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Square packing
Square packing is a packing problem where the objective is to determine how many congruent squares can be packed into some larger shape, often a square or circle.
Square packing in a square
Square packing in a square is the problem of determining the maximum number of unit squares (squares of side length one) that can be packed inside a larger square of side length $a$. If $a$ is an integer, the answer is $a^{2},$ but the precise – or even asymptotic – amount of unfilled space for an arbitrary non-integer $a$ is an open question.[1]
5 unit squares in a square of side length $2+1/{\sqrt {2}}\approx 2.707$
10 unit squares in a square of side length $3+1/{\sqrt {2}}\approx 3.707$
11 unit squares in a square of side length $a\approx 3.877084$
The smallest value of $a$ that allows the packing of $n$ unit squares is known when $n$ is a perfect square (in which case it is ${\sqrt {n}}$), as well as for $n={}$2, 3, 5, 6, 7, 8, 10, 13, 14, 15, 24, 34, 35, 46, 47, and 48. For most of these numbers (with the exceptions only of 5 and 10), the packing is the natural one with axis-aligned squares, and $a$ is $\lceil {\sqrt {n}}\,\rceil $, where $\lceil \,\ \rceil $ is the ceiling (round up) function.[2][3] The figure shows the optimal packings for 5 and 10 squares, the two smallest numbers of squares for which the optimal packing involves tilted squares.[4][5]
The smallest unresolved case involves packing 11 unit squares into a larger square. 11 unit squares cannot be packed in a square of side length less than $\textstyle 2+2{\sqrt {4/5}}\approx 3.789$. By contrast, the tightest known packing of 11 squares is inside a square of side length approximately 3.877084 found by Walter Trump.[6][4]
Asymptotic results
Unsolved problem in mathematics:
What is the asymptotic growth rate of wasted space for square packing in a half-integer square?
(more unsolved problems in mathematics)
For larger values of the side length $a$, the exact number of unit squares that can pack an $a\times a$ square remains unknown. It is always possible to pack a $\lfloor a\rfloor \!\times \!\lfloor a\rfloor $ grid of axis-aligned unit squares, but this may leave a large area, approximately $2a(a-\lfloor a\rfloor )$, uncovered and wasted.[4] Instead, Paul Erdős and Ronald Graham showed that for a different packing by tilted unit squares, the wasted space could be significantly reduced to $o(a^{7/11})$ (here written in little o notation).[7] Later, Graham and Fan Chung further reduced the wasted space to $O(a^{3/5})$.[8] However, as Klaus Roth and Bob Vaughan proved, all solutions must waste space at least $\Omega {\bigl (}a^{1/2}(a-\lfloor a\rfloor ){\bigr )}$. In particular, when $a$ is a half-integer, the wasted space is at least proportional to its square root.[9] The precise asymptotic growth rate of the wasted space, even for half-integer side lengths, remains an open problem.[1]
Some numbers of unit squares are never the optimal number in a packing. In particular, if a square of size $a\times a$ allows the packing of $n^{2}-2$ unit squares, then it must be the case that $a\geq n$ and that a packing of $n^{2}$ unit squares is also possible.[2]
Square packing in a circle
Square packing in a circle is a related problem of packing n unit squares into a circle with radius as small as possible. For this problem, good solutions are known for n up to 35. Here are minimum solutions for n up to 12:[10]
Number of squares Circle radius
1 0.707...
2 1.118...
3 1.288...
4 1.414...
5 1.581...
6 1.688...
7 1.802...
8 1.978...
9 2.077...
10 2.121...
11 2.214...
12 2.236...
See also
• Circle packing in a square
• Squaring the square
• Rectangle packing
• Moving sofa problem
References
1. Brass, Peter; Moser, William; Pach, János (2005), Research Problems in Discrete Geometry, New York: Springer, p. 45, ISBN 978-0387-23815-9, LCCN 2005924022, MR 2163782
2. Kearney, Michael J.; Shiu, Peter (2002), "Efficient packing of unit squares in a square", Electronic Journal of Combinatorics, 9 (1), Research Paper 14, 14 pp., MR 1912796
3. Bentz, Wolfram (2010), "Optimal packings of 13 and 46 unit squares in a square", The Electronic Journal of Combinatorics, 17 (R126), arXiv:1606.03746, doi:10.37236/398, MR 2729375
4. Friedman, Erich (2009), "Packing unit squares in squares: a survey and new results", Electronic Journal of Combinatorics, Dynamic Survey 7, MR 1668055
5. Stromquist, Walter (2003), "Packing 10 or 11 unit squares in a square", Electronic Journal of Combinatorics, 10, Research Paper 8, MR 2386538
6. Gensane, Thierry; Ryckelynck, Philippe (2005), "Improved dense packings of congruent squares in a square", Discrete & Computational Geometry, 34 (1): 97–109, doi:10.1007/s00454-004-1129-z, MR 2140885
7. Erdős, P.; Graham, R. L. (1975), "On packing squares with equal squares" (PDF), Journal of Combinatorial Theory, Series A, 19: 119–123, doi:10.1016/0097-3165(75)90099-0, MR 0370368
8. Chung, Fan; Graham, Ron (2020), "Efficient packings of unit squares in a large square" (PDF), Discrete & Computational Geometry, 64 (3): 690–699, doi:10.1007/s00454-019-00088-9
9. Roth, K. F.; Vaughan, R. C. (1978), "Inefficiency in packing squares with unit squares", Journal of Combinatorial Theory, Series A, 24 (2): 170–186, doi:10.1016/0097-3165(78)90005-5, MR 0487806
10. Friedman, Erich, Squares in Circles
External links
• Friedman, Erich, "Squares in Squares", Github, Erich's Packing Center
Packing problems
Abstract packing
• Bin
• Set
Circle packing
• In a circle / equilateral triangle / isosceles right triangle / square
• Apollonian gasket
• Circle packing theorem
• Tammes problem (on sphere)
Sphere packing
• Apollonian
• Finite
• In a sphere
• In a cube
• In a cylinder
• Close-packing
• Kissing number
• Sphere-packing (Hamming) bound
Other 2-D packing
• Square packing
Other 3-D packing
• Tetrahedron
• Ellipsoid
Puzzles
• Conway
• Slothouber–Graatsma
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Square principle
In mathematical set theory, a square principle is a combinatorial principle asserting the existence of a cohering sequence of short closed unbounded (club) sets so that no one (long) club set coheres with them all. As such they may be viewed as a kind of incompactness phenomenon.[1] They were introduced by Ronald Jensen in his analysis of the fine structure of the constructible universe L.
Definition
Define Sing to be the class of all limit ordinals which are not regular. Global square states that there is a system $(C_{\beta })_{\beta \in \mathrm {Sing} }$ satisfying:
1. $C_{\beta }$ is a club set of $\beta $.
2. ot$(C_{\beta })<\beta $
3. If $\gamma $ is a limit point of $C_{\beta }$ then $\gamma \in \mathrm {Sing} $ and $C_{\gamma }=C_{\beta }\cap \gamma $
Variant relative to a cardinal
Jensen introduced also a local version of the principle.[2] If $\kappa $ is an uncountable cardinal, then $\Box _{\kappa }$ asserts that there is a sequence $(C_{\beta }\mid \beta {\text{ a limit point of }}\kappa ^{+})$ satisfying:
1. $C_{\beta }$ is a club set of $\beta $.
2. If $cf\beta <\kappa $, then $|C_{\beta }|<\kappa $
3. If $\gamma $ is a limit point of $C_{\beta }$ then $C_{\gamma }=C_{\beta }\cap \gamma $
Jensen proved that this principle holds in the constructible universe for any uncountable cardinal κ.
Notes
1. Cummings, James (2005), "Notes on Singular Cardinal Combinatorics", Notre Dame Journal of Formal Logic, 46 (3): 251–282, doi:10.1305/ndjfl/1125409326 Section 4.
2. Jech, Thomas (2003), Set Theory: Third Millennium Edition, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-3-540-44085-7, p. 443.
• Jensen, R. Björn (1972), "The fine structure of the constructible hierarchy", Annals of Mathematical Logic, 4 (3): 229–308, doi:10.1016/0003-4843(72)90001-0, MR 0309729
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Octahedral pyramid
In 4-dimensional geometry, the octahedral pyramid is bounded by one octahedron on the base and 8 triangular pyramid cells which meet at the apex. Since an octahedron has a circumradius divided by edge length less than one,[1] the triangular pyramids can be made with regular faces (as regular tetrahedrons) by computing the appropriate height.
Octahedral pyramid
Schlegel diagram
Type Polyhedral pyramid
Schläfli symbol ( ) ∨ {3,4}
( ) ∨ r{3,3}
( ) ∨ s{2,6}
( ) ∨ [{4} + { }]
( ) ∨ [{ } + { } + { }]
Cells 9 1 {3,4}
8 ( ) ∨ {3}
Faces 20 {3}
Edges 18
Vertices 7
Dual Cubic pyramid
Symmetry group B3, [4,3,1], order 48
[3,3,1], order 24
[2+,6,1], order 12
[4,2,1], order 16
[2,2,1], order 8
Properties convex, regular-cells, Blind polytope
Having all regular cells, it is a Blind polytope. Two copies can be augmented to make an octahedral bipyramid which is also a Blind polytope.
Occurrences of the octahedral pyramid
The regular 16-cell has octahedral pyramids around every vertex, with the octahedron passing through the center of the 16-cell. Therefore placing two regular octahedral pyramids base to base constructs a 16-cell. The 16-cell tessellates 4-dimensional space as the 16-cell honeycomb.
Exactly 24 regular octahedral pyramids will fit together around a vertex in four-dimensional space (the apex of each pyramid). This construction yields a 24-cell with octahedral bounding cells, surrounding a central vertex with 24 edge-length long radii. The 4-dimensional content of a unit-edge-length 24-cell is 2, so the content of the regular octahedral pyramid is 1/12. The 24-cell tessellates 4-dimensional space as the 24-cell honeycomb.
The octahedral pyramid is the vertex figure for a truncated 5-orthoplex, .
The graph of the octahedral pyramid is the only possible minimal counterexample to Negami's conjecture, that the connected graphs with planar covers are themselves projective-planar.[2]
Example 4-dimensional coordinates, 6 points in first 3 coordinates for cube and 4th dimension for the apex.
(±1, 0, 0; 0)
( 0,±1, 0; 0)
( 0, 0,±1; 0)
( 0, 0, 0; 1)
Other polytopes
Cubic pyramid
The dual to the octahedral pyramid is a cubic pyramid, seen as a cubic base and 6 square pyramids meeting at an apex.
Example 4-dimensional coordinates, 8 points in first 3 coordinates for cube and 4th dimension for the apex.
(±1,±1,±1; 0)
( 0, 0, 0; 1)
Square-pyramidal pyramid
Square-pyramidal pyramid
Type Polyhedral pyramid
Schläfli symbol ( ) ∨ [( ) ∨ {4}]
[( )∨( )] ∨ {4} = { } ∨ {4}
{ } ∨ [{ } × { }]
{ } ∨ [{ } + { }]
Cells 6 2 ( )∨{4}
4 ( )∨{3}
Faces 12 {3}
1 {4}
Edges 13
Vertices 6
Dual Self-dual
Symmetry group [4,1,1], order 8
[4,2,1], order 16
[2,2,1], order 8
Properties convex, regular-faced
The square-pyramidal pyramid, ( ) ∨ [( ) ∨ {4}], is a bisected octahedral pyramid. It has a square pyramid base, and 4 tetrahedrons along with another one more square pyramid meeting at the apex. It can also be seen in an edge-centered projection as a square bipyramid with four tetrahedra wrapped around the common edge. If the height of the two apexes are the same, it can be given a higher symmetry name [( ) ∨ ( )] ∨ {4} = { } ∨ {4}, joining an edge to a perpendicular square.[3]
The square-pyramidal pyramid can be distorted into a rectangular-pyramidal pyramid, { } ∨ [{ } × { }] or a rhombic-pyramidal pyramid, { } ∨ [{ } + { }], or other lower symmetry forms.
The square-pyramidal pyramid exists as a vertex figure in uniform polytopes of the form , including the bitruncated 5-orthoplex and bitruncated tesseractic honeycomb.
Example 4-dimensional coordinates, 2 coordinates for square, and axial points for pyramidal points.
(±1,±1; 0; 0)
( 0, 0; 1; 0)
( 0, 0; 0; 1)
References
1. Klitzing, Richard. "3D convex uniform polyhedra x3o4o - oct". 1/sqrt(2) = 0.707107
2. Hliněný, Petr (2010), "20 years of Negami's planar cover conjecture" (PDF), Graphs and Combinatorics, 26 (4): 525–536, CiteSeerX 10.1.1.605.4932, doi:10.1007/s00373-010-0934-9, MR 2669457, S2CID 121645
3. Klitzing, Richard. "Segmentotope squasc, K-4.4".
External links
• Olshevsky, George. "Pyramid". Glossary for Hyperspace. Archived from the original on 4 February 2007.
• Klitzing, Richard. "4D Segmentotopes".
• Klitzing, Richard. "Segmentotope octpy, K-4.3".
• Richard Klitzing, Axial-Symmetrical Edge Facetings of Uniform Polyhedra
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Square Root Day
Square Root Day is an unofficial holiday celebrated on days when both the day of the month and the month are the square root of the last two digits of the year.[1] For example, the last Square Root Day was April 4, 2016 (4/4/16), and the next Square Root Day will be May 5, 2025 (5/5/25). The final Square Root Day of the century will occur on September 9, 2081. Square Root Days fall upon the same nine dates each century.
Ron Gordon, a Redwood City, California high school teacher, created the first Square Root Day for September 9, 1981 (9/9/81). Gordon remains the holiday's publicist, sending news releases to world media outlets.[2] Gordon's daughter set up a Facebook group where people can share how they were celebrating the day.[3]
One suggested way of celebrating the holiday is by eating radishes or other root vegetables cut into shapes with square cross sections (thus creating a "square root").[4]
Full list
Square Root Day occurs on the following dates each century:
• 1/1/01
• 2/2/04
• 3/3/09
• 4/4/16
• 5/5/25
• 6/6/36
• 7/7/49
• 8/8/64
• 9/9/81
Distribution
The number of years between consecutive Square Root Days in a century is consecutive odd numbers: 3, 5, 7, 9, 11, 13, 15, 17. This illustrates the fact that every odd number is the difference of two consecutive squares.
See also
• Mole Day
• Pi Day
• Sequential Day
References
1. Wong, Nicole C. (2004-02-02). "A day getting to the root". The Mercury News. Archived from the original on 2004-08-18. Retrieved 2007-02-20.
2. Hill, Angela (2009-03-02). "Have a rootin' tootin' Square Root Day". Oakland Tribune. Retrieved 2009-03-02.
3. "Math Fans to Celebrate Square Root Day". Yahoo News. 2009-03-02. Archived from the original on March 6, 2009. Retrieved 2009-03-02.
4. Musil, Steven (2009-03-02). "Square Root Day revelers to party like it's 3/3/09". cnet news. Retrieved 2009-03-02.
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Square root of 6
The square root of 6 is the positive real number that, when multiplied by itself, gives the natural number 6. It is more precisely called the principal square root of 6, to distinguish it from the negative number with the same property. This number appears in numerous geometric and number-theoretic contexts. It can be denoted in surd form as:[1]
${\sqrt {6}}\,,$
Square root of 6
RationalityIrrational
Representations
Decimal2.449489742783178098..._10
Algebraic form${\sqrt {6}}$
Continued fraction$2+{\cfrac {1}{2+{\cfrac {1}{4+{\cfrac {1}{2+{\cfrac {1}{4+\ddots }}}}}}}}$
Binary10.011100110001..._2
Hexadecimal2.7311c2812425c..._16
and in exponent form as:
$6^{\frac {1}{2}}.$
It is an irrational algebraic number.[2] The first sixty significant digits of its decimal expansion are:
2.44948974278317809819728407470589139196594748065667012843269....[3]
which can be rounded up to 2.45 to within about 99.98% accuracy (about 1 part in 4800); that is, it differs from the correct value by about 1/2,000. It takes two more digits (2.4495) to reduce the error by about half. The approximation 218/89 (≈ 2.449438...) is nearly ten times better: despite having a denominator of only 89, it differs from the correct value by less than 1/20,000, or less than one part in 47,000.
Since 6 is the product of 2 and 3, the square root of 6 is the geometric mean of 2 and 3, and is the product of the square root of 2 and the square root of 3, both of which are irrational algebraic numbers.
NASA has published more than a million decimal digits of the square root of six.[4]
Rational approximations
The square root of 6 can be expressed as the continued fraction
$[2;2,4,2,4,2,\ldots ]=2+{\cfrac {1}{2+{\cfrac {1}{4+{\cfrac {1}{2+{\cfrac {1}{4+\dots }}}}}}}}.$ (sequence A040003 in the OEIS)
The successive partial evaluations of the continued fraction, which are called its convergents, approach ${\sqrt {6}}$:
${\frac {2}{1}},{\frac {5}{2}},{\frac {22}{9}},{\frac {49}{20}},{\frac {218}{89}},{\frac {485}{198}},{\frac {2158}{881}},{\frac {4801}{1960}},\dots $
Their numerators are 2, 5, 22, 49, 218, 485, 2158, 4801, 21362, 47525, 211462, …(sequence A041006 in the OEIS), and their denominators are 1, 2, 9, 20, 89, 198, 881, 1960, 8721, 19402, 86329, …(sequence A041007 in the OEIS).[5]
Each convergent is a best rational approximation of ${\sqrt {6}}$; in other words, it is closer to ${\sqrt {6}}$ than any rational with a smaller denominator. Decimal equivalents improve linearly, at a rate of nearly one digit per convergent:
${\frac {2}{1}}=2.0,\quad {\frac {5}{2}}=2.5,\quad {\frac {22}{9}}=2.4444\dots ,\quad {\frac {49}{20}}=2.45,\quad {\frac {218}{89}}=2.44943...,\quad {\frac {485}{198}}=2.449494...,\quad \ldots $
The convergents, expressed as x/y, satisfy alternately the Pell's equations[5]
$x^{2}-6y^{2}=-2\quad \mathrm {and} \quad x^{2}-6y^{2}=1$
When ${\sqrt {6}}$ is approximated with the Babylonian method, starting with x0 = 2 and using xn+1 = 1/2(xn + 6/xn), the nth approximant xn is equal to the 2nth convergent of the continued fraction:
$x_{0}=2,\quad x_{1}={\frac {5}{2}}=2.5,\quad x_{2}={\frac {49}{20}}=2.45,\quad x_{3}={\frac {4801}{1960}}=2.449489796...,\quad x_{4}={\frac {46099201}{18819920}}=2.449489742783179...,\quad \dots $
The Babylonian method is equivalent to Newton's method for root finding applied to the polynomial $x^{2}-6$. The Newton's method update, $x_{n+1}=x_{n}-f(x_{n})/f'(x_{n}),$ is equal to $(x_{n}+6/x_{n})/2$ when $f(x)=x^{2}-6$. The method therefore converges quadratically.
Geometry
In plane geometry, the square root of 6 can be constructed via a sequence of dynamic rectangles, as illustrated here.[6][7][8]
In solid geometry, the square root of 6 appears as the longest distances between corners (vertices) of the double cube, as illustrated above. The square roots of all lower natural numbers appear as the distances between other vertex pairs in the double cube (including the vertices of the included two cubes).[8]
The edge length of a cube with total surface area of 1 is ${\frac {\sqrt {6}}{6}}$ or the reciprocal square root of 6. The edge lengths of a regular tetrahedron (t), a regular octahedron (o), and a cube (c) of equal total surface areas satisfy ${\frac {t\cdot o}{c^{2}}}={\sqrt {6}}$.[3][9]
The edge length of a regular octahedron is the square root of 6 times the radius of an inscribed sphere (that is, the distance from the center of the solid to the center of each face).[10]
The square root of 6 appears in various other geometry contexts, such as the side length ${\frac {{\sqrt {6}}+{\sqrt {2}}}{2}}$ for the square enclosing an equilateral triangle of side 2 (see figure).
Trigonometry
The square root of 6, with the square root of 2 added or subtracted, appears in several exact trigonometric values for angles at multiples of 15 degrees ($\pi /12$ radians).[11]
:{| class="wikitable" style="text-align: center;"
!Radians!!Degrees!!sin!!cos!!tan!!cot!!sec!!csc |- ! ${\frac {\pi }{12}}$!! $15^{\circ }$ |${\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}$|| ${\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}$|| $2-{\sqrt {3}}$|| $2+{\sqrt {3}}$|| ${\sqrt {6}}-{\sqrt {2}}$|| ${\sqrt {6}}+{\sqrt {2}}$ |- ! ${\frac {5\pi }{12}}$!! $75^{\circ }$ |${\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}$|| ${\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}$|| $2+{\sqrt {3}}$|| $2-{\sqrt {3}}$|| ${\sqrt {6}}+{\sqrt {2}}$|| ${\sqrt {6}}-{\sqrt {2}}$ |}
In culture
The square root of six (actually its reciprocal, "the square root of six over six") appears in Star Wars dialogue.[12]
The question of "whether the square root of six is three" has been posited as a question that might be answered by economic methods, if social issues can be so addressed.[13][14][15][16]
Villard de Honnecourt's 13th century construction of a Gothic "fifth-point arch" with circular arcs of radius 5 has a height of twice the square root of 6, as illustrated here.[17][18]
See also
• Square root
• Square root of 2
• Square root of 3
• Square root of 5
• Square root of 7
References
1. Ray, Joseph (1842). Ray's Eclectic Arithmetic on the Inductive and Analytic Methods of Instruction. Cincinnati: Truman and Smith. p. 217. Retrieved 20 March 2022.
2. O'Sullivan, Daniel (1872). The Principles of Arithmetic: A Comprehensive Text-Book. Dublin: Alexander Thom. p. 234. Retrieved 17 March 2022.
3. Sloane, N. J. A. (ed.). "Sequence A010464 (Decimal expansion of square root of 6)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
4. Robert Nemiroff; Jerry Bonnell. "the first 1 million digits of the square root of 6". nasa.gov. Retrieved 17 March 2022.
5. Conrad, Keith. "Pell's Equation II" (PDF). uconn.edu. Retrieved 17 March 2022. The continued fraction of √6 is [2; 2, 4], and the table of convergents below suggests (and it is true) that every other convergent provides a solution to x2 − 6y2 = 1.
6. Jay Hambidge (1920) [1920]. Dynamic Symmetry: The Greek Vase (Reprint of original Yale University Press ed.). Whitefish, MT: Kessinger Publishing. pp. 19–29. ISBN 0-7661-7679-7. Dynamic Symmetry root rectangles.
7. Matila Ghyka (1977). The Geometry of Art and Life. Courier Dover Publications. pp. 126–127. ISBN 9780486235424.
8. Fletcher, Rachel (2013). Infinite Measure: Learning to Design in Geometric Harmony with Art, Architecture, and Nature. George F Thompson Publishing. ISBN 978-1-938086-02-1.
9. Rechtman, Ana. "Un défi par semaine Avril 2016, 3e défi (Solution du 2e défi d'Avril)". Images des Mathématiques. Retrieved 23 March 2022.
10. S. C. & L. M. Gould (1890). The Bizarre Notes and Queries in History, Folk-lore, Mathematics, Mysticism, Art, Science, Etc, Volumes 7-8. Manchester, N. H. p. 342. Retrieved 19 March 2022. In the octahedron whose diameter is 2, the linear edge equals the square root of 6.{{cite book}}: CS1 maint: location missing publisher (link)
11. Abramowitz, Milton; Stegun, Irene A., eds. (1972). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover Publications. p. 74. ISBN 978-0-486-61272-0.
12. Hearne, Kevin (2015). Heir to the Jedi: Star Wars. Random House. p. 122. ISBN 9780345544872. Retrieved 20 March 2022. ...with the associated vectors square root of six over six times the vector one, one, two...
13. Sagoff, Mark (1981). "At the shrine of Our Lady of Fatima or why political questions are not all economic". Arizona Law Review. 23: 1283–1298. Retrieved 24 March 2022. (reprinted)
14. Tverdek, Edward F. (2015). The Moral Weight of Ecology: Public Goods, Cooperative Duties, and Environmental Politics. Lexington Books. p. 214. ISBN 9781498514545. Retrieved 20 March 2022.
15. Louis P. Pojman; Paul Pojman; Katie McShane (2016). Environmental Ethics: Readings in Theory and Application. Cengage. p. 448. ISBN 9781305687806. Retrieved 20 March 2022.
16. Lawrence Susskind; Bruno Verdini; Jessica Gordon; Yasmin Zaerpoor (2020). Environmental Problem-Solving: Balancing Science and Politics Using Consensus Building Tools. Anthem Press. p. 377. ISBN 9781785271328. Retrieved 20 March 2022.
17. Branner, Robert (1960). "Villard de Honnecourt, Archimedes, and Chartres". Journal of the Society of Architectural Historians. 19 (3): 91–96. doi:10.2307/988023. JSTOR 988023. Retrieved 25 March 2022.
18. Shelby, Lon R. (1969). "Setting Out the Keystones of Pointed Arches: A Note on Medieval 'Baugeometrie'". Technology and Culture. 10 (4): 537–548. doi:10.2307/3101574. JSTOR 3101574. Retrieved 25 March 2022.
Algebraic numbers
• Algebraic integer
• Chebyshev nodes
• Constructible number
• Conway's constant
• Cyclotomic field
• Eisenstein integer
• Gaussian integer
• Golden ratio (φ)
• Perron number
• Pisot–Vijayaraghavan number
• Quadratic irrational number
• Rational number
• Root of unity
• Salem number
• Silver ratio (δS)
• Square root of 2
• Square root of 3
• Square root of 5
• Square root of 6
• Square root of 7
• Doubling the cube
• Twelfth root of two
Mathematics portal
Irrational numbers
• Chaitin's (Ω)
• Liouville
• Prime (ρ)
• Omega
• Cahen
• Logarithm of 2
• Gauss's (G)
• Twelfth root of 2
• Apéry's (ζ(3))
• Plastic (ρ)
• Square root of 2
• Supergolden ratio (ψ)
• Erdős–Borwein (E)
• Golden ratio (φ)
• Square root of 3
• Square root of pi (√π)
• Square root of 5
• Silver ratio (δS)
• Square root of 6
• Square root of 7
• Euler's (e)
• Pi (π)
• Schizophrenic
• Transcendental
• Trigonometric
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Square root of 7
The square root of 7 is the positive real number that, when multiplied by itself, gives the prime number 7. It is more precisely called the principal square root of 7, to distinguish it from the negative number with the same property. This number appears in various geometric and number-theoretic contexts. It can be denoted in surd form as:[1]
${\sqrt {7}}\,,$
Square root of 7
RationalityIrrational
Representations
Decimal2.645751311064590590..._10
Algebraic form${\sqrt {7}}$
Continued fraction$2+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{4+\ddots }}}}}}}}$
Binary10.10100101010011111111..._2
Hexadecimal2.A54FF53A5F1D..._16
and in exponent form as:
$7^{\frac {1}{2}}.$
It is an irrational algebraic number. The first sixty significant digits of its decimal expansion are:
2.64575131106459059050161575363926042571025918308245018036833....[2]
which can be rounded up to 2.646 to within about 99.99% accuracy (about 1 part in 10000); that is, it differs from the correct value by about 1/4,000. The approximation 127/48 (≈ 2.645833...) is better: despite having a denominator of only 48, it differs from the correct value by less than 1/12,000, or less than one part in 33,000.
More than a million decimal digits of the square root of seven have been published.[3]
Rational approximations
The extraction of decimal-fraction approximations to square roots by various methods has used the square root of 7 as an example or exercise in textbooks, for hundreds of years. Different numbers of digits after the decimal point are shown: 5 in 1773[4] and 1852,[5] 3 in 1835,[6] 6 in 1808,[7] and 7 in 1797.[8] An extraction by Newton's method (approximately) was illustrated in 1922, concluding that it is 2.646 "to the nearest thousandth".[9]
For a family of good rational approximations, the square root of 7 can be expressed as the continued fraction
$[2;1,1,1,4,1,1,1,4,\ldots ]=2+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{4+{\cfrac {1}{1+\dots }}}}}}}}}}.$ (sequence A010121 in the OEIS)
The successive partial evaluations of the continued fraction, which are called its convergents, approach ${\sqrt {7}}$:
${\frac {2}{1}},{\frac {3}{1}},{\frac {5}{2}},{\frac {8}{3}},{\frac {37}{14}},{\frac {45}{17}},{\frac {82}{31}},{\frac {127}{48}},{\frac {590}{223}},{\frac {717}{271}},\dots $
Their numerators are 2, 3, 5, 8, 37, 45, 82, 127, 590, 717, 1307, 2024, 9403, 11427, 20830, 32257…(sequence A041008 in the OEIS) , and their denominators are 1, 1, 2, 3, 14, 17, 31, 48, 223, 271, 494, 765, 3554, 4319, 7873, 12192,…(sequence A041009 in the OEIS).
Each convergent is a best rational approximation of ${\sqrt {7}}$; in other words, it is closer to ${\sqrt {7}}$ than any rational with a smaller denominator. Approximate decimal equivalents improve linearly (number of digits proportional to convergent number) at a rate of less than one digit per step:
${\frac {2}{1}}=2.0,\quad {\frac {3}{1}}=3.0,\quad {\frac {5}{2}}=2.5,\quad {\frac {8}{3}}=2.66\dots ,\quad {\frac {37}{14}}=2.6429...,\quad {\frac {45}{17}}=2.64705...,\quad {\frac {82}{31}}=2.64516...,\quad {\frac {127}{48}}=2.645833...,\quad \ldots $
Every fourth convergent, starting with 8/3, expressed as x/y, satisfies the Pell's equation[10]
$x^{2}-7y^{2}=1.$
When ${\sqrt {7}}$ is approximated with the Babylonian method, starting with x1 = 3 and using xn+1 = 1/2(xn + 7/xn), the nth approximant xn is equal to the 2nth convergent of the continued fraction:
$x_{1}=3,\quad x_{2}={\frac {8}{3}}=2.66...,\quad x_{3}={\frac {127}{48}}=2.6458...,\quad x_{4}={\frac {32257}{12192}}=2.645751312...,\quad x_{5}={\frac {2081028097}{786554688}}=2.645751311064591...,\quad \dots $
All but the first of these satisfy the Pell's equation above.
The Babylonian method is equivalent to Newton's method for root finding applied to the polynomial $x^{2}-7$. The Newton's method update, $x_{n+1}=x_{n}-f(x_{n})/f'(x_{n}),$ is equal to $(x_{n}+7/x_{n})/2$ when $f(x)=x^{2}-7$. The method therefore converges quadratically (number of accurate decimal digits proportional to the square of the number of Newton or Babylonian steps).
Geometry
In plane geometry, the square root of 7 can be constructed via a sequence of dynamic rectangles, that is, as the largest diagonal of those rectangles illustrated here.[11][12][13]
The minimal enclosing rectangle of an equilateral triangle of edge length 2 has a diagonal of the square root of 7.[14]
Due to the Pythagorean theorem and Legendre's three-square theorem, ${\sqrt {7}}$ is the smallest square root of a natural number that cannot be the distance between any two points of a cubic integer lattice (or equivalently, the length of the space diagonal of a rectangular cuboid with integer side lengths). ${\sqrt {15}}$ is the next smallest such number.[15]
Outside of mathematics
On the reverse of the current US one-dollar bill, the "large inner box" has a length-to-width ratio of the square root of 7, and a diagonal of 6.0 inches, to within measurement accuracy.[16]
See also
• Square root
• Square root of 2
• Square root of 3
• Square root of 5
• Square root of 6
References
1. Darby, John (1843). The Practical Arithmetic, with Notes and Demonstrations to the Principal Rules, ... London: Whittaker & Company. p. 172. Retrieved 27 March 2022.
2. Sloane, N. J. A. (ed.). "Sequence A010465 (Decimal expansion of square root of 7)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
3. Robert Nemiroff; Jerry Bonnell (2008). The square root of 7. Retrieved 25 March 2022. {{cite book}}: |website= ignored (help)
4. Ewing, Alexander (1773). Institutes of Arithmetic: For the Use of Schools and Academies. Edinburgh: T. Caddell. p. 104.
5. Ray, Joseph (1852). Ray's Algebra, Part Second: An Analytical Treatise, Designed for High Schools and Academies, Part 2. Cincinnati: Sargent, Wilson & Hinkle. p. 132. Retrieved 27 March 2022.
6. Bailey, Ebenezer (1835). First Lessons in Algebra, Being an Easy Introduction to that Science... Russell, Shattuck & Company. pp. 212–213. Retrieved 27 March 2022.
7. Thompson, James (1808). The American Tutor's Guide: Being a Compendium of Arithmetic. In Six Parts. Albany: E. & E. Hosford. p. 122. Retrieved 27 March 2022.
8. Hawney, William (1797). The Complete Measurer: Or, the Whole Art of Measuring. In Two Parts. Part I. Teaching Decimal Arithmetic ... Part II. Teaching to Measure All Sorts of Superficies and Solids ... Thirteenth Edition. To which is Added an Appendix. 1. Of Gaging. 2. Of Land-measuring. London. pp. 59–60. Retrieved 27 March 2022.
9. George Wentworth, David Eugene Smith, Herbert Druery Harper (1922). Fundamentals of Practical Mathematics. Ginn and Company. p. 113. Retrieved 27 March 2022.{{cite book}}: CS1 maint: uses authors parameter (link)
10. Conrad, Keith. "Pell's Equation II" (PDF). uconn.edu. Retrieved 17 March 2022.
11. Jay Hambidge (1920) [1920]. Dynamic Symmetry: The Greek Vase (Reprint of original Yale University Press ed.). Whitefish, MT: Kessinger Publishing. pp. 19–29. ISBN 0-7661-7679-7. Dynamic Symmetry root rectangles.
12. Matila Ghyka (1977). The Geometry of Art and Life. Courier Dover Publications. pp. 126–127. ISBN 9780486235424.
13. Fletcher, Rachel (2013). Infinite Measure: Learning to Design in Geometric Harmony with Art, Architecture, and Nature. George F Thompson Publishing. ISBN 978-1-938086-02-1.
14. Blackwell, William (1984). Geometry in Architecture. Key Curriculum Press. p. 25. ISBN 9781559530187. Retrieved 26 March 2022.
15. Sloane, N. J. A. (ed.). "Sequence A005875". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
16. McGrath, Ken (2002). The Secret Geometry of the Dollar. AuthorHouse. pp. 47–49. ISBN 9780759611702. Retrieved 26 March 2022.
Algebraic numbers
• Algebraic integer
• Chebyshev nodes
• Constructible number
• Conway's constant
• Cyclotomic field
• Eisenstein integer
• Gaussian integer
• Golden ratio (φ)
• Perron number
• Pisot–Vijayaraghavan number
• Quadratic irrational number
• Rational number
• Root of unity
• Salem number
• Silver ratio (δS)
• Square root of 2
• Square root of 3
• Square root of 5
• Square root of 6
• Square root of 7
• Doubling the cube
• Twelfth root of two
Mathematics portal
Irrational numbers
• Chaitin's (Ω)
• Liouville
• Prime (ρ)
• Omega
• Cahen
• Logarithm of 2
• Gauss's (G)
• Twelfth root of 2
• Apéry's (ζ(3))
• Plastic (ρ)
• Square root of 2
• Supergolden ratio (ψ)
• Erdős–Borwein (E)
• Golden ratio (φ)
• Square root of 3
• Square root of pi (√π)
• Square root of 5
• Silver ratio (δS)
• Square root of 6
• Square root of 7
• Euler's (e)
• Pi (π)
• Schizophrenic
• Transcendental
• Trigonometric
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Wikipedia
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Square root of a 2 by 2 matrix
A square root of a 2×2 matrix M is another 2×2 matrix R such that M = R2, where R2 stands for the matrix product of R with itself. In general, there can be zero, two, four, or even an infinitude of square-root matrices. In many cases, such a matrix R can be obtained by an explicit formula.
Square roots that are not the all-zeros matrix come in pairs: if R is a square root of M, then −R is also a square root of M, since (−R)(−R) = (−1)(−1)(RR) = R2 = M.
A 2×2 matrix with two distinct nonzero eigenvalues has four square roots. A positive-definite matrix has precisely one positive-definite square root.
A general formula
The following is a general formula that applies to almost any 2 × 2 matrix.[1] Let the given matrix be
$M={\begin{pmatrix}A&B\\C&D\end{pmatrix}},$
where A, B, C, and D may be real or complex numbers. Furthermore, let τ = A + D be the trace of M, and δ = AD − BC be its determinant. Let s be such that s2 = δ, and t be such that t2 = τ + 2s. That is,
$s=\pm {\sqrt {\delta }},\qquad t=\pm {\sqrt {\tau +2s}}.$
Then, if t ≠ 0, a square root of M is
$R={\frac {1}{t}}{\begin{pmatrix}A+s&B\\C&D+s\end{pmatrix}}={\frac {1}{t}}\left(M+sI\right).$
Indeed, the square of R is
${\begin{aligned}R^{2}&={\frac {1}{t^{2}}}{\begin{pmatrix}A^{2}+BC+2sA+s^{2}&AB+BD+2sB\\CA+DC+2sC&CB+D^{2}+2sD+s^{2}\end{pmatrix}}\\[1ex]&={\frac {1}{t^{2}}}{\begin{pmatrix}A^{2}+BC+2sA+AD-BC&AB+BD+2sB\\AC+CD+2sC&BC+D^{2}+2sD+AD-BC\end{pmatrix}}\\[1ex]&={\frac {1}{A+D+2s}}{\begin{pmatrix}A(A+D+2s)&B(A+D+2s)\\C(A+D+2s)&D(A+D+2s)\end{pmatrix}}=M.\end{aligned}}$
Note that R may have complex entries even if M is a real matrix; this will be the case, in particular, if the determinant δ is negative.
The general case of this formula is when δ is nonzero, and τ2 ≠ 4δ, in which case s is nonzero, and t is nonzero for each choice of sign of s. Then the formula above will provide four distinct square roots R, one for each choice of signs for s and t.
Special cases of the formula
If the determinant δ is zero, but the trace τ is nonzero, the general formula above will give only two distinct solutions, corresponding to the two signs of t. Namely,
$R=\pm {\frac {1}{t}}{\begin{pmatrix}A&B\\C&D\end{pmatrix}}=\pm {\frac {1}{t}}M,$
where t is any square root of the trace τ.
The formula also gives only two distinct solutions if δ is nonzero, and τ2 = 4δ (the case of duplicate eigenvalues), in which case one of the choices for s will make the denominator t be zero. In that case, the two roots are
$R=\pm {\frac {1}{t}}{\begin{pmatrix}A+s&B\\C&D+s\end{pmatrix}}=\pm {\frac {1}{t}}\left(M+sI\right),$
where s is the square root of δ that makes τ − 2s nonzero, and t is any square root of τ − 2s.
The formula above fails completely if δ and τ are both zero; that is, if D = −A, and A2 = −BC, so that both the trace and the determinant of the matrix are zero. In this case, if M is the null matrix (with A = B = C = D = 0), then the null matrix is also a square root of M, as is any matrix
$R={\begin{pmatrix}0&0\\c&0\end{pmatrix}}\quad {\text{and}}\quad R={\begin{pmatrix}0&b\\0&0\end{pmatrix}},$
where b and c are arbitrary real or complex values. Otherwise M has no square root.
Formulas for special matrices
Idempotent matrix
If M is an idempotent matrix, meaning that MM = M, then if it is not the identity matrix, its determinant is zero, and its trace equals its rank, which (excluding the zero matrix) is 1. Then the above formula has s = 0 and τ = 1, giving M and −M as two square roots of M.
Exponential matrix
If the matrix M can be expressed as real multiple of the exponent of some matrix A, $M=r\exp(A)$, then two of its square roots are $\pm {\sqrt {r}}\exp \left({\tfrac {1}{2}}A\right)$. In this case the square root is real.[2]
Diagonal matrix
If M is diagonal (that is, B = C = 0), one can use the simplified formula
$R={\begin{pmatrix}a&0\\0&d\end{pmatrix}},$
where a = ±√A, and d = ±√D. This, for the various sign choices, gives four, two, or one distinct matrices, if none of, only one of, or both A and D are zero, respectively.
Identity matrix
Because it has duplicate eigenvalues, the 2×2 identity matrix $\left({\begin{smallmatrix}1&0\\0&1\end{smallmatrix}}\right)$ has infinitely many symmetric rational square roots given by
${\frac {1}{t}}{\begin{pmatrix}s&r\\r&-s\end{pmatrix}}{\text{ and }}{\begin{pmatrix}\pm 1&0\\0&\pm 1\end{pmatrix}},$
where (r, s, t) are any complex numbers such that $r^{2}+s^{2}=t^{2}.$[3]
Matrix with one off-diagonal zero
If B is zero, but A and D are not both zero, one can use
$R={\begin{pmatrix}a&0\\{\frac {C}{a+d}}&d\end{pmatrix}}.$
This formula will provide two solutions if A = D or A = 0 or D = 0, and four otherwise. A similar formula can be used when C is zero, but A and D are not both zero.
