace, to extend and generalize the methods of perturbation theory. These well-developed perturbation methods were adopted and adapted to solve new problems arising during the development of quantum mechanics in 20th century atomic and subatomic physics. Paul Dirac developed quantum perturbation theory in 1927 to evaluate when a particle would be emitted in radioactive elements. This was later named Fermi's golden rule. Perturbation theory in quantum mechanics is fairly accessible, mainly because quantum mechanics is limited to linear wave equations, but also since the quantum mechanical notation allows expressions to be written in fairly compact form, thus making them easier to comprehend. This resulted in an explosion of applications, ranging from the Zeeman effect to the hyperfine splitting in the hydrogen atom. Despite the simpler notation, perturbation theory applied to quantum field theory still easily gets out of hand. Richard Feynman developed the celebrated Feynman diagrams by observing that many terms repeat in a regular fashion. These terms can be replaced by dots, lines, squiggles and similar marks, each standing for a term, a denominator, an integral, and so on; thus complex integrals can be written as simple diagrams, with absolutely no ambiguity as to what they mean. The one-to-one correspondence between the diagrams, and specific integrals is what gives them their power. Although originally developed for quantum field theory, it turns out the diagrammatic technique is broadly applicable to many other perturbative series (although not always worthwhile). In the second half of the 20th century, as chaos theory developed, it became clear that unperturbed systems were in general completely integrable systems, while the perturbed systems were not. This promptly lead to the study of "nearly integrable systems", of which the KAM torus is the canonical example. At the same time, it was also discovered that many (rather special) non-linear systems, which were previously approachable only through perturbation theory, are in fact completely integrable. This discovery was quite dramatic, as it allowed exact solutions to be given. This, in turn, helped clarify the meaning of the perturbative series, as one could now compare the results of the series to the exact solutions. The improved understanding of dynamical systems coming from chaos theory helped shed light on what was termed the small denominator problem or small divisor problem. In the 19th century Poincaré observed (as perhaps had earlier mathematicians) that sometimes 2nd and higher order terms in the perturbative series have "small denominators": That is, they have the general form ψ n V ϕ m ( ω n − ω m ) {\displaystyle \ {\frac {\ \psi _{n}V\phi _{m}\ }{\ (\omega _{n}-\omega _{m})\ }}\ } where ψ n , {\displaystyle \ \psi _{n}\ ,} V , {\displaystyle \ V\ ,} and ϕ m {\displaystyle \ \phi _{m}\ } are some complicated expressions pertinent to the problem to be solved, and ω n {\displaystyle \ \omega _{n}\ } and ω m {\displaystyle \ \omega _{m}\ } are real numbers; very often they are the energy of normal modes. The small divisor problem arises when the difference ω n − ω m {\displaystyle \ \omega _{n}-\omega _{m}\ } is small, causing the perturbative correction to "blow up", becoming as large or maybe larger than the zeroth order term. This situation signals a breakdown of perturbation theory: It stops working at this point, and cannot be expanded or summed any further. In formal terms, the perturbative series is an asymptotic series: A useful approximation for a few terms, but at some point becomes less accurate if even more terms are added. The breakthrough from chaos theory was an explanation of why this happened: The small divisors occur whenever perturbation theory is applied to a chaotic system. The one signals the presence of the other. === Beginnings in the study of planetary motion === Since the planets are very remote from each other, and since their mass is small as compared to the mass of the Sun, the gravitational forces between the planets can be neglected, and the planetary motion is considered, to a first approximation, as taking place along Kepler's orbits, which are defined by the equations of the two-body problem, the two bodies being the planet and the Sun. Since astronomic data came to be known with much greater accuracy, it became necessary to consider how the motion of a planet around the Sun is affected by other planets. This was the origin of the three-body problem; thus, in