Poincarén.
/pwaŋˈkɑːreɪ/, /pwaŋˈkareɪ/, U.S. /pwɑŋˈkɑreɪ/ Mathematics and Physics.
1. attributive and in the genitive. Used to designate concepts arising from Poincaré's work.
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1904 Trans. Amer. Math. Soc. 5 255 (heading) Proof of Poincaré's Theorem.
1930 J. H. Jeans Universe around Us (ed. 2) v. 285 A mathematical theorem, generally known as Poincaré's theorem, proves to be of the utmost service in discussing the internal state of a gaseous star.
1965 Nuovo Cimento 39 350 Suppose that a real Poincaré algebra were embedded in a semisimple Lie algebra.
2. Poincaré sphere n. a sphere on which the polarization of a beam of light can be represented by a particular point, and which is also used in other optical calculations.
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the world > matter > physics > electromagnetic radiation > light > polarization > [noun] > analysis of > instrument or device
Wollaston('s) prism1890
Poincaré sphere1918
spectropolarimeter1926
chopper1955
1918 Amer. Math. Monthly Oct. 358 Illustration of non-Euclidean geometry on the Poincaré sphere.
1988 Optik 129 185/2 Let us see fig. 4 presenting a Poincaré sphere with two points representing different birefringent media, located on the opposite ends of a diameter.
3. Poincaré group n. a symmetry group that forms the basis of coordinate transformations.
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1929 Amer. Jrnl. Math. 51 305 It was pointed out to me by S. Lefschetz that the above problem is intimately bound up with the Poincaré group of f relative to its carrying complex projective plane.
2002 Sci. Amer. July 60/1 Take the very definition of a ‘particle’ in quantum field theory that originated with physicist Eugene Wigner: a particle is an ‘irreducible representation of the Poincaré group’.
4. Poincaré conjecture n. (also Poincaré's conjecture) the hypothesis that any simply connected compact three-dimensional manifold is topologically equivalent to a three-dimensional sphere.From the time of its formulation in 1904, the Poincaré conjecture was one of the most famous unsolved problems of mathematics. It was proved to great acclaim in 2003–6 by G. Perelman, a Russian mathematician.
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the world > relative properties > number > mathematics > [noun] > mathematical enquiry > proposition > specific conjecture
Goldbach1902
Riemann hypothesis1918
Waring's problem1920
Poincaré conjecture1938
Church's theorem1939
Church's thesis1954
1938 R. L. Wilder in Semicentennial Addresses Amer. Mathematical Soc. 173 An interesting by-product of these investigations..is the establishing of an equivalence between the Poincaré conjecture cited above and each of the..statements.
1961 M.-E. Hamstrom Regular Mappings & Space of Homeomorphisms on 3-Manifold 29 It is known that if each homotopy cell is a 3-cell, then the Poincaré conjecture is true and conversely.
1986 New Scientist 4 Sept. 43/1 Poincaré's Conjecture concerns just how close these family resemblances are and, in particular, how closely the 3-sphere resembles the 2-sphere.
2002 Sunday Tel. (Nexis) 14 Apr. 9 The Poincaré Conjecture is central to three-dimensional topology, so its proof would go far beyond mere ‘tidying up’.
2006 Nature 23 Aug. 859 A mathematician widely agreed to have derived a proof of the century-old Poincaré conjecture has declined to accept one of the 2006 Fields medals.
5. Poincaré recurrence n. (also Poincaré's recurrence) the recurrence of a given state within a dynamic system, posited as a certainty. Usually attributive, esp. in Poincaré recurrence theorem.
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1939 Amer. Jrnl. Math. 61 981 Assuming..that the conditions of §1 are satisfied, one can state Poincaré's recurrence theorem as follows.
1969 Science 7 Nov. 697 There are certain general theorems which such systems, when isolated, obey, one of them being the Poincaré recurrence theorem to the effect that..the system will return over and over again arbitrarily close to the initial state.
1988 J. D. Barrow & F. J. Tipler Anthropic Cosmol. Princ. (rev. ed.) iii. 176 Even closed universes, which do not expand forever, will avoid Poincaré recurrence because the momentum space is unbounded in this type of universe.
6. Poincaré cycle n. a single circuit of a simple closed curve in the phase space of a system, representing one possible solution of the governing differential equations.
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the world > relative properties > number > mathematics > [noun] > mathematical enquiry > result of > in specific case
saddle point1907
Poincaré cycle1943
1943 Rev. Mod. Physics 15 85 (title) Boltzmann's estimate of the period of a Poincaré cycle.
1988 Physics Lett. A. 126 311 Tube-like standing waves subject to a ‘final’ condition may resemble unparametrised orbits of the universe, with ‘quantum Poincaré cycles’ coinciding with its durations.
7.
a. Poincaré map n. a representation of the phase space of a dynamic system, indicating all possible trajectories; (also) = Poincaré section n. at sense 7c.
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1950 Proc. Royal Soc. A. 201 425 In this ‘Poincaré map’, lines are represented by semicircles (and lines) orthogonal to the thin central line.
1981 Sci. Amer. Nov. 24/2 This type of discrete representation of the fly's motion in a multidimensional space is called a Poincaré map.
1992 S. S. Hall Mapping Next Millennium (1993) xiv. 270 That pattern came to be called a Poincaré map, and it showed that a small change in initial conditions created patterns..that departed markedly from the previous initial conditions.
b. Poincaré surface n. (more fully Poincaré surface of section) the intersection of a Poincaré map with a given plane.
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1975 Jrnl. Chem. Physics 62 2119 (title) Semiclassical calculation of bound states in a multidimensional system. Use of Poincaré's surface of section.
1990 Courier-Mail (Brisbane) (Nexis) 31 May Poincare's surfaces are simple and beautiful, consisting of layer upon layer of hollow doughnuts containing particles spiralling around trying to get to a stationary surface which they can only reach when time runs out.
2004 K. Nakamura & T. Harayama Quantum Chaos & Quantum Dots ix. 163 To understand the classical dynamics of electrons, we analyze, as a Poincaré surface of section, the X–Z plane.
c. Poincaré section n. the intersection of a Poincaré map with a given line, plane, etc.
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1979 Progress Theoret. Physics 61 822 (caption) Poincaré section of the attractor in the chaotic region a=4, q=4.185.
1993 Sci. Amer. Aug. 66/3 One obtains information about the chaotic system by analyzing a slice of the chaotic attractor. After the information about this so-called Poincaré section has been gathered, one allows the system to run and waits until it comes near a desired periodic orbit in the section.
This entry has been updated (OED Third Edition, September 2006; most recently modified version published online March 2022).
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n.1904