References
1. Levinger, Bernard W. (September 1980), "The square root of a $2\times 2$ matrix", Mathematics Magazine, 53 (4): 222–224, doi:10.1080/0025570X.1980.11976858, JSTOR 2689616
2. Harkin, Anthony A.; Harkin, Joseph B. (2004), "Geometry of generalized complex numbers" (PDF), Mathematics Magazine, 77 (2): 118–129, doi:10.1080/0025570X.2004.11953236, JSTOR 3219099, MR 1573734
3. Mitchell, Douglas W. (November 2003), "87.57 Using Pythagorean triples to generate square roots of $I_{2}$", The Mathematical Gazette, 87 (510): 499–500, doi:10.1017/S0025557200173723, JSTOR 3621289
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Wikipedia
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Square root of a matrix
In mathematics, the square root of a matrix extends the notion of square root from numbers to matrices. A matrix B is said to be a square root of A if the matrix product BB is equal to A.[1]
Some authors use the name square root or the notation A1/2 only for the specific case when A is positive semidefinite, to denote the unique matrix B that is positive semidefinite and such that BB = BTB = A (for real-valued matrices, where BT is the transpose of B).
Less frequently, the name square root may be used for any factorization of a positive semidefinite matrix A as BTB = A, as in the Cholesky factorization, even if BB ≠ A. This distinct meaning is discussed in Positive definite matrix § Decomposition.
Examples
In general, a matrix can have several square roots. In particular, if $A=B^{2}$ then $A=(-B)^{2}$ as well.
The 2×2 identity matrix $\textstyle {\begin{pmatrix}1&0\\0&1\end{pmatrix}}$ has infinitely many square roots. They are given by
${\begin{pmatrix}\pm 1&0\\0&\pm 1\end{pmatrix}}$ and ${\begin{pmatrix}a&b\\c&-a\end{pmatrix}}$
where $(a,b,c)$ are any numbers (real or complex) such that $a^{2}+bc=1$. In particular if $(a,b,t)$ is any Pythagorean triple—that is, any set of positive integers such that $a^{2}+b^{2}=t^{2}$, then ${\frac {1}{t}}{\begin{pmatrix}a&b\\b&-a\end{pmatrix}}$ is a square root matrix of $I$ which is symmetric and has rational entries.[2] Thus
${\begin{pmatrix}1&0\\0&1\end{pmatrix}}={\begin{pmatrix}0&1\\1&0\end{pmatrix}}^{2}={\begin{pmatrix}{\frac {4}{5}}&{\frac {3}{5}}\\{\frac {3}{5}}&-{\frac {4}{5}}\end{pmatrix}}^{2}.$
Minus identity has a square root, for example:
$-{\begin{pmatrix}1&0\\0&1\end{pmatrix}}={\begin{pmatrix}0&-1\\1&0\end{pmatrix}}^{2},$
which can be used to represent the imaginary unit i and hence all complex numbers using 2×2 real matrices, see Matrix representation of complex numbers.
Just as with the real numbers, a real matrix may fail to have a real square root, but have a square root with complex-valued entries. Some matrices have no square root. An example is the matrix ${\begin{pmatrix}0&1\\0&0\end{pmatrix}}.$
While the square root of a nonnegative integer is either again an integer or an irrational number, in contrast an integer matrix can have a square root whose entries are rational, yet non-integral, as in examples above.
Positive semidefinite matrices
See also: Positive definite matrix § Decomposition
A symmetric real n × n matrix is called positive semidefinite if $x^{\textsf {T}}Ax\geq 0$ for all $x\in \mathbb {R} ^{n}$ (here $x^{\textsf {T}}$ denotes the transpose, changing a column vector x into a row vector). A square real matrix is positive semidefinite if and only if $A=B^{\textsf {T}}B$ for some matrix B. There can be many different such matrices B. A positive semidefinite matrix A can also have many matrices B such that $A=BB$. However, A always has precisely one square root B that is positive semidefinite (and hence symmetric). In particular, since B is required to be symmetric, $B=B^{\textsf {T}}$, so the two conditions $A=BB$ or $A=B^{\textsf {T}}B$ are equivalent.
For complex-valued matrices, the conjugate transpose $B^{*}$ is used instead and positive semidefinite matrices are Hermitian, meaning $B^{*}=B$.
Theorem[3] — Let A be a positive semidefinite matrix (real or complex). Then there is exactly one positive semidefinite matrix B such that $A=B^{*}B$.
This unique matrix is called the principal, non-negative, or positive square root (the latter in the case of positive definite matrices).
The principal square root of a real positive semidefinite matrix is real.[3] The principal square root of a positive definite matrix is positive definite; more generally, the rank of the principal square root of A is the same as the rank of A.[3]
The operation of taking the principal square root is continuous on this set of matrices.[4] These properties are consequences of the holomorphic functional calculus applied to matrices.[5][6] The existence and uniqueness of the principal square root can be deduced directly from the Jordan normal form (see below).
Matrices with distinct eigenvalues
An n×n matrix with n distinct nonzero eigenvalues has 2n square roots. Such a matrix, A, has an eigendecomposition VDV−1 where V is the matrix whose columns are eigenvectors of A and D is the diagonal matrix whose diagonal elements are the corresponding n eigenvalues λi. Thus the square roots of A are given by VD1/2 V−1, where D1/2 is any square root matrix of D, which, for distinct eigenvalues, must be diagonal with diagonal elements equal to square roots of the diagonal elements of D; since there are two possible choices for a square root of each diagonal element of D, there are 2n choices for the matrix D1/2.
This also leads to a proof of the above observation, that a positive-definite matrix has precisely one positive-definite square root: a positive definite matrix has only positive eigenvalues, and each of these eigenvalues has only one positive square root; and since the eigenvalues of the square root matrix are the diagonal elements of D1/2, for the square root matrix to be itself positive definite necessitates the use of only the unique positive square roots of the original eigenvalues.
Solutions in closed form
See also: Square root of a 2 by 2 matrix
If a matrix is idempotent, meaning $A^{2}=A$, then by definition one of its square roots is the matrix itself.
Diagonal and triangular matrices
If D is a diagonal n × n matrix $D=\operatorname {diag} (\lambda _{1},\dots ,\lambda _{n})$, then some of its square roots are diagonal matrices $\operatorname {diag} (\mu _{1},\dots ,\mu _{n})$, where $\mu _{i}=\pm {\sqrt {\lambda _{i}}}$. If the diagonal elements of D are real and non-negative then it is positive semidefinite, and if the square roots are taken with non-negative sign, the resulting matrix is the principal root of D. A diagonal matrix may have additional non-diagonal roots if some entries on the diagonal are equal, as exemplified by the identity matrix above.
If U is an upper triangular matrix (meaning its entries are $u_{i,j}=0$ for $i>j$) and at most one of its diagonal entries is zero, then one upper triangular solution of the equation $B^{2}=U$ can be found as follows. Since the equation $u_{i,i}=b_{i,i}^{2}$ should be satisfied, let $b_{i,i}$ be the principal square root of the complex number $u_{i,i}$. By the assumption $u_{i,i}\neq 0$, this guarantees that $b_{i,i}+b_{j,j}\neq 0$ for all i,j (because the principal square roots of complex numbers all lie on one half of the complex plane). From the equation
$u_{i,j}=b_{i,i}b_{i,j}+b_{i,i+1}b_{i+1,j}+b_{i,i+2}b_{i+2,j}+\dots +b_{i,j}b_{j,j}$
we deduce that $b_{i,j}$ can be computed recursively for $j-i$ increasing from 1 to n-1 as:
$b_{i,j}={\frac {1}{b_{i,i}+b_{j,j}}}\left(u_{i,j}-b_{i,i+1}b_{i+1,j}-b_{i,i+2}b_{i+2,j}-\dots -b_{i,j-1}b_{j-1,j}\right).$
If U is upper triangular but has multiple zeroes on the diagonal, then a square root might not exist, as exemplified by $\left({\begin{smallmatrix}0&1\\0&0\end{smallmatrix}}\right)$. Note the diagonal entries of a triangular matrix are precisely its eigenvalues (see Triangular matrix#Properties).
By diagonalization
An n × n matrix A is diagonalizable if there is a matrix V and a diagonal matrix D such that A = VDV−1. This happens if and only if A has n eigenvectors which constitute a basis for Cn. In this case, V can be chosen to be the matrix with the n eigenvectors as columns, and thus a square root of A is
$R=VSV^{-1}~,$
where S is any square root of D. Indeed,
$\left(VD^{\frac {1}{2}}V^{-1}\right)^{2}=VD^{\frac {1}{2}}\left(V^{-1}V\right)D^{\frac {1}{2}}V^{-1}=VDV^{-1}=A~.$
For example, the matrix $A=\left({\begin{smallmatrix}33&24\\48&57\end{smallmatrix}}\right)$ can be diagonalized as VDV−1, where
$V={\begin{pmatrix}1&1\\2&-1\end{pmatrix}}$ and $D={\begin{pmatrix}81&0\\0&9\end{pmatrix}}$.
D has principal square root
$D^{\frac {1}{2}}={\begin{pmatrix}9&0\\0&3\end{pmatrix}}$,
giving the square root
$A^{\frac {1}{2}}=VD^{\frac {1}{2}}V^{-1}={\begin{pmatrix}5&2\\4&7\end{pmatrix}}$.
When A is symmetric, the diagonalizing matrix V can be made an orthogonal matrix by suitably choosing the eigenvectors (see spectral theorem). Then the inverse of V is simply the transpose, so that
$R=VSV^{\textsf {T}}~.$
By Schur decomposition
Every complex-valued square matrix $A$, regardless of diagonalizability, has a Schur decomposition given by $A=QUQ^{*}$ where $U$ is upper triangular and $Q$ is unitary (meaning $Q^{*}=Q^{-1}$). The eigenvalues of $A$ are exactly the diagonal entries of $U$; if at most one of them is zero, then the following is a square root[7]
$A^{\frac {1}{2}}=QU^{\frac {1}{2}}Q^{*}.$
where a square root $U^{\frac {1}{2}}$ of the upper triangular matrix $U$ can be found as described above.
If $A$ is positive definite, then the eigenvalues are all positive reals, so the chosen diagonal of $U^{\frac {1}{2}}$ also consists of positive reals. Hence the eigenvalues of $QU^{\frac {1}{2}}Q^{*}$ are positive reals, which means the resulting matrix is the principal root of $A$.
By Jordan decomposition
Similarly as for the Schur decomposition, every square matrix $A$ can be decomposed as $A=P^{-1}JP$ where P is invertible and J is in Jordan normal form.
To see that any complex matrix with positive eigenvalues has a square root of the same form, it suffices to check this for a Jordan block. Any such block has the form λ(I + N) with λ > 0 and N nilpotent. If (1 + z)1/2 = 1 + a1 z + a2 z2 + ⋯ is the binomial expansion for the square root (valid in |z| < 1), then as a formal power series its square equals 1 + z. Substituting N for z, only finitely many terms will be non-zero and S = √λ (I + a1 N + a2 N2 + ⋯) gives a square root of the Jordan block with eigenvalue √λ.
It suffices to check uniqueness for a Jordan block with λ = 1. The square constructed above has the form S = I + L where L is polynomial in N without constant term. Any other square root T with positive eigenvalues has the form T = I + M with M nilpotent, commuting with N and hence L. But then 0 = S2 − T2 = 2(L − M)(I + (L + M)/2). Since L and M commute, the matrix L + M is nilpotent and I + (L + M)/2 is invertible with inverse given by a Neumann series. Hence L = M.
If A is a matrix with positive eigenvalues and minimal polynomial p(t), then the Jordan decomposition into generalized eigenspaces of A can be deduced from the partial fraction expansion of p(t)−1. The corresponding projections onto the generalized eigenspaces are given by real polynomials in A. On each eigenspace, A has the form λ(I + N) as above. The power series expression for the square root on the eigenspace show that the principal square root of A has the form q(A) where q(t) is a polynomial with real coefficients.
Power series
Recall the formal power series $ (1-z)^{\frac {1}{2}}=\sum _{n=0}^{\infty }(-1)^{n}{\binom {1/2}{n}}z^{n}$, which converges provided $\|z\|\leq 1$ (since the coefficients of the power series are summable). Plugging in $z=I-A$ into this expression yields
$A^{\frac {1}{2}}:=\sum _{n=0}^{\infty }(-1)^{n}{{\frac {1}{2}} \choose n}(I-A)^{n}$
provided that $ \limsup _{n}\|(I-A)^{n}\|^{\frac {1}{n}}<1$. By virtue of Gelfand formula, that condition is equivalent to the requirement that the spectrum of $A$ is contained within the disk $D(1,1)\subseteq \mathbb {C} $. This method of defining or computing $A^{\frac {1}{2}}$ is especially useful in the case where $A$ is positive semi-definite. In that case, we have $ \left\|I-{\frac {A}{\|A\|}}\right\|\leq 1$ and therefore $ \left\|\left(I-{\frac {A}{\|A\|}}\right)^{n}\right\|\leq \left\|I-{\frac {A}{\|A\|}}\right\|^{n}\leq 1$, so that the expression $ \|A\|^{\frac {1}{2}}=\left(\sum _{n=0}^{\infty }(-1)^{n}{\binom {1/2}{n}}\left(I-{\frac {A}{\|A\|}}\right)^{n}\right)$ defines a square root of $A$ which moreover turns out to be the unique positive semi-definite root. This method remains valid to define square roots of operators on infinite-dimensional Banach or Hilbert spaces or certain elements of (C*) Banach algebras.
Iterative solutions
By Denman–Beavers iteration
Another way to find the square root of an n × n matrix A is the Denman–Beavers square root iteration.[8]
Let Y0 = A and Z0 = I, where I is the n × n identity matrix. The iteration is defined by
${\begin{aligned}Y_{k+1}&={\frac {1}{2}}\left(Y_{k}+Z_{k}^{-1}\right),\\Z_{k+1}&={\frac {1}{2}}\left(Z_{k}+Y_{k}^{-1}\right).\end{aligned}}$
As this uses a pair of sequences of matrix inverses whose later elements change comparatively little, only the first elements have a high computational cost since the remainder can be computed from earlier elements with only a few passes of a variant of Newton's method for computing inverses,
$X_{n+1}=2X_{n}-X_{n}BX_{n}.$
With this, for later values of k one would set $X_{0}=Z_{k-1}^{-1}$ and $B=Z_{k},$ and then use $Z_{k}^{-1}=X_{n}$ for some small $n$ (perhaps just 1), and similarly for $Y_{k}^{-1}.$
Convergence is not guaranteed, even for matrices that do have square roots, but if the process converges, the matrix $Y_{k}$ converges quadratically to a square root A1/2, while $Z_{k}$ converges to its inverse, A−1/2.
By the Babylonian method
Yet another iterative method is obtained by taking the well-known formula of the Babylonian method for computing the square root of a real number, and applying it to matrices. Let X0 = I, where I is the identity matrix. The iteration is defined by
$X_{k+1}={\frac {1}{2}}\left(X_{k}+AX_{k}^{-1}\right)\,.$
Again, convergence is not guaranteed, but if the process converges, the matrix $X_{k}$ converges quadratically to a square root A1/2. Compared to Denman–Beavers iteration, an advantage of the Babylonian method is that only one matrix inverse need be computed per iteration step. On the other hand, as Denman–Beavers iteration uses a pair of sequences of matrix inverses whose later elements change comparatively little, only the first elements have a high computational cost since the remainder can be computed from earlier elements with only a few passes of a variant of Newton's method for computing inverses (see Denman–Beavers iteration above); of course, the same approach can be used to get the single sequence of inverses needed for the Babylonian method. However, unlike Denman–Beavers iteration, the Babylonian method is numerically unstable and more likely to fail to converge.[1]
The Babylonian method follows from Newton's method for the equation $X^{2}-A=0$ and using $AX_{k}=X_{k}A$ for all $k.$[9]
Square roots of positive operators
In linear algebra and operator theory, given a bounded positive semidefinite operator (a non-negative operator) T on a complex Hilbert space, B is a square root of T if T = B* B, where B* denotes the Hermitian adjoint of B. According to the spectral theorem, the continuous functional calculus can be applied to obtain an operator T1/2 such that T1/2 is itself positive and (T1/2)2 = T. The operator T1/2 is the unique non-negative square root of T.
A bounded non-negative operator on a complex Hilbert space is self adjoint by definition. So T = (T1/2)* T1/2. Conversely, it is trivially true that every operator of the form B* B is non-negative. Therefore, an operator T is non-negative if and only if T = B* B for some B (equivalently, T = CC* for some C).
The Cholesky factorization provides another particular example of square root, which should not be confused with the unique non-negative square root.
Unitary freedom of square roots
If T is a non-negative operator on a finite-dimensional Hilbert space, then all square roots of T are related by unitary transformations. More precisely, if T = A*A = B*B, then there exists a unitary U such that A = UB.
Indeed, take B = T1/2 to be the unique non-negative square root of T. If T is strictly positive, then B is invertible, and so U = AB−1 is unitary:
${\begin{aligned}U^{*}U&=\left(\left(B^{*}\right)^{-1}A^{*}\right)\left(AB^{-1}\right)=\left(B^{*}\right)^{-1}T\left(B^{-1}\right)\\&=\left(B^{*}\right)^{-1}B^{*}B\left(B^{-1}\right)=I.\end{aligned}}$
If T is non-negative without being strictly positive, then the inverse of B cannot be defined, but the Moore–Penrose pseudoinverse B+ can be. In that case, the operator B+A is a partial isometry, that is, a unitary operator from the range of T to itself. This can then be extended to a unitary operator U on the whole space by setting it equal to the identity on the kernel of T. More generally, this is true on an infinite-dimensional Hilbert space if, in addition, T has closed range. In general, if A, B are closed and densely defined operators on a Hilbert space H, and A* A = B* B, then A = UB where U is a partial isometry.
Some applications
Square roots, and the unitary freedom of square roots, have applications throughout functional analysis and linear algebra.
Polar decomposition
Main article: Polar decomposition
If A is an invertible operator on a finite-dimensional Hilbert space, then there is a unique unitary operator U and positive operator P such that
$A=UP;$
this is the polar decomposition of A. The positive operator P is the unique positive square root of the positive operator A∗A, and U is defined by U = AP−1.
If A is not invertible, then it still has a polar composition in which P is defined in the same way (and is unique). The unitary operator U is not unique. Rather it is possible to determine a "natural" unitary operator as follows: AP+ is a unitary operator from the range of A to itself, which can be extended by the identity on the kernel of A∗. The resulting unitary operator U then yields the polar decomposition of A.
Kraus operators
Main article: Choi's theorem on completely positive maps
By Choi's result, a linear map
$\Phi :C^{n\times n}\to C^{m\times m}$
is completely positive if and only if it is of the form
$\Phi (A)=\sum _{i}^{k}V_{i}AV_{i}^{*}$
where k ≤ nm. Let {Epq} ⊂ Cn × n be the n2 elementary matrix units. The positive matrix
$M_{\Phi }=\left(\Phi \left(E_{pq}\right)\right)_{pq}\in C^{nm\times nm}$
is called the Choi matrix of Φ. The Kraus operators correspond to the, not necessarily square, square roots of MΦ: For any square root B of MΦ, one can obtain a family of Kraus operators Vi by undoing the Vec operation to each column bi of B. Thus all sets of Kraus operators are related by partial isometries.
Mixed ensembles
Main article: Density matrix
In quantum physics, a density matrix for an n-level quantum system is an n × n complex matrix ρ that is positive semidefinite with trace 1. If ρ can be expressed as
$\rho =\sum _{i}p_{i}v_{i}v_{i}^{*}$
where $p_{i}>0$ and Σ pi = 1, the set
$\left\{p_{i},v_{i}\right\}$
is said to be an ensemble that describes the mixed state ρ. Notice {vi} is not required to be orthogonal. Different ensembles describing the state ρ are related by unitary operators, via the square roots of ρ. For instance, suppose
$\rho =\sum _{j}a_{j}a_{j}^{*}.$
The trace 1 condition means
$\sum _{j}a_{j}^{*}a_{j}=1.$
Let
$p_{i}=a_{i}^{*}a_{i},$
and vi be the normalized ai. We see that
$\left\{p_{i},v_{i}\right\}$
gives the mixed state ρ.
See also
• Matrix function
• Holomorphic functional calculus
• Logarithm of a matrix
• Sylvester's formula
• Square root of a 2 by 2 matrix
Notes
1. Higham, Nicholas J. (April 1986), "Newton's Method for the Matrix Square Root" (PDF), Mathematics of Computation, 46 (174): 537–549, doi:10.2307/2007992, JSTOR 2007992
2. Mitchell, Douglas W. (November 2003). "Using Pythagorean triples to generate square roots of $I_{2}$". The Mathematical Gazette. 87 (510): 499–500. doi:10.1017/s0025557200173723.
3. Horn & Johnson (2013), p. 439, Theorem 7.2.6 with $k=2$
4. Horn, Roger A.; Johnson, Charles R. (1990). Matrix analysis. Cambridge: Cambridge Univ. Press. p. 411. ISBN 9780521386326.
5. For analytic functions of matrices, see
• Higham 2008
• Horn & Johnson 1994
6. For the holomorphic functional calculus, see:
• Rudin 1991
• Bourbaki 2007
• Conway 1990
7. Deadman, Edvin; Higham, Nicholas J.; Ralha, Rui (2013), "Blocked Schur Algorithms for Computing the Matrix Square Root" (PDF), Applied Parallel and Scientific Computing, Springer Berlin Heidelberg, pp. 171–182, doi:10.1007/978-3-642-36803-5_12, ISBN 978-3-642-36802-8
8. Denman & Beavers 1976; Cheng et al. 2001
9. Higham, Nicholas J. (1997). "Stable iterations for the matrix square root". Numerical Algorithms. 15 (2): 227–242. Bibcode:1997NuAlg..15..227H. doi:10.1023/A:1019150005407.
References
• Bourbaki, Nicolas (2007), Théories spectrales, chapitres 1 et 2, Springer, ISBN 978-3540353317
• Conway, John B. (1990), A Course in Functional Analysis, Graduate Texts in Mathematics, vol. 96, Springer, pp. 199–205, ISBN 978-0387972459, Chapter IV, Reisz functional calculus
• Cheng, Sheung Hun; Higham, Nicholas J.; Kenney, Charles S.; Laub, Alan J. (2001), "Approximating the Logarithm of a Matrix to Specified Accuracy" (PDF), SIAM Journal on Matrix Analysis and Applications, 22 (4): 1112–1125, CiteSeerX 10.1.1.230.912, doi:10.1137/S0895479899364015, archived from the original (PDF) on 2011-08-09
• Burleson, Donald R., Computing the square root of a Markov matrix: eigenvalues and the Taylor series
• Denman, Eugene D.; Beavers, Alex N. (1976), "The matrix sign function and computations in systems", Applied Mathematics and Computation, 2 (1): 63–94, doi:10.1016/0096-3003(76)90020-5
• Higham, Nicholas (2008), Functions of Matrices. Theory and Computation, SIAM, ISBN 978-0-89871-646-7
• Horn, Roger A.; Johnson, Charles R. (2013). Matrix Analysis (2nd ed.). Cambridge University Press. ISBN 978-0-521-54823-6.
• Horn, Roger A.; Johnson, Charles R. (1994), Topics in Matrix Analysis, Cambridge University Press, ISBN 978-0521467131
• Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
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Wikipedia
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Imaginary number
An imaginary number is a real number multiplied by the imaginary unit i,[note 1] which is defined by its property i2 = −1.[1][2] The square of an imaginary number bi is −b2. For example, 5i is an imaginary number, and its square is −25. By definition, zero is considered to be both real and imaginary.[3]
All powers of i assume values
from blue area
i−3 = i
i−2 = −1
i−1 = −i
i0 = 1
i1 = i
i2 = −1
i3 = −i
i4 = 1
i5 = i
i6 = −1
i is a 4th root of unity
Originally coined in the 17th century by René Descartes[4] as a derogatory term and regarded as fictitious or useless, the concept gained wide acceptance following the work of Leonhard Euler (in the 18th century) and Augustin-Louis Cauchy and Carl Friedrich Gauss (in the early 19th century).
An imaginary number bi can be added to a real number a to form a complex number of the form a + bi, where the real numbers a and b are called, respectively, the real part and the imaginary part of the complex number.[5]
History
Main article: History of complex numbers
Although the Greek mathematician and engineer Hero of Alexandria is noted as the first to present a calculation involving the square root of a negative number,[6][7] it was Rafael Bombelli who first set down the rules for multiplication of complex numbers in 1572. The concept had appeared in print earlier, such as in work by Gerolamo Cardano. At the time, imaginary numbers and negative numbers were poorly understood and were regarded by some as fictitious or useless, much as zero once was. Many other mathematicians were slow to adopt the use of imaginary numbers, including René Descartes, who wrote about them in his La Géométrie in which he coined the term imaginary and meant it to be derogatory.[8][9] The use of imaginary numbers was not widely accepted until the work of Leonhard Euler (1707–1783) and Carl Friedrich Gauss (1777–1855). The geometric significance of complex numbers as points in a plane was first described by Caspar Wessel (1745–1818).[10]
In 1843, William Rowan Hamilton extended the idea of an axis of imaginary numbers in the plane to a four-dimensional space of quaternion imaginaries in which three of the dimensions are analogous to the imaginary numbers in the complex field.
Geometric interpretation
Geometrically, imaginary numbers are found on the vertical axis of the complex number plane, which allows them to be presented perpendicular to the real axis. One way of viewing imaginary numbers is to consider a standard number line positively increasing in magnitude to the right and negatively increasing in magnitude to the left. At 0 on the x-axis, a y-axis can be drawn with "positive" direction going up; "positive" imaginary numbers then increase in magnitude upwards, and "negative" imaginary numbers increase in magnitude downwards. This vertical axis is often called the "imaginary axis"[11] and is denoted $i\mathbb {R} ,$ $\mathbb {I} ,$ or ℑ.[12]
In this representation, multiplication by –1 corresponds to a rotation of 180 degrees about the origin, which is a half circle. Multiplication by i corresponds to a rotation of 90 degrees about the origin which is a quarter of a circle. Both these numbers are roots of $1$: $(-1)^{2}=1$, $i^{4}=1$. In the field of complex numbers, for every $n\in \mathbb {N} $, $1$ has nth roots $\varphi _{n}$, meaning $\varphi _{n}^{n}=1$, called roots of unity. Multiplying by the first $n$th root of unity causes a rotation of ${\frac {360}{n}}$ degrees about the origin.
Multiplying by a complex number is the same as rotating around the origin by the complex number's argument, followed by a scaling by its magnitude.[13]
Square roots of negative numbers
Care must be used when working with imaginary numbers that are expressed as the principal values of the square roots of negative numbers:[14]
$6={\sqrt {36}}={\sqrt {(-4)(-9)}}\neq {\sqrt {-4}}{\sqrt {-9}}=(2i)(3i)=6i^{2}=-6.$
That is sometimes written as:
$-1=i^{2}={\sqrt {-1}}{\sqrt {-1}}{\stackrel {\text{ (fallacy) }}{=}}{\sqrt {(-1)(-1)}}={\sqrt {1}}=1.$
The fallacy occurs as the equality ${\sqrt {xy}}={\sqrt {x}}{\sqrt {y}}$ fails when the variables are not suitably constrained. In that case, the equality fails to hold as the numbers are both negative, which can be demonstrated by:
${\sqrt {-x}}{\sqrt {-y}}=i{\sqrt {x}}\ i{\sqrt {y}}=i^{2}{\sqrt {x}}{\sqrt {y}}=-{\sqrt {xy}}\neq {\sqrt {xy}},$
where both x and y are positive real numbers.
See also
• Octonion
• −1
Number systems
Complex $:\;\mathbb {C} $ :\;\mathbb {C} }
Real $:\;\mathbb {R} $ :\;\mathbb {R} }
Rational $:\;\mathbb {Q} $ :\;\mathbb {Q} }
Integer $:\;\mathbb {Z} $ :\;\mathbb {Z} }
Natural $:\;\mathbb {N} $ :\;\mathbb {N} }
Zero: 0
One: 1
Prime numbers
Composite numbers
Negative integers
Fraction
Finite decimal
Dyadic (finite binary)
Repeating decimal
Irrational
Algebraic irrational
Transcendental
Imaginary
Notes
1. j is usually used in engineering contexts where i has other meanings (such as electrical current)
References
1. Uno Ingard, K. (1988). "Chapter 2". Fundamentals of Waves and Oscillations. Cambridge University Press. p. 38. ISBN 0-521-33957-X.
2. Weisstein, Eric W. "Imaginary Number". mathworld.wolfram.com. Retrieved 2020-08-10.
3. Sinha, K.C. (2008). A Text Book of Mathematics Class XI (Second ed.). Rastogi Publications. p. 11.2. ISBN 978-81-7133-912-9.
4. Giaquinta, Mariano; Modica, Giuseppe (2004). Mathematical Analysis: Approximation and Discrete Processes (illustrated ed.). Springer Science & Business Media. p. 121. ISBN 978-0-8176-4337-9. Extract of page 121
5. Aufmann, Richard; Barker, Vernon C.; Nation, Richard (2009). College Algebra: Enhanced Edition (6th ed.). Cengage Learning. p. 66. ISBN 978-1-4390-4379-0.
6. Hargittai, István (1992). Fivefold Symmetry (2 ed.). World Scientific. p. 153. ISBN 981-02-0600-3.
7. Roy, Stephen Campbell (2007). Complex Numbers: lattice simulation and zeta function applications. Horwood. p. 1. ISBN 978-1-904275-25-1.
8. Descartes, René, Discours de la méthode (Leiden, (Netherlands): Jan Maire, 1637), appended book: La Géométrie, book three, p. 380. From page 380: "Au reste tant les vrayes racines que les fausses ne sont pas tousjours reelles; mais quelquefois seulement imaginaires; c'est a dire qu'on peut bien tousjours en imaginer autant que jay dit en chasque Equation; mais qu'il n'y a quelquefois aucune quantité, qui corresponde a celles qu'on imagine, comme encore qu'on en puisse imaginer trois en celle cy, x3 – 6xx + 13x – 10 = 0, il n'y en a toutefois qu'une reelle, qui est 2, & pour les deux autres, quoy qu'on les augmente, ou diminue, ou multiplie en la façon que je viens d'expliquer, on ne sçauroit les rendre autres qu'imaginaires." (Moreover, the true roots as well as the false [roots] are not always real; but sometimes only imaginary [quantities]; that is to say, one can always imagine as many of them in each equation as I said; but there is sometimes no quantity that corresponds to what one imagines, just as although one can imagine three of them in this [equation], x3 – 6xx + 13x – 10 = 0, only one of them however is real, which is 2, and regarding the other two, although one increase, or decrease, or multiply them in the manner that I just explained, one would not be able to make them other than imaginary [quantities].)
9. Martinez, Albert A. (2006), Negative Math: How Mathematical Rules Can Be Positively Bent, Princeton: Princeton University Press, ISBN 0-691-12309-8, discusses ambiguities of meaning in imaginary expressions in historical context.
10. Rozenfeld, Boris Abramovich (1988). "Chapter 10". A History of Non-Euclidean Geometry: Evolution of the Concept of a Geometric Space. Springer. p. 382. ISBN 0-387-96458-4.
11. von Meier, Alexandra (2006). Electric Power Systems – A Conceptual Introduction. John Wiley & Sons. pp. 61–62. ISBN 0-471-17859-4. Retrieved 2022-01-13.
12. Webb, Stephen (2018). "5. Meaningless marks on paper". Clash of Symbols – A Ride Through the Riches of Glyphs. Springer Science+Business Media. pp. 204–205. doi:10.1007/978-3-319-71350-2_5. ISBN 978-3-319-71350-2.
13. Kuipers, J. B. (1999). Quaternions and Rotation Sequences: A Primer with Applications to Orbits, Aerospace, and Virtual Reality. Princeton University Press. pp. 10–11. ISBN 0-691-10298-8. Retrieved 2022-01-13.
14. Nahin, Paul J. (2010). An Imaginary Tale: The Story of "i" [the square root of minus one]. Princeton University Press. p. 12. ISBN 978-1-4008-3029-9. Extract of page 12
Bibliography
• Nahin, Paul (1998). An Imaginary Tale: the Story of the Square Root of −1. Princeton: Princeton University Press. ISBN 0-691-02795-1., explains many applications of imaginary expressions.
External links
Look up imaginary number in Wiktionary, the free dictionary.
• How can one show that imaginary numbers really do exist? – an article that discusses the existence of imaginary numbers.
• 5Numbers programme 4 BBC Radio 4 programme
• Why Use Imaginary Numbers? Basic Explanation and Uses of Imaginary Numbers
Complex numbers
• Complex conjugate
• Complex plane
• Imaginary number
• Real number
• Unit complex number
Number systems
Sets of definable numbers
• Natural numbers ($\mathbb {N} $)
• Integers ($\mathbb {Z} $)
• Rational numbers ($\mathbb {Q} $)
• Constructible numbers
• Algebraic numbers ($\mathbb {A} $)
• Closed-form numbers
• Periods
• Computable numbers
• Arithmetical numbers
• Set-theoretically definable numbers
• Gaussian integers
Composition algebras
• Division algebras: Real numbers ($\mathbb {R} $)
• Complex numbers ($\mathbb {C} $)
• Quaternions ($\mathbb {H} $)
• Octonions ($\mathbb {O} $)
Split
types
• Over $\mathbb {R} $:
• Split-complex numbers
• Split-quaternions
• Split-octonions
Over $\mathbb {C} $:
• Bicomplex numbers
• Biquaternions
• Bioctonions
Other hypercomplex
• Dual numbers
• Dual quaternions
• Dual-complex numbers
• Hyperbolic quaternions
• Sedenions ($\mathbb {S} $)
• Split-biquaternions
• Multicomplex numbers
• Geometric algebra/Clifford algebra
• Algebra of physical space
• Spacetime algebra
Other types
• Cardinal numbers
• Extended natural numbers
• Irrational numbers
• Fuzzy numbers
• Hyperreal numbers
• Levi-Civita field
• Surreal numbers
• Transcendental numbers
• Ordinal numbers
• p-adic numbers (p-adic solenoids)
• Supernatural numbers
• Profinite integers
• Superreal numbers
• Normal numbers
• Classification
• List
Authority control: National
• Germany
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Wikipedia
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Square root
In mathematics, a square root of a number x is a number y such that $y^{2}=x$; in other words, a number y whose square (the result of multiplying the number by itself, or $y\cdot y$) is x.[1] For example, 4 and −4 are square roots of 16 because $4^{2}=(-4)^{2}=16$.
Every nonnegative real number x has a unique nonnegative square root, called the principal square root, which is denoted by ${\sqrt {x}},$ where the symbol "${\sqrt {~^{~}}}$" is called the radical sign[2] or radix. For example, to express the fact that the principal square root of 9 is 3, we write ${\sqrt {9}}=3$. The term (or number) whose square root is being considered is known as the radicand. The radicand is the number or expression underneath the radical sign, in this case, 9. For non-negative x, the principal square root can also be written in exponent notation, as $x^{1/2}$.
Every positive number x has two square roots: ${\sqrt {x}}$ (which is positive) and $-{\sqrt {x}}$ (which is negative). The two roots can be written more concisely using the ± sign as $\pm {\sqrt {x}}$. Although the principal square root of a positive number is only one of its two square roots, the designation "the square root" is often used to refer to the principal square root.[3][4]
Square roots of negative numbers can be discussed within the framework of complex numbers. More generally, square roots can be considered in any context in which a notion of the "square" of a mathematical object is defined. These include function spaces and square matrices, among other mathematical structures.
History
The Yale Babylonian Collection clay tablet YBC 7289 was created between 1800 BC and 1600 BC, showing ${\sqrt {2}}$ and $ {\frac {\sqrt {2}}{2}}={\frac {1}{\sqrt {2}}}$ respectively as 1;24,51,10 and 0;42,25,35 base 60 numbers on a square crossed by two diagonals.[5] (1;24,51,10) base 60 corresponds to 1.41421296, which is correct to 5 decimal points (1.41421356...).
The Rhind Mathematical Papyrus is a copy from 1650 BC of an earlier Berlin Papyrus and other texts – possibly the Kahun Papyrus – that shows how the Egyptians extracted square roots by an inverse proportion method.[6]
In Ancient India, the knowledge of theoretical and applied aspects of square and square root was at least as old as the Sulba Sutras, dated around 800–500 BC (possibly much earlier).[7] A method for finding very good approximations to the square roots of 2 and 3 are given in the Baudhayana Sulba Sutra.[8] Aryabhata, in the Aryabhatiya (section 2.4), has given a method for finding the square root of numbers having many digits.
It was known to the ancient Greeks that square roots of positive integers that are not perfect squares are always irrational numbers: numbers not expressible as a ratio of two integers (that is, they cannot be written exactly as ${\frac {m}{n}}$, where m and n are integers). This is the theorem Euclid X, 9, almost certainly due to Theaetetus dating back to circa 380 BC.[9] The discovery of irrational numbers, including the particular case of the square root of 2, is widely associated with the Pythagorean school.[10][11] Although some accounts attribute the discovery to Hippasus, the specific contributor remains uncertain due to the scarcity of primary sources and the secretive nature of the brotherhood.[12][13] It is exactly the length of the diagonal of a square with side length 1.
In the Chinese mathematical work Writings on Reckoning, written between 202 BC and 186 BC during the early Han Dynasty, the square root is approximated by using an "excess and deficiency" method, which says to "...combine the excess and deficiency as the divisor; (taking) the deficiency numerator multiplied by the excess denominator and the excess numerator times the deficiency denominator, combine them as the dividend."[14]
A symbol for square roots, written as an elaborate R, was invented by Regiomontanus (1436–1476). An R was also used for radix to indicate square roots in Gerolamo Cardano's Ars Magna.[15]
According to historian of mathematics D.E. Smith, Aryabhata's method for finding the square root was first introduced in Europe by Cataneo—in 1546.
According to Jeffrey A. Oaks, Arabs used the letter jīm/ĝīm (ج), the first letter of the word "جذر" (variously transliterated as jaḏr, jiḏr, ǧaḏr or ǧiḏr, "root"), placed in its initial form (ﺟ) over a number to indicate its square root. The letter jīm resembles the present square root shape. Its usage goes as far as the end of the twelfth century in the works of the Moroccan mathematician Ibn al-Yasamin.[16]
The symbol "√" for the square root was first used in print in 1525, in Christoph Rudolff's Coss.[17]
Properties and uses
The principal square root function $f(x)={\sqrt {x}}$ (usually just referred to as the "square root function") is a function that maps the set of nonnegative real numbers onto itself. In geometrical terms, the square root function maps the area of a square to its side length.
The square root of x is rational if and only if x is a rational number that can be represented as a ratio of two perfect squares. (See square root of 2 for proofs that this is an irrational number, and quadratic irrational for a proof for all non-square natural numbers.) The square root function maps rational numbers into algebraic numbers, the latter being a superset of the rational numbers).
For all real numbers x,
${\sqrt {x^{2}}}=\left|x\right|={\begin{cases}x,&{\text{if }}x\geq 0\\-x,&{\text{if }}x<0.\end{cases}}$
(see absolute value).
For all nonnegative real numbers x and y,
${\sqrt {xy}}={\sqrt {x}}{\sqrt {y}}$
and
${\sqrt {x}}=x^{1/2}.$
The square root function is continuous for all nonnegative x, and differentiable for all positive x. If f denotes the square root function, whose derivative is given by:
$f'(x)={\frac {1}{2{\sqrt {x}}}}.$
The Taylor series of ${\sqrt {1+x}}$ about x = 0 converges for |x| ≤ 1, and is given by
${\sqrt {1+x}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(2n)!}{(1-2n)(n!)^{2}(4^{n})}}x^{n}=1+{\frac {1}{2}}x-{\frac {1}{8}}x^{2}+{\frac {1}{16}}x^{3}-{\frac {5}{128}}x^{4}+\cdots ,$
The square root of a nonnegative number is used in the definition of Euclidean norm (and distance), as well as in generalizations such as Hilbert spaces. It defines an important concept of standard deviation used in probability theory and statistics. It has a major use in the formula for roots of a quadratic equation; quadratic fields and rings of quadratic integers, which are based on square roots, are important in algebra and have uses in geometry. Square roots frequently appear in mathematical formulas elsewhere, as well as in many physical laws.
Square roots of positive integers
A positive number has two square roots, one positive, and one negative, which are opposite to each other. When talking of the square root of a positive integer, it is usually the positive square root that is meant.
The square roots of an integer are algebraic integers—more specifically quadratic integers.