studying the system Moon-Earth-Sun, the mass ratio between the Moon and the Earth was chosen as the "small parameter". Lagrange and Laplace were the first to advance the view that the so-called "constants" which describe the motion of a planet around the Sun gradually change: They are "perturbed", as it were, by the motion of other planets and vary as a function of time; hence the name "perturbation theory". Perturbation theory was investigated by the classical scholars – Laplace, Siméon Denis Poisson, Carl Friedrich Gauss – as a result of which the computations could be performed with a very high accuracy. The discovery of the planet Neptune in 1848 by Urbain Le Verrier, based on the deviations in motion of the planet Uranus. He sent the coordinates to J.G. Galle who successfully observed Neptune through his telescope – a triumph of perturbation theory. == Perturbation orders == The standard exposition of perturbation theory is given in terms of the order to which the perturbation is carried out: first-order perturbation theory or second-order perturbation theory, and whether the perturbed states are degenerate, which requires singular perturbation. In the singular case extra care must be taken, and the theory is slightly more elaborate. == In chemistry == Many of the ab initio quantum chemistry methods use perturbation theory directly or are closely related methods. Implicit perturbation theory works with the complete Hamiltonian from the very beginning and never specifies a perturbation operator as such. Møller–Plesset perturbation theory uses the difference between the Hartree–Fock Hamiltonian and the exact non-relativistic Hamiltonian as the perturbation. The zero-order energy is the sum of orbital energies. The first-order energy is the Hartree–Fock energy and electron correlation is included at second-order or higher. Calculations to second, third or fourth order are very common and the code is included in most ab initio quantum chemistry programs. A related but more accurate method is the coupled cluster method. == Shell-crossing == A shell-crossing (sc) occurs in perturbation theory when matter trajectories intersect, forming a singularity. This limits the predictive power of physical simulations at small scales. == See also == == References == == External links == van den Eijnden, Eric. "Introduction to regular perturbation theory" (PDF). Archived (PDF) from the original on 2004-09-20. Chow, Carson C. (23 October 2007). "Perturbation method of multiple scales". Scholarpedia. 2 (10): 1617. doi:10.4249/scholarpedia.1617. Alternative approach to quantum perturbation theory Martínez-Carranza, J.; Soto-Eguibar, F.; Moya-Cessa, H. (2012). "Alternative analysis to perturbation theory in quantum mechanics". The European Physical Journal D. 66 (1): 22. arXiv:1110.0723. Bibcode:2012EPJD...66...22M. doi:10.1140/epjd/e2011-20654-5. S2CID 117362666.
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Wikipedia:Peter Benner#0
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Peter Benner (born May 25, 1967) is a German mathematician specialized in dynamical systems and numerical analysis. He was managing director at the Max Planck Institute for Dynamics of Complex Technical Systems in Magdeburg, Germany. == Education and career == Benner was born in Kirchen (Sieg). After graduating from high school in 1986 at the Freiherr vom Stein Gymnasium in Betzdorf, Benner studied mathematics with a minor in economics/operations research at the RWTH Aachen University from 1987. In 1993 he received the Springorum commemorative coin from RWTH Aachen for his diploma thesis. He received his doctorate in 1997 in the field of mathematics from the Chemnitz University of Technology under the supervision of Volker Mehrmann. From 1997 to 2001 he worked as a research assistant at the Center for Technomathematics at the University of Bremen. He completed his habilitation there in 2001. From 2001 to 2003, Benner taught as a senior assistant at the Institute for Mathematics at Technische Universität Berlin. In the meantime he worked as a visiting professor at the Hamburg University of Technology. In 2003 he accepted a professorship for mathematics in industry and technology at Chemnitz University of Technology. Benner was a visiting scientist and professor at the University of Kansas, the Università di Modena e Reggio Emilia, the Lawrence Berkeley National Laboratory, the Courant Institute of Mathematical Sciences of New York University, Virginia Tech, the Université du Littoral Côte d'Opale in Calais and Shanghai University employed. In 2018 he was a J. Tinsley Oden Faculty Fellow at the University of Texas at Austin. In 2010, Benner was appointed director and scientific member at the Max Planck Institute for Dynamics of Complex Technical Systems. He began research there with his specialist group Computational Methods in Systems and Control Theory at the Max Planck Institute on May 1, 2010. In the same year he was a visiting professor at the Université du Littoral Côte d'Opale in Calais, France. At the beginning of 2011 he was appointed honorary professor of mathematics at the Otto von Guericke University Magdeburg. In 2015, he was Distinguished Professor at Shanghai University. Since 2017 he has been a Fellow of the Society for Industrial and Applied Mathematics (SIAM) since 2017. Benner is co-editor of various publications and mathematical journals, including the SIAM Journal on Matrix Analysis and Applications and co-author of various software packages. He is involved in mathematical societies, including SIAM and the Society for Applied Mathematics and Mechanics (GAMM). He has also been a member of the Niconet e.V. association since 2006. V., who develops and maintains the software library “Subroutine Library in Systems and Control” (SLICOT). == Bibliography == Benner, Peter; Sorensen, Danny C.; Mehrmann, Volker, eds. (2005). Dimension Reduction of Large-Scale Systems: Proceedings of a Workshop held in Oberwolfach, Germany, October 19–25, 2003. Lecture Notes in Computational Science and Engineering. Vol. 45. Berlin, Heidelberg: Springer Berlin Heidelberg. doi:10.1007/3-540-27909-1. ISBN 978-3-540-24545-2. Benner, Peter; Hinze, Michael; ter Maten, E. Jan W., eds. (2011). Model Reduction for Circuit Simulation. Lecture Notes in Electrical Engineering. Vol. 74. Dordrecht: Springer Netherlands. doi:10.1007/978-94-007-0089-5. ISBN 978-94-007-0088-8. Benner, Peter; Findeisen, Rolf; Flockerzi, Dietrich; Reichl, Udo; Sundmacher, Kai, eds. (2014). Large-Scale Networks in Engineering and Life Sciences. Modeling and Simulation in Science, Engineering and Technology. Cham: Springer International Publishing. doi:10.1007/978-3-319-08437-4. ISBN 978-3-319-08436-7. Benner, Peter; Bollhöfer, Matthias; Kressner, Daniel; Mehl, Christian; Stykel, Tatjana, eds. (2015). Numerical Algebra, Matrix Theory, Differential-Algebraic Equations and Control Theory: Festschrift in Honor of Volker Mehrmann. Cham: Springer International Publishing. doi:10.1007/978-3-319-15260-8. ISBN 978-3-319-15259-2. Benner, Peter; Ohlberger, Mario; Patera, Anthony; Rozza, Gianluigi; Urban, Karsten, eds. (2017). Model Reduction of Parametrized Systems. MS&A. Vol. 17. Cham: Springer International Publishing. doi:10.1007/978-3-319-58786-8. ISBN 978-3-319-58785-1. Benner, Peter, ed. (2017). System Reduction for Nanoscale IC Design. Mathematics in Industry. Vol. 20. Cham: Springer International Publishing. doi:10.1007/978-3-319-07236-4. ISBN 978-3-319-07235-7. Benner, Peter; Breiten, Tobias; Faßbender, Heike; Hinze, Michael; Stykel, Tatjana; Zimmermann, Ralf, eds. (2021). Model Reduction of Complex Dynamical Systems. International Series of Numerical Mathematics. Vol. 171. Cham: Springer International Publishing. doi:10.1007/978-3-030-72983-7. ISBN 978-3-030-72982-0. == References ==
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Wikipedia:Peter Borwein#0
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Peter Benjamin Borwein (born St. Andrews, Scotland, May 10, 1953 – 23 August 2020) was a Canadian mathematician and a professor at Simon Fraser University. He is known as a co-author of the paper which presented the Bailey–Borwein–Plouffe algorithm (discovered by Simon Plouffe) for computing π. == First interest in mathematics == Borwein was born into a Jewish family. He became interested in number theory and classical analysis during his second year of university. He had not previously been interested in math, although his father was the head of the University of Western Ontario's mathematics department and his mother is associate dean of medicine there. Borwein and his two siblings majored in mathematics. == Academic career == After completing a Bachelor of Science in Honours Math at the University of Western Ontario in 1974, he went on to complete an MSc and Ph.D. at the University of British Columbia. He joined the Department of Mathematics at Dalhousie University. While he was there, he, his brother Jonathan Borwein and David H. Bailey of NASA wrote the 1989 paper that