The square root of a positive integer is the product of the roots of its prime factors, because the square root of a product is the product of the square roots of the factors. Since $ {\sqrt {p^{2k}}}=p^{k},$ only roots of those primes having an odd power in the factorization are necessary. More precisely, the square root of a prime factorization is
${\sqrt {p_{1}^{2e_{1}+1}\cdots p_{k}^{2e_{k}+1}p_{k+1}^{2e_{k+1}}\dots p_{n}^{2e_{n}}}}=p_{1}^{e_{1}}\dots p_{n}^{e_{n}}{\sqrt {p_{1}\dots p_{k}}}.$
As decimal expansions
The square roots of the perfect squares (e.g., 0, 1, 4, 9, 16) are integers. In all other cases, the square roots of positive integers are irrational numbers, and hence have non-repeating decimals in their decimal representations. Decimal approximations of the square roots of the first few natural numbers are given in the following table.
n${\sqrt {n}},$ truncated to 50 decimal places
00
11
21.41421356237309504880168872420969807856967187537694
31.73205080756887729352744634150587236694280525381038
42
52.23606797749978969640917366873127623544061835961152
62.44948974278317809819728407470589139196594748065667
72.64575131106459059050161575363926042571025918308245
82.82842712474619009760337744841939615713934375075389
93
103.16227766016837933199889354443271853371955513932521
As expansions in other numeral systems
As with before, the square roots of the perfect squares (e.g., 0, 1, 4, 9, 16) are integers. In all other cases, the square roots of positive integers are irrational numbers, and therefore have non-repeating digits in any standard positional notation system.
The square roots of small integers are used in both the SHA-1 and SHA-2 hash function designs to provide nothing up my sleeve numbers.
As periodic continued fractions
One of the most intriguing results from the study of irrational numbers as continued fractions was obtained by Joseph Louis Lagrange c. 1780. Lagrange found that the representation of the square root of any non-square positive integer as a continued fraction is periodic. That is, a certain pattern of partial denominators repeats indefinitely in the continued fraction. In a sense these square roots are the very simplest irrational numbers, because they can be represented with a simple repeating pattern of integers.
${\sqrt {2}}$= [1; 2, 2, ...]
${\sqrt {3}}$= [1; 1, 2, 1, 2, ...]
${\sqrt {4}}$= [2]
${\sqrt {5}}$= [2; 4, 4, ...]
${\sqrt {6}}$= [2; 2, 4, 2, 4, ...]
${\sqrt {7}}$= [2; 1, 1, 1, 4, 1, 1, 1, 4, ...]
${\sqrt {8}}$= [2; 1, 4, 1, 4, ...]
${\sqrt {9}}$= [3]
${\sqrt {10}}$= [3; 6, 6, ...]
${\sqrt {11}}$= [3; 3, 6, 3, 6, ...]
${\sqrt {12}}$= [3; 2, 6, 2, 6, ...]
${\sqrt {13}}$= [3; 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, ...]
${\sqrt {14}}$= [3; 1, 2, 1, 6, 1, 2, 1, 6, ...]
${\sqrt {15}}$= [3; 1, 6, 1, 6, ...]
${\sqrt {16}}$= [4]
${\sqrt {17}}$= [4; 8, 8, ...]
${\sqrt {18}}$= [4; 4, 8, 4, 8, ...]
${\sqrt {19}}$= [4; 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, ...]
${\sqrt {20}}$= [4; 2, 8, 2, 8, ...]
The square bracket notation used above is a short form for a continued fraction. Written in the more suggestive algebraic form, the simple continued fraction for the square root of 11, [3; 3, 6, 3, 6, ...], looks like this:
${\sqrt {11}}=3+{\cfrac {1}{3+{\cfrac {1}{6+{\cfrac {1}{3+{\cfrac {1}{6+{\cfrac {1}{3+\ddots }}}}}}}}}}$
where the two-digit pattern {3, 6} repeats over and over again in the partial denominators. Since 11 = 32 + 2, the above is also identical to the following generalized continued fractions:
${\sqrt {11}}=3+{\cfrac {2}{6+{\cfrac {2}{6+{\cfrac {2}{6+{\cfrac {2}{6+{\cfrac {2}{6+\ddots }}}}}}}}}}=3+{\cfrac {6}{20-1-{\cfrac {1}{20-{\cfrac {1}{20-{\cfrac {1}{20-{\cfrac {1}{20-\ddots }}}}}}}}}}.$
Computation
Main article: Methods of computing square roots
Square roots of positive numbers are not in general rational numbers, and so cannot be written as a terminating or recurring decimal expression. Therefore in general any attempt to compute a square root expressed in decimal form can only yield an approximation, though a sequence of increasingly accurate approximations can be obtained.
Most pocket calculators have a square root key. Computer spreadsheets and other software are also frequently used to calculate square roots. Pocket calculators typically implement efficient routines, such as the Newton's method (frequently with an initial guess of 1), to compute the square root of a positive real number.[18][19] When computing square roots with logarithm tables or slide rules, one can exploit the identities
${\sqrt {a}}=e^{(\ln a)/2}=10^{(\log _{10}a)/2},$
where ln and log10 are the natural and base-10 logarithms.
By trial-and-error,[20] one can square an estimate for ${\sqrt {a}}$ and raise or lower the estimate until it agrees to sufficient accuracy. For this technique it is prudent to use the identity
$(x+c)^{2}=x^{2}+2xc+c^{2},$
as it allows one to adjust the estimate x by some amount c and measure the square of the adjustment in terms of the original estimate and its square. Furthermore, $(x+c)^{2}\approx x^{2}+2xc$ when c is close to 0, because the tangent line to the graph of $x^{2}+2xc+c^{2}$ at $c=0$, as a function of c alone, is $y=2xc+x^{2}$. Thus, small adjustments to x can be planned out by setting $2xc$ to a, or $c={\frac {a}{2x}}$.
The most common iterative method of square root calculation by hand is known as the "Babylonian method" or "Heron's method" after the first-century Greek philosopher Heron of Alexandria, who first described it.[21] The method uses the same iterative scheme as the Newton–Raphson method yields when applied to the function y = f(x) = x2 − a, using the fact that its slope at any point is dy/dx = f′(x) = 2x, but predates it by many centuries.[22] The algorithm is to repeat a simple calculation that results in a number closer to the actual square root each time it is repeated with its result as the new input. The motivation is that if x is an overestimate to the square root of a nonnegative real number a then a/x will be an underestimate and so the average of these two numbers is a better approximation than either of them. However, the inequality of arithmetic and geometric means shows this average is always an overestimate of the square root (as noted below), and so it can serve as a new overestimate with which to repeat the process, which converges as a consequence of the successive overestimates and underestimates being closer to each other after each iteration. To find x:
1. Start with an arbitrary positive start value x. The closer to the square root of a, the fewer the iterations that will be needed to achieve the desired precision.
2. Replace x by the average (x + a/x) / 2 between x and a/x.
3. Repeat from step 2, using this average as the new value of x.
That is, if an arbitrary guess for ${\sqrt {a}}$ is x0, and xn + 1 = (xn + a/xn) / 2, then each xn is an approximation of ${\sqrt {a}}$ which is better for large n than for small n. If a is positive, the convergence is quadratic, which means that in approaching the limit, the number of correct digits roughly doubles in each next iteration. If a = 0, the convergence is only linear.
Using the identity
${\sqrt {a}}=2^{-n}{\sqrt {4^{n}a}},$
the computation of the square root of a positive number can be reduced to that of a number in the range [1, 4). This simplifies finding a start value for the iterative method that is close to the square root, for which a polynomial or piecewise-linear approximation can be used.
The time complexity for computing a square root with n digits of precision is equivalent to that of multiplying two n-digit numbers.
Another useful method for calculating the square root is the shifting nth root algorithm, applied for n = 2.
The name of the square root function varies from programming language to programming language, with sqrt[23] (often pronounced "squirt" [24]) being common, used in C, C++, and derived languages like JavaScript, PHP, and Python.
Square roots of negative and complex numbers
First leaf of the complex square root
Second leaf of the complex square root
Using the Riemann surface of the square root, it is shown how the two leaves fit together
The square of any positive or negative number is positive, and the square of 0 is 0. Therefore, no negative number can have a real square root. However, it is possible to work with a more inclusive set of numbers, called the complex numbers, that does contain solutions to the square root of a negative number. This is done by introducing a new number, denoted by i (sometimes by j, especially in the context of electricity where i traditionally represents electric current) and called the imaginary unit, which is defined such that i2 = −1. Using this notation, we can think of i as the square root of −1, but we also have (−i)2 = i2 = −1 and so −i is also a square root of −1. By convention, the principal square root of −1 is i, or more generally, if x is any nonnegative number, then the principal square root of −x is
${\sqrt {-x}}=i{\sqrt {x}}.$
The right side (as well as its negative) is indeed a square root of −x, since
$(i{\sqrt {x}})^{2}=i^{2}({\sqrt {x}})^{2}=(-1)x=-x.$
For every non-zero complex number z there exist precisely two numbers w such that w2 = z: the principal square root of z (defined below), and its negative.
Principal square root of a complex number
To find a definition for the square root that allows us to consistently choose a single value, called the principal value, we start by observing that any complex number $x+iy$ can be viewed as a point in the plane, $(x,y),$ expressed using Cartesian coordinates. The same point may be reinterpreted using polar coordinates as the pair $(r,\varphi ),$ where $r\geq 0$ is the distance of the point from the origin, and $\varphi $ is the angle that the line from the origin to the point makes with the positive real ($x$) axis. In complex analysis, the location of this point is conventionally written $re^{i\varphi }.$ If
$z=re^{i\varphi }{\text{ with }}-\pi <\varphi \leq \pi ,$
then the principal square root of $z$ is defined to be the following:
${\sqrt {z}}={\sqrt {r}}e^{i\varphi /2}.$
The principal square root function is thus defined using the non-positive real axis as a branch cut. If $z$ is a non-negative real number (which happens if and only if $\varphi =0$) then the principal square root of $z$ is ${\sqrt {r}}e^{i(0)/2}={\sqrt {r}};$ in other words, the principal square root of a non-negative real number is just the usual non-negative square root. It is important that $-\pi <\varphi \leq \pi $ because if, for example, $z=-2i$ (so $\varphi =-\pi /2$) then the principal square root is
${\sqrt {-2i}}={\sqrt {2e^{i\varphi }}}={\sqrt {2}}e^{i\varphi /2}={\sqrt {2}}e^{i(-\pi /4)}=1-i$
but using ${\tilde {\varphi }}:=\varphi +2\pi =3\pi /2$ would instead produce the other square root ${\sqrt {2}}e^{i{\tilde {\varphi }}/2}={\sqrt {2}}e^{i(3\pi /4)}=-1+i=-{\sqrt {-2i}}.$
The principal square root function is holomorphic everywhere except on the set of non-positive real numbers (on strictly negative reals it is not even continuous). The above Taylor series for ${\sqrt {1+x}}$ remains valid for complex numbers $x$ with $|x|<1.$
The above can also be expressed in terms of trigonometric functions:
${\sqrt {r\left(\cos \varphi +i\sin \varphi \right)}}={\sqrt {r}}\left(\cos {\frac {\varphi }{2}}+i\sin {\frac {\varphi }{2}}\right).$
Algebraic formula
When the number is expressed using its real and imaginary parts, the following formula can be used for the principal square root:[25][26]
${\sqrt {x+iy}}={\sqrt {{\tfrac {1}{2}}{\bigl (}{\sqrt x^{2}+y^{2}}}+x{\bigr )}}}+i\operatorname {sgn}(y){\sqrt {{\tfrac {1}{2}}{\bigl (}{\sqrt x^{2}+y^{2}}}-x{\bigr )}}},$
where sgn(y) is the sign of y (except that, here, sgn(0) = 1). In particular, the imaginary parts of the original number and the principal value of its square root have the same sign. The real part of the principal value of the square root is always nonnegative.
For example, the principal square roots of ±i are given by:
${\sqrt {i}}={\frac {1+i}{\sqrt {2}}},\qquad {\sqrt {-i}}={\frac {1-i}{\sqrt {2}}}.$
Notes
In the following, the complex z and w may be expressed as:
• $z=|z|e^{i\theta _{z}}$
• $w=|w|e^{i\theta _{w}}$
where $-\pi <\theta _{z}\leq \pi $ and $-\pi <\theta _{w}\leq \pi $.
Because of the discontinuous nature of the square root function in the complex plane, the following laws are not true in general.
• ${\sqrt {zw}}={\sqrt {z}}{\sqrt {w}}$
Counterexample for the principal square root: z = −1 and w = −1
This equality is valid only when $-\pi <\theta _{z}+\theta _{w}\leq \pi $
• ${\frac {\sqrt {w}}{\sqrt {z}}}={\sqrt {\frac {w}{z}}}$
Counterexample for the principal square root: w = 1 and z = −1
This equality is valid only when $-\pi <\theta _{w}-\theta _{z}\leq \pi $
• ${\sqrt {z^{*}}}=\left({\sqrt {z}}\right)^{*}$
Counterexample for the principal square root: z = −1)
This equality is valid only when $\theta _{z}\neq \pi $
A similar problem appears with other complex functions with branch cuts, e.g., the complex logarithm and the relations logz + logw = log(zw) or log(z*) = log(z)* which are not true in general.
Wrongly assuming one of these laws underlies several faulty "proofs", for instance the following one showing that −1 = 1:
${\begin{aligned}-1&=i\cdot i\\&={\sqrt {-1}}\cdot {\sqrt {-1}}\\&={\sqrt {\left(-1\right)\cdot \left(-1\right)}}\\&={\sqrt {1}}\\&=1.\end{aligned}}$
The third equality cannot be justified (see invalid proof).[27]: Chapter VI, Section I, Subsection 2 The fallacy that +1 = -1 It can be made to hold by changing the meaning of √ so that this no longer represents the principal square root (see above) but selects a branch for the square root that contains ${\sqrt {1}}\cdot {\sqrt {-1}}.$ The left-hand side becomes either
${\sqrt {-1}}\cdot {\sqrt {-1}}=i\cdot i=-1$
if the branch includes +i or
${\sqrt {-1}}\cdot {\sqrt {-1}}=(-i)\cdot (-i)=-1$
if the branch includes −i, while the right-hand side becomes
${\sqrt {\left(-1\right)\cdot \left(-1\right)}}={\sqrt {1}}=-1,$
where the last equality, ${\sqrt {1}}=-1,$ is a consequence of the choice of branch in the redefinition of √.
N-th roots and polynomial roots
The definition of a square root of $x$ as a number $y$ such that $y^{2}=x$ has been generalized in the following way.
A cube root of $x$ is a number $y$ such that $y^{3}=x$; it is denoted ${\sqrt[{3}]{x}}.$
If n is an integer greater than two, a n-th root of $x$ is a number $y$ such that $y^{n}=x$; it is denoted ${\sqrt[{n}]{x}}.$
Given any polynomial p, a root of p is a number y such that p(y) = 0. For example, the nth roots of x are the roots of the polynomial (in y) $y^{n}-x.$
Abel–Ruffini theorem states that, in general, the roots of a polynomial of degree five or higher cannot be expressed in terms of n-th roots.
Square roots of matrices and operators
Main article: Square root of a matrix
See also: Square root of a 2 by 2 matrix
If A is a positive-definite matrix or operator, then there exists precisely one positive definite matrix or operator B with B2 = A; we then define A1/2 = B. In general matrices may have multiple square roots or even an infinitude of them. For example, the 2 × 2 identity matrix has an infinity of square roots,[28] though only one of them is positive definite.
In integral domains, including fields
Each element of an integral domain has no more than 2 square roots. The difference of two squares identity u2 − v2 = (u − v)(u + v) is proved using the commutativity of multiplication. If u and v are square roots of the same element, then u2 − v2 = 0. Because there are no zero divisors this implies u = v or u + v = 0, where the latter means that two roots are additive inverses of each other. In other words if an element a square root u of an element a exists, then the only square roots of a are u and −u. The only square root of 0 in an integral domain is 0 itself.
In a field of characteristic 2, an element either has one square root or does not have any at all, because each element is its own additive inverse, so that −u = u. If the field is finite of characteristic 2 then every element has a unique square root. In a field of any other characteristic, any non-zero element either has two square roots, as explained above, or does not have any.
Given an odd prime number p, let q = pe for some positive integer e. A non-zero element of the field Fq with q elements is a quadratic residue if it has a square root in Fq. Otherwise, it is a quadratic non-residue. There are (q − 1)/2 quadratic residues and (q − 1)/2 quadratic non-residues; zero is not counted in either class. The quadratic residues form a group under multiplication. The properties of quadratic residues are widely used in number theory.
In rings in general
Unlike in an integral domain, a square root in an arbitrary (unital) ring need not be unique up to sign. For example, in the ring $\mathbb {Z} /8\mathbb {Z} $ of integers modulo 8 (which is commutative, but has zero divisors), the element 1 has four distinct square roots: ±1 and ±3.
Another example is provided by the ring of quaternions $\mathbb {H} ,$ which has no zero divisors, but is not commutative. Here, the element −1 has infinitely many square roots, including ±i, ±j, and ±k. In fact, the set of square roots of −1 is exactly
$\{ai+bj+ck\mid a^{2}+b^{2}+c^{2}=1\}.$
A square root of 0 is either 0 or a zero divisor. Thus in rings where zero divisors do not exist, it is uniquely 0. However, rings with zero divisors may have multiple square roots of 0. For example, in $\mathbb {Z} /n^{2}\mathbb {Z} ,$ any multiple of n is a square root of 0.
Geometric construction of the square root
The square root of a positive number is usually defined as the side length of a square with the area equal to the given number. But the square shape is not necessary for it: if one of two similar planar Euclidean objects has the area a times greater than another, then the ratio of their linear sizes is ${\sqrt {a}}$.
A square root can be constructed with a compass and straightedge. In his Elements, Euclid (fl. 300 BC) gave the construction of the geometric mean of two quantities in two different places: Proposition II.14 and Proposition VI.13. Since the geometric mean of a and b is ${\sqrt {ab}}$, one can construct ${\sqrt {a}}$ simply by taking b = 1.
The construction is also given by Descartes in his La Géométrie, see figure 2 on page 2. However, Descartes made no claim to originality and his audience would have been quite familiar with Euclid.
Euclid's second proof in Book VI depends on the theory of similar triangles. Let AHB be a line segment of length a + b with AH = a and HB = b. Construct the circle with AB as diameter and let C be one of the two intersections of the perpendicular chord at H with the circle and denote the length CH as h. Then, using Thales' theorem and, as in the proof of Pythagoras' theorem by similar triangles, triangle AHC is similar to triangle CHB (as indeed both are to triangle ACB, though we don't need that, but it is the essence of the proof of Pythagoras' theorem) so that AH:CH is as HC:HB, i.e. a/h = h/b, from which we conclude by cross-multiplication that h2 = ab, and finally that $h={\sqrt {ab}}$. When marking the midpoint O of the line segment AB and drawing the radius OC of length (a + b)/2, then clearly OC > CH, i.e. $ {\frac {a+b}{2}}\geq {\sqrt {ab}}$ (with equality if and only if a = b), which is the arithmetic–geometric mean inequality for two variables and, as noted above, is the basis of the Ancient Greek understanding of "Heron's method".
Another method of geometric construction uses right triangles and induction: ${\sqrt {1}}$ can be constructed, and once ${\sqrt {x}}$ has been constructed, the right triangle with legs 1 and ${\sqrt {x}}$ has a hypotenuse of ${\sqrt {x+1}}$. Constructing successive square roots in this manner yields the Spiral of Theodorus depicted above.
See also
• Apotome (mathematics)
• Cube root
• Functional square root
• Integer square root
• Nested radical
• Nth root
• Root of unity
• Solving quadratic equations with continued fractions
• Square root principle
• Quantum gate § Square root of NOT gate (√NOT)
Notes
1. Gel'fand, p. 120 Archived 2016-09-02 at the Wayback Machine
2. "Squares and Square Roots". www.mathsisfun.com. Retrieved 2020-08-28.
3. Zill, Dennis G.; Shanahan, Patrick (2008). A First Course in Complex Analysis With Applications (2nd ed.). Jones & Bartlett Learning. p. 78. ISBN 978-0-7637-5772-4. Archived from the original on 2016-09-01. Extract of page 78 Archived 2016-09-01 at the Wayback Machine
4. Weisstein, Eric W. "Square Root". mathworld.wolfram.com. Retrieved 2020-08-28.
5. "Analysis of YBC 7289". ubc.ca. Retrieved 19 January 2015.
6. Anglin, W.S. (1994). Mathematics: A Concise History and Philosophy. New York: Springer-Verlag.
7. Seidenberg, A. (1961). "The ritual origin of geometry". Archive for History of Exact Sciences. 1 (5): 488–527. doi:10.1007/bf00327767. ISSN 0003-9519. S2CID 119992603. Seidenberg (pp. 501-505) proposes: "It is the distinction between use and origin." [By analogy] "KEPLER needed the ellipse to describe the paths of the planets around the sun; he did not, however invent the ellipse, but made use of a curve that had been lying around for nearly 2000 years". In this manner Seidenberg argues: "Although the date of a manuscript or text cannot give us the age of the practices it discloses, nonetheless the evidence is contained in manuscripts." Seidenberg quotes Thibaut from 1875: "Regarding the time in which the Sulvasutras may have been composed, it is impossible to give more accurate information than we are able to give about the date of the Kalpasutras. But whatever the period may have been during which Kalpasutras and Sulvasutras were composed in the form now before us, we must keep in view that they only give a systematically arranged description of sacrificial rites, which had been practiced during long preceding ages." Lastly, Seidenberg summarizes: "In 1899, THIBAUT ventured to assign the fourth or the third centuries B.C. as the latest possible date for the composition of the Sulvasutras (it being understood that this refers to a codification of far older material)."
8. Joseph, ch.8.
9. Heath, Sir Thomas L. (1908). The Thirteen Books of The Elements, Vol. 3. Cambridge University Press. p. 3.
10. Craig Smorynski (2007). History of Mathematics: A Supplement (illustrated, annotated ed.). Springer Science & Business Media. p. 49. ISBN 978-0-387-75480-2. Extract of page 49
11. Brian E. Blank; Steven George Krantz (2006). Calculus: Single Variable, Volume 1 (illustrated ed.). Springer Science & Business Media. p. 71. ISBN 978-1-931914-59-8. Extract of page 71
12. Boyer, Carl B.; Merzbach, Uta C. (2011). A History of Mathematics (3rd ed.). Hoboken, NJ: John Wiley & Sons. pp. 51–53. ISBN 978-0470525487.
13. Stillwell, John (2010). Mathematics and Its History (3rd ed.). New York, NY: Springer. pp. 14–15. ISBN 978-1441960528.
14. Dauben (2007), p. 210.
15. "The Development of Algebra - 2". maths.org. Archived from the original on 24 November 2014. Retrieved 19 January 2015.
16. Oaks, Jeffrey A. (2012). Algebraic Symbolism in Medieval Arabic Algebra (PDF) (Thesis). Philosophica. p. 36. Archived (PDF) from the original on 2016-12-03.
17. Manguel, Alberto (2006). "Done on paper: the dual nature of numbers and the page". The Life of Numbers. ISBN 84-86882-14-1.
18. Parkhurst, David F. (2006). Introduction to Applied Mathematics for Environmental Science. Springer. pp. 241. ISBN 9780387342283.
19. Solow, Anita E. (1993). Learning by Discovery: A Lab Manual for Calculus. Cambridge University Press. pp. 48. ISBN 9780883850831.
20. Aitken, Mike; Broadhurst, Bill; Hladky, Stephen (2009). Mathematics for Biological Scientists. Garland Science. p. 41. ISBN 978-1-136-84393-8. Archived from the original on 2017-03-01. Extract of page 41 Archived 2017-03-01 at the Wayback Machine
21. Heath, Sir Thomas L. (1921). A History of Greek Mathematics, Vol. 2. Oxford: Clarendon Press. pp. 323–324.
22. Muller, Jean-Mic (2006). Elementary functions: algorithms and implementation. Springer. pp. 92–93. ISBN 0-8176-4372-9., Chapter 5, p 92 Archived 2016-09-01 at the Wayback Machine
23. "Function sqrt". CPlusPlus.com. The C++ Resources Network. 2016. Archived from the original on November 22, 2012. Retrieved June 24, 2016.
24. Overland, Brian (2013). C++ for the Impatient. Addison-Wesley. p. 338. ISBN 9780133257120. OCLC 850705706. Archived from the original on September 1, 2016. Retrieved June 24, 2016.
25. Abramowitz, Milton; Stegun, Irene A. (1964). Handbook of mathematical functions with formulas, graphs, and mathematical tables. Courier Dover Publications. p. 17. ISBN 0-486-61272-4. Archived from the original on 2016-04-23., Section 3.7.27, p. 17 Archived 2009-09-10 at the Wayback Machine
26. Cooke, Roger (2008). Classical algebra: its nature, origins, and uses. John Wiley and Sons. p. 59. ISBN 978-0-470-25952-8. Archived from the original on 2016-04-23.
27. Maxwell, E. A. (1959). Fallacies in Mathematics. Cambridge University Press.
28. Mitchell, Douglas W., "Using Pythagorean triples to generate square roots of I2", Mathematical Gazette 87, November 2003, 499–500.
References
• Dauben, Joseph W. (2007). "Chinese Mathematics I". In Katz, Victor J. (ed.). The Mathematics of Egypt, Mesopotamia, China, India, and Islam. Princeton: Princeton University Press. ISBN 978-0-691-11485-9.
• Gel'fand, Izrael M.; Shen, Alexander (1993). Algebra (3rd ed.). Birkhäuser. p. 120. ISBN 0-8176-3677-3.
• Joseph, George (2000). The Crest of the Peacock. Princeton: Princeton University Press. ISBN 0-691-00659-8.
• Smith, David (1958). History of Mathematics. Vol. 2. New York: Dover Publications. ISBN 978-0-486-20430-7.
• Selin, Helaine (2008), Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures, Springer, Bibcode:2008ehst.book.....S, ISBN 978-1-4020-4559-2.
External links
Wikimedia Commons has media related to Square root.
• Algorithms, implementations, and more – Paul Hsieh's square roots webpage
• How to manually find a square root
• AMS Featured Column, Galileo's Arithmetic by Tony Philips – includes a section on how Galileo found square roots
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Wikipedia
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Square tiling
In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane. It has Schläfli symbol of {4,4}, meaning it has 4 squares around every vertex. Conway called it a quadrille.
Square tiling
TypeRegular tiling
Vertex configuration4.4.4.4 (or 44)
Face configurationV4.4.4.4 (or V44)
Schläfli symbol(s){4,4}
{∞}×{∞}
Wythoff symbol(s)4 | 2 4
Coxeter diagram(s)
Symmetryp4m, [4,4], (*442)
Rotation symmetryp4, [4,4]+, (442)
Dualself-dual
PropertiesVertex-transitive, edge-transitive, face-transitive
The internal angle of the square is 90 degrees so four squares at a point make a full 360 degrees. It is one of three regular tilings of the plane. The other two are the triangular tiling and the hexagonal tiling.
Uniform colorings
There are 9 distinct uniform colorings of a square tiling. Naming the colors by indices on the 4 squares around a vertex: 1111, 1112(i), 1112(ii), 1122, 1123(i), 1123(ii), 1212, 1213, 1234. (i) cases have simple reflection symmetry, and (ii) glide reflection symmetry. Three can be seen in the same symmetry domain as reduced colorings: 1112i from 1213, 1123i from 1234, and 1112ii reduced from 1123ii.
9 uniform colorings
1111121212131112i1122
p4m (*442) p4m (*442) pmm (*2222)
12341123i1123ii1112ii
pmm (*2222) cmm (2*22)
Related polyhedra and tilings
This tiling is topologically related as a part of sequence of regular polyhedra and tilings, extending into the hyperbolic plane: {4,p}, p=3,4,5...
*n42 symmetry mutation of regular tilings: {4,n}
Spherical Euclidean Compact hyperbolic Paracompact
{4,3}
{4,4}
{4,5}
{4,6}
{4,7}
{4,8}...
{4,∞}
This tiling is also topologically related as a part of sequence of regular polyhedra and tilings with four faces per vertex, starting with the octahedron, with Schläfli symbol {n,4}, and Coxeter diagram , with n progressing to infinity.
*n42 symmetry mutation of regular tilings: {n,4}
Spherical Euclidean Hyperbolic tilings
24 34 44 54 64 74 84 ...∞4
*n42 symmetry mutations of quasiregular dual tilings: V(4.n)2
Symmetry
*4n2
[n,4]
Spherical Euclidean Compact hyperbolic Paracompact Noncompact
*342
[3,4]
*442
[4,4]
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]...
*∞42
[∞,4]
[iπ/λ,4]
Tiling
Conf.
V4.3.4.3
V4.4.4.4
V4.5.4.5
V4.6.4.6
V4.7.4.7
V4.8.4.8
V4.∞.4.∞
V4.∞.4.∞
*n42 symmetry mutation of expanded tilings: n.4.4.4
Symmetry
[n,4], (*n42)
Spherical Euclidean Compact hyperbolic Paracomp.
*342
[3,4]
*442
[4,4]
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]
*∞42
[∞,4]
Expanded
figures
Config. 3.4.4.4 4.4.4.4 5.4.4.4 6.4.4.4 7.4.4.4 8.4.4.4 ∞.4.4.4
Rhombic
figures
config.
V3.4.4.4
V4.4.4.4
V5.4.4.4
V6.4.4.4
V7.4.4.4
V8.4.4.4
V∞.4.4.4
Wythoff constructions from square tiling
Like the uniform polyhedra there are eight uniform tilings that can be based from the regular square tiling.
Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, all 8 forms are distinct. However treating faces identically, there are only three topologically distinct forms: square tiling, truncated square tiling, snub square tiling.
Uniform tilings based on square tiling symmetry
Symmetry: [4,4], (*442) [4,4]+, (442) [4,4+], (4*2)
{4,4} t{4,4} r{4,4} t{4,4} {4,4} rr{4,4} tr{4,4} sr{4,4} s{4,4}
Uniform duals
V4.4.4.4 V4.8.8 V4.4.4.4 V4.8.8 V4.4.4.4 V4.4.4.4 V4.8.8 V3.3.4.3.4
Topologically equivalent tilings
Other quadrilateral tilings can be made which are topologically equivalent to the square tiling (4 quads around every vertex).
Isohedral tilings have identical faces (face-transitivity) and vertex-transitivity, there are 18 variations, with 6 identified as triangles that do not connect edge-to-edge, or as quadrilateral with two collinear edges. Symmetry given assumes all faces are the same color.[1]
Isohedral quadrilateral tilings
Square
p4m, (*442)
Quadrilateral
p4g, (4*2)
Rectangle
pmm, (*2222)
Parallelogram
p2, (2222)
Parallelogram
pmg, (22*)
Rhombus
cmm, (2*22)
Rhombus
pmg, (22*)
Trapezoid
cmm, (2*22)
Quadrilateral
pgg, (22×)
Kite
pmg, (22*)
Quadrilateral
pgg, (22×)
Quadrilateral
p2, (2222)
Degenerate quadrilaterals or non-edge-to-edge triangles
Isosceles
pmg, (22*)
Isosceles
pgg, (22×)
Scalene
pgg, (22×)
Scalene
p2, (2222)
Circle packing
The square tiling can be used as a circle packing, placing equal diameter circles at the center of every point. Every circle is in contact with 4 other circles in the packing (kissing number).[2] The packing density is π/4=78.54% coverage. There are 4 uniform colorings of the circle packings.
Related regular complex apeirogons
There are 3 regular complex apeirogons, sharing the vertices of the square tiling. Regular complex apeirogons have vertices and edges, where edges can contain 2 or more vertices. Regular apeirogons p{q}r are constrained by: 1/p + 2/q + 1/r = 1. Edges have p vertices, and vertex figures are r-gonal.[3]
Self-dualDuals
4{4}4 or 2{8}4 or 4{8}2 or
See also
Wikimedia Commons has media related to Order-4 square tiling.
• Checkerboard
• List of regular polytopes
• List of uniform tilings
• Square lattice
• Tilings of regular polygons
References
1. Tilings and Patterns, from list of 107 isohedral tilings, p.473-481
2. Order in Space: A design source book, Keith Critchlow, p.74-75, circle pattern 3
3. Coxeter, Regular Complex Polytopes, pp. 111-112, p. 136.
• Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p. 296, Table II: Regular honeycombs
• Klitzing, Richard. "2D Euclidean tilings o4o4x - squat - O1".
• Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. p36
• Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 0-7167-1193-1. (Chapter 2.1: Regular and uniform tilings, p. 58-65)
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5
External links
• Weisstein, Eric W. "Square Grid". MathWorld.
• Weisstein, Eric W. "Regular tessellation". MathWorld.
• Weisstein, Eric W. "Uniform tessellation". MathWorld.
Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space Family ${\tilde {A}}_{n-1}$ ${\tilde {C}}_{n-1}$ ${\tilde {B}}_{n-1}$ ${\tilde {D}}_{n-1}$ ${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
E2 Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6
E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222
E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133 • 331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152 • 251 • 521
E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10
E10 Uniform 10-honeycomb {3[11]} δ11 hδ11 qδ11
En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k2 • 2k1 • k21
Tessellation
Periodic
• Pythagorean
• Rhombille
• Schwarz triangle
• Rectangle
• Domino
• Uniform tiling and honeycomb
• Coloring
• Convex
• Kisrhombille
• Wallpaper group
• Wythoff
Aperiodic
• Ammann–Beenker
• Aperiodic set of prototiles
• List
• Einstein problem
• Socolar–Taylor
• Gilbert
• Penrose
• Pentagonal
• Pinwheel
• Quaquaversal
• Rep-tile and Self-tiling
• Sphinx
• Socolar
• Truchet
Other
• Anisohedral and Isohedral
• Architectonic and catoptric
• Circle Limit III
• Computer graphics
• Honeycomb
• Isotoxal
• List
• Packing
• Problems
• Domino
• Wang
• Heesch's
• Squaring
• Dividing a square into similar rectangles
• Prototile
• Conway criterion
• Girih
• Regular Division of the Plane
• Regular grid
• Substitution
• Voronoi
• Voderberg
By vertex type
Spherical
• 2n
• 33.n
• V33.n
• 42.n
• V42.n
Regular
• 2∞
• 36
• 44
• 63
Semi-
regular
• 32.4.3.4
• V32.4.3.4
• 33.42
• 33.∞
• 34.6
• V34.6
• 3.4.6.4
• (3.6)2
• 3.122
• 42.∞
• 4.6.12
• 4.82
Hyper-
bolic
• 32.4.3.5
• 32.4.3.6
• 32.4.3.7
• 32.4.3.8
• 32.4.3.∞
• 32.5.3.5
• 32.5.3.6
• 32.6.3.6
• 32.6.3.8
• 32.7.3.7
• 32.8.3.8
• 33.4.3.4
• 32.∞.3.∞
• 34.7
• 34.8
• 34.∞
• 35.4
• 37
• 38
• 3∞
• (3.4)3
• (3.4)4
• 3.4.62.4
• 3.4.7.4
• 3.4.8.4
• 3.4.∞.4
• 3.6.4.6
• (3.7)2
• (3.8)2
• 3.142
• 3.162
• (3.∞)2
• 3.∞2
• 42.5.4
• 42.6.4
• 42.7.4
• 42.8.4
• 42.∞.4
• 45
• 46
• 47
• 48
• 4∞
• (4.5)2
• (4.6)2
• 4.6.12
• 4.6.14
• V4.6.14
• 4.6.16
• V4.6.16
• 4.6.∞
• (4.7)2
• (4.8)2
• 4.8.10
• V4.8.10
• 4.8.12
• 4.8.14
• 4.8.16
• 4.8.∞
• 4.102
• 4.10.12
• 4.122
• 4.12.16
• 4.142
• 4.162
• 4.∞2
• (4.∞)2
• 54
• 55
• 56
• 5∞
• 5.4.6.4
• (5.6)2
• 5.82
• 5.102
• 5.122
• (5.∞)2
• 64
• 65
• 66
• 68
• 6.4.8.4
• (6.8)2
• 6.82
• 6.102
• 6.122
• 6.162
• 73
• 74
• 77
• 7.62
• 7.82
• 7.142
• 83
• 84
• 86
• 88
• 8.62
• 8.122
• 8.162
• ∞3
• ∞4
• ∞5
• ∞∞
• ∞.62
• ∞.82
|
Wikipedia
|
Squaring the circle
Squaring the circle is a problem in geometry first proposed in Greek mathematics. It is the challenge of constructing a square with the area of a circle by using only a finite number of steps with a compass and straightedge. The difficulty of the problem raised the question of whether specified axioms of Euclidean geometry concerning the existence of lines and circles implied the existence of such a square.
Part of a series of articles on the
mathematical constant π
3.1415926535897932384626433...
Uses
• Area of a circle
• Circumference
• Use in other formulae
Properties
• Irrationality
• Transcendence
Value
• Less than 22/7
• Approximations
• Madhava's correction term
• Memorization
People
• Archimedes
• Liu Hui
• Zu Chongzhi
• Aryabhata
• Madhava
• Jamshīd al-Kāshī
• Ludolph van Ceulen
• François Viète
• Seki Takakazu
• Takebe Kenko
• William Jones
• John Machin
• William Shanks
• Srinivasa Ramanujan
• John Wrench
• Chudnovsky brothers
• Yasumasa Kanada
History
• Chronology
• A History of Pi
In culture
• Indiana Pi Bill
• Pi Day
Related topics
• Squaring the circle
• Basel problem
• Six nines in π
• Other topics related to π
In 1882, the task was proven to be impossible, as a consequence of the Lindemann–Weierstrass theorem, which proves that pi ($\pi $) is a transcendental number. That is, $\pi $ is not the root of any polynomial with rational coefficients. It had been known for decades that the construction would be impossible if $\pi $ were transcendental, but that fact was not proven until 1882. Approximate constructions with any given non-perfect accuracy exist, and many such constructions have been found.
Despite the proof that it is impossible, attempts to square the circle have been common in pseudomathematics (i.e. the work of mathematical cranks). The expression "squaring the circle" is sometimes used as a metaphor for trying to do the impossible.[1] The term quadrature of the circle is sometimes used as a synonym for squaring the circle, but it may also refer to approximate or numerical methods for finding the area of a circle.
History
Methods to calculate the approximate area of a given circle, which can be thought of as a precursor problem to squaring the circle, were known already in many ancient cultures. These methods can be summarized by stating the approximation to π that they produce. In around 2000 BCE, the Babylonian mathematicians used the approximation $\pi \approx {\tfrac {25}{8}}=3.125$, and at approximately the same time the ancient Egyptian mathematicians used $\pi \approx {\tfrac {256}{81}}\approx 3.16$. Over 1000 years later, the Old Testament Books of Kings used the simpler approximation $\pi \approx 3$.[2] Ancient Indian mathematics, as recorded in the Shatapatha Brahmana and Shulba Sutras, used several different approximations to $\pi $.[3] Archimedes proved a formula for the area of a circle, according to which $3\,{\tfrac {10}{71}}\approx 3.141<\pi <3\,{\tfrac {1}{7}}\approx 3.143$.[2] In Chinese mathematics, in the third century CE, Liu Hui found even more accurate approximations using a method similar to that of Archimedes, and in the fifth century Zu Chongzhi found $\pi \approx 355/113\approx 3.141593$, an approximation known as Milü.[4]
The problem of constructing a square whose area is exactly that of a circle, rather than an approximation to it, comes from Greek mathematics. Greek mathematicians found compass and straightedge constructions to convert any polygon into a square of equivalent area.[5] They used this construction to compare areas of polygons geometrically, rather than by the numerical computation of area that would be more typical in modern mathematics. As Proclus wrote many centuries later, this motivated the search for methods that would allow comparisons with non-polygonal shapes:
Having taken their lead from this problem, I believe, the ancients also sought the quadrature of the circle. For if a parallelogram is found equal to any rectilinear figure, it is worthy of investigation whether one can prove that rectilinear figures are equal to figures bound by circular arcs.[6]
The first known Greek to study the problem was Anaxagoras, who worked on it while in prison. Hippocrates of Chios attacked the problem by finding a shape bounded by circular arcs, the lune of Hippocrates, that could be squared. Antiphon the Sophist believed that inscribing regular polygons within a circle and doubling the number of sides would eventually fill up the area of the circle (this is the method of exhaustion). Since any polygon can be squared,[5] he argued, the circle can be squared. In contrast, Eudemus argued that magnitudes cannot be divided up without limit, so the area of the circle would never be used up.[7] Contemporaneously with Antiphon, Bryson of Heraclea argued that, since larger and smaller circles both exist, there must be a circle of equal area; this principle can be seen as a form of the modern intermediate value theorem.[8] The more general goal of carrying out all geometric constructions using only a compass and straightedge has often been attributed to Oenopides, but the evidence for this is circumstantial.[9]
The problem of finding the area under an arbitrary curve, now known as integration in calculus, or quadrature in numerical analysis, was known as squaring before the invention of calculus.[10] Since the techniques of calculus were unknown, it was generally presumed that a squaring should be done via geometric constructions, that is, by compass and straightedge. For example, Newton wrote to Oldenburg in 1676 "I believe M. Leibnitz will not dislike the theorem towards the beginning of my letter pag. 4 for squaring curve lines geometrically".[11] In modern mathematics the terms have diverged in meaning, with quadrature generally used when methods from calculus are allowed, while squaring the curve retains the idea of using only restricted geometric methods.