outlined and popularized a proof for computing one billion digits of π. The authors won the 1993 Chauvenet Prize and Merten M. Hasse Prize for this paper. In 1993, he moved to Simon Fraser University, joining his brother Jonathan in establishing the Centre for Experimental and Constructive Mathematics (CECM) where he developed the Inverse Symbolic Calculator. == Research == In 1995, the Borweins collaborated with Yasumasa Kanada of the University of Tokyo to compute π to more than four billion digits. Borwein has developed an algorithm that applies Chebyshev polynomials to the Dirichlet eta function to produce a very rapidly convergent series suitable for high precision numerical calculations, which he published on the occasion of the awarding of an honorary doctorate to his brother, Jonathan. Peter Borwein also collaborated with NASA's David Bailey and the Université du Québec's Simon Plouffe to calculate the individual hexadecimal digits of π. This provided a way for mathematicians to determine the nth digit of π without calculating preceding digits. In 2007 with Tamás Erdélyi, Ronald Ferguson, and Richard Lockhart he settled Littlewood's Problem 22. == Affiliations == A former professor at Simon Fraser University, Peter Borwein was affiliated with Interdisciplinary Research in the Mathematical and Computational Sciences (IRMACS), Centre for Experimental and Constructive Mathematics (CECM), Mathematics of Information Technology and Complex Systems (MITACS), and Pacific Institute for the Mathematical Sciences (PIMS). == Personal life and death == Borwein was diagnosed with multiple sclerosis prior to 2000. He died on 23 August 2020 of pneumonia as a result of his MS. == Publications == As a co-author, Borwein has written Pi: A Source Book (with Lennart Berggren and Jonathan Borwein, 2000), Polynomials and Polynomial Inequalities (with Tamas Erdelyi, 1998), Pi and the AGM (1987; reprinted in 1998), A Dictionary of Real Numbers (with Jonathan Borwein), Computational Excursions in Analysis and Number Theory (2002), The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike (with Stephen Choi, Brendan Rooney, and Andrea Weirathmueller, 2007). He and his brother, Jonathan, co-edited the Canadian Mathematical Society/Springer-Verlag series of Books in Mathematics. In 2002 Peter Borwein, with Loki Jorgenson, won a Lester R. Ford Award for their expository article Visible Structures in Number Theory. == See also == Bailey–Borwein–Plouffe formula Erdős–Borwein constant David Borwein (father and mathematician) Jonathan Borwein (brother and mathematician) == References == == External links == Science.ca profile Peter Borwein's research interests Simon Fraser University Centre for Systems Science bio Bailey, David H. (April 2023). "Peter Borwein: A Visionary Mathematician" (PDF). Notices of the American Mathematical Society. 70 (4): 610–613. doi:10.1090/noti2675. SFU news release on Borwein siblings Borwein's website Peter Borwein at the Mathematics Genealogy Project Tamas Erdelyi's website
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Wikipedia:Peter Bouwknegt#0
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Pier Gerard "Peter" Bouwknegt (born 20 April 1961, Geldrop) is professor of theoretical physics and mathematics at the Australian National University (ANU), and deputy director of their Mathematical Sciences Institute. He is an adjunct professor at University of Adelaide. == Biography == He studied Theoretical Physics and Mathematics at the University of Utrecht, Netherlands, and at the University of Amsterdam under the direction of Prof F.A. Bais, obtaining his PhD in 1988. After that, he became a postdoctoral fellow at MIT, CERN, and the University of Southern California. He moved to Australia in 1995 and worked at the University of Adelaide as an ARC QEII Fellow and subsequently as an ARC Senior Research Fellow. In 2005, he was appointed Professor of Theoretical Physics and Mathematics at the Australian National University. == Awards == In 2001, he received the 2001 Australian Mathematical Society Medal, and from 2009 to 2011, he served on the Australian Research Council's College of Experts. He was formerly director of the Mathematical Sciences Institute at ANU, where is now deputy director. == Academic work == Bowknegt specializes in the mathematical foundations of String Theory and Conformal Field Theory. According to his web site at ANU, his specific interests are "the investigation of mathematical aspects of physical theories, in particular quantum field theories. Main expertise is the structure of two-dimensional conformal field theory and their applications in diverse areas such as condensed matter physics, integrable models of statistical mechanics and string theory, as well as the mathematical structures underlying string theory and D-branes, using mathematical techniques such as K-theory and gerbes." == References == == Sources == P.G. Bouwknegt, 1961 - at the University of Amsterdam Album Academicum website == External links == K-theory (physics) Tpsrv.anu.edu.au Peter Bouwknegt on INSPIRE-HEP