James Gregory attempted a proof of the impossibility of squaring the circle in Vera Circuli et Hyperbolae Quadratura (The True Squaring of the Circle and of the Hyperbola) in 1667. Although his proof was faulty, it was the first paper to attempt to solve the problem using algebraic properties of $\pi $.[12][13] Johann Heinrich Lambert proved in 1761 that $\pi $ is an irrational number.[14][15] It was not until 1882 that Ferdinand von Lindemann succeeded in proving more strongly that π is a transcendental number, and by doing so also proved the impossibility of squaring the circle with compass and straightedge.[16][17]
After Lindemann's impossibility proof, the problem was considered to be settled by professional mathematicians, and its subsequent mathematical history is dominated by pseudomathematical attempts at circle-squaring constructions, largely by amateurs, and by the debunking of these efforts.[19] As well, several later mathematicians including Srinivasa Ramanujan developed compass and straightedge constructions that approximate the problem accurately in few steps.[20][21]
Two other classical problems of antiquity, famed for their impossibility, were doubling the cube and trisecting the angle. Like squaring the circle, these cannot be solved by compass and straightedge. However, they have a different character than squaring the circle, in that their solution involves the root of a cubic equation, rather than being transcendental. Therefore, more powerful methods than compass and straightedge constructions, such as neusis construction or mathematical paper folding, can be used to construct solutions to these problems.[22][23]
Impossibility
The solution of the problem of squaring the circle by compass and straightedge requires the construction of the number ${\sqrt {\pi }}$, the length of the side of a square whose area equals that of a unit circle. If ${\sqrt {\pi }}$ were a constructible number, it would follow from standard compass and straightedge constructions that $\pi $ would also be constructible. In 1837, Pierre Wantzel showed that lengths that could be constructed with compass and straightedge had to be solutions of certain polynomial equations with rational coefficients.[24][25] Thus, constructible lengths must be algebraic numbers. If the circle could be squared using only compass and straightedge, then $\pi $ would have to be an algebraic number. It was not until 1882 that Ferdinand von Lindemann proved the transcendence of $\pi $ and so showed the impossibility of this construction. Lindemann's idea was to combine the proof of transcendence of Euler's number $e$, shown by Charles Hermite in 1873, with Euler's identity
$e^{i\pi }=-1.$
This identity immediately shows that $\pi $ is an irrational number, because a rational power of a transcendental number remains transcendental. Lindemann was able to extend this argument, through the Lindemann–Weierstrass theorem on linear independence of algebraic powers of $e$, to show that $\pi $ is transcendental and therefore that squaring the circle is impossible.[16][17]
Bending the rules by introducing a supplemental tool, allowing an infinite number of compass-and-straightedge operations or by performing the operations in certain non-Euclidean geometries makes squaring the circle possible in some sense. For example, Dinostratus' theorem uses the quadratrix of Hippias to square the circle, meaning that if this curve is somehow already given, then a square and circle of equal areas can be constructed from it. The Archimedean spiral can be used for another similar construction.[26] Although the circle cannot be squared in Euclidean space, it sometimes can be in hyperbolic geometry under suitable interpretations of the terms. The hyperbolic plane does not contain squares (quadrilaterals with four right angles and four equal sides), but instead it contains regular quadrilaterals, shapes with four equal sides and four equal angles sharper than right angles. There exist in the hyperbolic plane (countably) infinitely many pairs of constructible circles and constructible regular quadrilaterals of equal area, which, however, are constructed simultaneously. There is no method for starting with an arbitrary regular quadrilateral and constructing the circle of equal area. Symmetrically, there is no method for starting with an arbitrary circle and constructing a regular quadrilateral of equal area, and for sufficiently large circles no such quadrilateral exists.[27][28]
Approximate constructions
Although squaring the circle exactly with compass and straightedge is impossible, approximations to squaring the circle can be given by constructing lengths close to $\pi $. It takes only elementary geometry to convert any given rational approximation of $\pi $ into a corresponding compass and straightedge construction, but such constructions tend to be very long-winded in comparison to the accuracy they achieve. After the exact problem was proven unsolvable, some mathematicians applied their ingenuity to finding approximations to squaring the circle that are particularly simple among other imaginable constructions that give similar precision.
Construction by Kochański
Kochański's approximate construction
Continuation with equal-area circle and square; $r$ denotes the initial radius
One of many early historical approximate compass-and-straightedge constructions is from a 1685 paper by Polish Jesuit Adam Adamandy Kochański, producing an approximation diverging from $\pi $ in the 5th decimal place. Although much more precise numerical approximations to $\pi $ were already known, Kochański's construction has the advantage of being quite simple.[29] In the left diagram
$|P_{3}P_{9}|=|P_{1}P_{2}|{\sqrt {{\frac {40}{3}}-2{\sqrt {3}}}}\approx 3.141\,5{\color {red}33\,338}\cdot |P_{1}P_{2}|\approx \pi r.$
In the same work, Kochański also derived a sequence of increasingly accurate rational approximations for $\pi $.[30]
Constructions using 355/113
Jacob de Gelder's 355/113 construction
Ramanujan's 355/113 construction
Jacob de Gelder published in 1849 a construction based on the approximation
$\pi \approx {\frac {355}{113}}=3.141\;592{\color {red}\;920\;\ldots }$
This value is accurate to six decimal places and has been known in China since the 5th century as Milü, and in Europe since the 17th century.[31]
Gelder did not construct the side of the square; it was enough for him to find the value
${\overline {AH}}={\frac {4^{2}}{7^{2}+8^{2}}}.$
The illustration shows de Gelder's construction.
In 1914, Indian mathematician Srinivasa Ramanujan gave another geometric construction for the same approximation.[20][21]
Constructions using the golden ratio
Hobson's golden ratio construction
Dixon's golden ratio construction
Beatrix's 13-step construction
An approximate construction by E. W. Hobson in 1913[31] is accurate to three decimal places. Hobson's construction corresponds to an approximate value of
${\frac {6}{5}}\cdot \left(1+\varphi \right)=3.141\;{\color {red}640\;\ldots },$
where $\varphi $ is the golden ratio, $\varphi =(1+{\sqrt {5}})/2$.
The same approximate value appears in a 1991 construction by Robert Dixon.[32] In 2022 Frédéric Beatrix presented a geometrographic construction in 13 steps.[33]
Second construction by Ramanujan
Squaring the circle, approximate construction according to Ramanujan of 1914, with continuation of the construction (dashed lines, mean proportional red line), see animation.
Sketch of "Manuscript book 1 of Srinivasa Ramanujan" p. 54, Ramanujan's 355/113 construction
In 1914, Ramanujan gave a construction which was equivalent to taking the approximate value for $\pi $ to be
$\left(9^{2}+{\frac {19^{2}}{22}}\right)^{\frac {1}{4}}={\sqrt[{4}]{\frac {2143}{22}}}=3.141\;592\;65{\color {red}2\;582\;\ldots }$
giving eight decimal places of $\pi $.[20][21] He describes the construction of line segment OS as follows.[20]
Let AB (Fig.2) be a diameter of a circle whose centre is O. Bisect the arc ACB at C and trisect AO at T. Join BC and cut off from it CM and MN equal to AT. Join AM and AN and cut off from the latter AP equal to AM. Through P draw PQ parallel to MN and meeting AM at Q. Join OQ and through T draw TR, parallel to OQ and meeting AQ at R. Draw AS perpendicular to AO and equal to AR, and join OS. Then the mean proportional between OS and OB will be very nearly equal to a sixth of the circumference, the error being less than a twelfth of an inch when the diameter is 8000 miles long.
Incorrect constructions
In his old age, the English philosopher Thomas Hobbes convinced himself that he had succeeded in squaring the circle, a claim refuted by John Wallis as part of the Hobbes–Wallis controversy.[34] During the 18th and 19th century, the false notions that the problem of squaring the circle was somehow related to the longitude problem, and that a large reward would be given for a solution, became prevalent among would-be circle squarers.[35][36] In 1851, John Parker published a book Quadrature of the Circle in which he claimed to have squared the circle. His method actually produced an approximation of $\pi $ accurate to six digits.[37][38][39]
The Victorian-age mathematician, logician, and writer Charles Lutwidge Dodgson, better known by his pseudonym Lewis Carroll, also expressed interest in debunking illogical circle-squaring theories. In one of his diary entries for 1855, Dodgson listed books he hoped to write, including one called "Plain Facts for Circle-Squarers". In the introduction to "A New Theory of Parallels", Dodgson recounted an attempt to demonstrate logical errors to a couple of circle-squarers, stating:[40]
The first of these two misguided visionaries filled me with a great ambition to do a feat I have never heard of as accomplished by man, namely to convince a circle squarer of his error! The value my friend selected for Pi was 3.2: the enormous error tempted me with the idea that it could be easily demonstrated to BE an error. More than a score of letters were interchanged before I became sadly convinced that I had no chance.
A ridiculing of circle squaring appears in Augustus De Morgan's book A Budget of Paradoxes, published posthumously by his widow in 1872. Having originally published the work as a series of articles in The Athenæum, he was revising it for publication at the time of his death. Circle squaring declined in popularity after the nineteenth century, and it is believed that De Morgan's work helped bring this about.[19]
Even after it had been proved impossible, in 1894, amateur mathematician Edwin J. Goodwin claimed that he had developed a method to square the circle. The technique he developed did not accurately square the circle, and provided an incorrect area of the circle which essentially redefined $\pi $ as equal to 3.2. Goodwin then proposed the Indiana Pi Bill in the Indiana state legislature allowing the state to use his method in education without paying royalties to him. The bill passed with no objections in the state house, but the bill was tabled and never voted on in the Senate, amid increasing ridicule from the press.[41]
The mathematical crank Carl Theodore Heisel also claimed to have squared the circle in his 1934 book, "Behold! : the grand problem no longer unsolved: the circle squared beyond refutation."[42] Paul Halmos referred to the book as a "classic crank book."[43]
In literature
The problem of squaring the circle has been mentioned over a wide range of literary eras, with a variety of metaphorical meanings.[44] Its literary use dates back at least to 414 BC, when the play The Birds by Aristophanes was first performed. In it, the character Meton of Athens mentions squaring the circle, possibly to indicate the paradoxical nature of his utopian city.[45]
Dante's Paradise, canto XXXIII, lines 133–135, contain the verse:
As the geometer his mind applies
To square the circle, nor for all his wit
Finds the right formula, howe'er he tries
For Dante, squaring the circle represents a task beyond human comprehension, which he compares to his own inability to comprehend Paradise.[46] Dante's image also calls to mind a passage from Vitruvius, famously illustrated later in Leonardo da Vinci's Vitruvian Man, of a man simultaneously inscribed in a circle and a square.[47] Dante uses the circle as a symbol for God, and may have mentioned this combination of shapes in reference to the simultaneous divine and human nature of Jesus.[44][47] Earlier, in canto XIII, Dante calls out Greek circle-squarer Bryson as having sought knowledge instead of wisdom.[44]
Several works of 17th-century poet Margaret Cavendish elaborate on the circle-squaring problem and its metaphorical meanings, including a contrast between unity of truth and factionalism, and the impossibility of rationalizing "fancy and female nature".[44] By 1742, when Alexander Pope published the fourth book of his Dunciad, attempts at circle-squaring had come to be seen as "wild and fruitless":[38]
Mad Mathesis alone was unconfined,
Too mad for mere material chains to bind,
Now to pure space lifts her ecstatic stare,
Now, running round the circle, finds it square.
Similarly, the Gilbert and Sullivan comic opera Princess Ida features a song which satirically lists the impossible goals of the women's university run by the title character, such as finding perpetual motion. One of these goals is "And the circle – they will square it/Some fine day."[48]
The sestina, a poetic form first used in the 12th century by Arnaut Daniel, has been said to metaphorically square the circle in its use of a square number of lines (six stanzas of six lines each) with a circular scheme of six repeated words. Spanos (1978) writes that this form invokes a symbolic meaning in which the circle stands for heaven and the square stands for the earth.[49] A similar metaphor was used in "Squaring the Circle", a 1908 short story by O. Henry, about a long-running family feud. In the title of this story, the circle represents the natural world, while the square represents the city, the world of man.[50]
In later works, circle-squarers such as Leopold Bloom in James Joyce's novel Ulysses and Lawyer Paravant in Thomas Mann's The Magic Mountain are seen as sadly deluded or as unworldly dreamers, unaware of its mathematical impossibility and making grandiose plans for a result they will never attain.[51][52]
See also
• Mrs. Miniver's problem – Problem on areas of intersecting circles
• Round square copula – Philosophical treatment of oxymoronsPages displaying short descriptions of redirect targets
• Squircle – Shape between a square and a circle
• Tarski's circle-squaring problem – Problem of cutting and reassembling a disk into a square
References
1. Ammer, Christine. "Square the Circle. Dictionary.com. The American Heritage® Dictionary of Idioms". Houghton Mifflin Company. Retrieved 16 April 2012.
2. Bailey, D. H.; Borwein, J. M.; Borwein, P. B.; Plouffe, S. (1997). "The quest for pi". The Mathematical Intelligencer. 19 (1): 50–57. doi:10.1007/BF03024340. MR 1439159. S2CID 14318695.
3. Plofker, Kim (2009). Mathematics in India. Princeton University Press. p. 27. ISBN 978-0691120676.
4. Lam, Lay Yong; Ang, Tian Se (1986). "Circle measurements in ancient China". Historia Mathematica. 13 (4): 325–340. doi:10.1016/0315-0860(86)90055-8. MR 0875525. Reprinted in Berggren, J. L.; Borwein, Jonathan M.; Borwein, Peter, eds. (2004). Pi: A Source Book. Springer. pp. 20–35. ISBN 978-0387205717.
5. The construction of a square equal in area to a given polygon is Proposition 14 of Euclid's Elements, Book II.
6. Translation from Knorr (1986), p. 25
7. Heath, Thomas (1921). History of Greek Mathematics. The Clarendon Press. See in particular Anaxagoras, pp. 172–174; Lunes of Hippocrates, pp. 183–200; Later work, including Antiphon, Eudemus, and Aristophanes, pp. 220–235.
8. Bos, Henk J. M. (2001). "The legitimation of geometrical procedures before 1590". Redefining Geometrical Exactness: Descartes' Transformation of the Early Modern Concept of Construction. Sources and Studies in the History of Mathematics and Physical Sciences. New York: Springer. pp. 23–36. doi:10.1007/978-1-4613-0087-8_2. MR 1800805.
9. Knorr, Wilbur Richard (1986). The Ancient Tradition of Geometric Problems. Boston: Birkhäuser. pp. 15–16. ISBN 0-8176-3148-8. MR 0884893.
10. Guicciardini, Niccolò (2009). Isaac Newton on Mathematical Certainty and Method. Transformations. Vol. 4. MIT Press. p. 10. ISBN 9780262013178.
11. Cotes, Roger (1850). Correspondence of Sir Isaac Newton and Professor Cotes: Including letters of other eminent men.
12. Gregory, James (1667). Vera Circuli et Hyperbolæ Quadratura … [The true squaring of the circle and of the hyperbola …]. Padua: Giacomo Cadorino. Available at: ETH Bibliothek (Zürich, Switzerland)
13. Crippa, Davide (2019). "James Gregory and the impossibility of squaring the central conic sections". The Impossibility of Squaring the Circle in the 17th Century. Springer International Publishing. pp. 35–91. doi:10.1007/978-3-030-01638-8_2. S2CID 132820288.
14. Lambert, Johann Heinrich (1761). "Mémoire sur quelques propriétés remarquables des quantités transcendentes circulaires et logarithmiques" [Memoir on some remarkable properties of circular transcendental and logarithmic quantities]. Histoire de l'Académie Royale des Sciences et des Belles-Lettres de Berlin (in French) (published 1768). 17: 265–322.
15. Laczkovich, M. (1997). "On Lambert's proof of the irrationality of π". The American Mathematical Monthly. 104 (5): 439–443. doi:10.1080/00029890.1997.11990661. JSTOR 2974737. MR 1447977.
16. Lindemann, F. (1882). "Über die Zahl π" [On the number π]. Mathematische Annalen (in German). 20: 213–225. doi:10.1007/bf01446522. S2CID 120469397.
17. Fritsch, Rudolf (1984). "The transcendence of π has been known for about a century—but who was the man who discovered it?". Results in Mathematics. 7 (2): 164–183. doi:10.1007/BF03322501. MR 0774394. S2CID 119986449.
18. Cajori, Florian (1919). A History of Mathematics (2nd ed.). New York: The Macmillan Company. p. 143.
19. Dudley, Underwood (1987). A Budget of Trisections. Springer-Verlag. pp. xi–xii. ISBN 0-387-96568-8. Reprinted as The Trisectors.
20. Ramanujan, S. (1914). "Modular equations and approximations to π" (PDF). Quarterly Journal of Mathematics. 45: 350–372.
21. Castellanos, Dario (April 1988). "The ubiquitous π". Mathematics Magazine. 61 (2): 67–98. doi:10.1080/0025570X.1988.11977350. JSTOR 2690037.
22. Alperin, Roger C. (2005). "Trisections and totally real origami". The American Mathematical Monthly. 112 (3): 200–211. arXiv:math/0408159. doi:10.2307/30037438. JSTOR 30037438. MR 2125383.
23. Fuchs, Clemens (2011). "Angle trisection with origami and related topics". Elemente der Mathematik. 66 (3): 121–131. doi:10.4171/EM/179. MR 2824428.
24. Wantzel, L. (1837). "Recherches sur les moyens de reconnaître si un problème de géométrie peut se résoudre avec la règle et le compas" [Investigations into means of knowing if a problem of geometry can be solved with a straightedge and compass]. Journal de Mathématiques Pures et Appliquées (in French). 2: 366–372.
25. Cajori, Florian (1918). "Pierre Laurent Wantzel". Bulletin of the American Mathematical Society. 24 (7): 339–347. doi:10.1090/s0002-9904-1918-03088-7. MR 1560082.
26. Boyer, Carl B.; Merzbach, Uta C. (11 January 2011). A History of Mathematics. John Wiley & Sons. pp. 62–63, 113–115. ISBN 978-0-470-52548-7. OCLC 839010064.
27. Jagy, William C. (1995). "Squaring circles in the hyperbolic plane" (PDF). The Mathematical Intelligencer. 17 (2): 31–36. doi:10.1007/BF03024895. S2CID 120481094.
28. Greenberg, Marvin Jay (2008). Euclidean and Non-Euclidean Geometries (Fourth ed.). W H Freeman. pp. 520–528. ISBN 978-0-7167-9948-1.
29. Więsław, Witold (2001). "Squaring the circle in XVI–XVIII centuries". In Fuchs, Eduard (ed.). Mathematics throughout the ages. Including papers from the 10th and 11th Novembertagung on the History of Mathematics held in Holbæk, October 28–31, 1999 and in Brno, November 2–5, 2000. Dějiny Matematiky/History of Mathematics. Vol. 17. Prague: Prometheus. pp. 7–20. MR 1872936.
30. Fukś, Henryk (2012). "Adam Adamandy Kochański's approximations of π: reconstruction of the algorithm". The Mathematical Intelligencer. 34 (4): 40–45. arXiv:1111.1739. doi:10.1007/s00283-012-9312-1. MR 3029928. S2CID 123623596.
31. Hobson, Ernest William (1913). Squaring the Circle: A History of the Problem. Cambridge University Press. pp. 34–35.
32. Dixon, Robert A. (1987). "Squaring the circle". Mathographics. Blackwell. pp. 44–47. Reprinted by Dover Publications, 1991
33. Beatrix, Frédéric (2022). "Squaring the circle like a medieval master mason". Parabola. UNSW School of Mathematics and Statistics. 58 (2).
34. Bird, Alexander (1996). "Squaring the Circle: Hobbes on Philosophy and Geometry". Journal of the History of Ideas. 57 (2): 217–231. doi:10.1353/jhi.1996.0012. S2CID 171077338.
35. De Morgan, Augustus (1872). A Budget of Paradoxes. p. 96.
36. Board of Longitude / Vol V / Confirmed Minutes. Cambridge University Library: Royal Observatory. 1737–1779. p. 48. Retrieved 1 August 2021.
37. Beckmann, Petr (2015). A History of Pi. St. Martin's Press. p. 178. ISBN 9781466887169.
38. Schepler, Herman C. (1950). "The chronology of pi". Mathematics Magazine. 23 (3): 165–170, 216–228, 279–283. doi:10.2307/3029284. JSTOR 3029832. MR 0037596.
39. Abeles, Francine F. (1993). "Charles L. Dodgson's geometric approach to arctangent relations for pi". Historia Mathematica. 20 (2): 151–159. doi:10.1006/hmat.1993.1013. MR 1221681.
40. Gardner, Martin (1996). The Universe in a Handkerchief: Lewis Carroll's Mathematical Recreations, Games, Puzzles, and Word Plays. New York: Copernicus. pp. 29–31. doi:10.1007/0-387-28952-6. ISBN 0-387-94673-X.
41. Singmaster, David (1985). "The legal values of pi". The Mathematical Intelligencer. 7 (2): 69–72. doi:10.1007/BF03024180. MR 0784946. S2CID 122137198. Reprinted in Berggren, Lennart; Borwein, Jonathan; Borwein, Peter (2004). Pi: a source book (Third ed.). New York: Springer-Verlag. pp. 236–239. doi:10.1007/978-1-4757-4217-6_27. ISBN 0-387-20571-3. MR 2065455.
42. Heisel, Carl Theodore (1934). Behold! : the grand problem the circle squared beyond refutation no longer unsolved.
43. Paul R. Halmos (1970). "How to Write Mathematics". L'Enseignement mathématique. 16 (2): 123–152. — Pdf
44. Tubbs, Robert (December 2020). "Squaring the circle: A literary history". In Tubbs, Robert; Jenkins, Alice; Engelhardt, Nina (eds.). The Palgrave Handbook of Literature and Mathematics. Springer International Publishing. pp. 169–185. doi:10.1007/978-3-030-55478-1_10. MR 4272388. S2CID 234128826.
45. Amati, Matthew (2010). "Meton's star-city: Geometry and utopia in Aristophanes' Birds". The Classical Journal. 105 (3): 213–222. doi:10.5184/classicalj.105.3.213. JSTOR 10.5184/classicalj.105.3.213.
46. Herzman, Ronald B.; Towsley, Gary B. (1994). "Squaring the circle: Paradiso 33 and the poetics of geometry". Traditio. 49: 95–125. doi:10.1017/S0362152900013015. JSTOR 27831895. S2CID 155844205.
47. Kay, Richard (July 2005). "Vitruvius and Dante's Imago dei ". Word & Image. 21 (3): 252–260. doi:10.1080/02666286.2005.10462116. S2CID 194056860.
48. Dolid, William A. (1980). "Vivie Warren and the Tripos". The Shaw Review. 23 (2): 52–56. JSTOR 40682600. Dolid contrasts Vivie Warren, a fictional female mathematics student in Mrs. Warren's Profession by George Bernard Shaw, with the satire of college women presented by Gilbert and Sullivan. He writes that "Vivie naturally knew better than to try to square circles."
49. Spanos, Margaret (1978). "The Sestina: An Exploration of the Dynamics of Poetic Structure". Speculum. 53 (3): 545–557. doi:10.2307/2855144. JSTOR 2855144. S2CID 162823092.
50. Bloom, Harold (1987). Twentieth-century American literature. Chelsea House Publishers. p. 1848. ISBN 9780877548034. Similarly, the story "Squaring the Circle" is permeated with the integrating image: nature is a circle, the city a square.
51. Pendrick, Gerard (1994). "Two notes on "Ulysses"". James Joyce Quarterly. 32 (1): 105–107. JSTOR 25473619.
52. Goggin, Joyce (1997). The Big Deal: Card Games in 20th-Century Fiction (PhD). University of Montréal. p. 196.
Further reading and external links
Wikimedia Commons has media related to Squaring the circle.
Wikisource has original text related to this article:
Squaring the circle
• Bogomolny, Alexander. "Squaring the Circle". cut-the-knot.
• Grime, James. "Squaring the Circle". Numberphile. Brady Haran – via YouTube.
• Harper, Suzanne; Driskell, Shannon (August 2010). "An Investigation of Historical Geometric Constructions". Convergence. Mathematical Association of America.
• O'Connor, J J; Robertson, E F (April 1999). "Squaring the circle". MacTutor History of Mathematics archive.
• Otero, Daniel E. (July 2010). "The Quadrature of the Circle and Hippocrates' Lunes". Convergence. Mathematical Association of America.
• Polster, Burkard. "2000 years unsolved: Why is doubling cubes and squaring circles impossible?". Mathologer – via YouTube.
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Clifford torus
In geometric topology, the Clifford torus is the simplest and most symmetric flat embedding of the Cartesian product of two circles S1
a
and S1
b
(in the same sense that the surface of a cylinder is "flat"). It is named after William Kingdon Clifford. It resides in R4, as opposed to in R3. To see why R4 is necessary, note that if S1
a
and S1
b
each exists in its own independent embedding space R2
a
and R2
b
, the resulting product space will be R4 rather than R3. The historically popular view that the Cartesian product of two circles is an R3 torus in contrast requires the highly asymmetric application of a rotation operator to the second circle, since that circle will only have one independent axis z available to it after the first circle consumes x and y.
Stated another way, a torus embedded in R3 is an asymmetric reduced-dimension projection of the maximally symmetric Clifford torus embedded in R4. The relationship is similar to that of projecting the edges of a cube onto a sheet of paper. Such a projection creates a lower-dimensional image that accurately captures the connectivity of the cube edges, but also requires the arbitrary selection and removal of one of the three fully symmetric and interchangeable axes of the cube.
If S1
a
and S1
b
each has a radius of $\textstyle {\sqrt {1/2}}$, their Clifford torus product will fit perfectly within the unit 3-sphere S3, which is a 3-dimensional submanifold of R4. When mathematically convenient, the Clifford torus can be viewed as residing inside the complex coordinate space C2, since C2 is topologically equivalent to R4.
The Clifford torus is an example of a square torus, because it is isometric to a square with opposite sides identified. It is further known as a Euclidean 2-torus (the "2" is its topological dimension); figures drawn on it obey Euclidean geometry as if it were flat, whereas the surface of a common "doughnut"-shaped torus is positively curved on the outer rim and negatively curved on the inner. Although having a different geometry than the standard embedding of a torus in three-dimensional Euclidean space, the square torus can also be embedded into three-dimensional space, by the Nash embedding theorem; one possible embedding modifies the standard torus by a fractal set of ripples running in two perpendicular directions along the surface.[1]
Formal definition
The unit circle S1 in R2 can be parameterized by an angle coordinate:
$S^{1}=\{(\cos \theta ,\sin \theta )\mid 0\leq \theta <2\pi \}.$
In another copy of R2, take another copy of the unit circle
$S^{1}=\{(\cos \varphi ,\sin \varphi )\mid 0\leq \varphi <2\pi \}.$
Then the Clifford torus is
${\frac {1}{\sqrt {2}}}S^{1}\times {\frac {1}{\sqrt {2}}}S^{1}=\left\{{\frac {1}{\sqrt {2}}}(\cos \theta ,\sin \theta ,\cos \varphi ,\sin \varphi )\mid 0\leq \theta <2\pi ,0\leq \varphi <2\pi \right\}.$
Since each copy of S1 is an embedded submanifold of R2, the Clifford torus is an embedded torus in R2 × R2 = R4.
If R4 is given by coordinates (x1, y1, x2, y2), then the Clifford torus is given by
$x_{1}^{2}+y_{1}^{2}={\frac {1}{2}}=x_{2}^{2}+y_{2}^{2}.$
This shows that in R4 the Clifford torus is a submanifold of the unit 3-sphere S3.
It is easy to verify that the Clifford torus is a minimal surface in S3.
Alternative derivation using complex numbers
It is also common to consider the Clifford torus as an embedded torus in C2. In two copies of C, we have the following unit circles (still parametrized by an angle coordinate):
$S^{1}=\left\{e^{i\theta }\mid 0\leq \theta <2\pi \right\}$
and
$S^{1}=\left\{e^{i\varphi }\mid 0\leq \varphi <2\pi \right\}.$
Now the Clifford torus appears as
${\frac {1}{\sqrt {2}}}S^{1}\times {\frac {1}{\sqrt {2}}}S^{1}=\left\{{\frac {1}{\sqrt {2}}}\left(e^{i\theta },e^{i\varphi }\right)\,|\,0\leq \theta <2\pi ,0\leq \varphi <2\pi \right\}.$
As before, this is an embedded submanifold, in the unit sphere S3 in C2.
If C2 is given by coordinates (z1, z2), then the Clifford torus is given by
$\left|z_{1}\right|^{2}={\frac {1}{2}}=\left|z_{2}\right|^{2}.$
In the Clifford torus as defined above, the distance of any point of the Clifford torus to the origin of C2 is
${\sqrt {{\frac {1}{2}}\left|e^{i\theta }\right|^{2}+{\frac {1}{2}}\left|e^{i\varphi }\right|^{2}}}=1.$
The set of all points at a distance of 1 from the origin of C2 is the unit 3-sphere, and so the Clifford torus sits inside this 3-sphere. In fact, the Clifford torus divides this 3-sphere into two congruent solid tori (see Heegaard splitting[2]).
Since O(4) acts on R4 by orthogonal transformations, we can move the "standard" Clifford torus defined above to other equivalent tori via rigid rotations. These are all called "Clifford tori". The six-dimensional group O(4) acts transitively on the space of all such Clifford tori sitting inside the 3-sphere. However, this action has a two-dimensional stabilizer (see group action) since rotation in the meridional and longitudinal directions of a torus preserves the torus (as opposed to moving it to a different torus). Hence, there is actually a four-dimensional space of Clifford tori.[2] In fact, there is a one-to-one correspondence between Clifford tori in the unit 3-sphere and pairs of polar great circles (i.e., great circles that are maximally separated). Given a Clifford torus, the associated polar great circles are the core circles of each of the two complementary regions. Conversely, given any pair of polar great circles, the associated Clifford torus is the locus of points of the 3-sphere that are equidistant from the two circles.
More general definition of Clifford tori
The flat tori in the unit 3-sphere S3 that are the product of circles of radius r in one 2-plane R2 and radius √1 − r2 in another 2-plane R2 are sometimes also called "Clifford tori".
The same circles may be thought of as having radii that are cos(θ) and sin(θ) for some angle θ in the range 0 ≤ θ ≤ π/2 (where we include the degenerate cases θ = 0 and θ = π/2).
The union for 0 ≤ θ ≤ π/2 of all of these tori of form
$T_{\theta }=S(\cos \theta )\times S(\sin \theta )$
(where S(r) denotes the circle in the plane R2 defined by having center (0, 0) and radius r) is the 3-sphere S3. (Note that we must include the two degenerate cases θ = 0 and θ = π/2, each of which corresponds to a great circle of S3, and which together constitute a pair of polar great circles.)
This torus Tθ is readily seen to have area
$\operatorname {area} (T_{\theta })=4\pi ^{2}\cos \theta \sin \theta =2\pi ^{2}\sin 2\theta ,$
so only the torus Tπ/4 has the maximum possible area of 2π2. This torus Tπ/4 is the torus Tθ that is most commonly called the "Clifford torus" – and it is also the only one of the Tθ that is a minimal surface in S3.
Still more general definition of Clifford tori in higher dimensions
Any unit sphere S2n−1 in an even-dimensional euclidean space R2n = Cn may be expressed in terms of the complex coordinates as follows:
$S^{2n-1}=\left\{(z_{1},\ldots ,z_{n})\in \mathbf {C} ^{n}:|z_{1}|^{2}+\cdots +|z_{n}|^{2}=1\right\}.$
Then, for any non-negative numbers r1, ..., rn such that r12 + ... + rn2 = 1, we may define a generalized Clifford torus as follows:
$T_{r_{1},\ldots ,r_{n}}=\left\{(z_{1},\ldots ,z_{n})\in \mathbf {C} ^{n}:|z_{k}|=r_{k},~1\leqslant k\leqslant n\right\}.$
These generalized Clifford tori are all disjoint from one another. We may once again conclude that the union of each one of these tori Tr1, ..., rn is the unit (2n − 1)-sphere S2n−1 (where we must again include the degenerate cases where at least one of the radii rk = 0).
Properties
• The Clifford torus is "flat"; it can be flattened out to a plane without stretching, unlike the standard torus of revolution.
• The Clifford torus divides the 3-sphere into two congruent solid tori. (In a stereographic projection, the Clifford torus appears as a standard torus of revolution. The fact that it divides the 3-sphere equally means that the interior of the projected torus is equivalent to the exterior, which is not easily visualized).
Uses in mathematics
In symplectic geometry, the Clifford torus gives an example of an embedded Lagrangian submanifold of C2 with the standard symplectic structure. (Of course, any product of embedded circles in C gives a Lagrangian torus of C2, so these need not be Clifford tori.)
The Lawson conjecture states that every minimally embedded torus in the 3-sphere with the round metric must be a Clifford torus. This conjecture was proved by Simon Brendle in 2012.
Clifford tori and their images under conformal transformations are the global minimizers of the Willmore functional.
See also
• Duocylinder
• Hopf fibration
• Clifford parallel and Clifford surface
• William Kingdom Clifford
References
1. Borrelli, V.; Jabrane, S.; Lazarus, F.; Thibert, B. (April 2012), "Flat tori in three-dimensional space and convex integration", Proceedings of the National Academy of Sciences, 109 (19): 7218–7223, doi:10.1073/pnas.1118478109, PMC 3358891, PMID 22523238.
2. Norbs, P (September 2005). "The 12th problem" (PDF). The Australian Mathematical Society Gazette. 32 (4): 244–246.
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Wikipedia
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Square triangular number
In mathematics, a square triangular number (or triangular square number) is a number which is both a triangular number and a square number. There are infinitely many square triangular numbers; the first few are:
0, 1, 36, 1225, 41616, 1413721, 48024900, 1631432881, 55420693056, 1882672131025 (sequence A001110 in the OEIS)
For squares of triangular numbers, see squared triangular number.
Explicit formulas
Write Nk for the kth square triangular number, and write sk and tk for the sides of the corresponding square and triangle, so that
$N_{k}=s_{k}^{2}={\frac {t_{k}(t_{k}+1)}{2}}.$
Define the triangular root of a triangular number N = n(n + 1)/2 to be n. From this definition and the quadratic formula,
$n={\frac {{\sqrt {8N+1}}-1}{2}}.$
Therefore, N is triangular (n is an integer) if and only if 8N + 1 is square. Consequently, a square number M2 is also triangular if and only if 8M2 + 1 is square, that is, there are numbers x and y such that x2 − 8y2 = 1. This is an instance of the Pell equation with n = 8. All Pell equations have the trivial solution x = 1, y = 0 for any n; this is called the zeroth solution, and indexed as (x0, y0) = (1,0). If (xk, yk) denotes the kth nontrivial solution to any Pell equation for a particular n, it can be shown by the method of descent that
${\begin{aligned}x_{k+1}&=2x_{k}x_{1}-x_{k-1},\\y_{k+1}&=2y_{k}x_{1}-y_{k-1}.\end{aligned}}$
Hence there are an infinity of solutions to any Pell equation for which there is one non-trivial one, which holds whenever n is not a square. The first non-trivial solution when n = 8 is easy to find: it is (3,1). A solution (xk, yk) to the Pell equation for n = 8 yields a square triangular number and its square and triangular roots as follows:
$s_{k}=y_{k},\quad t_{k}={\frac {x_{k}-1}{2}},\quad N_{k}=y_{k}^{2}.$
Hence, the first square triangular number, derived from (3,1), is 1, and the next, derived from 6 × (3,1) − (1,0) = (17,6), is 36.
The sequences Nk, sk and tk are the OEIS sequences OEIS: A001110, OEIS: A001109, and OEIS: A001108 respectively.
In 1778 Leonhard Euler determined the explicit formula[1][2]: 12–13
$N_{k}=\left({\frac {\left(3+2{\sqrt {2}}\right)^{k}-\left(3-2{\sqrt {2}}\right)^{k}}{4{\sqrt {2}}}}\right)^{2}.$
Other equivalent formulas (obtained by expanding this formula) that may be convenient include
${\begin{aligned}N_{k}&={\tfrac {1}{32}}\left(\left(1+{\sqrt {2}}\right)^{2k}-\left(1-{\sqrt {2}}\right)^{2k}\right)^{2}\\&={\tfrac {1}{32}}\left(\left(1+{\sqrt {2}}\right)^{4k}-2+\left(1-{\sqrt {2}}\right)^{4k}\right)\\&={\tfrac {1}{32}}\left(\left(17+12{\sqrt {2}}\right)^{k}-2+\left(17-12{\sqrt {2}}\right)^{k}\right).\end{aligned}}$
The corresponding explicit formulas for sk and tk are:[2]: 13
${\begin{aligned}s_{k}&={\frac {\left(3+2{\sqrt {2}}\right)^{k}-\left(3-2{\sqrt {2}}\right)^{k}}{4{\sqrt {2}}}},\\t_{k}&={\frac {\left(3+2{\sqrt {2}}\right)^{k}+\left(3-2{\sqrt {2}}\right)^{k}-2}{4}}.\end{aligned}}$
Pell's equation
The problem of finding square triangular numbers reduces to Pell's equation in the following way.[3]
Every triangular number is of the form t(t + 1)/2. Therefore we seek integers t, s such that
${\frac {t(t+1)}{2}}=s^{2}.$
Rearranging, this becomes
$\left(2t+1\right)^{2}=8s^{2}+1,$
and then letting x = 2t + 1 and y = 2s, we get the Diophantine equation
$x^{2}-2y^{2}=1,$
which is an instance of Pell's equation. This particular equation is solved by the Pell numbers Pk as[4]
$x=P_{2k}+P_{2k-1},\quad y=P_{2k};$
and therefore all solutions are given by
$s_{k}={\frac {P_{2k}}{2}},\quad t_{k}={\frac {P_{2k}+P_{2k-1}-1}{2}},\quad N_{k}=\left({\frac {P_{2k}}{2}}\right)^{2}.$
There are many identities about the Pell numbers, and these translate into identities about the square triangular numbers.
Recurrence relations
There are recurrence relations for the square triangular numbers, as well as for the sides of the square and triangle involved. We have[5]: (12)
${\begin{aligned}N_{k}&=34N_{k-1}-N_{k-2}+2,&{\text{with }}N_{0}&=0{\text{ and }}N_{1}=1;\\N_{k}&=\left(6{\sqrt {N_{k-1}}}-{\sqrt {N_{k-2}}}\right)^{2},&{\text{with }}N_{0}&=0{\text{ and }}N_{1}=1.\end{aligned}}$
We have[1][2]: 13
${\begin{aligned}s_{k}&=6s_{k-1}-s_{k-2},&{\text{with }}s_{0}&=0{\text{ and }}s_{1}=1;\\t_{k}&=6t_{k-1}-t_{k-2}+2,&{\text{with }}t_{0}&=0{\text{ and }}t_{1}=1.\end{aligned}}$
Other characterizations
All square triangular numbers have the form b2c2, where b/c is a convergent to the continued fraction expansion of √2.[6]
A. V. Sylwester gave a short proof that there are an infinity of square triangular numbers:[7] If the nth triangular number n(n + 1)/2 is square, then so is the larger 4n(n + 1)th triangular number, since:
${\frac {{\bigl (}4n(n+1){\bigr )}{\bigl (}4n(n+1)+1{\bigr )}}{2}}=4\,{\frac {n(n+1)}{2}}\,\left(2n+1\right)^{2}.$
As the product of three squares, the right hand side is square. The triangular roots tk are alternately simultaneously one less than a square and twice a square if k is even, and simultaneously a square and one less than twice a square if k is odd. Thus,
49 = 72 = 2 × 52 − 1,
288 = 172 − 1 = 2 × 122, and
1681 = 412 = 2 × 292 − 1.
In each case, the two square roots involved multiply to give sk: 5 × 7 = 35, 12 × 17 = 204, and 29 × 41 = 1189.
Additionally:
$N_{k}-N_{k-1}=s_{2k-1};$
36 − 1 = 35, 1225 − 36 = 1189, and 41616 − 1225 = 40391. In other words, the difference between two consecutive square triangular numbers is the square root of another square triangular number.
The generating function for the square triangular numbers is:[8]
${\frac {1+z}{(1-z)\left(z^{2}-34z+1\right)}}=1+36z+1225z^{2}+\cdots $
Numerical data
As k becomes larger, the ratio tk/sk approaches √2 ≈ 1.41421356, and the ratio of successive square triangular numbers approaches (1 + √2)4 = 17 + 12√2 ≈ 33.970562748. The table below shows values of k between 0 and 11, which comprehend all square triangular numbers up to 1016.
k Nk sk tk tk/sk Nk/Nk − 1
0 0 0 0
1 1 1 1 1
2 36 6 8 1.33333333 36
3 1225 35 49 1.4 34.027777778
4 41616 204 288 1.41176471 33.972244898
5 1413721 1189 1681 1.41379310 33.970612265
6 48024900 6930 9800 1.41414141 33.970564206
7 1631432881 40391 57121 1.41420118 33.970562791
8 55420693056 235416 332928 1.41421144 33.970562750
9 1882672131025 1372105 1940449 1.41421320 33.970562749
10 63955431761796 7997214 11309768 1.41421350 33.970562748
11 2172602007770041 46611179 65918161 1.41421355 33.970562748
See also
• Cannonball problem, on numbers that are simultaneously square and square pyramidal
• Sixth power, numbers that are simultaneously square and cubical
Notes
1. Dickson, Leonard Eugene (1999) [1920]. History of the Theory of Numbers. Vol. 2. Providence: American Mathematical Society. p. 16. ISBN 978-0-8218-1935-7.
2. Euler, Leonhard (1813). "Regula facilis problemata Diophantea per numeros integros expedite resolvendi (An easy rule for Diophantine problems which are to be resolved quickly by integral numbers)". Mémoires de l'Académie des Sciences de St.-Pétersbourg (in Latin). 4: 3–17. Retrieved 2009-05-11. According to the records, it was presented to the St. Petersburg Academy on May 4, 1778.
3. Barbeau, Edward (2003). Pell's Equation. Problem Books in Mathematics. New York: Springer. pp. 16–17. ISBN 978-0-387-95529-2. Retrieved 2009-05-10.
4. Hardy, G. H.; Wright, E. M. (1979). An Introduction to the Theory of Numbers (5th ed.). Oxford University Press. p. 210. ISBN 0-19-853171-0. Theorem 244
5. Weisstein, Eric W. "Square Triangular Number". MathWorld.
6. Ball, W. W. Rouse; Coxeter, H. S. M. (1987). Mathematical Recreations and Essays. New York: Dover Publications. p. 59. ISBN 978-0-486-25357-2.
7. Pietenpol, J. L.; Sylwester, A. V.; Just, Erwin; Warten, R. M. (February 1962). "Elementary Problems and Solutions: E 1473, Square Triangular Numbers". American Mathematical Monthly. Mathematical Association of America. 69 (2): 168–169. doi:10.2307/2312558. ISSN 0002-9890. JSTOR 2312558.
8. Plouffe, Simon (August 1992). "1031 Generating Functions" (PDF). University of Quebec, Laboratoire de combinatoire et d'informatique mathématique. p. A.129. Archived from the original (PDF) on 2012-08-20. Retrieved 2009-05-11.
External links
• Triangular numbers that are also square at cut-the-knot
• Weisstein, Eric W. "Square Triangular Number". MathWorld.
• Michael Dummett's solution
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Wikipedia
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Square trisection
In geometry, a square trisection is a type of dissection problem which consists of cutting a square into pieces that can be rearranged to form three identical squares.
History
The dissection of a square in three congruent partitions is a geometrical problem that dates back to the Islamic Golden Age. Craftsman who mastered the art of zellige needed innovative techniques to achieve their fabulous mosaics with complex geometric figures. The first solution to this problem was proposed in the 10th century AD by the Persian mathematician Abu'l-Wafa' (940-998) in his treatise "On the geometric constructions necessary for the artisan".[1] Abu'l-Wafa' also used his dissection to demonstrate the Pythagorean theorem.[2] This geometrical proof of Pythagoras' theorem would be rediscovered in the years 1835 - 1840 [3] by Henry Perigal and published in 1875.[4]
Search of optimality
The beauty of a dissection depends on several parameters. However, it is usual to search for solutions with the minimum number of parts. Far from being minimal, the square trisection proposed by Abu'l-Wafa' uses 9 pieces. In the 14th century Abu Bakr al-Khalil gave two solutions, one of which uses 8 pieces.[5] In the late 17th century Jacques Ozanam came back to this issue [6] and in the 19th century, solutions using 8 and 7 pieces were found, including one given by the mathematician Édouard Lucas.[7] In 1891 Henry Perigal published the first known solution with only 6 pieces [8] (see illustration below). Nowadays, new dissections are still found [9] (see illustration above) and the conjecture that 6 is the minimal number of necessary pieces remains unproved.
See also
• Proofs by dissection and rearrangement of Pythagorean theorem
• Dissection puzzle
• Tangram
Bibliography
• Frederickson, Greg N. (1997). Dissections: Plane and Fancy. Cambridge University Press. ISBN 0-521-57197-9.
• Frederickson, Greg N. (2002). Hinged Dissections: Swinging and Twisting. Cambridge University Press. ISBN 0-521-81192-9.
• Frederickson, Greg N. (2006). Piano-hinged Dissections: Time to Fold!. en:A K Peters. ISBN 1-56881-299-X.
References
1. Alpay Özdural (1995). Omar Khayyam, Mathematicians, and “conversazioni” with Artisans. Journal of the Society of Architectural Vol. 54, No. 1, Mar., 1995
2. Reza Sarhangi, Slavik Jablan (2006). Elementary Constructions of Persian Mosaics. Towson University and The Mathematical Institute. online Archived 2011-07-28 at the Wayback Machine
3. See appendix of L. J. Rogers (1897). Biography of Henry Perigal: On certain Regular Polygons in Modular Network. Proceedings London Mathematical Society. Volume s1-29, Appendix pp. 732-735.
4. Henry Perigal (1875). On Geometric Dissections and Transformations, Messenger of Mathematics, No 19, 1875.
5. Alpay Özdural (2000). Mathematics and Arts: Connections between Theory and Practice in the Medieval Islamic World, Historia Mathematica, Volume 27, Issue 2, May 2000, Pages 171-201.
6. (fr) Jean-Etienne Montucla (1778), completed and re-edited by Jacques Ozanam (1640-1717) Récréations mathématiques, Tome 1 (1694), p. 297 Pl.15.
7. (fr) Edouard Lucas (1883). Récréations Mathématiques, Volume 2. Paris, Gauthier-Villars. Second of four volumes. Second edition (1893) reprinted by Blanchard in 1960. See pp. 151 and 152 in Volume 2 of this edition. online (pp. 145-147).
8. Henry Perigal (1891). Geometric Dissections and Transpositions, Association for the Improvement of Geometrical Teaching. wikisource
9. Christian Blanvillain, János Pach (2010). Square Trisection. Bulletin d'Informatique Approfondie et Applications N°86 - Juin 2010 Archived 2011-07-24 at the Wayback Machine also at EPFL: oai:infoscience.epfl.ch:161493.
External links
• Greg N. Frederickson web site
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Wikipedia
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Square wheel
A square wheel is a wheel that, instead of being circular, has the shape of a square. While literal square wheels exist, a more common use is as an idiom meaning meaning stereotypically bad or naive engineering (see reinventing the wheel).
A square wheel can roll smoothly if the ground consists of evenly shaped inverted catenaries of the right size and curvature.[1][2][3]
A different type of square-wheeled vehicle was invented in 2006 by Jason Winckler of Global Composites, Inc. in the United States. This has square wheels, linked together and offset by 22.5°, rolling on a flat surface. The prototype appears ungainly, but the inventor proposes that the system may be useful in microscopic-sized machines (MEMS).[4] In 1997 Macalester College mathematics professor Stan Wagon constructed the first prototype of a catenary tricycle. An improved model made out of modern materials was built when the original vehicle wore out in April, 2004.[5]
In 2012, MythBusters experimented with modifying vehicles with square tires, determining that, with speed, a truck fitted with square wheels can deliver a relatively smooth ride.
See also
• Reuleaux triangle
• Non-circular gear
References
1. Peterson, Ivars (4 April 2004), "Riding on Square Wheels", Science News, vol. 165, no. 14, archived from the original on July 2, 2008, retrieved 2009-05-03
2. A Catenary Road and Square Wheels, New Trier High School, Winnetka, Illinois, archived from the original on September 30, 2006, retrieved 2009-05-03
3. Non-Circular Wheels, vol. Physics and Astronomy Lecture Demonstrations, University of Iowa, retrieved 2009-05-03
4. Derby, Stephen J.; Anderson, Kurt; Winckler, Steven; Winckler, Jason (2006). "Motion Characteristics of a Square Wheel Car". Volume 2: 30th Annual Mechanisms and Robotics Conference, Parts A and B. Vol. 2006. Philadelphia, Pennsylvania, USA: ASME. pp. 811–816. doi:10.1115/DETC2006-99140. ISBN 9780791842560.
5. Wagon, Stan. "Untitled". Retrieved 19 May 2010.
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Wikipedia
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Squaregraph
In graph theory, a branch of mathematics, a squaregraph is a type of undirected graph that can be drawn in the plane in such a way that every bounded face is a quadrilateral and every vertex with three or fewer neighbors is incident to an unbounded face.
Related graph classes
The squaregraphs include as special cases trees, grid graphs, gear graphs, and the graphs of polyominos.
As well as being planar graphs, squaregraphs are median graphs, meaning that for every three vertices u, v, and w there is a unique median vertex m(u,v,w) that lies on shortest paths between each pair of the three vertices.[1] As with median graphs more generally, squaregraphs are also partial cubes: their vertices can be labeled with binary strings such that the Hamming distance between strings is equal to the shortest path distance between vertices.
The graph obtained from a squaregraph by making a vertex for each zone (an equivalence class of parallel edges of quadrilaterals) and an edge for each two zones that meet in a quadrilateral is a circle graph determined by a triangle-free chord diagram of the unit disk.[2]
Characterization
Squaregraphs may be characterized in several ways other than via their planar embeddings:[2]
• They are the median graphs that do not contain as an induced subgraph any member of an infinite family of forbidden graphs. These forbidden graphs are the cube (the simplex graph of K3), the Cartesian product of an edge and a claw K1,3 (the simplex graph of a claw), and the graphs formed from a gear graph by adding one more vertex connected to the hub of the wheel (the simplex graph of the disjoint union of a cycle with an isolated vertex).
• They are the graphs that are connected and bipartite, such that (if an arbitrary vertex r is picked as a root) every vertex has at most two neighbors closer to r, and such that at every vertex v, the link of v (a graph with a vertex for each edge incident to v and an edge for each 4-cycle containing v) is either a cycle of length greater than three or a disjoint union of paths.
• They are the dual graphs of arrangements of lines in the hyperbolic plane that do not have three mutually-crossing lines.
Algorithms
The characterization of squaregraphs in terms of distance from a root and links of vertices can be used together with breadth first search as part of a linear time algorithm for testing whether a given graph is a squaregraph, without any need to use the more complex linear-time algorithms for planarity testing of arbitrary graphs.[2]
Several algorithmic problems on squaregraphs may be computed more efficiently than in more general planar or median graphs; for instance, Chepoi, Dragan & Vaxès (2002) and Chepoi, Fanciullini & Vaxès (2004) present linear time algorithms for computing the diameter of squaregraphs, and for finding a vertex minimizing the maximum distance to all other vertices.
Notes
1. Soltan, Zambitskii & Prisǎcaru (1973). See Peterin (2006) for a discussion of planar median graphs more generally.
2. Bandelt, Chepoi & Eppstein (2010).
References
• Bandelt, Hans-Jürgen; Chepoi, Victor; Eppstein, David (2010), "Combinatorics and geometry of finite and infinite squaregraphs", SIAM Journal on Discrete Mathematics, 24 (4): 1399–1440, arXiv:0905.4537, doi:10.1137/090760301, S2CID 10788524.
• Chepoi, Victor; Dragan, Feodor; Vaxès, Yann (2002), "Center and diameter problem in planar quadrangulations and triangulations", Proc. 13th Annu. ACM–SIAM Symp. on Discrete Algorithms (SODA 2002), pp. 346–355.
• Chepoi, Victor; Fanciullini, Clémentine; Vaxès, Yann (2004), "Median problem in some plane triangulations and quadrangulations", Computational Geometry, 27 (3): 193–210, doi:10.1016/j.comgeo.2003.11.002.
• Peterin, Iztok (2006), "A characterization of planar median graphs", Discussiones Mathematicae Graph Theory, 26 (1): 41–48, doi:10.7151/dmgt.1299
• Soltan, P.; Zambitskii, D.; Prisǎcaru, C. (1973), Extremal Problems on Graphs and Algorithms of their Solution (in Russian), Chişinǎu, Moldova: Ştiinţa.
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Wikipedia
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Square pyramidal number
In mathematics, a pyramid number, or square pyramidal number, is a natural number that counts the number of stacked spheres in a pyramid with a square base. The study of these numbers goes back to Archimedes and Fibonacci. They are part of a broader topic of figurate numbers representing the numbers of points forming regular patterns within different shapes.
As well as counting spheres in a pyramid, these numbers can be described algebraically as a sum of the first $n$ positive square numbers, or as the values of a cubic polynomial. They can be used to solve several other counting problems, including counting squares in a square grid and counting acute triangles formed from the vertices of an odd regular polygon. They equal the sums of consecutive tetrahedral numbers, and are one-fourth of a larger tetrahedral number. The sum of two consecutive square pyramidal numbers is an octahedral number.
History
The pyramidal numbers were one of the few types of three-dimensional figurate numbers studied in Greek mathematics, in works by Nicomachus, Theon of Smyrna, and Iamblichus.[1] Formulas for summing consecutive squares to give a cubic polynomial, whose values are the square pyramidal numbers, are given by Archimedes, who used this sum as a lemma as part of a study of the volume of a cone,[2] and by Fibonacci, as part of a more general solution to the problem of finding formulas for sums of progressions of squares.[3] The square pyramidal numbers were also one of the families of figurate numbers studied by Japanese mathematicians of the wasan period, who named them "kirei saijō suida" (with modern kanji, 奇零 再乗 蓑深).[4]
The same problem, formulated as one of counting the cannonballs in a square pyramid, was posed by Walter Raleigh to mathematician Thomas Harriot in the late 1500s, while both were on a sea voyage. The cannonball problem, asking whether there are any square pyramidal numbers that are also square numbers other than 1 and 4900, is said to have developed out of this exchange. Édouard Lucas found the 4900-ball pyramid with a square number of balls, and in making the cannonball problem more widely known, suggested that it was the only nontrivial solution.[5] After incomplete proofs by Lucas and Claude-Séraphin Moret-Blanc, the first complete proof that no other such numbers exist was given by G. N. Watson in 1918.[6]
Formula
If spheres are packed into square pyramids whose number of layers is 1, 2, 3, etc., then the square pyramidal numbers giving the numbers of spheres in each pyramid are:[7][8]
1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506, 650, 819, ... .
These numbers can be calculated algebraically, as follows. If a pyramid of spheres is decomposed into its square layers with a square number of spheres in each, then the total number $P_{n}$ of spheres can be counted as the sum of the number of spheres in each square,
$P_{n}=\sum _{k=1}^{n}k^{2}=1+4+9+\cdots +n^{2},$
and this summation can be solved to give a cubic polynomial, which can be written in several equivalent ways:
$P_{n}={\frac {n(n+1)(2n+1)}{6}}={\frac {2n^{3}+3n^{2}+n}{6}}={\frac {n^{3}}{3}}+{\frac {n^{2}}{2}}+{\frac {n}{6}}.$
This equation for a sum of squares is a special case of Faulhaber's formula for sums of powers, and may be proved by mathematical induction.[9]
More generally, figurate numbers count the numbers of geometric points arranged in regular patterns within certain shapes. The centers of the spheres in a pyramid of spheres form one of these patterns, but for many other types of figurate numbers it does not make sense to think of the points as being centers of spheres.[8] In modern mathematics, related problems of counting points in integer polyhedra are formalized by the Ehrhart polynomials. These differ from figurate numbers in that, for Ehrhart polynomials, the points are always arranged in an integer lattice rather than having an arrangement that is more carefully fitted to the shape in question, and the shape they fit into is a polyhedron with lattice points as its vertices. Specifically, the Ehrhart polynomial L(P,t) of an integer polyhedron P is a polynomial that counts the number of integer points in a copy of P that is expanded by multiplying all its coordinates by the number t. The usual symmetric form of a square pyramid, with a unit square as its base, is not an integer polyhedron, because the topmost point of the pyramid, its apex, is not an integer point. Instead, the Ehrhart polynomial can be applied to an asymmetric square pyramid P with a unit square base and an apex that can be any integer point one unit above the base plane. For this choice of P, the Ehrhart polynomial of a pyramid is (t + 1)(t + 2)(2t + 3)/6 = Pt + 1.[10]
Geometric enumeration
As well as counting spheres in a pyramid, these numbers can be used to solve several other counting problems. For example, a common mathematical puzzle involves finding the number of squares in a large n by n square grid.[11] This number can be derived as follows:
• The number of 1 × 1 squares found in the grid is n2.
• The number of 2 × 2 squares found in the grid is (n − 1)2. These can be counted by counting all of the possible upper-left corners of 2 × 2 squares.
• The number of k × k squares (1 ≤ k ≤ n) found in the grid is (n − k + 1)2. These can be counted by counting all of the possible upper-left corners of k × k squares.
It follows that the number of squares in an n × n square grid is:[12]
$n^{2}+(n-1)^{2}+(n-2)^{2}+(n-3)^{2}+\ldots ={\frac {n(n+1)(2n+1)}{6}}.$
That is, the solution to the puzzle is given by the nth square pyramidal number.[7] The number of rectangles in a square grid is given by the squared triangular numbers.[13]
The square pyramidal number $P_{n}$ also counts the number of acute triangles formed from the vertices of a $(2n+1)$-sided regular polygon. For instance, an equilateral triangle contains only one acute triangle (itself), a regular pentagon has five acute golden triangles within it, a regular heptagon has 14 acute triangles of two shapes, etc.[7] More abstractly, when permutations of the rows or columns of a matrix are considered as equivalent, the number of $2\times 2$ matrices with non-negative integer coefficients summing to $n$, for odd values of $n$, is a square pyramidal number.[14]
Relations to other figurate numbers
The cannonball problem asks for the sizes of pyramids of cannonballs that can also be spread out to form a square array, or equivalently, which numbers are both square and square pyramidal. Besides 1, there is only one other number that has this property: 4900, which is both the 70th square number and the 24th square pyramidal number.[6]
The square pyramidal numbers can be expressed as sums of binomial coefficients:[15][16]
$P_{n}={\binom {n+2}{3}}+{\binom {n+1}{3}}.$
The binomial coefficients occurring in this representation are tetrahedral numbers, and this formula expresses a square pyramidal number as the sum of two tetrahedral numbers in the same way as square numbers are the sums of two consecutive triangular numbers.[8][15] If a tetrahedron is reflected across one of its faces, the two copies form a triangular bipyramid. The square pyramidal numbers are also the figurate numbers of the triangular bipyramids, and this formula can be interpreted as an equality between the square pyramidal numbers and the triangular bipyramidal numbers.[7] Analogously, reflecting a square pyramid across its base produces an octahedron, from which it follows that each octahedral number is the sum of two consecutive square pyramidal numbers.[17]
Square pyramidal numbers are also related to tetrahedral numbers in a different way: the points from four copies of the same square pyramid can be rearranged to form a single tetrahedron with twice as many points along each edge. That is,[18]
$4P_{n}=Te_{2n}={\binom {2n+2}{3}}.$
To see this, arrange each square pyramid so that each layer is directly above the previous layer, e.g. the heights are
4321
3321
2221
1111
Four of these can then be joined by the height 4 pillar to make an even square pyramid, with layers $4,16,36,\dots $.
Each layer is the sum of consecutive triangular numbers, i.e. $(1+3),(6+10),(15+21),\dots $, which, when totalled, sum to the tetrahedral number.
Other properties
The alternating series of unit fractions with the square pyramidal numbers as denominators is closely related to the Leibniz formula for π, although it converges more quickly. It is:[19]
${\begin{aligned}\sum _{i=1}^{\infty }&(-1)^{i-1}{\frac {1}{P_{i}}}\\&=1-{\frac {1}{5}}+{\frac {1}{14}}-{\frac {1}{30}}+{\frac {1}{55}}-{\frac {1}{91}}+{\frac {1}{140}}-{\frac {1}{204}}+\cdots \\&=6(\pi -3)\\&\approx 0.849556.\\\end{aligned}}$
In approximation theory, the sequences of odd numbers, sums of odd numbers (square numbers), sums of square numbers (square pyramidal numbers), etc., form the coefficients in a method for converting Chebyshev approximations into polynomials.[20]
References
1. Federico, Pasquale Joseph (1982), "Pyramidal numbers", Descartes on Polyhedra: A Study of the "De solidorum elementis", Sources in the History of Mathematics and Physical Sciences, vol. 4, Springer, pp. 89–91, doi:10.1007/978-1-4612-5759-2, ISBN 978-1-4612-5761-5
2. Archimedes, On Conoids and Spheroids, Lemma to Prop. 2, and On Spirals, Prop. 10. See "Lemma to Proposition 2", The Works of Archimedes, translated by T. L. Heath, Cambridge University Press, 1897, pp. 107–109
3. Fibonacci (1202), Liber Abaci, ch. II.12. See Fibonacci's Liber Abaci, translated by Laurence E. Sigler, Springer-Verlag, 2002, pp. 260–261, ISBN 0-387-95419-8
4. Yanagihara, Kitizi (November 1918), "On the Dajutu or the arithmetic series of higher orders as studied by wasanists", Tohoku Mathematical Journal, 14 (3–4): 305–324
5. Parker, Matt (2015), "Ship shape", Things to Make and Do in the Fourth Dimension: A Mathematician's Journey Through Narcissistic Numbers, Optimal Dating Algorithms, at Least Two Kinds of Infinity, and More, New York: Farrar, Straus and Giroux, pp. 56–59, ISBN 978-0-374-53563-6, MR 3753642
6. Anglin, W. S. (1990), "The square pyramid puzzle", The American Mathematical Monthly, 97 (2): 120–124, doi:10.1080/00029890.1990.11995558, JSTOR 2323911
7. Sloane, N. J. A. (ed.), "Sequence A000330 (Square pyramidal numbers)", The On-Line Encyclopedia of Integer Sequences, OEIS Foundation
8. Beiler, A. H. (1964), Recreations in the Theory of Numbers, Dover, pp. 194–195, ISBN 0-486-21096-0
9. Hopcroft, John E.; Motwani, Rajeev; Ullman, Jeffrey D. (2007), Introduction to Automata Theory, Languages, and Computation (3 ed.), Pearson/Addison Wesley, p. 20, ISBN 9780321455369
10. Beck, M.; De Loera, J. A.; Develin, M.; Pfeifle, J.; Stanley, R. P. (2005), "Coefficients and roots of Ehrhart polynomials", Integer Points in Polyhedra—Geometry, Number Theory, Algebra, Optimization, Contemporary Mathematics, vol. 374, Providence, Rhode Island, pp. 15–36, arXiv:math/0402148, MR 2134759{{citation}}: CS1 maint: location missing publisher (link)
11. Duffin, Janet; Patchett, Mary; Adamson, Ann; Simmons, Neil (November 1984), "Old squares new faces", Mathematics in School, 13 (5): 2–4, JSTOR 30216270
12. Robitaille, David F. (May 1974), "Mathematics and chess", The Arithmetic Teacher, 21 (5): 396–400, doi:10.5951/AT.21.5.0396, JSTOR 41190919
13. Stein, Robert G. (1971), "A combinatorial proof that $\textstyle \sum k^{3}=(\sum k)^{2}$", Mathematics Magazine, 44 (3): 161–162, doi:10.2307/2688231, JSTOR 2688231
14. Babcock, Ben; Van Tuyl, Adam (2013), "Revisiting the spreading and covering numbers", The Australasian Journal of Combinatorics, 56: 77–84, arXiv:1109.5847, MR 3097709
15. Conway, John H.; Guy, Richard (1998), "Square pyramid numbers", The Book of Numbers, Springer, pp. 47–49, ISBN 978-0-387-97993-9
16. Grassl, Richard (July 1995), "79.33 The squares do fit!", The Mathematical Gazette, 79 (485): 361–364, doi:10.2307/3618315, JSTOR 3618315, S2CID 187946568
17. Caglayan, Günhan; Buddoo, Horace (September 2014), "Tetrahedral numbers", The Mathematics Teacher, 108 (2): 92–97, doi:10.5951/mathteacher.108.2.0092, JSTOR 10.5951/mathteacher.108.2.0092
18. Alsina, Claudi; Nelsen, Roger B. (2015), "Challenge 2.13", A Mathematical Space Odyssey: Solid Geometry in the 21st Century, The Dolciani Mathematical Expositions, vol. 50, Washington, DC: Mathematical Association of America, pp. 43, 234, ISBN 978-0-88385-358-0, MR 3379535
19. Fearnehough, Alan (November 2006), "90.67 A series for the 'bit'", Notes, The Mathematical Gazette, 90 (519): 460–461, doi:10.1017/S0025557200180337, JSTOR 40378200, S2CID 113711266
20. Men'šikov, G. G.; Zaezdnyĭ, A. M. (1966), "Recurrence formulae simplifying the construction of approximating power polynomials", Žurnal Vyčislitel' noĭ Matematiki i Matematičeskoĭ Fiziki, 6: 360–363, MR 0196353; translated into English as Zaezdnyi, A. M.; Men'shikov, G. G. (January 1966), "Recurrence formulae simplifying the construction of approximating power polynomials", USSR Computational Mathematics and Mathematical Physics, 6 (2): 234–238, doi:10.1016/0041-5553(66)90072-3
External links
• Weisstein, Eric W., "Square Pyramidal Number", MathWorld
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Wikipedia
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Square wave
A square wave is a non-sinusoidal periodic waveform in which the amplitude alternates at a steady frequency between fixed minimum and maximum values, with the same duration at minimum and maximum. In an ideal square wave, the transitions between minimum and maximum are instantaneous.
Square wave
Sine, square, triangle, and sawtooth waveforms
General information
General definition$x(t)=4\left\lfloor t\right\rfloor -2\left\lfloor 2t\right\rfloor +1,2t\notin \mathbb {Z} $
Fields of applicationElectronics, synthesizers
Domain, Codomain and Image
Domain$\mathbb {R} \setminus \left\{{\tfrac {n}{2}}\right\},n\in \mathbb {Z} $
Codomain$\left\{-1,1\right\}$
Basic features
ParityOdd
Period1
AntiderivativeTriangle wave
Fourier series$x(t)={\frac {4}{\pi }}\sum _{k=1}^{\infty }{\frac {1}{2k-1}}\sin \left(2\pi \left(2k-1\right)t\right)$
The square wave is a special case of a pulse wave which allows arbitrary durations at minimum and maximum amplitudes. The ratio of the high period to the total period of a pulse wave is called the duty cycle. A true square wave has a 50% duty cycle (equal high and low periods).
Square waves are often encountered in electronics and signal processing, particularly digital electronics and digital signal processing. Its stochastic counterpart is a two-state trajectory.
Origin and uses
Square waves are universally encountered in digital switching circuits and are naturally generated by binary (two-level) logic devices. They are used as timing references or "clock signals", because their fast transitions are suitable for triggering synchronous logic circuits at precisely determined intervals. However, as the frequency-domain graph shows, square waves contain a wide range of harmonics; these can generate electromagnetic radiation or pulses of current that interfere with other nearby circuits, causing noise or errors. To avoid this problem in very sensitive circuits such as precision analog-to-digital converters, sine waves are used instead of square waves as timing references.
In musical terms, they are often described as sounding hollow, and are therefore used as the basis for wind instrument sounds created using subtractive synthesis. Additionally, the distortion effect used on electric guitars clips the outermost regions of the waveform, causing it to increasingly resemble a square wave as more distortion is applied.
Simple two-level Rademacher functions are square waves.
Definitions
The square wave in mathematics has many definitions, which are equivalent except at the discontinuities:
It can be defined as simply the sign function of a sinusoid:
${\begin{aligned}x(t)&=\operatorname {sgn} \left(\sin {\frac {2\pi t}{T}}\right)=\operatorname {sgn}(\sin 2\pi ft)\\v(t)&=\operatorname {sgn} \left(\cos {\frac {2\pi t}{T}}\right)=\operatorname {sgn}(\cos 2\pi ft),\end{aligned}}$
which will be 1 when the sinusoid is positive, −1 when the sinusoid is negative, and 0 at the discontinuities. Here, T is the period of the square wave and f is its frequency, which are related by the equation f = 1/T.
A square wave can also be defined with respect to the Heaviside step function u(t) or the rectangular function Π(t):
${\begin{aligned}x(t)&=2\left[\sum _{n=-\infty }^{\infty }\Pi \left({\frac {2(t-nT)}{T}}-{\frac {1}{2}}\right)\right]-1\\&=2\sum _{n=-\infty }^{\infty }\left[u\left({\frac {t}{T}}-n\right)-u\left({\frac {t}{T}}-n-{\frac {1}{2}}\right)\right]-1.\end{aligned}}$
A square wave can also be generated using the floor function directly:
$x(t)=2\left(2\lfloor ft\rfloor -\lfloor 2ft\rfloor \right)+1$
and indirectly:
$x(t)=\left(-1\right)^{\lfloor 2ft\rfloor }.$
Using the fourier series (below) one can show that the floor function may be written in trigonometric form[1]
${\frac {2}{\pi }}\arctan \left(\tan \left({\frac {\pi ft}{2}}\right)\right)+{\frac {2}{\pi }}\arctan \left(\cot \left({\frac {\pi ft}{2}}\right)\right)$
Fourier analysis
Using Fourier expansion with cycle frequency f over time t, an ideal square wave with an amplitude of 1 can be represented as an infinite sum of sinusoidal waves:
${\begin{aligned}x(t)&={\frac {4}{\pi }}\sum _{k=1}^{\infty }{\frac {\sin \left(2\pi (2k-1)ft\right)}{2k-1}}\\&={\frac {4}{\pi }}\left(\sin(\omega t)+{\frac {1}{3}}\sin(3\omega t)+{\frac {1}{5}}\sin(5\omega t)+\ldots \right),&{\text{where }}\omega =2\pi f.\end{aligned}}$
The ideal square wave contains only components of odd-integer harmonic frequencies (of the form 2π(2k − 1)f).
A curiosity of the convergence of the Fourier series representation of the square wave is the Gibbs phenomenon. Ringing artifacts in non-ideal square waves can be shown to be related to this phenomenon. The Gibbs phenomenon can be prevented by the use of σ-approximation, which uses the Lanczos sigma factors to help the sequence converge more smoothly.
An ideal mathematical square wave changes between the high and the low state instantaneously, and without under- or over-shooting. This is impossible to achieve in physical systems, as it would require infinite bandwidth.
Square waves in physical systems have only finite bandwidth and often exhibit ringing effects similar to those of the Gibbs phenomenon or ripple effects similar to those of the σ-approximation.
For a reasonable approximation to the square-wave shape, at least the fundamental and third harmonic need to be present, with the fifth harmonic being desirable. These bandwidth requirements are important in digital electronics, where finite-bandwidth analog approximations to square-wave-like waveforms are used. (The ringing transients are an important electronic consideration here, as they may go beyond the electrical rating limits of a circuit or cause a badly positioned threshold to be crossed multiple times.)
Characteristics of imperfect square waves
As already mentioned, an ideal square wave has instantaneous transitions between the high and low levels. In practice, this is never achieved because of physical limitations of the system that generates the waveform. The times taken for the signal to rise from the low level to the high level and back again are called the rise time and the fall time respectively.
If the system is overdamped, then the waveform may never actually reach the theoretical high and low levels, and if the system is underdamped, it will oscillate about the high and low levels before settling down. In these cases, the rise and fall times are measured between specified intermediate levels, such as 5% and 95%, or 10% and 90%. The bandwidth of a system is related to the transition times of the waveform; there are formulas allowing one to be determined approximately from the other.
See also
• List of periodic functions
• Rectangular function
• Pulse wave
• Sine wave
• Triangle wave
• Sawtooth wave
• Waveform
• Sound
• Multivibrator
• Ronchi ruling, a square-wave stripe target used in imaging.
• Cross sea
• Clarinet, a musical instrument that produces odd overtones approximating a square wave.
References
1. "Partial sum formula". www.wolframalpha.com. Archived from the original on 22 January 2023. Retrieved 9 July 2023.
External links
• Fourier decomposition of a square wave Interactive demo of square wave synthesis using sine waves, from GeoGebra site.
• Square Wave Approximated by Sines Interactive demo of square wave synthesis using sine waves.
• Flash applets Square wave.
Waveforms
• Sine wave
• Non-sinusoidal
• Rectangular wave
• Sawtooth wave
• Square wave
• Triangle wave
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Wikipedia
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Squeeze operator
In quantum physics, the squeeze operator for a single mode of the electromagnetic field is[1]
${\hat {S}}(z)=\exp \left({1 \over 2}(z^{*}{\hat {a}}^{2}-z{\hat {a}}^{\dagger 2})\right),\qquad z=r\,e^{i\theta }$
where the operators inside the exponential are the ladder operators. It is a unitary operator and therefore obeys $S(\zeta )S^{\dagger }(\zeta )=S^{\dagger }(\zeta )S(\zeta )={\hat {1}}$, where ${\hat {1}}$ is the identity operator.
Its action on the annihilation and creation operators produces
${\hat {S}}^{\dagger }(z){\hat {a}}{\hat {S}}(z)={\hat {a}}\cosh r-e^{i\theta }{\hat {a}}^{\dagger }\sinh r\qquad {\text{and}}\qquad {\hat {S}}^{\dagger }(z){\hat {a}}^{\dagger }{\hat {S}}(z)={\hat {a}}^{\dagger }\cosh r-e^{-i\theta }{\hat {a}}\sinh r$
The squeeze operator is ubiquitous in quantum optics and can operate on any state. For example, when acting upon the vacuum, the squeezing operator produces the squeezed vacuum state.
The squeezing operator can also act on coherent states and produce squeezed coherent states. The squeezing operator does not commute with the displacement operator:
${\hat {S}}(z){\hat {D}}(\alpha )\neq {\hat {D}}(\alpha ){\hat {S}}(z),$
nor does it commute with the ladder operators, so one must pay close attention to how the operators are used. There is, however, a simple braiding relation, ${\hat {D}}(\alpha ){\hat {S}}(z)={\hat {S}}(z){\hat {S}}^{\dagger }(z){\hat {D}}(\alpha ){\hat {S}}(z)={\hat {S}}(z){\hat {D}}(\gamma ),\qquad {\text{where}}\qquad \gamma =\alpha \cosh r+\alpha ^{*}e^{i\theta }\sinh r$ [2]
Application of both operators above on the vacuum produces squeezed coherent states:
${\hat {D}}(\alpha ){\hat {S}}(r)|0\rangle =|\alpha ,r\rangle $.
Derivation of action on creation operator
As mentioned above, the action of the squeeze operator $S(z)$ on the annihilation operator $a$ can be written as
$S^{\dagger }(z)aS(z)=\cosh(|z|)a-{\frac {z}{|z|}}\sinh(|z|)a^{\dagger }.$
To derive this equality, let us define the (skew-Hermitian) operator $A\equiv (za^{\dagger 2}-z^{*}a^{2})/2$, so that $S^{\dagger }=e^{A}$. The left hand side of the equality is thus $e^{A}ae^{-A}$. We can now make use of the general equality
$e^{A}Be^{-A}=\sum _{k=0}^{\infty }{\frac {1}{k!}}[\underbrace {A,[A,\dots ,[A} _{k\,{\text{times}}},B]\dots ]],$
which holds true for any pair of operators $A$ and $B$. To compute $e^{A}ae^{-A}$ thus reduces to the problem of computing the repeated commutators between $A$ and $a$. As can be readily verified, we have
$[A,a]={\frac {1}{2}}[za^{\dagger 2}-z^{*}a^{2},a]={\frac {z}{2}}[a^{\dagger 2},a]=-za^{\dagger },$
$[A,a^{\dagger }]={\frac {1}{2}}[za^{\dagger 2}-z^{*}a^{2},a^{\dagger }]=-{\frac {z^{*}}{2}}[a^{2},a^{\dagger }]=-z^{*}a.$
Using these equalities, we obtain
$[\underbrace {A,[A,\dots ,[A} _{n},a]\dots ]]={\begin{cases}|z|^{n}a,&{\text{ for }}n{\text{ even}},\\-z|z|^{n-1}a^{\dagger },&{\text{ for }}n{\text{ odd}}.\end{cases}}$
so that finally we get
$e^{A}ae^{-A}=a\sum _{k=0}^{\infty }{\frac {|z|^{2k}}{(2k)!}}-a^{\dagger }{\frac {z}{|z|}}\sum _{k=0}^{\infty }{\frac {|z|^{2k+1}}{(2k+1)!}}=a\cosh |z|-a^{\dagger }e^{i\theta }\sinh |z|.$
See also
• Squeezed coherent state
References
1. Gerry, C.C. & Knight, P.L. (2005). Introductory quantum optics. Cambridge University Press. p. 182. ISBN 978-0-521-52735-4.
2. M. M. Nieto and D. Truax (1995), Nieto, Michael Martin; Truax, D. Rodney (1997). "Holstein‐Primakoff/Bogoliubov Transformations and the Multiboson System". Fortschritte der Physik/Progress of Physics. 45 (2): 145–156. arXiv:quant-ph/9506025. doi:10.1002/prop.2190450204. S2CID 14213781. Eqn (15). Note that in this reference, the definition of the squeeze operator (eqn. 12) differs by a minus sign inside the exponential, therefore the expression of $\gamma $ is modified accordingly ($\theta \rightarrow \theta +\pi $).
Operators in physics
General
Space and time
• d'Alembertian
• Parity
• Time
Particles
• C-symmetry
Operators for operators
• Anti-symmetric operator
• Ladder operator
Quantum
Fundamental
• Momentum
• Position
• Rotation
Energy
• Total energy
• Hamiltonian
• Kinetic energy
Angular momentum
• Total
• Orbital
• Spin
Electromagnetism
• Transition dipole moment
Optics
• Displacement
• Hanbury Brown and Twiss effect
• Quantum correlator
• Squeeze
Particle physics
• Casimir invariant
• Creation and annihilation
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Wikipedia
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Squircle
A squircle is a shape intermediate between a square and a circle. There are at least two definitions of "squircle" in use, the most common of which is based on the superellipse. The word "squircle" is a portmanteau of the words "square" and "circle". Squircles have been applied in design and optics.
Superellipse-based squircle
In a Cartesian coordinate system, the superellipse is defined by the equation
$\left|{\frac {x-a}{r_{a}}}\right|^{n}+\left|{\frac {y-b}{r_{b}}}\right|^{n}=1,$
where ra and rb are the semi-major and semi-minor axes, a and b are the x and y coordinates of the centre of the ellipse, and n is a positive number. The squircle is then defined as the superellipse with ra = rb and n = 4. Its equation is:[1]
$\left(x-a\right)^{4}+\left(y-b\right)^{4}=r^{4}$
where r is the minor radius of the squircle. Compare this to the equation of a circle. When the squircle is centred at the origin, then a = b = 0, and it is called Lamé's special quartic.
The area inside the squircle can be expressed in terms of the gamma function Γ as[1]
$\mathrm {Area} =4r^{2}{\frac {\left(\operatorname {\Gamma } \left(1+{\frac {1}{4}}\right)\right)^{2}}{\operatorname {\Gamma } \left(1+{\frac {2}{4}}\right)}}={\frac {8r^{2}\left(\operatorname {\Gamma } \left({\frac {5}{4}}\right)\right)^{2}}{\sqrt {\pi }}}=\varpi {\sqrt {2}}\,r^{2}\approx 3.708149\,r^{2},$
where r is the minor radius of the squircle, and $\varpi $ is the lemniscate constant.
p-norm notation
In terms of the p-norm ‖ · ‖p on R2, the squircle can be expressed as:
$\left\|\mathbf {x} -\mathbf {x} _{c}\right\|_{p}=r$
where p = 4, xc = (a, b) is the vector denoting the centre of the squircle, and x = (x, y). Effectively, this is still a "circle" of points at a distance r from the centre, but distance is defined differently. For comparison, the usual circle is the case p = 2, whereas the square is given by the p → ∞ case (the supremum norm), and a rotated square is given by p = 1 (the taxicab norm). This allows a straightforward generalization to a spherical cube, or sphube, in R3, or hypersphubes in higher dimensions.[2]
Fernández-Guasti squircle
Another squircle comes from work in optics.[3][4] It may be called the Fernández-Guasti squircle, after one of its authors, to distinguish it from the superellipse-related squircle above.[2] This kind of squircle, centred at the origin, can be defined by the equation:
$x^{2}+y^{2}-{\frac {s^{2}}{r^{2}}}x^{2}y^{2}=r^{2}$
where r is the minor radius of the squircle, s is the squareness parameter, and x and y are in the interval [−r, r]. If s = 0, the equation is a circle; if s = 1, this is a square. This equation allows a smooth parametrization of the transition from a circle to a square, without involving infinity.
Similar shapes
A shape similar to a squircle, called a rounded square, may be generated by separating four quarters of a circle and connecting their loose ends with straight lines, or by separating the four sides of a square and connecting them with quarter-circles. Such a shape is very similar but not identical to the squircle. Although constructing a rounded square may be conceptually and physically simpler, the squircle has the simpler equation and can be generalised much more easily. One consequence of this is that the squircle and other superellipses can be scaled up or down quite easily. This is useful where, for example, one wishes to create nested squircles.