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Wikipedia:Peter Bühlmann#0
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Peter Lukas Bühlmann (born 12 April 1965 in Zürich) is a Swiss mathematician and statistician. == Biography == Bühlmann studied mathematics from 1985 at the ETH Zurich with Diplom in 1990 and doctorate in 1993. His thesis The Blockwise Bootstrap in Time Series and Empirical Processes was written under the supervision of Hans-Rudolf Künsch and Erwin Bolthausen. At the University of California, Berkeley, Bühlmann was from 1994 to 1995 a postdoctoral research fellow and from 1995 to 1997 Neyman Assistant Professor. At ETH Zurich he became assistant professor in 1997 and is a full professor from 2004 to the present. From 2013 to 2017 he chaired the Department of Mathematics. His research deals with statistics, machine learning, and computational biology. He is married and has four children. Bühlmann is a frequent mountaineer in the Alps. == Honors and awards == Bühlmann is a Fellow of the Institute of Mathematical Statistics, of the American Statistical Association, and Elected Member of the International Statistical Institute. He received the Wald Memorial Award and Lecture from the Institute of Mathematical Statistics (2024), From 2022 to 2023, he was President of the Institute of Mathematical Statistics. Since 2022, he is a member of the German National Academy of Sciences Leopoldina. He is an honorary doctor of the Catholic University of Louvain and a recipient of Guy Medal in Silver from the Royal Statistical Society (2018). He presented the Neyman Lecture from the Institute of Mathematical Statistics (2018), was Rothschild Fellow and Lecturer at the Isaac Newton Institute (2018), invited speaker at the International Congress of Mathematicians in Rio de Janeiro (2018) and a Plenary Speaker at the 8th European Congress of Mathematics in Portoroz (2021). From 2014 to 2020 he was a Highly Cited Researcher at Thomson Reuters/Clarivate Analytics. From 2010 to 2012 he was a co-editor of the Annals of Statistics. == Selected publications == === Books === with Sara van de Geer: Statistics for high-dimensional data. Methods, Theory and Applications, Springer 2011 as editor with P. Drineas, M. Kane, M. van der Laan: Handbook of Big Data, Chapman and Hall 2016 as editor with others: Statistical Analysis for High-Dimensional Data. The Abel Symposium 2014, Springer 2016 === Articles === with N. Meinshausen: High-dimensional graphs and variable selection with the lasso, Annals of Statistics, vol. 34, 2006, pp. 1436–1462, Arxiv with N. Meinshausen: Stability selection, Journal of the Royal Statistical Society, Series B, vol. 72, 2010, pp. 417–473 doi:10.1111/j.1467-9868.2010.00740.x with L. Meier, S. Van de Geer: The group lasso for logistic regression, Journal of the Royal Statistical Society, Series B, vol. 70, 2008, pp. 53–71 doi:10.1111/j.1467-9868.2007.00627.x with A. Prelić et al.: A systematic comparison and evaluation of biclustering methods for gene expression data, Bioinformatics, vol. 22, 2006, pp. 1122–1129 doi:10.1093/bioinformatics/btl060 with B. Yu: Boosting with the L2 loss: regression and classification, Journal of the American Statistical Association, vol. 98, 2003, pp. 324–339 doi:10.1198/016214503000125 with J. J. Goeman: Analyzing gene expression data in terms of gene sets: methodological issues, Bioinformatics, vol. 23, 2007, pp. 980–987 doi:10.1093/bioinformatics/btm051 with B. Yu: Analyzing bagging, Annals of Statistics, vol. 30, 2002, pp. 927–961 doi:10.1214/aos/1031689014 with T. Hothorn: Boosting algorithms: Regularization, prediction and model fitting, Statistical Science, vol. 22, 2007, pp. 477–505 doi:10.1214/07-STS242 with S. van de Geer: On the conditions used to prove oracle results for the Lasso, Electronic Journal of Statistics, vol. 3, 2009, pp. 1360–1392 doi:10.1214/09-EJS506 == References ==
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