Another similar shape is a truncated circle, the boundary of the intersection of the regions enclosed by a square and by a concentric circle whose diameter is both greater than the length of the side of the square and less than the length of the diagonal of the square (so that each figure has interior points that are not in the interior of the other). Such shapes lack the tangent continuity possessed by both superellipses and rounded squares.
A rounded cube can be defined in terms of superellipsoids.
Uses
Squircles are useful in optics. If light is passed through a two-dimensional square aperture, the central spot in the diffraction pattern can be closely modelled by a squircle or supercircle. If a rectangular aperture is used, the spot can be approximated by a superellipse.[4]
Squircles have also been used to construct dinner plates. A squircular plate has a larger area (and can thus hold more food) than a circular one with the same radius, but still occupies the same amount of space in a rectangular or square cupboard.[5]
Many Nokia phone models have been designed with a squircle-shaped touchpad button,[6][7] as was the second generation Microsoft Zune.[8] Apple uses an approximation of a squircle (actually a quintic superellipse) for icons in iOS, iPadOS, macOS, and the home buttons of some Apple hardware.[9] One of the shapes for adaptive icons introduced in the Android "Oreo" operating system is a squircle.[10] Samsung uses squircle-shaped icons in their Android software overlay One UI, and in Samsung Experience and TouchWiz.[11]
Italian car manufacturer Fiat used numerous squircles in the interior and exterior design of the third generation Panda.[12]
See also
• Astroid
• Ellipse
• Ellipsoid
• Lp spaces
• Oval
• Squround
• Superegg
References
1. Weisstein, Eric W. "Squircle". MathWorld.
2. Chamberlain Fong (2016). "Squircular Calculations". arXiv:1604.02174. Bibcode:2016arXiv160402174F. {{cite journal}}: Cite journal requires |journal= (help)
3. M. Fernández Guasti (1992). "Analytic Geometry of Some Rectilinear Figures". Int. J. Educ. Sci. Technol. 23: 895–901.
4. M. Fernández Guasti; A. Meléndez Cobarrubias; F.J. Renero Carrillo; A. Cornejo Rodríguez (2005). "LCD pixel shape and far-field diffraction patterns" (PDF). Optik. 116 (6): 265–269. Bibcode:2005Optik.116..265F. doi:10.1016/j.ijleo.2005.01.018. Retrieved 20 November 2006.
5. "Squircle Plate". Kitchen Contraptions. Archived from the original on 1 November 2006. Retrieved 20 November 2006.
6. Nokia Designer Mark Delaney mentions the squircle in a video regarding classic Nokia phone designs:
Nokia 6700 – The little black dress of phones. Archived from the original on 6 January 2010. Retrieved 9 December 2009. See 3:13 in video
7. "Clayton Miller evaluates shapes on mobile phone platforms". Retrieved 2 July 2011.
8. Marsal, Katie. "Microsoft discontinues hard drives, "squircle" from Zune lineup". Apple Insider. Retrieved 25 August 2022.
9. "The Hunt for the Squircle". Retrieved 23 May 2022.
10. "Adaptive Icons". Retrieved 15 January 2018.
11. "OneUI". Samsung Developers. Retrieved 2022-04-14.
12. "PANDA DESIGN STORY" (PDF). Retrieved 30 December 2018.
External links
Wikimedia Commons has media related to Squircle.
• What is the area of a Squircle? on YouTube by Matt Parker
• Online Calculator for supercircle and super-ellipse
• Web based supercircle generator
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Srinivasa Ramanujan Medal
The Srinivasa Ramanujan Medal, named after the Indian mathematician Srinivasa Ramanujan, is awarded by the Indian National Science Academy for work in the mathematical sciences.
Past recipients include:
• 1962 S. Chandrasekhar
• 1964 B. P. Pal
• 1966 K. S. Chandrasekharan
• 1968 P. C. Mahalanobis
• 1972 G. N. Ramachandran
• 1974 Harish-Chandra
• 1979 R. P. Bambah
• 1982 S. Chowla
• 1985 C. S. Seshadri
• 1988 M. S. Narasimhan
• 1991 M. S. Raghunathan
• 1997 K. Ramachandra
• 2003 C. R. Rao
• 2006 R. Parimala
• 2008 S. Ramanan[1]
• 2013 K. R. Parthasarathy
• 2016 Tyakal Nanjundiah Venkataramana
• 2019 K. B. Sinha
See also
• List of mathematics awards
References
1. "Sundararaman Ramanan". Chennai Mathematical Institute. Retrieved 17 April 2018.
• Srinivasa Ramanujan Medal recipients
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Cantellated 6-orthoplexes
In six-dimensional geometry, a cantellated 6-orthoplex is a convex uniform 6-polytope, being a cantellation of the regular 6-orthoplex.
6-orthoplex
Cantellated 6-orthoplex
Bicantellated 6-orthoplex
6-cube
Cantellated 6-cube
Bicantellated 6-cube
Cantitruncated 6-orthoplex
Bicantitruncated 6-orthoplex
Bicantitruncated 6-cube
Cantitruncated 6-cube
Orthogonal projections in B6 Coxeter plane
There are 8 cantellation for the 6-orthoplex including truncations. Half of them are more easily constructed from the dual 5-cube
Cantellated 6-orthoplex
Cantellated 6-orthoplex
Typeuniform 6-polytope
Schläfli symbolt0,2{3,3,3,3,4}
rr{3,3,3,3,4}
Coxeter-Dynkin diagrams
=
5-faces136
4-faces1656
Cells5040
Faces6400
Edges3360
Vertices480
Vertex figure
Coxeter groupsB6, [3,3,3,3,4]
D6, [33,1,1]
Propertiesconvex
Alternate names
• Cantellated hexacross
• Small rhombated hexacontatetrapeton (acronym: srog) (Jonathan Bowers)[1]
Construction
There are two Coxeter groups associated with the cantellated 6-orthoplex, one with the B6 or [4,3,3,3,3] Coxeter group, and a lower symmetry with the D6 or [33,1,1] Coxeter group.
Coordinates
Cartesian coordinates for the 480 vertices of a cantellated 6-orthoplex, centered at the origin, are all the sign and coordinate permutations of
(2,1,1,0,0,0)
Images
orthographic projections
Coxeter plane B6 B5 B4
Graph
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]
Bicantellated 6-orthoplex
Bicantellated 6-orthoplex
Typeuniform 6-polytope
Schläfli symbolt1,3{3,3,3,3,4}
2rr{3,3,3,3,4}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges8640
Vertices1440
Vertex figure
Coxeter groupsB6, [3,3,3,3,4]
D6, [33,1,1]
Propertiesconvex
Alternate names
• Bicantellated hexacross, bicantellated hexacontatetrapeton
• Small birhombated hexacontatetrapeton (acronym: siborg) (Jonathan Bowers)[2]
Construction
There are two Coxeter groups associated with the bicantellated 6-orthoplex, one with the B6 or [4,3,3,3,3] Coxeter group, and a lower symmetry with the D6 or [33,1,1] Coxeter group.
Coordinates
Cartesian coordinates for the 1440 vertices of a bicantellated 6-orthoplex, centered at the origin, are all the sign and coordinate permutations of
(2,2,1,1,0,0)
Images
orthographic projections
Coxeter plane B6 B5 B4
Graph
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]
Cantitruncated 6-orthoplex
Cantitruncated 6-orthoplex
Typeuniform 6-polytope
Schläfli symbolt0,1,2{3,3,3,3,4}
tr{3,3,3,3,4}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges3840
Vertices960
Vertex figure
Coxeter groupsB6, [3,3,3,3,4]
D6, [33,1,1]
Propertiesconvex
Alternate names
• Cantitruncated hexacross, cantitruncated hexacontatetrapeton
• Great rhombihexacontatetrapeton (acronym: grog) (Jonathan Bowers)[3]
Construction
There are two Coxeter groups associated with the cantitruncated 6-orthoplex, one with the B6 or [4,3,3,3,3] Coxeter group, and a lower symmetry with the D6 or [33,1,1] Coxeter group.
Coordinates
Cartesian coordinates for the 960 vertices of a cantitruncated 6-orthoplex, centered at the origin, are all the sign and coordinate permutations of
(3,2,1,0,0,0)
Images
orthographic projections
Coxeter plane B6 B5 B4
Graph
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]
Bicantitruncated 6-orthoplex
Bicantitruncated 6-orthoplex
Typeuniform 6-polytope
Schläfli symbolt1,2,3{3,3,3,3,4}
2tr{3,3,3,3,4}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges10080
Vertices2880
Vertex figure
Coxeter groupsB6, [3,3,3,3,4]
D6, [33,1,1]
Propertiesconvex
Alternate names
• Bicantitruncated hexacross, bicantitruncated hexacontatetrapeton
• Great birhombihexacontatetrapeton (acronym: gaborg) (Jonathan Bowers)[4]
Construction
There are two Coxeter groups associated with the bicantitruncated 6-orthoplex, one with the B6 or [4,3,3,3,3] Coxeter group, and a lower symmetry with the D6 or [33,1,1] Coxeter group.
Coordinates
Cartesian coordinates for the 2880 vertices of a bicantitruncated 6-orthoplex, centered at the origin, are all the sign and coordinate permutations of
(3,3,2,1,0,0)
Images
orthographic projections
Coxeter plane B6 B5 B4
Graph
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]
Related polytopes
These polytopes are part of a set of 63 uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex.
B6 polytopes
β6
t1β6
t2β6
t2γ6
t1γ6
γ6
t0,1β6
t0,2β6
t1,2β6
t0,3β6
t1,3β6
t2,3γ6
t0,4β6
t1,4γ6
t1,3γ6
t1,2γ6
t0,5γ6
t0,4γ6
t0,3γ6
t0,2γ6
t0,1γ6
t0,1,2β6
t0,1,3β6
t0,2,3β6
t1,2,3β6
t0,1,4β6
t0,2,4β6
t1,2,4β6
t0,3,4β6
t1,2,4γ6
t1,2,3γ6
t0,1,5β6
t0,2,5β6
t0,3,4γ6
t0,2,5γ6
t0,2,4γ6
t0,2,3γ6
t0,1,5γ6
t0,1,4γ6
t0,1,3γ6
t0,1,2γ6
t0,1,2,3β6
t0,1,2,4β6
t0,1,3,4β6
t0,2,3,4β6
t1,2,3,4γ6
t0,1,2,5β6
t0,1,3,5β6
t0,2,3,5γ6
t0,2,3,4γ6
t0,1,4,5γ6
t0,1,3,5γ6
t0,1,3,4γ6
t0,1,2,5γ6
t0,1,2,4γ6
t0,1,2,3γ6
t0,1,2,3,4β6
t0,1,2,3,5β6
t0,1,2,4,5β6
t0,1,2,4,5γ6
t0,1,2,3,5γ6
t0,1,2,3,4γ6
t0,1,2,3,4,5γ6
Notes
1. Klitzing, (x3o3x3o3o4o - srog)
2. Klitzing, (o3x3o3x3o4o - siborg)
3. Klitzing, (x3x3x3o3o4o - grog)
4. Klitzing, (o3x3x3x3o4o - gaborg)
References
• H.S.M. Coxeter:
• H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
• (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
• (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
• Klitzing, Richard. "6D uniform polytopes (polypeta)". x3o3x3o3o4o - srog, o3x3o3x3o4o - siborg, x3x3x3o3o4o - grog, o3x3x3x3o4o - gaborg
External links
• Polytopes of Various Dimensions
• Multi-dimensional Glossary
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
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Small rhombihexahedron
In geometry, the small rhombihexahedron (or small rhombicube) is a nonconvex uniform polyhedron, indexed as U18. It has 18 faces (12 squares and 6 octagons), 48 edges, and 24 vertices. Its vertex figure is an antiparallelogram.
Small rhombihexahedron
TypeUniform star polyhedron
ElementsF = 18, E = 48
V = 24 (χ = −6)
Faces by sides12{4}+6{8}
Coxeter diagram (with extra double-covered triangles)
(with extra double-covered squares)
Wythoff symbol2 4 (3/2 4/2) |
Symmetry groupOh, [4,3], *432
Index referencesU18, C60, W86
Dual polyhedronSmall rhombihexacron
Vertex figure
4.8.4/3.8/7
Bowers acronymSroh
Related polyhedra
This polyhedron shares the vertex arrangement with the stellated truncated hexahedron. It additionally shares its edge arrangement with the convex rhombicuboctahedron (having 12 square faces in common) and with the small cubicuboctahedron (having the octagonal faces in common).
Rhombicuboctahedron
Small cubicuboctahedron
Small rhombihexahedron
Stellated truncated hexahedron
It may be constructed as the exclusive or (blend) of three octagonal prisms.
External links
• Eric W. Weisstein, Small rhombihexahedron (Uniform polyhedron) at MathWorld.
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Srđan Ognjanović
Srđan Ognjanović (Serbian Cyrillic: Срђан Огњановић, English alternatives: Srdjan Ognjanovic, and Srdan Ognjanovic) is a Serbian mathematician. He was a principal of Mathematical Grammar School in Belgrade.
Srđan Ognjanović
Срђан Огњановић
Incumbent
Assumed office
2008
Preceded byVladimir Dragović, dr sci in mathematics, Faculty of Mathematics of University of Belgrade, and Faculty of Mechanics and Mathematics, Department for Higher Geometry and Topology, Moscow State University
Personal details
Alma materUniversity of Belgrade
Career
He received his degrees in the field of Mathematical Sciences from the Faculty of Mathematics and Natural Sciences, University of Belgrade. Prior to that, Ognjanović was a student of Mathematical Gymnasium Belgrade, from which he graduated in 1972, in A-division.[1] Ognjanović started his professional career as a teacher of mathematics at Mathematical Gymnasium Belgrade (Serbian: "Matematička Gimnazija") while still a student of mathematics at Faculty of Mathematics and Natural Sciences, University of Belgrade, continued his career after graduation also in Mathematical Gymnasium Belgrade, and devoted his career to teaching mathematics in the same school, now being a professor in Mathematical Gymnasium Belgrade for more than 30 years.
Students of professor Ognjanović won numerous prizes at International Science Olympiads in Mathematics, Physics, Informatics, Astronomy, Astrophysics, and Earth Sciences, also at other prestigious competitions around the world, and, accordingly, won many full scholarships at top-ranked universities.[2][3]
Mr Ognjanović is the author of numerous books and collections of problems for elementary and secondary schools, as well as special collections of assignments for preparation for mathematics competitions and mathematics workbooks used as a preparation for admission to faculties.[4]
Awards and legacy
Ognjanović was listed among "300 most powerful people in Serbia" in a list published annually by "Blic" daily newspaper (14 February 2011), member of Axel Springer AG. The criteria were easiness in achieving goals, public awareness, financial and political influence, personal integrity and authority, respectiveness of the institution the person represents, and personal charisma.
Among latest awards for his published works Mr Ognjanović received (in 2010): Grand Prize at 16th International Book Fair, in Novi Sad, from a Business Chamber of Vojvodina,[5] and "Stojan Novaković" Prize for the best textbook and set of textbooks published by Zavod - Serbian State Company of Textbooks.[6]
References
1. Математичка гимназијa - бивши ученици - list of former students Archived 2011-01-28 at the Wayback Machine (in Serbian)
2. Achievements and awards
3. International competitions - last 5 years
4. Virtual Library of Serbia - COBISS-OPAC Search Engine; search string: "Srđan Ognjanović"
5. Grand Prize - Business Chamber of Vojvodina (in Serbian)
6. Prize for the best textbook and set of textbooks published by Zavod - Serbian State Company of Textbooks. (in Serbian)
External links
Wikimedia Commons has media related to Srđan Ognjanović.
• Mathematical Gymnasium Belgrade homepage
• Principal of Mathematical Gymnasium Belgrade homepage
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st-planar graph
In graph theory, an st-planar graph is a bipolar orientation of a plane graph for which both the source and the sink of the orientation are on the outer face of the graph. That is, it is a directed graph drawn without crossings in the plane, in such a way that there are no directed cycles in the graph, exactly one graph vertex has no incoming edges, exactly one graph vertex has no outgoing edges, and these two special vertices both lie on the outer face of the graph.[1]
Within the drawing, each face of the graph must have the same structure: there is one vertex that acts as the source of the face, one vertex that acts as the sink of the face, and all edges within the face are directed along two paths from the source to the sink. If one draws an additional edge from the sink of an st-planar graph back to the source, through the outer face, and then constructs the dual graph (oriented each dual edge clockwise with respect to its primal edge) then the result is again an st-planar graph, augmented with an extra edge in the same way.[1]
Order theory
These graphs are closely related to partially ordered sets and lattices. The Hasse diagram of a partially ordered set is a directed acyclic graph whose vertices are the set elements, with an edge from x to y for each pair x, y of elements for which x ≤ y in the partial order but for which there does not exist z with x ≤ y ≤ z. A partially ordered set forms a complete lattice if and only if every subset of elements has a unique greatest lower bound and a unique least upper bound, and the order dimension of a partially ordered set is the least number of total orders on the same set of elements whose intersection is the given partial order. If the vertices of an st-planar graph are partially ordered by reachability, then this ordering always forms a two-dimensional complete lattice, whose Hasse diagram is the transitive reduction of the given graph. Conversely, the Hasse diagram of every two-dimensional complete lattice is always an st-planar graph.[2]
Graph drawing
Based on this two-dimensional partial order property, every st-planar graph can be given a dominance drawing, in which for every two vertices u and v there exists a path from u to v if and only if both coordinates of u are smaller than the corresponding coordinates of v.[3] The coordinates of such a drawing may also be used as a data structure that can be used to test whether one vertex of an st-planar graph can reach another in constant time per query. Rotating such a drawing by 45° gives an upward planar drawing of the graph. A directed acyclic graph G has an upward planar drawing if and only if G is a subgraph of an st-planar graph.[4]
References
1. Di Battista, Giuseppe; Eades, Peter; Tamassia, Roberto; Tollis, Ioannis G. (1998), "4.2 Properties of Planar Acyclic Digraphs", Graph Drawing: Algorithms for the Visualization of Graphs, Prentice Hall, pp. 89–96, ISBN 978-0-13-301615-4.
2. Platt, C. R. (1976), "Planar lattices and planar graphs", Journal of Combinatorial Theory, Ser. B, 21 (1): 30–39, doi:10.1016/0095-8956(76)90024-1.
3. Di Battista et al. (1998), 4.7 Dominance Drawings, pp. 112–127.
4. Di Battista, Giuseppe; Tamassia, Roberto (1988), "Algorithms for plane representations of acyclic digraphs", Theoretical Computer Science, 61 (2–3): 175–198, doi:10.1016/0304-3975(88)90123-5.
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Stål Aanderaa
Stål Aanderaa (born 1 February 1931) is a Norwegian mathematician.
Stål Aanderaa
Born (1931-02-01) February 1, 1931
Beitstad
OccupationMathematician
Years active1959-
Biography
Aanderaa was born in Beitstad. He completed the mag.scient. degree in 1959 and his doctorate at Harvard University in 1966. He was a professor at the University of Oslo from 1978 to his retirement in 2001.[1]
Aanderaa is a member of the Norwegian Academy of Science and Letters.[2]
Work
Aanderaa is one of the namesakes of the Aanderaa–Karp–Rosenberg conjecture.
References
1. Henriksen, Petter, ed. (2007). "Stål Aanderaa". Store norske leksikon (in Norwegian). Oslo: Kunnskapsforlaget. Retrieved 29 October 2009.
2. "Gruppe 1: Matematiske fag" (in Norwegian). Norwegian Academy of Science and Letters. Retrieved 28 October 2009.
Authority control: Academics
• DBLP
• MathSciNet
• Mathematics Genealogy Project
• zbMATH
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Stability (algebraic geometry)
In mathematics, and especially algebraic geometry, stability is a notion which characterises when a geometric object, for example a point, an algebraic variety, a vector bundle, or a sheaf, has some desirable properties for the purpose of classifying them. The exact characterisation of what it means to be stable depends on the type of geometric object, but all such examples share the property of having a minimal amount of internal symmetry, that is such stable objects have few automorphisms. This is related to the concept of simplicity in mathematics, which measures when some mathematical object has few subobjects inside it (see for example simple groups, which have no non-trivial normal subgroups). In addition to stability, some objects may be described with terms such as semi-stable (having a small but not minimal amount of symmetry), polystable (being made out of stable objects), or unstable (having too much symmetry, the opposite of stable).
Background
In many areas of mathematics, and indeed within geometry itself, it is often very desirable to have highly symmetric objects, and these objects are often regarded as aesthetically pleasing. However, high amounts of symmetry are not desirable when one is attempting to classify geometric objects by constructing moduli spaces of them, because the symmetries of these objects cause the formation of singularities, and obstruct the existence of universal families.
The concept of stability was first introduced in its modern form by David Mumford in 1965 in the context of geometric invariant theory, a theory which explains how to take quotients of algebraic varieties by group actions, and obtain a quotient space that is still an algebraic variety, a so-called categorical quotient.[1] However the ideas behind Mumford's work go back to the invariant theory of David Hilbert in 1893, and the fundamental concepts involved date back even to the work of Bernhard Riemann on constructing moduli spaces of Riemann surfaces.[2] Since the work of Mumford, stability has appeared in many forms throughout algebraic geometry, often with various notions of stability either derived from geometric invariant theory, or inspired by it. A completely general theory of stability does not exist (although one attempt to form such a theory is Bridgeland stability), and this article serves to summarise and compare the different manifestations of stability in geometry and the relations between them.
In addition to its use in classification and forming quotients in algebraic geometry, stability also finds significant use in differential geometry and geometric analysis, due to the general principle which states that stable algebraic geometric objects correspond to extremal differential geometric objects. Here extremal is generally meant in the sense of the calculus of variations, in that such objects minimize some functional. The prototypical example of this principle is the Kempf–Ness theorem, which relates GIT quotients to symplectic quotients by showing that stable points minimize the energy functional of the moment map. Due to this general principle, stability has found use as a key tool in constructing the existence of solutions to many important partial differential equations in geometry, such as the Yang–Mills equations and the Kähler–Einstein equations. More examples of this correspondence in action include Kobayashi–Hitchin correspondence, the nonabelian Hodge correspondence, the Yau–Tian–Donaldson conjecture for Kähler–Einstein manifolds, and even the uniformization theorem.
Stability conditions
• Gieseker stability
• Slope stability
• Bridgeland stability
• K-stability
References
1. Mumford, D., Fogarty, J. and Kirwan, F., 1994. Geometric invariant theory (Vol. 34). Springer Science & Business Media.
2. Hilbert, D., 1893. Ueber die vollen Invariantensysteme. Mathematische Annalen, 42(3), pp.313-373.
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Stability spectrum
In model theory, a branch of mathematical logic, a complete first-order theory T is called stable in λ (an infinite cardinal number), if the Stone space of every model of T of size ≤ λ has itself size ≤ λ. T is called a stable theory if there is no upper bound for the cardinals κ such that T is stable in κ. The stability spectrum of T is the class of all cardinals κ such that T is stable in κ.
For countable theories there are only four possible stability spectra. The corresponding dividing lines are those for total transcendentality, superstability and stability. This result is due to Saharon Shelah, who also defined stability and superstability.
The stability spectrum theorem for countable theories
Theorem. Every countable complete first-order theory T falls into one of the following classes:
• T is stable in λ for all infinite cardinals λ—T is totally transcendental.
• T is stable in λ exactly for all cardinals λ with λ ≥ 2ω—T is superstable but not totally transcendental.
• T is stable in λ exactly for all cardinals λ that satisfy λ = λω—T is stable but not superstable.
• T is not stable in any infinite cardinal λ—T is unstable.
The condition on λ in the third case holds for cardinals of the form λ = κω, but not for cardinals λ of cofinality ω (because λ < λcof λ).
Totally transcendental theories
Main article: Totally transcendental theory
A complete first-order theory T is called totally transcendental if every formula has bounded Morley rank, i.e. if RM(φ) < ∞ for every formula φ(x) with parameters in a model of T, where x may be a tuple of variables. It is sufficient to check that RM(x=x) < ∞, where x is a single variable.
For countable theories total transcendence is equivalent to stability in ω, and therefore countable totally transcendental theories are often called ω-stable for brevity. A totally transcendental theory is stable in every λ ≥ |T|, hence a countable ω-stable theory is stable in all infinite cardinals.
Every uncountably categorical countable theory is totally transcendental. This includes complete theories of vector spaces or algebraically closed fields. The theories of groups of finite Morley rank are another important example of totally transcendental theories.
Superstable theories
Main article: Superstable theory
A complete first-order theory T is superstable if there is a rank function on complete types that has essentially the same properties as Morley rank in a totally transcendental theory. Every totally transcendental theory is superstable. A theory T is superstable if and only if it is stable in all cardinals λ ≥ 2|T|.
Stable theories
Main article: Stable theory
A theory that is stable in one cardinal λ ≥ |T| is stable in all cardinals λ that satisfy λ = λ|T|. Therefore a theory is stable if and only if it is stable in some cardinal λ ≥ |T|.
Unstable theories
Most mathematically interesting theories fall into this category, including complicated theories such as any complete extension of ZF set theory, and relatively tame theories such as the theory of real closed fields. This shows that the stability spectrum is a relatively blunt tool. To get somewhat finer results one can look at the exact cardinalities of the Stone spaces over models of size ≤ λ, rather than just asking whether they are at most λ.
The uncountable case
For a general stable theory T in a possibly uncountable language, the stability spectrum is determined by two cardinals κ and λ0, such that T is stable in λ exactly when λ ≥ λ0 and λμ = λ for all μ<κ. So λ0 is the smallest infinite cardinal for which T is stable. These invariants satisfy the inequalities
• κ ≤ |T|+
• κ ≤ λ0
• λ0 ≤ 2|T|
• If λ0 > |T|, then λ0 ≥ 2ω
When |T| is countable the 4 possibilities for its stability spectrum correspond to the following values of these cardinals:
• κ and λ0 are not defined: T is unstable.
• λ0 is 2ω, κ is ω1: T is stable but not superstable
• λ0 is 2ω, κ is ω: T is superstable but not ω-stable.
• λ0 is ω, κ is ω: T is totally transcendental (or ω-stable)
See also
• Spectrum of a theory
References
• Poizat, Bruno (2000), A course in model theory. An introduction to contemporary mathematical logic, Universitext, New York: Springer, pp. xxxii+443, ISBN 0-387-98655-3, MR 1757487 Translated from the French
• Shelah, Saharon (1990) [1978], Classification theory and the number of nonisomorphic models, Studies in Logic and the Foundations of Mathematics (2nd ed.), Elsevier, ISBN 978-0-444-70260-9
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Wikipedia
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Stability theory
In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. The heat equation, for example, is a stable partial differential equation because small perturbations of initial data lead to small variations in temperature at a later time as a result of the maximum principle. In partial differential equations one may measure the distances between functions using Lp norms or the sup norm, while in differential geometry one may measure the distance between spaces using the Gromov–Hausdorff distance.
For the branch of model theory, see stable theory.
In dynamical systems, an orbit is called Lyapunov stable if the forward orbit of any point is in a small enough neighborhood or it stays in a small (but perhaps, larger) neighborhood. Various criteria have been developed to prove stability or instability of an orbit. Under favorable circumstances, the question may be reduced to a well-studied problem involving eigenvalues of matrices. A more general method involves Lyapunov functions. In practice, any one of a number of different stability criteria are applied.
Overview in dynamical systems
Many parts of the qualitative theory of differential equations and dynamical systems deal with asymptotic properties of solutions and the trajectories—what happens with the system after a long period of time. The simplest kind of behavior is exhibited by equilibrium points, or fixed points, and by periodic orbits. If a particular orbit is well understood, it is natural to ask next whether a small change in the initial condition will lead to similar behavior. Stability theory addresses the following questions: Will a nearby orbit indefinitely stay close to a given orbit? Will it converge to the given orbit? In the former case, the orbit is called stable; in the latter case, it is called asymptotically stable and the given orbit is said to be attracting.
An equilibrium solution $f_{e}$ to an autonomous system of first order ordinary differential equations is called:
• stable if for every (small) $\epsilon >0$, there exists a $\delta >0$ such that every solution $f(t)$ having initial conditions within distance $\delta $ i.e. $\|f(t_{0})-f_{e}\|<\delta $ of the equilibrium remains within distance $\epsilon $ i.e. $\|f(t)-f_{e}\|<\epsilon $ for all $t\geq t_{0}$.
• asymptotically stable if it is stable and, in addition, there exists $\delta _{0}>0$ such that whenever $\|f(t_{0})-f_{e}\|<\delta _{0}$ then $f(t)\rightarrow f_{e}$as $t\rightarrow \infty $.
Stability means that the trajectories do not change too much under small perturbations. The opposite situation, where a nearby orbit is getting repelled from the given orbit, is also of interest. In general, perturbing the initial state in some directions results in the trajectory asymptotically approaching the given one and in other directions to the trajectory getting away from it. There may also be directions for which the behavior of the perturbed orbit is more complicated (neither converging nor escaping completely), and then stability theory does not give sufficient information about the dynamics.
One of the key ideas in stability theory is that the qualitative behavior of an orbit under perturbations can be analyzed using the linearization of the system near the orbit. In particular, at each equilibrium of a smooth dynamical system with an n-dimensional phase space, there is a certain n×n matrix A whose eigenvalues characterize the behavior of the nearby points (Hartman–Grobman theorem). More precisely, if all eigenvalues are negative real numbers or complex numbers with negative real parts then the point is a stable attracting fixed point, and the nearby points converge to it at an exponential rate, cf Lyapunov stability and exponential stability. If none of the eigenvalues are purely imaginary (or zero) then the attracting and repelling directions are related to the eigenspaces of the matrix A with eigenvalues whose real part is negative and, respectively, positive. Analogous statements are known for perturbations of more complicated orbits.
Stability of fixed points in 2D
The paradigmatic case is the stability of the origin under the linear autonomous differential equation ${\dot {X}}=AX$ where $X={\begin{bmatrix}x\\y\end{bmatrix}}$ and $A$ is a 2-by-2 matrix.
We would sometimes perform change-of-basis by $X'=CX$ for some invertible matrix $C$, which gives ${\dot {X}}'=C^{-1}ACX'$. We say $C^{-1}AC$ is "$A$ in the new basis". Since $\det A=\det C^{-1}AC$ and $\operatorname {tr} A=\operatorname {tr} C^{-1}AC$, we can classify the stability of origin using $\det A$ and $\operatorname {tr} A$, while freely using change-of-basis.
Classification of stability types
If $\det A=0$, then the rank of $A$ is zero or one.
• If the rank is zero, then $A=0$, and there is no flow.
• If the rank is one, then $\ker A$ and $\operatorname {im} A$ are both one-dimensional.
• If $\ker A=\operatorname {im} A$, then let $v$ span $\ker A$, and let $w$ be a preimage of $v$, then in $\{v,w\}$ basis, $A={\begin{bmatrix}0&1\\0&0\end{bmatrix}}$, and so the flow is a shearing along the $v$ direction. In this case, $\operatorname {tr} A=0$.
• If $\ker A\neq \operatorname {im} A$, then let $v$ span $\ker A$ and let $w$ span $\operatorname {im} A$, then in $\{v,w\}$ basis, $A={\begin{bmatrix}0&0\\0&a\end{bmatrix}}$ for some nonzero real number $a$.
• If $\operatorname {tr} A>0$, then it is unstable, diverging at a rate of $a$ from $\ker A$ along parallel translates of $\operatorname {im} A$.
• If $\operatorname {tr} A<0$, then it is stable, converging at a rate of $a$ to $\ker A$ along parallel translates of $\operatorname {im} A$.
If $\det A\neq 0$, we first find the Jordan normal form of the matrix, to obtain a basis $\{v,w\}$ in which $A$ is one of three possible forms:
• ${\begin{bmatrix}a&0\\0&b\end{bmatrix}}$ where $a,b\neq 0$.
• If $a,b>0$, then ${\begin{cases}4\det A-(\operatorname {tr} A)^{2}=-(a-b)^{2}\leq 0\\\det A=ab>0\end{cases}}$. The origin is a source, with integral curves of form $y=cx^{b/a}$
• Similarly for $a,b<0$. The origin is a sink.
• If $a>0>b$ or $a<0<b$, then $\det A<0$, and the origin is a saddle point. with integral curves of form $y=cx^{-|b/a|}$.
• ${\begin{bmatrix}a&1\\0&a\end{bmatrix}}$ where $a\neq 0$. This can be further simplified by a change-of-basis with $C={\begin{bmatrix}1/a&0\\0&1\end{bmatrix}}$, after which $A=a{\begin{bmatrix}1&1\\0&1\end{bmatrix}}$. We can explicitly solve for ${\dot {X}}=AX$ with $A=a{\begin{bmatrix}1&1\\0&1\end{bmatrix}}$. The solution is $X(t)=e^{At}X(0)$ with $e^{At}=e^{at}{\begin{bmatrix}1&at\\0&1\end{bmatrix}}$. This case is called the "degenerate node". The integral curves in this basis are central dilations of $x=y\ln y$, plus the x-axis.
• If $\operatorname {tr} A>0$, then the origin is an degenerate source. Otherwise it is a degenerate sink.
• In both cases, $4\det A-(\operatorname {tr} A)^{2}=0$
• $a{\begin{bmatrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \end{bmatrix}}$ where $a>0,\theta \in (-\pi ,\pi ]$. In this case, $4\det A-(\operatorname {tr} A)^{2}=(2a\sin \theta )^{2}\geq 0$.
• If $\theta \in (-\pi ,-\pi /2)\cup (\pi /2,\pi ]$, then this is a spiral sink. In this case, ${\begin{cases}4\det A-(\operatorname {tr} A)^{2}>0\\\operatorname {tr} A<0\end{cases}}$. The integral lines are logarithmic spirals.
• If $\theta \in (-\pi /2,\pi /2)$, then this is a spiral source. In this case, ${\begin{cases}4\det A-(\operatorname {tr} A)^{2}>0\\\operatorname {tr} A>0\end{cases}}$. The integral lines are logarithmic spirals.
• If $\theta =-\pi /2,\pi /2$, then this is a rotation ("neutral stability") at a rate of $a$, moving neither towards nor away from origin. In this case, $\operatorname {tr} A=0$. The integral lines are circles.
The summary is shown in the stability diagram on the right. In each case, except the case of $4\det A-(\operatorname {tr} A)^{2}=0$, the values $(\operatorname {tr} A,\det A)$ allows unique classification of the type of flow.
For the special case of $4\det A-(\operatorname {tr} A)^{2}=0$, there are two cases that cannot be distinguished by $(\operatorname {tr} A,\det A)$. In both cases, $A$ has only one eigenvalue, with algebraic multiplicity 2.
• If the eigenvalue has a two-dimensional eigenspace (geometric multiplicity 2), then the system is a central node (sometimes called a "star", or "dicritical node") which is either a source (when $\operatorname {tr} A>0$) or a sink (when $\operatorname {tr} A<0$).[2]
• If it has a one-dimensional eigenspace (geometric multiplicity 1), then the system is a degenerate node (if $\det A>0$) or a shearing flow (if $\det A=0$).
Area-preserving flow
When $\operatorname {tr} A=0$, we have $\det e^{At}=e^{\operatorname {tr} (A)t}=I$, so the flow is area-preserving. In this case, the type of flow is classified by $\det A$.
• If $\det A>0$, then it is a rotation ("neutral stability") around the origin.
• If $\det A=0$, then it is a shearing flow.
• If $\det A<0$, then the origin is a saddle point.
Stability of fixed points
The simplest kind of an orbit is a fixed point, or an equilibrium. If a mechanical system is in a stable equilibrium state then a small push will result in a localized motion, for example, small oscillations as in the case of a pendulum. In a system with damping, a stable equilibrium state is moreover asymptotically stable. On the other hand, for an unstable equilibrium, such as a ball resting on a top of a hill, certain small pushes will result in a motion with a large amplitude that may or may not converge to the original state.
There are useful tests of stability for the case of a linear system. Stability of a nonlinear system can often be inferred from the stability of its linearization.
Maps
Let f: R → R be a continuously differentiable function with a fixed point a, f(a) = a. Consider the dynamical system obtained by iterating the function f:
$x_{n+1}=f(x_{n}),\quad n=0,1,2,\ldots .$
The fixed point a is stable if the absolute value of the derivative of f at a is strictly less than 1, and unstable if it is strictly greater than 1. This is because near the point a, the function f has a linear approximation with slope f'(a):
$f(x)\approx f(a)+f'(a)(x-a).$
Thus
$x_{n+1}-a=f(x_{n})-a\simeq f(a)+f'(a)(x_{n}-a)-a=a+f'(a)(x_{n}-a)-a=f'(a)(x_{n}-a)\to {\frac {x_{n+1}-a}{x_{n}-a}}=f'(a)$
which means that the derivative measures the rate at which the successive iterates approach the fixed point a or diverge from it. If the derivative at a is exactly 1 or −1, then more information is needed in order to decide stability.
There is an analogous criterion for a continuously differentiable map f: Rn → Rn with a fixed point a, expressed in terms of its Jacobian matrix at a, Ja(f). If all eigenvalues of J are real or complex numbers with absolute value strictly less than 1 then a is a stable fixed point; if at least one of them has absolute value strictly greater than 1 then a is unstable. Just as for n=1, the case of the largest absolute value being 1 needs to be investigated further — the Jacobian matrix test is inconclusive. The same criterion holds more generally for diffeomorphisms of a smooth manifold.
Linear autonomous systems
The stability of fixed points of a system of constant coefficient linear differential equations of first order can be analyzed using the eigenvalues of the corresponding matrix.
An autonomous system
$x'=Ax,$
where x(t) ∈ Rn and A is an n×n matrix with real entries, has a constant solution
$x(t)=0.$
(In a different language, the origin 0 ∈ Rn is an equilibrium point of the corresponding dynamical system.) This solution is asymptotically stable as t → ∞ ("in the future") if and only if for all eigenvalues λ of A, Re(λ) < 0. Similarly, it is asymptotically stable as t → −∞ ("in the past") if and only if for all eigenvalues λ of A, Re(λ) > 0. If there exists an eigenvalue λ of A with Re(λ) > 0 then the solution is unstable for t → ∞.
Application of this result in practice, in order to decide the stability of the origin for a linear system, is facilitated by the Routh–Hurwitz stability criterion. The eigenvalues of a matrix are the roots of its characteristic polynomial. A polynomial in one variable with real coefficients is called a Hurwitz polynomial if the real parts of all roots are strictly negative. The Routh–Hurwitz theorem implies a characterization of Hurwitz polynomials by means of an algorithm that avoids computing the roots.
Non-linear autonomous systems
Asymptotic stability of fixed points of a non-linear system can often be established using the Hartman–Grobman theorem.
Suppose that v is a C1-vector field in Rn which vanishes at a point p, v(p) = 0. Then the corresponding autonomous system
$x'=v(x)$
has a constant solution
$x(t)=p.$
Let Jp(v) be the n×n Jacobian matrix of the vector field v at the point p. If all eigenvalues of J have strictly negative real part then the solution is asymptotically stable. This condition can be tested using the Routh–Hurwitz criterion.
Lyapunov function for general dynamical systems
A general way to establish Lyapunov stability or asymptotic stability of a dynamical system is by means of Lyapunov functions.
See also
• Chaos theory
• Lyapunov stability
• Hyperstability
• Linear stability
• Orbital stability
• Stability criterion
• Stability radius
• Structural stability
• von Neumann stability analysis
References
1. Egwald Mathematics - Linear Algebra: Systems of Linear Differential Equations: Linear Stability Analysis Accessed 10 October 2019.
2. "Node - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2023-03-30.
• Philip Holmes and Eric T. Shea-Brown (ed.). "Stability". Scholarpedia.
External links
• Stable Equilibria by Michael Schreiber, The Wolfram Demonstrations Project.
Chaos theory
Concepts
Core
• Attractor
• Bifurcation
• Fractal
• Limit set
• Lyapunov exponent
• Orbit
• Periodic point
• Phase space
• Anosov diffeomorphism
• Arnold tongue
• axiom A dynamical system
• Bifurcation diagram
• Box-counting dimension
• Correlation dimension
• Conservative system
• Ergodicity
• False nearest neighbors
• Hausdorff dimension
• Invariant measure
• Lyapunov stability
• Measure-preserving dynamical system
• Mixing
• Poincaré section
• Recurrence plot
• SRB measure
• Stable manifold
• Topological conjugacy
Theorems
• Ergodic theorem
• Liouville's theorem
• Krylov–Bogolyubov theorem
• Poincaré–Bendixson theorem
• Poincaré recurrence theorem
• Stable manifold theorem
• Takens's theorem
Theoretical
branches
• Bifurcation theory
• Control of chaos
• Dynamical system
• Ergodic theory
• Quantum chaos
• Stability theory
• Synchronization of chaos
Chaotic
maps (list)
Discrete
• Arnold's cat map
• Baker's map
• Complex quadratic map
• Coupled map lattice
• Duffing map
• Dyadic transformation
• Dynamical billiards
• outer
• Exponential map
• Gauss map
• Gingerbreadman map
• Hénon map
• Horseshoe map
• Ikeda map
• Interval exchange map
• Irrational rotation
• Kaplan–Yorke map
• Langton's ant
• Logistic map
• Standard map
• Tent map
• Tinkerbell map
• Zaslavskii map
Continuous
• Double scroll attractor
• Duffing equation
• Lorenz system
• Lotka–Volterra equations
• Mackey–Glass equations
• Rabinovich–Fabrikant equations
• Rössler attractor
• Three-body problem
• Van der Pol oscillator
Physical
systems
• Chua's circuit
• Convection
• Double pendulum
• Elastic pendulum
• FPUT problem
• Hénon–Heiles system
• Kicked rotator
• Multiscroll attractor
• Population dynamics
• Swinging Atwood's machine
• Tilt-A-Whirl
• Weather
Chaos
theorists
• Michael Berry
• Rufus Bowen
• Mary Cartwright
• Chen Guanrong
• Leon O. Chua
• Mitchell Feigenbaum
• Peter Grassberger
• Celso Grebogi
• Martin Gutzwiller
• Brosl Hasslacher
• Michel Hénon
• Svetlana Jitomirskaya
• Bryna Kra
• Edward Norton Lorenz
• Aleksandr Lyapunov
• Benoît Mandelbrot
• Hee Oh
• Edward Ott
• Henri Poincaré
• Mary Rees
• Otto Rössler
• David Ruelle
• Caroline Series
• Yakov Sinai
• Oleksandr Mykolayovych Sharkovsky
• Nina Snaith
• Floris Takens
• Audrey Terras
• Mary Tsingou
• Marcelo Viana
• Amie Wilkinson
• James A. Yorke
• Lai-Sang Young
Related
articles
• Butterfly effect
• Complexity
• Edge of chaos
• Predictability
• Santa Fe Institute
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Wikipedia
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Stabilizer code
The theory of quantum error correction plays a prominent role in the practical realization and engineering of quantum computing and quantum communication devices. The first quantum error-correcting codes are strikingly similar to classical block codes in their operation and performance. Quantum error-correcting codes restore a noisy, decohered quantum state to a pure quantum state. A stabilizer quantum error-correcting code appends ancilla qubits to qubits that we want to protect. A unitary encoding circuit rotates the global state into a subspace of a larger Hilbert space. This highly entangled, encoded state corrects for local noisy errors. A quantum error-correcting code makes quantum computation and quantum communication practical by providing a way for a sender and receiver to simulate a noiseless qubit channel given a noisy qubit channel whose noise conforms to a particular error model.
The stabilizer theory of quantum error correction allows one to import some classical binary or quaternary codes for use as a quantum code. However, when importing the classical code, it must satisfy the dual-containing (or self-orthogonality) constraint. Researchers have found many examples of classical codes satisfying this constraint, but most classical codes do not. Nevertheless, it is still useful to import classical codes in this way (though, see how the entanglement-assisted stabilizer formalism overcomes this difficulty).
Mathematical background
The stabilizer formalism exploits elements of the Pauli group $\Pi $ in formulating quantum error-correcting codes. The set $\Pi =\left\{I,X,Y,Z\right\}$ consists of the Pauli operators:
$I\equiv {\begin{bmatrix}1&0\\0&1\end{bmatrix}},\ X\equiv {\begin{bmatrix}0&1\\1&0\end{bmatrix}},\ Y\equiv {\begin{bmatrix}0&-i\\i&0\end{bmatrix}},\ Z\equiv {\begin{bmatrix}1&0\\0&-1\end{bmatrix}}.$
The above operators act on a single qubit – a state represented by a vector in a two-dimensional Hilbert space. Operators in $\Pi $ have eigenvalues $\pm 1$ and either commute or anti-commute. The set $\Pi ^{n}$ consists of $n$-fold tensor products of Pauli operators:
$\Pi ^{n}=\left\{{\begin{array}{c}e^{i\phi }A_{1}\otimes \cdots \otimes A_{n}:\forall j\in \left\{1,\ldots ,n\right\}A_{j}\in \Pi ,\ \ \phi \in \left\{0,\pi /2,\pi ,3\pi /2\right\}\end{array}}\right\}.$
Elements of $\Pi ^{n}$ act on a quantum register of $n$ qubits. We occasionally omit tensor product symbols in what follows so that
$A_{1}\cdots A_{n}\equiv A_{1}\otimes \cdots \otimes A_{n}.$
The $n$-fold Pauli group $\Pi ^{n}$ plays an important role for both the encoding circuit and the error-correction procedure of a quantum stabilizer code over $n$ qubits.
Definition
Let us define an $\left[n,k\right]$ stabilizer quantum error-correcting code to encode $k$ logical qubits into $n$ physical qubits. The rate of such a code is $k/n$. Its stabilizer ${\mathcal {S}}$ is an abelian subgroup of the $n$-fold Pauli group $\Pi ^{n}$. ${\mathcal {S}}$ does not contain the operator $-I^{\otimes n}$. The simultaneous $+1$-eigenspace of the operators constitutes the codespace. The codespace has dimension $2^{k}$ so that we can encode $k$ qubits into it. The stabilizer ${\mathcal {S}}$ has a minimal representation in terms of $n-k$ independent generators
$\left\{g_{1},\ldots ,g_{n-k}\ |\ \forall i\in \left\{1,\ldots ,n-k\right\},\ g_{i}\in {\mathcal {S}}\right\}.$
The generators are independent in the sense that none of them is a product of any other two (up to a global phase). The operators $g_{1},\ldots ,g_{n-k}$ function in the same way as a parity check matrix does for a classical linear block code.
Stabilizer error-correction conditions
One of the fundamental notions in quantum error correction theory is that it suffices to correct a discrete error set with support in the Pauli group $\Pi ^{n}$. Suppose that the errors affecting an encoded quantum state are a subset ${\mathcal {E}}$ of the Pauli group $\Pi ^{n}$:
${\mathcal {E}}\subset \Pi ^{n}.$
Because ${\mathcal {E}}$ and ${\mathcal {S}}$ are both subsets of $\Pi ^{n}$, an error $E\in {\mathcal {E}}$ that affects an encoded quantum state either commutes or anticommutes with any particular element $g$ in ${\mathcal {S}}$. The error $E$ is correctable if it anticommutes with an element $g$ in ${\mathcal {S}}$. An anticommuting error $E$ is detectable by measuring each element $g$ in ${\mathcal {S}}$ and computing a syndrome $\mathbf {r} $ identifying $E$. The syndrome is a binary vector $\mathbf {r} $ with length $n-k$ whose elements identify whether the error $E$ commutes or anticommutes with each $g\in {\mathcal {S}}$. An error $E$ that commutes with every element $g$ in ${\mathcal {S}}$ is correctable if and only if it is in ${\mathcal {S}}$. It corrupts the encoded state if it commutes with every element of ${\mathcal {S}}$ but does not lie in ${\mathcal {S}}$. So we compactly summarize the stabilizer error-correcting conditions: a stabilizer code can correct any errors $E_{1},E_{2}$ in ${\mathcal {E}}$ if
$E_{1}^{\dagger }E_{2}\notin {\mathcal {Z}}\left({\mathcal {S}}\right)$
or
$E_{1}^{\dagger }E_{2}\in {\mathcal {S}}$
where ${\mathcal {Z}}\left({\mathcal {S}}\right)$ is the centralizer of ${\mathcal {S}}$ (i.e., the subgroup of elements that commute with all members of ${\mathcal {S}}$, also known as the commutant).
Simple example of a stabilizer code
A simple example of a stabilizer code is a three qubit $\left[[3,1,3\right]]$ stabilizer code. It encodes $k=1$ logical qubit into $n=3$ physical qubits and protects against a single-bit flip error in the set $\left\{X_{i}\right\}$. This does not protect against other Pauli errors such as phase flip errors in the set $\left\{Y_{i}\right\}$.or $\left\{Z_{i}\right\}$. This has code distance $d=3$. Its stabilizer consists of $n-k=2$ Pauli operators:
${\begin{array}{ccc}g_{1}&=&Z&Z&I\\g_{2}&=&I&Z&Z\\\end{array}}$
If there are no bit-flip errors, both operators $g_{1}$ and $g_{2}$ commute, the syndrome is +1,+1, and no errors are detected.
If there is a bit-flip error on the first encoded qubit, operator $g_{1}$ will anti-commute and $g_{2}$ commute, the syndrome is -1,+1, and the error is detected. If there is a bit-flip error on the second encoded qubit, operator $g_{1}$ will anti-commute and $g_{2}$ anti-commute, the syndrome is -1,-1, and the error is detected. If there is a bit-flip error on the third encoded qubit, operator $g_{1}$ will commute and $g_{2}$ anti-commute, the syndrome is +1,-1, and the error is detected.
Example of a stabilizer code
An example of a stabilizer code is the five qubit $\left[[5,1,3\right]]$ stabilizer code. It encodes $k=1$ logical qubit into $n=5$ physical qubits and protects against an arbitrary single-qubit error. It has code distance $d=3$. Its stabilizer consists of $n-k=4$ Pauli operators:
${\begin{array}{ccccccc}g_{1}&=&X&Z&Z&X&I\\g_{2}&=&I&X&Z&Z&X\\g_{3}&=&X&I&X&Z&Z\\g_{4}&=&Z&X&I&X&Z\end{array}}$
The above operators commute. Therefore, the codespace is the simultaneous +1-eigenspace of the above operators. Suppose a single-qubit error occurs on the encoded quantum register. A single-qubit error is in the set $\left\{X_{i},Y_{i},Z_{i}\right\}$ where $A_{i}$ denotes a Pauli error on qubit $i$. It is straightforward to verify that any arbitrary single-qubit error has a unique syndrome. The receiver corrects any single-qubit error by identifying the syndrome via a parity measurement and applying a corrective operation.
Relation between Pauli group and binary vectors
A simple but useful mapping exists between elements of $\Pi $ and the binary vector space $\left(\mathbb {Z} _{2}\right)^{2}$. This mapping gives a simplification of quantum error correction theory. It represents quantum codes with binary vectors and binary operations rather than with Pauli operators and matrix operations respectively.
We first give the mapping for the one-qubit case. Suppose $\left[A\right]$ is a set of equivalence classes of an operator $A$ that have the same phase:
$\left[A\right]=\left\{\beta A\ |\ \beta \in \mathbb {C} ,\ \left\vert \beta \right\vert =1\right\}.$
Let $\left[\Pi \right]$ be the set of phase-free Pauli operators where $\left[\Pi \right]=\left\{\left[A\right]\ |\ A\in \Pi \right\}$. Define the map $N:\left(\mathbb {Z} _{2}\right)^{2}\rightarrow \Pi $ as
$00\to I,\,\,01\to X,\,\,11\to Y,\,\,10\to Z$
Suppose $u,v\in \left(\mathbb {Z} _{2}\right)^{2}$. Let us employ the shorthand $u=\left(z|x\right)$ and $v=\left(z^{\prime }|x^{\prime }\right)$ where $z$, $x$, $z^{\prime }$, $x^{\prime }\in \mathbb {Z} _{2}$. For example, suppose $u=\left(0|1\right)$. Then $N\left(u\right)=X$. The map $N$ induces an isomorphism $\left[N\right]:\left(\mathbb {Z} _{2}\right)^{2}\rightarrow \left[\Pi \right]$ because addition of vectors in $\left(\mathbb {Z} _{2}\right)^{2}$ is equivalent to multiplication of Pauli operators up to a global phase:
$\left[N\left(u+v\right)\right]=\left[N\left(u\right)\right]\left[N\left(v\right)\right].$
Let $\odot $ denote the symplectic product between two elements $u,v\in \left(\mathbb {Z} _{2}\right)^{2}$:
$u\odot v\equiv zx^{\prime }-xz^{\prime }.$
The symplectic product $\odot $ gives the commutation relations of elements of $\Pi $:
$N\left(u\right)N\left(v\right)=\left(-1\right)^{\left(u\odot v\right)}N\left(v\right)N\left(u\right).$
The symplectic product and the mapping $N$ thus give a useful way to phrase Pauli relations in terms of binary algebra. The extension of the above definitions and mapping $N$ to multiple qubits is straightforward. Let $\mathbf {A} =A_{1}\otimes \cdots \otimes A_{n}$ denote an arbitrary element of $\Pi ^{n}$. We can similarly define the phase-free $n$-qubit Pauli group $\left[\Pi ^{n}\right]=\left\{\left[\mathbf {A} \right]\ |\ \mathbf {A} \in \Pi ^{n}\right\}$ where
$\left[\mathbf {A} \right]=\left\{\beta \mathbf {A} \ |\ \beta \in \mathbb {C} ,\ \left\vert \beta \right\vert =1\right\}.$
The group operation $\ast $ for the above equivalence class is as follows:
$\left[\mathbf {A} \right]\ast \left[\mathbf {B} \right]\equiv \left[A_{1}\right]\ast \left[B_{1}\right]\otimes \cdots \otimes \left[A_{n}\right]\ast \left[B_{n}\right]=\left[A_{1}B_{1}\right]\otimes \cdots \otimes \left[A_{n}B_{n}\right]=\left[\mathbf {AB} \right].$
The equivalence class $\left[\Pi ^{n}\right]$ forms a commutative group under operation $\ast $. Consider the $2n$-dimensional vector space
$\left(\mathbb {Z} _{2}\right)^{2n}=\left\{\left(\mathbf {z,x} \right):\mathbf {z} ,\mathbf {x} \in \left(\mathbb {Z} _{2}\right)^{n}\right\}.$
It forms the commutative group $(\left(\mathbb {Z} _{2}\right)^{2n},+)$ with operation $+$ defined as binary vector addition. We employ the notation $\mathbf {u} =\left(\mathbf {z} |\mathbf {x} \right),\mathbf {v} =\left(\mathbf {z} ^{\prime }|\mathbf {x} ^{\prime }\right)$ to represent any vectors $\mathbf {u,v} \in \left(\mathbb {Z} _{2}\right)^{2n}$ respectively. Each vector $\mathbf {z} $ and $\mathbf {x} $ has elements $\left(z_{1},\ldots ,z_{n}\right)$ and $\left(x_{1},\ldots ,x_{n}\right)$ respectively with similar representations for $\mathbf {z} ^{\prime }$ and $\mathbf {x} ^{\prime }$. The symplectic product $\odot $ of $\mathbf {u} $ and $\mathbf {v} $ is
$\mathbf {u} \odot \mathbf {v\equiv } \sum _{i=1}^{n}z_{i}x_{i}^{\prime }-x_{i}z_{i}^{\prime },$
or
$\mathbf {u} \odot \mathbf {v\equiv } \sum _{i=1}^{n}u_{i}\odot v_{i},$
where $u_{i}=\left(z_{i}|x_{i}\right)$ and $v_{i}=\left(z_{i}^{\prime }|x_{i}^{\prime }\right)$. Let us define a map $\mathbf {N} :\left(\mathbb {Z} _{2}\right)^{2n}\rightarrow \Pi ^{n}$ :\left(\mathbb {Z} _{2}\right)^{2n}\rightarrow \Pi ^{n}} as follows:
$\mathbf {N} \left(\mathbf {u} \right)\equiv N\left(u_{1}\right)\otimes \cdots \otimes N\left(u_{n}\right).$
Let
$\mathbf {X} \left(\mathbf {x} \right)\equiv X^{x_{1}}\otimes \cdots \otimes X^{x_{n}},\,\,\,\,\,\,\,\mathbf {Z} \left(\mathbf {z} \right)\equiv Z^{z_{1}}\otimes \cdots \otimes Z^{z_{n}},$
so that $\mathbf {N} \left(\mathbf {u} \right)$ and $\mathbf {Z} \left(\mathbf {z} \right)\mathbf {X} \left(\mathbf {x} \right)$ belong to the same equivalence class:
$\left[\mathbf {N} \left(\mathbf {u} \right)\right]=\left[\mathbf {Z} \left(\mathbf {z} \right)\mathbf {X} \left(\mathbf {x} \right)\right].$
The map $\left[\mathbf {N} \right]:\left(\mathbb {Z} _{2}\right)^{2n}\rightarrow \left[\Pi ^{n}\right]$ is an isomorphism for the same reason given as in the previous case:
$\left[\mathbf {N} \left(\mathbf {u+v} \right)\right]=\left[\mathbf {N} \left(\mathbf {u} \right)\right]\left[\mathbf {N} \left(\mathbf {v} \right)\right],$
where $\mathbf {u,v} \in \left(\mathbb {Z} _{2}\right)^{2n}$. The symplectic product captures the commutation relations of any operators $\mathbf {N} \left(\mathbf {u} \right)$ and $\mathbf {N} \left(\mathbf {v} \right)$:
$\mathbf {N\left(\mathbf {u} \right)N} \left(\mathbf {v} \right)=\left(-1\right)^{\left(\mathbf {u} \odot \mathbf {v} \right)}\mathbf {N} \left(\mathbf {v} \right)\mathbf {N} \left(\mathbf {u} \right).$
The above binary representation and symplectic algebra are useful in making the relation between classical linear error correction and quantum error correction more explicit.
By comparing quantum error correcting codes in this language to symplectic vector spaces, we can see the following. A symplectic subspace corresponds to a direct sum of Pauli algebras (i.e., encoded qubits), while an isotropic subspace corresponds to a set of stabilizers.
References
• D. Gottesman, "Stabilizer codes and quantum error correction," quant-ph/9705052, Caltech Ph.D. thesis. https://arxiv.org/abs/quant-ph/9705052
• Shor, Peter W. (1995-10-01). "Scheme for reducing decoherence in quantum computer memory". Physical Review A. American Physical Society (APS). 52 (4): R2493–R2496. Bibcode:1995PhRvA..52.2493S. doi:10.1103/physreva.52.r2493. ISSN 1050-2947. PMID 9912632.
• Calderbank, A. R.; Shor, Peter W. (1996-08-01). "Good quantum error-correcting codes exist". Physical Review A. American Physical Society (APS). 54 (2): 1098–1105. arXiv:quant-ph/9512032. Bibcode:1996PhRvA..54.1098C. doi:10.1103/physreva.54.1098. ISSN 1050-2947. PMID 9913578. S2CID 11524969.
• Steane, A. M. (1996-07-29). "Error Correcting Codes in Quantum Theory". Physical Review Letters. American Physical Society (APS). 77 (5): 793–797. Bibcode:1996PhRvL..77..793S. doi:10.1103/physrevlett.77.793. ISSN 0031-9007. PMID 10062908.
• A. Calderbank, E. Rains, P. Shor, and N. Sloane, “Quantum error correction via codes over GF(4),” IEEE Trans. Inf. Theory, vol. 44, pp. 1369–1387, 1998. Available at https://arxiv.org/abs/quant-ph/9608006
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Wikipedia
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Stability (probability)
In probability theory, the stability of a random variable is the property that a linear combination of two independent copies of the variable has the same distribution, up to location and scale parameters.[1] The distributions of random variables having this property are said to be "stable distributions". Results available in probability theory show that all possible distributions having this property are members of a four-parameter family of distributions. The article on the stable distribution describes this family together with some of the properties of these distributions.
The importance in probability theory of "stability" and of the stable family of probability distributions is that they are "attractors" for properly normed sums of independent and identically distributed random variables.
Important special cases of stable distributions are the normal distribution, the Cauchy distribution and the Lévy distribution. For details see stable distribution.
Definition
There are several basic definitions for what is meant by stability. Some are based on summations of random variables and others on properties of characteristic functions.
Definition via distribution functions
Feller[2] makes the following basic definition. A random variable X is called stable (has a stable distribution) if, for n independent copies Xi of X, there exist constants cn > 0 and dn such that
$X_{1}+X_{2}+\ldots +X_{n}{\stackrel {d}{=}}c_{n}X+d_{n},$
where this equality refers to equality of distributions. A conclusion drawn from this starting point is that the sequence of constants cn must be of the form
$c_{n}=n^{1/\alpha }\,$ for $0<\alpha \leq 2.$
A further conclusion is that it is enough for the above distributional identity to hold for n=2 and n=3 only.[3]
Stability in probability theory
There are a number of mathematical results that can be derived for distributions which have the stability property. That is, all possible families of distributions which have the property of being closed under convolution are being considered.[4] It is convenient here to call these stable distributions, without meaning specifically the distribution described in the article named stable distribution, or to say that a distribution is stable if it is assumed that it has the stability property. The following results can be obtained for univariate distributions which are stable.
• Stable distributions are always infinitely divisible.[5]
• All stable distributions are absolutely continuous.[6]
• All stable distributions are unimodal.[7]
Other types of stability
The above concept of stability is based on the idea of a class of distributions being closed under a given set of operations on random variables, where the operation is "summation" or "averaging". Other operations that have been considered include:
• geometric stability: here the operation is to take the sum of a random number of random variables, where the number has a geometric distribution.[8] The counterpart of the stable distribution in this case is the geometric stable distribution
• Max-stability: here the operation is to take the maximum of a number of random variables. The counterpart of the stable distribution in this case is the generalized extreme value distribution, and the theory for this case is dealt with as extreme value theory. See also the stability postulate. A version of this case in which the minimum is taken instead of the maximum is available by a simple extension.
See also
• Infinite divisibility
• Indecomposable distribution
Notes
1. Lukacs, E. (1970) Section 5.7
2. Feller (1971), Section VI.1
3. Feller (1971), Problem VI.13.3
4. Lukacs, E. (1970) Section 5.7
5. Lukacs, E. (1970) Theorem 5.7.1
6. Lukacs, E. (1970) Theorem 5.8.1
7. Lukacs, E. (1970) Theorem 5.10.1
8. Klebanov et al. (1984)
References
• Lukacs, E. (1970) Characteristic Functions. Griffin, London.
• Feller, W. (1971) An Introduction to Probability Theory and Its Applications, Volume 2. Wiley. ISBN 0-471-25709-5
• Klebanov, L.B., Maniya, G.M., Melamed, I.A. (1984) "A problem of V. M. Zolotarev and analogues of infinitely divisible and stable distributions in a scheme for summation of a random number of random variables". Theory Probab. Appl., 29, 791–794
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Wikipedia
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Stable count distribution
In probability theory, the stable count distribution is the conjugate prior of a one-sided stable distribution. This distribution was discovered by Stephen Lihn (Chinese: 藺鴻圖) in his 2017 study of daily distributions of the S&P 500 and the VIX.[1] The stable distribution family is also sometimes referred to as the Lévy alpha-stable distribution, after Paul Lévy, the first mathematician to have studied it.[2]
Stable count
Probability density function
Cumulative distribution function
Parameters
$\alpha $ ∈ (0, 1) — stability parameter
$\theta $ ∈ (0, ∞) — scale parameter
$\nu _{0}$ ∈ (−∞, ∞) — location parameter
Support x ∈ R and x ∈ [$\nu _{0}$, ∞)
PDF ${\mathfrak {N}}_{\alpha }(x;\nu _{0},\theta )={\frac {1}{\Gamma ({\frac {1}{\alpha }}+1)}}{\frac {1}{x-\nu _{0}}}L_{\alpha }({\frac {\theta }{x-\nu _{0}}})$
CDF integral form exists
Mean ${\frac {\Gamma ({\frac {2}{\alpha }})}{\Gamma ({\frac {1}{\alpha }})}}$
Median not analytically expressible
Mode not analytically expressible
Variance ${\frac {\Gamma ({\frac {3}{\alpha }})}{2\Gamma ({\frac {1}{\alpha }})}}-\left[{\frac {\Gamma ({\frac {2}{\alpha }})}{\Gamma ({\frac {1}{\alpha }})}}\right]^{2}$
Skewness TBD
Ex. kurtosis TBD
MGF Fox-Wright representation exists
Of the three parameters defining the distribution, the stability parameter $\alpha $ is most important. Stable count distributions have $0<\alpha <1$. The known analytical case of $\alpha =1/2$ is related to the VIX distribution (See Section 7 of [1]). All the moments are finite for the distribution.
Definition
Its standard distribution is defined as
${\mathfrak {N}}_{\alpha }(\nu )={\frac {1}{\Gamma ({\frac {1}{\alpha }}+1)}}{\frac {1}{\nu }}L_{\alpha }\left({\frac {1}{\nu }}\right),$
where $\nu >0$ and $0<\alpha <1.$
Its location-scale family is defined as
${\mathfrak {N}}_{\alpha }(\nu ;\nu _{0},\theta )={\frac {1}{\Gamma ({\frac {1}{\alpha }}+1)}}{\frac {1}{\nu -\nu _{0}}}L_{\alpha }\left({\frac {\theta }{\nu -\nu _{0}}}\right),$ ;\nu _{0},\theta )={\frac {1}{\Gamma ({\frac {1}{\alpha }}+1)}}{\frac {1}{\nu -\nu _{0}}}L_{\alpha }\left({\frac {\theta }{\nu -\nu _{0}}}\right),}
where $\nu >\nu _{0}$, $\theta >0$, and $0<\alpha <1.$
In the above expression, $L_{\alpha }(x)$ is a one-sided stable distribution,[3] which is defined as following.
Let $X$ be a standard stable random variable whose distribution is characterized by $f(x;\alpha ,\beta ,c,\mu )$, then we have
$L_{\alpha }(x)=f(x;\alpha ,1,\cos \left({\frac {\pi \alpha }{2}}\right)^{1/\alpha },0),$
where $0<\alpha <1$.
Consider the Lévy sum $Y=\sum _{i=1}^{N}X_{i}$ where $X_{i}\sim L_{\alpha }(x)$, then $Y$ has the density $ {\frac {1}{\nu }}L_{\alpha }\left({\frac {x}{\nu }}\right)$ where $ \nu =N^{1/\alpha }$. Set $x=1$, we arrive at ${\mathfrak {N}}_{\alpha }(\nu )$ without the normalization constant.
The reason why this distribution is called "stable count" can be understood by the relation $\nu =N^{1/\alpha }$. Note that $N$ is the "count" of the Lévy sum. Given a fixed $\alpha $, this distribution gives the probability of taking $N$ steps to travel one unit of distance.
Integral form
Based on the integral form of $L_{\alpha }(x)$ and $q=\exp(-i\alpha \pi /2)$, we have the integral form of ${\mathfrak {N}}_{\alpha }(\nu )$ as
${\begin{aligned}{\mathfrak {N}}_{\alpha }(\nu )&={\frac {2}{\pi \Gamma ({\frac {1}{\alpha }}+1)}}\int _{0}^{\infty }e^{-{\text{Re}}(q)\,t^{\alpha }}{\frac {1}{\nu }}\sin({\frac {t}{\nu }})\sin(-{\text{Im}}(q)\,t^{\alpha })\,dt,{\text{ or }}\\&={\frac {2}{\pi \Gamma ({\frac {1}{\alpha }}+1)}}\int _{0}^{\infty }e^{-{\text{Re}}(q)\,t^{\alpha }}{\frac {1}{\nu }}\cos({\frac {t}{\nu }})\cos({\text{Im}}(q)\,t^{\alpha })\,dt.\\\end{aligned}}$
Based on the double-sine integral above, it leads to the integral form of the standard CDF:
${\begin{aligned}\Phi _{\alpha }(x)&={\frac {2}{\pi \Gamma ({\frac {1}{\alpha }}+1)}}\int _{0}^{x}\int _{0}^{\infty }e^{-{\text{Re}}(q)\,t^{\alpha }}{\frac {1}{\nu }}\sin({\frac {t}{\nu }})\sin(-{\text{Im}}(q)\,t^{\alpha })\,dt\,d\nu \\&=1-{\frac {2}{\pi \Gamma ({\frac {1}{\alpha }}+1)}}\int _{0}^{\infty }e^{-{\text{Re}}(q)\,t^{\alpha }}\sin(-{\text{Im}}(q)\,t^{\alpha })\,{\text{Si}}({\frac {t}{x}})\,dt,\\\end{aligned}}$
where ${\text{Si}}(x)=\int _{0}^{x}{\frac {\sin(x)}{x}}\,dx$ is the sine integral function.
The Wright representation
In "Series representation", it is shown that the stable count distribution is a special case of the Wright function (See Section 4 of [4]):
${\mathfrak {N}}_{\alpha }(\nu )={\frac {1}{\Gamma \left({\frac {1}{\alpha }}+1\right)}}W_{-\alpha ,0}(-\nu ^{\alpha }),\,{\text{where}}\,\,W_{\lambda ,\mu }(z)=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!\,\Gamma (\lambda n+\mu )}}.$
This leads to the Hankel integral: (based on (1.4.3) of [5])
${\mathfrak {N}}_{\alpha }(\nu )={\frac {1}{\Gamma \left({\frac {1}{\alpha }}+1\right)}}{\frac {1}{2\pi i}}\int _{Ha}e^{t-(\nu t)^{\alpha }}\,dt,\,$where Ha represents a Hankel contour.
Alternative derivation – lambda decomposition
Another approach to derive the stable count distribution is to use the Laplace transform of the one-sided stable distribution, (Section 2.4 of [1])
$\int _{0}^{\infty }e^{-zx}L_{\alpha }(x)\,dx=e^{-z^{\alpha }},$where $0<\alpha <1$.
Let $x=1/\nu $, and one can decompose the integral on the left hand side as a product distribution of a standard Laplace distribution and a standard stable count distribution,
${\frac {1}{2}}{\frac {1}{\Gamma ({\frac {1}{\alpha }}+1)}}e^{-|z|^{\alpha }}=\int _{0}^{\infty }{\frac {1}{\nu }}\left({\frac {1}{2}}e^{-|z|/\nu }\right)\left({\frac {1}{\Gamma ({\frac {1}{\alpha }}+1)}}{\frac {1}{\nu }}L_{\alpha }\left({\frac {1}{\nu }}\right)\right)\,d\nu ,$
where $z\in {\mathsf {R}}$.
This is called the "lambda decomposition" (See Section 4 of [1]) since the LHS was named as "symmetric lambda distribution" in Lihn's former works. However, it has several more popular names such as "exponential power distribution", or the "generalized error/normal distribution", often referred to when $\alpha >1$. It is also the Weibull survival function in Reliability engineering.
Lambda decomposition is the foundation of Lihn's framework of asset returns under the stable law. The LHS is the distribution of asset returns. On the RHS, the Laplace distribution represents the lepkurtotic noise, and the stable count distribution represents the volatility.
Stable Vol distribution
A variant of the stable count distribution is called the stable vol distribution $V_{\alpha }(s)$. It can be derived from lambda decomposition by a change of variable (See Section 6 of [4]). The Laplace transform of $e^{-|z|^{\alpha }}$ is expressed in terms of a Gaussian mixture such that
${\frac {1}{2}}{\frac {1}{\Gamma ({\frac {1}{\alpha }}+1)}}e^{-|z|^{\alpha }}={\frac {1}{2}}{\frac {1}{\Gamma ({\frac {1}{\alpha }}+1)}}e^{-(z^{2})^{\alpha /2}}=\int _{0}^{\infty }{\frac {1}{s}}\left({\frac {1}{\sqrt {2\pi }}}e^{-{\frac {1}{2}}(z/s)^{2}}\right)V_{\alpha }(s)\,ds,$
where
$V_{\alpha }(s)={\frac {{\sqrt {2\pi }}\,\Gamma ({\frac {2}{\alpha }}+1)}{\Gamma ({\frac {1}{\alpha }}+1)}}\,{\mathfrak {N}}_{\frac {\alpha }{2}}(2s^{2}),0<\alpha \leq 2.$
This transformation is named generalized Gauss transmutation since it generalizes the Gauss-Laplace transmutation, which is equivalent to $V_{1}(s)=2{\sqrt {2\pi }}\,{\mathfrak {N}}_{\frac {1}{2}}(2s^{2})=s\,e^{-s^{2}/2}$.
Connection to Gamma and Poisson distributions
The shape parameter of the Gamma and Poisson Distributions is connected to the inverse of Lévy's stability parameter $1/\alpha $. The upper regularized gamma function $Q(s,x)$ can be expressed as an incomplete integral of $e^{-{u^{\alpha }}}$ as
$Q({\frac {1}{\alpha }},z^{\alpha })={\frac {1}{\Gamma ({\frac {1}{\alpha }}+1)}}\displaystyle \int _{z}^{\infty }e^{-{u^{\alpha }}}\,du.$
By replacing $e^{-{u^{\alpha }}}$ with the decomposition and carrying out one integral, we have:
$Q({\frac {1}{\alpha }},z^{\alpha })=\displaystyle \int _{z}^{\infty }\,du\displaystyle \int _{0}^{\infty }{\frac {1}{\nu }}\left(e^{-u/\nu }\right)\,{\mathfrak {N}}_{\alpha }\left(\nu \right)\,d\nu =\displaystyle \int _{0}^{\infty }\left(e^{-z/\nu }\right)\,{\mathfrak {N}}_{\alpha }\left(\nu \right)\,d\nu .$
Reverting $({\frac {1}{\alpha }},z^{\alpha })$ back to $(s,x)$, we arrive at the decomposition of $Q(s,x)$ in terms of a stable count:
$Q(s,x)=\displaystyle \int _{0}^{\infty }e^{\left(-{x^{s}}/{\nu }\right)}\,{\mathfrak {N}}_{{1}/{s}}\left(\nu \right)\,d\nu .\,\,(s>1)$
Differentiate $Q(s,x)$ by $x$, we arrive at the desired formula:
${\begin{aligned}{\frac {1}{\Gamma (s)}}x^{s-1}e^{-x}&=\displaystyle \int _{0}^{\infty }{\frac {1}{\nu }}\left[s\,x^{s-1}e^{\left(-{x^{s}}/{\nu }\right)}\right]\,{\mathfrak {N}}_{{1}/{s}}\left(\nu \right)\,d\nu \\&=\displaystyle \int _{0}^{\infty }{\frac {1}{t}}\left[s\,{\left({\frac {x}{t}}\right)}^{s-1}e^{-{\left(x/t\right)}^{s}}\right]\,\left[{\mathfrak {N}}_{{1}/{s}}\left(t^{s}\right)\,s\,t^{s-1}\right]\,dt\,\,\,(\nu =t^{s})\\&=\displaystyle \int _{0}^{\infty }{\frac {1}{t}}\,{\text{Weibull}}\left({\frac {x}{t}};s\right)\,\left[{\mathfrak {N}}_{{1}/{s}}\left(t^{s}\right)\,s\,t^{s-1}\right]\,dt\end{aligned}}$
This is in the form of a product distribution. The term $\left[s\,{\left({\frac {x}{t}}\right)}^{s-1}e^{-{\left(x/t\right)}^{s}}\right]$ in the RHS is associated with a Weibull distribution of shape $s$. Hence, this formula connects the stable count distribution to the probability density function of a Gamma distribution (here) and the probability mass function of a Poisson distribution (here, $s\rightarrow s+1$). And the shape parameter $s$ can be regarded as inverse of Lévy's stability parameter $1/\alpha $.
Connection to Chi and Chi-squared distributions
The degrees of freedom $k$ in the chi and chi-squared Distributions can be shown to be related to $2/\alpha $. Hence, the original idea of viewing $\lambda =2/\alpha $ as an integer index in the lambda decomposition is justified here.
For the chi-squared distribution, it is straightforward since the chi-squared distribution is a special case of the gamma distribution, in that $\chi _{k}^{2}\sim {\text{Gamma}}\left({\frac {k}{2}},\theta =2\right)$. And from above, the shape parameter of a gamma distribution is $1/\alpha $.
For the chi distribution, we begin with its CDF $P\left({\frac {k}{2}},{\frac {x^{2}}{2}}\right)$, where $P(s,x)=1-Q(s,x)$. Differentiate $P\left({\frac {k}{2}},{\frac {x^{2}}{2}}\right)$ by $x$ , we have its density function as
${\begin{aligned}\chi _{k}(x)={\frac {x^{k-1}e^{-x^{2}/2}}{2^{{\frac {k}{2}}-1}\Gamma \left({\frac {k}{2}}\right)}}&=\displaystyle \int _{0}^{\infty }{\frac {1}{\nu }}\left[2^{-{\frac {k}{2}}}\,k\,x^{k-1}e^{\left(-2^{-{\frac {k}{2}}}\,{x^{k}}/{\nu }\right)}\right]\,{\mathfrak {N}}_{\frac {2}{k}}\left(\nu \right)\,d\nu \\&=\displaystyle \int _{0}^{\infty }{\frac {1}{t}}\left[k\,{\left({\frac {x}{t}}\right)}^{k-1}e^{-{\left(x/t\right)}^{k}}\right]\,\left[{\mathfrak {N}}_{\frac {2}{k}}\left(2^{-{\frac {k}{2}}}t^{k}\right)\,2^{-{\frac {k}{2}}}\,k\,t^{k-1}\right]\,dt,\,\,\,(\nu =2^{-{\frac {k}{2}}}t^{k})\\&=\displaystyle \int _{0}^{\infty }{\frac {1}{t}}\,{\text{Weibull}}\left({\frac {x}{t}};k\right)\,\left[{\mathfrak {N}}_{\frac {2}{k}}\left(2^{-{\frac {k}{2}}}t^{k}\right)\,2^{-{\frac {k}{2}}}\,k\,t^{k-1}\right]\,dt\end{aligned}}$
This formula connects $2/k$ with $\alpha $ through the ${\mathfrak {N}}_{\frac {2}{k}}\left(\cdot \right)$ term.
Connection to generalized Gamma distributions
The generalized gamma distribution is a probability distribution with two shape parameters, and is the super set of the gamma distribution, the Weibull distribution, the exponential distribution, and the half-normal distribution. Its CDF is in the form of $P(s,x^{c})=1-Q(s,x^{c})$. (Note: We use $s$ instead of $a$ for consistency and to avoid confusion with $\alpha $.) Differentiate $P(s,x^{c})$ by $x$, we arrive at the product-distribution formula:
${\begin{aligned}{\text{GenGamma}}(x;s,c)&=\displaystyle \int _{0}^{\infty }{\frac {1}{t}}\,{\text{Weibull}}\left({\frac {x}{t}};sc\right)\,\left[{\mathfrak {N}}_{\frac {1}{s}}\left(t^{sc}\right)\,sc\,t^{sc-1}\right]\,dt\,\,(s\geq 1)\end{aligned}}$
where ${\text{GenGamma}}(x;s,c)$ denotes the PDF of a generalized gamma distribution, whose CDF is parametrized as $P(s,x^{c})$. This formula connects $1/s$ with $\alpha $ through the ${\mathfrak {N}}_{\frac {1}{s}}\left(\cdot \right)$ term. The $sc$ term is an exponent representing the second degree of freedom in the shape-parameter space.
This formula is singular for the case of a Weibull distribution since $s$ must be one for ${\text{GenGamma}}(x;1,c)={\text{Weibull}}(x;c)$; but for ${\mathfrak {N}}_{\frac {1}{s}}\left(\nu \right)$ to exist, $s$ must be greater than one. When $s\rightarrow 1$, ${\mathfrak {N}}_{\frac {1}{s}}\left(\nu \right)$ is a delta function and this formula becomes trivial. The Weibull distribution has its distinct way of decomposition as following.
Connection to Weibull distribution
For a Weibull distribution whose CDF is $F(x;k,\lambda )=1-e^{-(x/\lambda )^{k}}\,\,(x>0)$, its shape parameter $k$ is equivalent to Lévy's stability parameter $\alpha $.
A similar expression of product distribution can be derived, such that the kernel is either a one-sided Laplace distribution $F(x;1,\sigma )$ or a Rayleigh distribution $F(x;2,{\sqrt {2}}\sigma )$. It begins with the complementary CDF, which comes from Lambda decomposition:
$1-F(x;k,1)={\begin{cases}\displaystyle \int _{0}^{\infty }{\frac {1}{\nu }}\,(1-F(x;1,\nu ))\left[\Gamma \left({\frac {1}{k}}+1\right){\mathfrak {N}}_{k}(\nu )\right]\,d\nu ,&1\geq k>0;{\text{or }}\\\displaystyle \int _{0}^{\infty }{\frac {1}{s}}\,(1-F(x;2,{\sqrt {2}}s))\left[{\sqrt {\frac {2}{\pi }}}\,\Gamma \left({\frac {1}{k}}+1\right)V_{k}(s)\right]\,ds,&2\geq k>0.\end{cases}}$
By taking derivative on $x$, we obtain the product distribution form of a Weibull distribution PDF ${\text{Weibull}}(x;k)$ as
${\text{Weibull}}(x;k)={\begin{cases}\displaystyle \int _{0}^{\infty }{\frac {1}{\nu }}\,{\text{Laplace}}({\frac {x}{\nu }})\left[\Gamma \left({\frac {1}{k}}+1\right){\frac {1}{\nu }}{\mathfrak {N}}_{k}(\nu )\right]\,d\nu ,&1\geq k>0;{\text{or }}\\\displaystyle \int _{0}^{\infty }{\frac {1}{s}}\,{\text{Rayleigh}}({\frac {x}{s}})\left[{\sqrt {\frac {2}{\pi }}}\,\Gamma \left({\frac {1}{k}}+1\right){\frac {1}{s}}V_{k}(s)\right]\,ds,&2\geq k>0.\end{cases}}$
where ${\text{Laplace}}(x)=e^{-x}$ and ${\text{Rayleigh}}(x)={\frac {1}{x}}e^{-x^{2}/2}$. it is clear that $k=\alpha $ from the ${\mathfrak {N}}_{k}(\nu )$ and $V_{k}(s)$ terms.
Asymptotic properties
For stable distribution family, it is essential to understand its asymptotic behaviors. From,[3] for small $\nu $,
${\begin{aligned}{\mathfrak {N}}_{\alpha }(\nu )&\rightarrow B(\alpha )\,\nu ^{\alpha },{\text{ for }}\nu \rightarrow 0{\text{ and }}B(\alpha )>0.\\\end{aligned}}$
This confirms ${\mathfrak {N}}_{\alpha }(0)=0$.
For large $\nu $,
${\begin{aligned}{\mathfrak {N}}_{\alpha }(\nu )&\rightarrow \nu ^{\frac {\alpha }{2(1-\alpha )}}e^{-A(\alpha )\,\nu ^{\frac {\alpha }{1-\alpha }}},{\text{ for }}\nu \rightarrow \infty {\text{ and }}A(\alpha )>0.\\\end{aligned}}$
This shows that the tail of ${\mathfrak {N}}_{\alpha }(\nu )$ decays exponentially at infinity. The larger $\alpha $ is, the stronger the decay.
This tail is in the form of a generalized gamma distribution, where in its $f(x;a,d,p)$ parametrization, $p={\frac {\alpha }{1-\alpha }}$, $a=A(\alpha )^{-1/p}$, and $d=1+{\frac {p}{2}}$. Hence, it is equivalent to ${\text{GenGamma}}({\frac {x}{a}};s={\frac {1}{\alpha }}-{\frac {1}{2}},c=p)$, whose CDF is parametrized as $P\left(s,\left({\frac {x}{a}}\right)^{c}\right)$.
Moments
The n-th moment $m_{n}$ of ${\mathfrak {N}}_{\alpha }(\nu )$ is the $-(n+1)$-th moment of $L_{\alpha }(x)$. All positive moments are finite. This in a way solves the thorny issue of diverging moments in the stable distribution. (See Section 2.4 of [1])
${\begin{aligned}m_{n}&=\int _{0}^{\infty }\nu ^{n}{\mathfrak {N}}_{\alpha }(\nu )d\nu ={\frac {1}{\Gamma ({\frac {1}{\alpha }}+1)}}\int _{0}^{\infty }{\frac {1}{t^{n+1}}}L_{\alpha }(t)\,dt.\\\end{aligned}}$
The analytic solution of moments is obtained through the Wright function:
${\begin{aligned}m_{n}&={\frac {1}{\Gamma ({\frac {1}{\alpha }}+1)}}\int _{0}^{\infty }\nu ^{n}W_{-\alpha ,0}(-\nu ^{\alpha })\,d\nu \\&={\frac {\Gamma ({\frac {n+1}{\alpha }})}{\Gamma (n+1)\Gamma ({\frac {1}{\alpha }})}},\,n\geq -1.\\\end{aligned}}$
where $\int _{0}^{\infty }r^{\delta }W_{-\nu ,\mu }(-r)\,dr={\frac {\Gamma (\delta +1)}{\Gamma (\nu \delta +\nu +\mu )}},\,\delta >-1,0<\nu <1,\mu >0.$(See (1.4.28) of [5])
Thus, the mean of ${\mathfrak {N}}_{\alpha }(\nu )$ is
$m_{1}={\frac {\Gamma ({\frac {2}{\alpha }})}{\Gamma ({\frac {1}{\alpha }})}}$
The variance is
$\sigma ^{2}={\frac {\Gamma ({\frac {3}{\alpha }})}{2\Gamma ({\frac {1}{\alpha }})}}-\left[{\frac {\Gamma ({\frac {2}{\alpha }})}{\Gamma ({\frac {1}{\alpha }})}}\right]^{2}$
And the lowest moment is $m_{-1}={\frac {1}{\Gamma ({\frac {1}{\alpha }}+1)}}$.
Moment generating function
The MGF can be expressed by a Fox-Wright function or Fox H-function:
${\begin{aligned}M_{\alpha }(s)&=\sum _{n=0}^{\infty }{\frac {m_{n}\,s^{n}}{n!}}={\frac {1}{\Gamma ({\frac {1}{\alpha }})}}\sum _{n=0}^{\infty }{\frac {\Gamma ({\frac {n+1}{\alpha }})\,s^{n}}{\Gamma (n+1)^{2}}}\\&={\frac {1}{\Gamma ({\frac {1}{\alpha }})}}{}_{1}\Psi _{1}\left[({\frac {1}{\alpha }},{\frac {1}{\alpha }});(1,1);s\right],\,\,{\text{or}}\\&={\frac {1}{\Gamma ({\frac {1}{\alpha }})}}H_{1,2}^{1,1}\left[-s{\bigl |}{\begin{matrix}(1-{\frac {1}{\alpha }},{\frac {1}{\alpha }})\\(0,1);(0,1)\end{matrix}}\right]\\\end{aligned}}$
As a verification, at $\alpha ={\frac {1}{2}}$, $M_{\frac {1}{2}}(s)=(1-4s)^{-{\frac {3}{2}}}$ (see below) can be Taylor-expanded to ${}_{1}\Psi _{1}\left[(2,2);(1,1);s\right]=\sum _{n=0}^{\infty }{\frac {\Gamma (2n+2)\,s^{n}}{\Gamma (n+1)^{2}}}$ via $\Gamma ({\frac {1}{2}}-n)={\sqrt {\pi }}{\frac {(-4)^{n}n!}{(2n)!}}$.
Known analytical case – quartic stable count
When $\alpha ={\frac {1}{2}}$, $L_{1/2}(x)$ is the Lévy distribution which is an inverse gamma distribution. Thus ${\mathfrak {N}}_{1/2}(\nu ;\nu _{0},\theta )$ ;\nu _{0},\theta )} is a shifted gamma distribution of shape 3/2 and scale $4\theta $,
${\mathfrak {N}}_{\frac {1}{2}}(\nu ;\nu _{0},\theta )={\frac {1}{4{\sqrt {\pi }}\theta ^{3/2}}}(\nu -\nu _{0})^{1/2}e^{-(\nu -\nu _{0})/4\theta },$ ;\nu _{0},\theta )={\frac {1}{4{\sqrt {\pi }}\theta ^{3/2}}}(\nu -\nu _{0})^{1/2}e^{-(\nu -\nu _{0})/4\theta },}
where $\nu >\nu _{0}$, $\theta >0$.
Its mean is $\nu _{0}+6\theta $ and its standard deviation is ${\sqrt {24}}\theta $. This called "quartic stable count distribution". The word "quartic" comes from Lihn's former work on the lambda distribution[6] where $\lambda =2/\alpha =4$. At this setting, many facets of stable count distribution have elegant analytical solutions.
The p-th central moments are ${\frac {2\Gamma (p+3/2)}{\Gamma (3/2)}}4^{p}\theta ^{p}$. The CDF is $ {\frac {2}{\sqrt {\pi }}}\gamma \left({\frac {3}{2}},{\frac {\nu -\nu _{0}}{4\theta }}\right)$ where $\gamma (s,x)$ is the lower incomplete gamma function. And the MGF is $M_{\frac {1}{2}}(s)=e^{s\nu _{0}}(1-4s\theta )^{-{\frac {3}{2}}}$. (See Section 3 of [1])
Special case when α → 1
As $\alpha $ becomes larger, the peak of the distribution becomes sharper. A special case of ${\mathfrak {N}}_{\alpha }(\nu )$ is when $\alpha \rightarrow 1$. The distribution behaves like a Dirac delta function,
${\mathfrak {N}}_{\alpha \rightarrow 1}(\nu )\rightarrow \delta (\nu -1),$
where $\delta (x)={\begin{cases}\infty ,&{\text{if }}x=0\\0,&{\text{if }}x\neq 0\end{cases}}$, and $\int _{0_{-}}^{0_{+}}\delta (x)dx=1$.
Series representation
Based on the series representation of the one-sided stable distribution, we have:
${\begin{aligned}{\mathfrak {N}}_{\alpha }(x)&={\frac {1}{\pi \Gamma ({\frac {1}{\alpha }}+1)}}\sum _{n=1}^{\infty }{\frac {-\sin(n(\alpha +1)\pi )}{n!}}{x}^{\alpha n}\Gamma (\alpha n+1)\\&={\frac {1}{\pi \Gamma ({\frac {1}{\alpha }}+1)}}\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}\sin(n\alpha \pi )}{n!}}{x}^{\alpha n}\Gamma (\alpha n+1)\\\end{aligned}}$.
This series representation has two interpretations:
• First, a similar form of this series was first given in Pollard (1948),[7] and in "Relation to Mittag-Leffler function", it is stated that ${\mathfrak {N}}_{\alpha }(x)={\frac {\alpha ^{2}x^{\alpha }}{\Gamma \left({\frac {1}{\alpha }}\right)}}H_{\alpha }(x^{\alpha }),$ where $H_{\alpha }(k)$ is the Laplace transform of the Mittag-Leffler function $E_{\alpha }(-x)$ .
• Secondly, this series is a special case of the Wright function $W_{\lambda ,\mu }(z)$: (See Section 1.4 of [5])
${\begin{aligned}{\mathfrak {N}}_{\alpha }(x)&={\frac {1}{\pi \Gamma ({\frac {1}{\alpha }}+1)}}\sum _{n=1}^{\infty }{\frac {(-1)^{n}{x}^{\alpha n}}{n!}}\,\sin((\alpha n+1)\pi )\Gamma (\alpha n+1)\\&={\frac {1}{\Gamma \left({\frac {1}{\alpha }}+1\right)}}W_{-\alpha ,0}(-x^{\alpha }),\,{\text{where}}\,\,W_{\lambda ,\mu }(z)=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!\,\Gamma (\lambda n+\mu )}},\lambda >-1.\\\end{aligned}}$
The proof is obtained by the reflection formula of the Gamma function: $\sin((\alpha n+1)\pi )\Gamma (\alpha n+1)=\pi /\Gamma (-\alpha n)$, which admits the mapping: $\lambda =-\alpha ,\mu =0,z=-x^{\alpha }$ in $W_{\lambda ,\mu }(z)$. The Wright representation leads to analytical solutions for many statistical properties of the stable count distribution and establish another connection to fractional calculus.
Applications
Stable count distribution can represent the daily distribution of VIX quite well. It is hypothesized that VIX is distributed like ${\mathfrak {N}}_{\frac {1}{2}}(\nu ;\nu _{0},\theta )$ ;\nu _{0},\theta )} with $\nu _{0}=10.4$ and $\theta =1.6$ (See Section 7 of [1]). Thus the stable count distribution is the first-order marginal distribution of a volatility process. In this context, $\nu _{0}$ is called the "floor volatility". In practice, VIX rarely drops below 10. This phenomenon justifies the concept of "floor volatility". A sample of the fit is shown below:
One form of mean-reverting SDE for ${\mathfrak {N}}_{\frac {1}{2}}(\nu ;\nu _{0},\theta )$ ;\nu _{0},\theta )} is based on a modified Cox–Ingersoll–Ross (CIR) model. Assume $S_{t}$ is the volatility process, we have
$dS_{t}={\frac {\sigma ^{2}}{8\theta }}(6\theta +\nu _{0}-S_{t})\,dt+\sigma {\sqrt {S_{t}-\nu _{0}}}\,dW,$
where $\sigma $ is the so-called "vol of vol". The "vol of vol" for VIX is called VVIX, which has a typical value of about 85.[8]
This SDE is analytically tractable and satisfies the Feller condition, thus $S_{t}$ would never go below $\nu _{0}$. But there is a subtle issue between theory and practice. There has been about 0.6% probability that VIX did go below $\nu _{0}$. This is called "spillover". To address it, one can replace the square root term with ${\sqrt {\max(S_{t}-\nu _{0},\delta \nu _{0})}}$, where $\delta \nu _{0}\approx 0.01\,\nu _{0}$ provides a small leakage channel for $S_{t}$ to drift slightly below $\nu _{0}$.
Extremely low VIX reading indicates a very complacent market. Thus the spillover condition, $S_{t}<\nu _{0}$, carries a certain significance - When it occurs, it usually indicates the calm before the storm in the business cycle.
Generation of Random Variables
As the modified CIR model above shows, it takes another input parameter $\sigma $ to simulate sequences of stable count random variables. The mean-reverting stochastic process takes the form of
$dS_{t}=\sigma ^{2}\mu _{\alpha }\left({\frac {S_{t}}{\theta }}\right)\,dt+\sigma {\sqrt {S_{t}}}\,dW,$
which should produce $\{S_{t}\}$ that distributes like ${\mathfrak {N}}_{\alpha }(\nu ;\theta )$ ;\theta )} as $t\rightarrow \infty $. And $\sigma $ is a user-specified preference for how fast $S_{t}$ should change.
By solving the Fokker-Planck equation, the solution for $\mu _{\alpha }(x)$ in terms of ${\mathfrak {N}}_{\alpha }(x)$ is
${\begin{array}{lcl}\mu _{\alpha }(x)&=&\displaystyle {\frac {1}{2}}{\frac {\left(x{d \over dx}+1\right){\mathfrak {N}}_{\alpha }(x)}{{\mathfrak {N}}_{\alpha }(x)}}\\&=&\displaystyle {\frac {1}{2}}\left[x{d \over dx}\left(\log {\mathfrak {N}}_{\alpha }(x)\right)+1\right]\end{array}}$
It can also be written as a ratio of two Wright functions,
${\begin{array}{lcl}\mu _{\alpha }(x)&=&\displaystyle -{\frac {1}{2}}{\frac {W_{-\alpha ,-1}(-x^{\alpha })}{\Gamma ({\frac {1}{\alpha }}+1)\,{\mathfrak {N}}_{\alpha }(x)}}\\&=&\displaystyle -{\frac {1}{2}}{\frac {W_{-\alpha ,-1}(-x^{\alpha })}{W_{-\alpha ,0}(-x^{\alpha })}}\end{array}}$
When $\alpha =1/2$, this process is reduced to the modified CIR model where $\mu _{1/2}(x)={\frac {1}{8}}(6-x)$. This is the only special case where $\mu _{\alpha }(x)$ is a straight line.
Stable Extension of the CIR Model
By relaxing the rigid relation between the $\sigma ^{2}$ term and the $\sigma $ term above, the stable extension of the CIR model can be constructed as
$dr_{t}=a\,\left[{\frac {8b}{6}}\,\mu _{\alpha }\left({\frac {6}{b}}r_{t}\right)\right]\,dt+\sigma {\sqrt {r_{t}}}\,dW,$
which is reduced to the original CIR model at $\alpha =1/2$: $dr_{t}=a\left(b-r_{t}\right)dt+\sigma {\sqrt {r_{t}}}\,dW$. Hence, the parameter $a$ controls the mean-reverting speed, the location parameter $b$ sets where the mean is, $\sigma $ is the volatility parameter, and $\alpha $ is the shape parameter for the stable law.
By solving the Fokker-Planck equation, the solution for the PDF $p(x)$ at $r_{\infty }$ is
${\begin{array}{lcl}p(x)&\propto &\displaystyle \exp \left[\int ^{x}{\frac {dx}{x}}\left(2D\,\mu _{\alpha }\left({\frac {6}{b}}x\right)-1\right)\right],{\text{ where }}D={\frac {4ab}{3\sigma ^{2}}}\\&=&\displaystyle {\mathfrak {N}}_{\alpha }\left({\frac {6}{b}}x\right)^{D}\,x^{D-1}\end{array}}$
To make sense of this solution, consider asymptotically for large $x$, $p(x)$'s tail is still in the form of a generalized gamma distribution, where in its $f(x;a',d,p)$ parametrization, $p={\frac {\alpha }{1-\alpha }}$, $a'={\frac {b}{6}}(D\,A(\alpha ))^{-1/p}$, and $d=D\left(1+{\frac {p}{2}}\right)$. It is reduced to the original CIR model at $\alpha =1/2$ where $p(x)\propto x^{d-1}e^{-x/a'}$ with $d={\frac {2ab}{\sigma ^{2}}}$ and $A(\alpha )={\frac {1}{4}}$; hence ${\frac {1}{a'}}={\frac {6}{b}}\left({\frac {D}{4}}\right)={\frac {2a}{\sigma ^{2}}}$.
Fractional calculus
Relation to Mittag-Leffler function
From Section 4 of,[9] the inverse Laplace transform $H_{\alpha }(k)$ of the Mittag-Leffler function $E_{\alpha }(-x)$ is ($k>0$)
$H_{\alpha }(k)={\mathcal {L}}^{-1}\{E_{\alpha }(-x)\}(k)={\frac {2}{\pi }}\int _{0}^{\infty }E_{2\alpha }(-t^{2})\cos(kt)\,dt.$
On the other hand, the following relation was given by Pollard (1948),[7]
$H_{\alpha }(k)={\frac {1}{\alpha }}{\frac {1}{k^{1+1/\alpha }}}L_{\alpha }\left({\frac {1}{k^{1/\alpha }}}\right).$
Thus by $k=\nu ^{\alpha }$, we obtain the relation between stable count distribution and Mittag-Leffter function:
${\mathfrak {N}}_{\alpha }(\nu )={\frac {\alpha ^{2}\nu ^{\alpha }}{\Gamma \left({\frac {1}{\alpha }}\right)}}H_{\alpha }(\nu ^{\alpha }).$
This relation can be verified quickly at $\alpha ={\frac {1}{2}}$ where $H_{\frac {1}{2}}(k)={\frac {1}{\sqrt {\pi }}}\,e^{-k^{2}/4}$ and $k^{2}=\nu $. This leads to the well-known quartic stable count result:
${\mathfrak {N}}_{\frac {1}{2}}(\nu )={\frac {\nu ^{1/2}}{4\,\Gamma (2)}}\times {\frac {1}{\sqrt {\pi }}}\,e^{-\nu /4}={\frac {1}{4\,{\sqrt {\pi }}}}\nu ^{1/2}\,e^{-\nu /4}.$
Relation to time-fractional Fokker-Planck equation
The ordinary Fokker-Planck equation (FPE) is ${\frac {\partial P_{1}(x,t)}{\partial t}}=K_{1}\,{\tilde {L}}_{FP}P_{1}(x,t)$, where ${\tilde {L}}_{FP}={\frac {\partial }{\partial x}}{\frac {F(x)}{T}}+{\frac {\partial ^{2}}{\partial x^{2}}}$ is the Fokker-Planck space operator, $K_{1}$ is the diffusion coefficient, $T$ is the temperature, and $F(x)$ is the external field. The time-fractional FPE introduces the additional fractional derivative $\,_{0}D_{t}^{1-\alpha }$ such that ${\frac {\partial P_{\alpha }(x,t)}{\partial t}}=K_{\alpha }\,_{0}D_{t}^{1-\alpha }{\tilde {L}}_{FP}P_{\alpha }(x,t)$, where $K_{\alpha }$ is the fractional diffusion coefficient.
Let $k=s/t^{\alpha }$ in $H_{\alpha }(k)$, we obtain the kernel for the time-fractional FPE (Eq (16) of [10])
$n(s,t)={\frac {1}{\alpha }}{\frac {t}{s^{1+1/\alpha }}}L_{\alpha }\left({\frac {t}{s^{1/\alpha }}}\right)$
from which the fractional density $P_{\alpha }(x,t)$ can be calculated from an ordinary solution $P_{1}(x,t)$ via
$P_{\alpha }(x,t)=\int _{0}^{\infty }n\left({\frac {s}{K}},t\right)\,P_{1}(x,s)\,ds,{\text{ where }}K={\frac {K_{\alpha }}{K_{1}}}.$
Since $n({\frac {s}{K}},t)\,ds=\Gamma \left({\frac {1}{\alpha }}+1\right){\frac {1}{\nu }}\,{\mathfrak {N}}_{\alpha }(\nu ;\theta =K^{1/\alpha })\,d\nu $ ;\theta =K^{1/\alpha })\,d\nu } via change of variable $\nu t=s^{1/\alpha }$, the above integral becomes the product distribution with ${\mathfrak {N}}_{\alpha }(\nu )$, similar to the "lambda decomposition" concept, and scaling of time $t\Rightarrow (\nu t)^{\alpha }$:
$P_{\alpha }(x,t)=\Gamma \left({\frac {1}{\alpha }}+1\right)\int _{0}^{\infty }{\frac {1}{\nu }}\,{\mathfrak {N}}_{\alpha }(\nu ;\theta =K^{1/\alpha })\,P_{1}(x,(\nu t)^{\alpha })\,d\nu .$ ;\theta =K^{1/\alpha })\,P_{1}(x,(\nu t)^{\alpha })\,d\nu .}
Here ${\mathfrak {N}}_{\alpha }(\nu ;\theta =K^{1/\alpha })$ ;\theta =K^{1/\alpha })} is interpreted as the distribution of impurity, expressed in the unit of $K^{1/\alpha }$, that causes the anomalous diffusion.
See also
• Lévy flight
• Lévy process
• Fractional calculus
• Anomalous diffusion
• Incomplete gamma function and Gamma distribution
• Poisson distribution
References
1. Lihn, Stephen (2017). "A Theory of Asset Return and Volatility Under Stable Law and Stable Lambda Distribution". SSRN 3046732.
2. Paul Lévy, Calcul des probabilités 1925
3. Penson, K. A.; Górska, K. (2010-11-17). "Exact and Explicit Probability Densities for One-Sided Lévy Stable Distributions". Physical Review Letters. 105 (21): 210604. arXiv:1007.0193. Bibcode:2010PhRvL.105u0604P. doi:10.1103/PhysRevLett.105.210604. PMID 21231282. S2CID 27497684.
4. Lihn, Stephen (2020). "Stable Count Distribution for the Volatility Indices and Space-Time Generalized Stable Characteristic Function". SSRN 3659383.
5. Mathai, A.M.; Haubold, H.J. (2017). Fractional and Multivariable Calculus. Springer Optimization and Its Applications. Vol. 122. Cham: Springer International Publishing. doi:10.1007/978-3-319-59993-9. ISBN 9783319599922.
6. Lihn, Stephen H. T. (2017-01-26). "From Volatility Smile to Risk Neutral Probability and Closed Form Solution of Local Volatility Function". Rochester, NY. doi:10.2139/ssrn.2906522. S2CID 157746678. SSRN 2906522. {{cite journal}}: Cite journal requires |journal= (help)
7. Pollard, Harry (1948-12-01). "The completely monotonic character of the Mittag-Leffler function $E_a \left( { - x} \right)$". Bulletin of the American Mathematical Society. 54 (12): 1115–1117. doi:10.1090/S0002-9904-1948-09132-7. ISSN 0002-9904.
8. "DOUBLE THE FUN WITH CBOE's VVIX Index" (PDF). www.cboe.com. Retrieved 2019-08-09.
9. Saxena, R. K.; Mathai, A. M.; Haubold, H. J. (2009-09-01). "Mittag-Leffler Functions and Their Applications". arXiv:0909.0230 [math.CA].
10. Barkai, E. (2001-03-29). "Fractional Fokker-Planck equation, solution, and application". Physical Review E. 63 (4): 046118. Bibcode:2001PhRvE..63d6118B. doi:10.1103/PhysRevE.63.046118. ISSN 1063-651X. PMID 11308923. S2CID 18112355.
External links
• R Package 'stabledist' by Diethelm Wuertz, Martin Maechler and Rmetrics core team members. Computes stable density, probability, quantiles, and random numbers. Updated Sept. 12, 2016.
Probability distributions (list)
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Continuous
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Wikipedia
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Linear stability
In mathematics, in the theory of differential equations and dynamical systems, a particular stationary or quasistationary solution to a nonlinear system is called linearly unstable if the linearization of the equation at this solution has the form $dr/dt=Ar$, where r is the perturbation to the steady state, A is a linear operator whose spectrum contains eigenvalues with positive real part. If all the eigenvalues have negative real part, then the solution is called linearly stable. Other names for linear stability include exponential stability or stability in terms of first approximation.[1][2] If there exist an eigenvalue with zero real part then the question about stability cannot be solved on the basis of the first approximation and we approach the so-called "centre and focus problem".[3]
Examples
Ordinary differential equation
The differential equation
${\frac {dx}{dt}}=x-x^{2}$
has two stationary (time-independent) solutions: x = 0 and x = 1. The linearization at x = 0 has the form ${\frac {dx}{dt}}=x$. The linearized operator is A0 = 1. The only eigenvalue is $\lambda =1$. The solutions to this equation grow exponentially; the stationary point x = 0 is linearly unstable.
To derive the linearization at x = 1, one writes ${\frac {dr}{dt}}=(1+r)-(1+r)^{2}=-r-r^{2}$, where r = x − 1. The linearized equation is then ${\frac {dr}{dt}}=-r$; the linearized operator is A1 = −1, the only eigenvalue is $\lambda =-1$, hence this stationary point is linearly stable.
Nonlinear Schrödinger Equation
The nonlinear Schrödinger equation
$i{\frac {\partial u}{\partial t}}=-{\frac {\partial ^{2}u}{\partial x^{2}}}-|u|^{2k}u,$
where u(x,t) ∈ C and k > 0, has solitary wave solutions of the form $\phi (x)e^{-i\omega t}$.[4]
To derive the linearization at a solitary wave, one considers the solution in the form $u(x,t)=(\phi (x)+r(x,t))e^{-i\omega t}$. The linearized equation on $r(x,t)$ is given by
${\frac {\partial }{\partial t}}{\begin{bmatrix}{\text{Re}}\,r\\{\text{Im}}\,r\end{bmatrix}}=A{\begin{bmatrix}{\text{Re}}\,r\\{\text{Im}}\,r\end{bmatrix}},$
where
$A={\begin{bmatrix}0&L_{0}\\-L_{1}&0\end{bmatrix}},$
with
$L_{0}=-{\frac {\partial }{\partial x^{2}}}-k\phi ^{2}-\omega $
and
$L_{1}=-{\frac {\partial }{\partial x^{2}}}-(2k+1)\phi ^{2}-\omega $
the differential operators. According to Vakhitov–Kolokolov stability criterion,[5] when k > 2, the spectrum of A has positive point eigenvalues, so that the linearized equation is linearly (exponentially) unstable; for 0 < k ≤ 2, the spectrum of A is purely imaginary, so that the corresponding solitary waves are linearly stable.
It should be mentioned that linear stability does not automatically imply stability; in particular, when k = 2, the solitary waves are unstable. On the other hand, for 0 < k < 2, the solitary waves are not only linearly stable but also orbitally stable.[6]
See also
• Asymptotic stability
• Linearization (stability analysis)
• Lyapunov stability
• Orbital stability
• Stability theory
• Vakhitov–Kolokolov stability criterion
References
1. V.I. Arnold, Ordinary Differential Equations. MIT Press, Cambridge, MA (1973)
2. P. Glendinning, Stability, instability and chaos: an introduction to the theory of nonlinear differential equations. Cambridge university press, 1994.
3. V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations", Princeton Univ. Press (1960)
4. H. Berestycki and P.-L. Lions (1983). "Nonlinear scalar field equations. I. Existence of a ground state". Arch. Rational Mech. Anal. 82 (4): 313–345. Bibcode:1983ArRMA..82..313B. doi:10.1007/BF00250555. S2CID 123081616.
5. N.G. Vakhitov and A.A. Kolokolov (1973). "Stationary solutions of the wave equation in the medium with nonlinearity saturation". Radiophys. Quantum Electron. 16 (7): 783–789. Bibcode:1973R&QE...16..783V. doi:10.1007/BF01031343. S2CID 123386885.
6. Manoussos Grillakis, Jalal Shatah, and Walter Strauss (1987). "Stability theory of solitary waves in the presence of symmetry. I". J. Funct. Anal. 74: 160–197. doi:10.1016/0022-1236(87)90044-9.{{cite journal}}: CS1 maint: multiple names: authors list (link)
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Wikipedia
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Fractional matching
In graph theory, a fractional matching is a generalization of a matching in which, intuitively, each vertex may be broken into fractions that are matched to different neighbor vertices.
Definition
Given a graph G = (V, E), a fractional matching in G is a function that assigns, to each edge e in E, a fraction f(e) in [0, 1], such that for every vertex v in V, the sum of fractions of edges adjacent to v is at most 1:[1]
$\forall v\in V:\sum _{e\ni v}f(e)\leq 1$
A matching in the traditional sense is a special case of a fractional matching, in which the fraction of every edge is either 0 or 1: f(e) = 1 if e is in the matching, and f(e) = 0 if it is not. For this reason, in the context of fractional matchings, usual matchings are sometimes called integral matchings.
The size of an integral matching is the number of edges in the matching, and the matching number $\nu (G)$ of a graph G is the largest size of a matching in G. Analogously, the size of a fractional matching is the sum of fractions of all edges. The fractional matching number of a graph G is the largest size of a fractional matching in G. It is often denoted by $\nu ^{*}(G)$.[2] Since a matching is a special case of a fractional matching, for every graph G one has that the integral matching number of G is less than or equal to the fractional matching number of G; in symbols:
$\nu (G)\leq \nu ^{*}(G).$
A graph in which $\nu (G)=\nu ^{*}(G)$ is called a stable graph.[3] Every bipartite graph is stable; this means that in every bipartite graph, the fractional matching number is an integer and it equals the integral matching number.
In a general graph, $\nu (G)>{\frac {2}{3}}\nu ^{*}(G).$ The fractional matching number is either an integer or a half-integer.[4]
Matrix presentation
For a bipartite graph G = (X+Y, E), a fractional matching can be presented as a matrix with |X| rows and |Y| columns. The value of the entry in row x and column y is the fraction of the edge (x,y).
Perfect fractional matching
A fractional matching is called perfect if the sum of weights adjacent to each vertex is exactly 1. The size of a perfect matching is exactly |V|/2.
In a bipartite graph G = (X+Y, E), a fractional matching is called X-perfect if the sum of weights adjacent to each vertex of X is exactly 1. The size of an X-perfect fractional matching is exactly |X|.
For a bipartite graph G = (X+Y, E), the following are equivalent:
• G admits an X-perfect integral matching,
• G admits an X-perfect fractional matching, and
• G satisfies the condition to Hall's marriage theorem.
The first condition implies the second because an integral matching is a fractional matching. The second implies the third because, for each subset W of X, the sum of weights near vertices of W is |W|, so the edges adjacent to them are necessarily adjacent to at least |W| vertices of Y. By Hall's marriage theorem, the last condition implies the first one.[5]
In a general graph, the above conditions are not equivalent - the largest fractional matching can be larger than the largest integral matching. For example, a 3-cycle admits a perfect fractional matching of size 3/2 (the fraction of every edge is 1/2), but does not admit perfect integral matching - the largest integral matching is of size 1.
Algorithmic aspects
A largest fractional matching in a graph can be easily found by linear programming, or alternatively by a maximum flow algorithm. In a bipartite graph, it is possible to convert a maximum fractional matching to a maximum integral matching of the same size. This leads to a simple polynomial-time algorithm for finding a maximum matching in a bipartite graph.[6]
If G is a bipartite graph with |X| = |Y| = n, and M is a perfect fractional matching, then the matrix representation of M is a doubly stochastic matrix - the sum of elements in each row and each column is 1. Birkhoff's algorithm can be used to decompose the matrix into a convex sum of at most n2-2n+2 permutation matrices. This corresponds to decomposing M into a convex sum of at most n2-2n+2 perfect matchings.
Maximum-cardinality fractional matching
A fractional matching of maximum cardinality (i.e., maximum sum of fractions) can be found by linear programming. There is also a strongly-polynomial time algorithm,[7] using augmenting paths, that runs in time $O(|V||E|)$.
Maximum-weight fractional matching
Suppose each edge on the graph has a weight. A fractional matching of maximum weight in a graph can be found by linear programming. In a bipartite graph, it is possible to convert a maximum-weight fractional matching to a maximum-weight integral matching of the same size, in the following way:[8]
• Let f be the fractional matching.
• Let H be a subgraph of G containing only the edges e with non-integral fraction, 0<f(e)<1.
• If H is empty, then we are done.
• if H has a cycle, then it must be even-length (since the graph is bipartite), so we can construct a new fractional matching f1 by transferring a small fraction ε from even edges to odd edges, and a new fractional matching f2 by transferring ε from odd edges to even edges. Since f is the average of f1 and f2, the weight of f is the average between the weight of f1 and of f2. Since f has maximum weight, all three matchings must have the same weight. There exists a choice of ε for which at least one of f1 or f2 has less non-integral fractions. Continuing in the same way leads to an integral matching of the same weight.
• Suppose H has no cycle, and let P be a longest path in H. The fraction of every edge adjacent to the first or last vertex in P must be 0 (if it is 1 - the first / last edge in P violates the fractional matching condition; if it is in (0,1) - then P is not the longest). Therefore, we can construct new fractional matchings f1 and f2 by transferring ε from odd edges to even edges or vice versa. Again f1 and f2 must have maximum weight, and at least one of them has less non-integral fractions.
Fractional matching polytope
Main article: Matching polytope
Given a graph G = (V,E), the fractional matching polytope of G is a convex polytope that represents all possible fractional matchings of G. It is a polytope in R|E| - the |E|-dimensional Euclidean space. Each point (x1,...,x|E|) in the polytope represents a matching in which the fraction of each edge e is xe. The polytope is defined by |E| non-negativity constraints (xe ≥ 0 for all e in E) and |V| vertex constraints (the sum of xe, for all edges e that are adjacent to a vertex v, is at most 1). In a bipartite graph, the vertices of the fractional matching polytope are all integral.
References
1. Aharoni, Ron; Kessler, Ofra (1990-10-15). "On a possible extension of Hall's theorem to bipartite hypergraphs". Discrete Mathematics. 84 (3): 309–313. doi:10.1016/0012-365X(90)90136-6. ISSN 0012-365X.
2. Liu, Yan; Liu, Guizhen (2002). "The fractional matching numbers of graphs". Networks. 40 (4): 228–231. doi:10.1002/net.10047. ISSN 1097-0037. S2CID 43698695.
3. Beckenbach, Isabel; Borndörfer, Ralf (2018-10-01). "Hall's and Kőnig's theorem in graphs and hypergraphs". Discrete Mathematics. 341 (10): 2753–2761. doi:10.1016/j.disc.2018.06.013. ISSN 0012-365X. S2CID 52067804.
4. Füredi, Zoltán (1981-06-01). "Maximum degree and fractional matchings in uniform hypergraphs". Combinatorica. 1 (2): 155–162. doi:10.1007/BF02579271. ISSN 1439-6912. S2CID 10530732.
5. "co.combinatorics - Fractional Matching version of Hall's Marriage theorem". MathOverflow. Retrieved 2020-06-29.
6. Gärtner, Bernd; Matoušek, Jiří (2006). Understanding and Using Linear Programming. Berlin: Springer. ISBN 3-540-30697-8.
7. Bourjolly, Jean-Marie; Pulleyblank, William R. (1989-01-01). "König-Egerváry graphs, 2-bicritical graphs and fractional matchings". Discrete Applied Mathematics. 24 (1): 63–82. doi:10.1016/0166-218X(92)90273-D. ISSN 0166-218X.
8. Vazirani, Umesh (2012). "Maximum Weighted Matchings" (PDF). U. C. Berkeley.{{cite web}}: CS1 maint: url-status (link)
See also
• Fractional coloring
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Wikipedia
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Stable homotopy theory
In mathematics, stable homotopy theory is the part of homotopy theory (and thus algebraic topology) concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor. A founding result was the Freudenthal suspension theorem, which states that given any pointed space $X$, the homotopy groups $\pi _{n+k}(\Sigma ^{n}X)$ stabilize for $n$ sufficiently large. In particular, the homotopy groups of spheres $\pi _{n+k}(S^{n})$ stabilize for $n\geq k+2$. For example,
$\langle {\text{id}}_{S^{1}}\rangle =\mathbb {Z} =\pi _{1}(S^{1})\cong \pi _{2}(S^{2})\cong \pi _{3}(S^{3})\cong \cdots $
$\langle \eta \rangle =\mathbb {Z} =\pi _{3}(S^{2})\to \pi _{4}(S^{3})\cong \pi _{5}(S^{4})\cong \cdots $
In the two examples above all the maps between homotopy groups are applications of the suspension functor. The first example is a standard corollary of the Hurewicz theorem, that $\pi _{n}(S^{n})\cong \mathbb {Z} $. In the second example the Hopf map, $\eta $, is mapped to its suspension $\Sigma \eta $, which generates $\pi _{4}(S^{3})\cong \mathbb {Z} /2$.
One of the most important problems in stable homotopy theory is the computation of stable homotopy groups of spheres. According to Freudenthal's theorem, in the stable range the homotopy groups of spheres depend not on the specific dimensions of the spheres in the domain and target, but on the difference in those dimensions. With this in mind the k-th stable stem is
$\pi _{k}^{s}:=\lim _{n}\pi _{n+k}(S^{n})$.
This is an abelian group for all k. It is a theorem of Jean-Pierre Serre[1] that these groups are finite for $k\neq 0$. In fact, composition makes $\pi _{*}^{S}$ into a graded ring. A theorem of Goro Nishida[2] states that all elements of positive grading in this ring are nilpotent. Thus the only prime ideals are the primes in $\pi _{0}^{s}\cong \mathbb {Z} $. So the structure of $\pi _{*}^{s}$ is quite complicated.
In the modern treatment of stable homotopy theory, spaces are typically replaced by spectra. Following this line of thought, an entire stable homotopy category can be created. This category has many nice properties that are not present in the (unstable) homotopy category of spaces, following from the fact that the suspension functor becomes invertible. For example, the notion of cofibration sequence and fibration sequence are equivalent.
See also
• Adams filtration
• Adams spectral sequence
• Chromatic homotopy theory
• Equivariant stable homotopy theory
• Nilpotence theorem
References
1. Serre, Jean-Pierre (1953). "Groupes d'homotopie et classes de groupes abelien". Annals of Mathematics. 58 (2): 258–295. doi:10.2307/1969789. JSTOR 1969789.
2. Nishida, Goro (1973), "The nilpotency of elements of the stable homotopy groups of spheres", Journal of the Mathematical Society of Japan, 25 (4): 707–732, doi:10.2969/jmsj/02540707, hdl:2433/220059, ISSN 0025-5645, MR 0341485
• Adams, J. Frank (1966), Stable homotopy theory, Second revised edition. Lectures delivered at the University of California at Berkeley, vol. 1961, Berlin, New York: Springer-Verlag, MR 0196742
• May, J. Peter (1999), "Stable Algebraic Topology, 1945–1966" (PDF), Stable algebraic topology, 1945--1966, Amsterdam: North-Holland, pp. 665–723, CiteSeerX 10.1.1.30.6299, doi:10.1016/B978-044482375-5/50025-0, ISBN 9780444823755, MR 1721119
• Ravenel, Douglas C. (1992), Nilpotence and periodicity in stable homotopy theory, Annals of Mathematics Studies, vol. 128, Princeton University Press, ISBN 978-0-691-02572-8, MR 1192553
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Stable ∞-category
In category theory, a branch of mathematics, a stable ∞-category is an ∞-category such that[1]
• (i) It has a zero object.
• (ii) Every morphism in it admits a fiber and cofiber.
• (iii) A triangle in it is a fiber sequence if and only if it is a cofiber sequence.
The homotopy category of a stable ∞-category is triangulated.[2] A stable ∞-category admits finite limits and colimits.[3]
Examples: the derived category of an abelian category and the ∞-category of spectra are both stable.
A stabilization of an ∞-category C having finite limits and base point is a functor from the stable ∞-category S to C. It preserves limit. The objects in the image have the structure of infinite loop spaces; whence, the notion is a generalization of the corresponding notion (stabilization (topology)) in classical algebraic topology.
By definition, the t-structure of a stable ∞-category is the t-structure of its homotopy category. Let C be a stable ∞-category with a t-structure. Then every filtered object $X(i),i\in \mathbb {Z} $ in C gives rise to a spectral sequence $E_{r}^{p,q}$, which, under some conditions, converges to $\pi _{p+q}\operatorname {colim} X(i).$[4] By the Dold–Kan correspondence, this generalizes the construction of the spectral sequence associated to a filtered chain complex of abelian groups.
Notes
1. Lurie, Definition 1.1.1.9.
2. Lurie, Theorem 1.1.2.14.
3. Lurie, Proposition 1.1.3.4.
4. Lurie, Construction 1.2.2.6.
References
• Lurie, J. "Higher Algebra" (PDF). last updated August 2017
Category theory
Key concepts
Key concepts
• Category
• Adjoint functors
• CCC
• Commutative diagram
• Concrete category
• End
• Exponential
• Functor
• Kan extension
• Morphism
• Natural transformation
• Universal property
Universal constructions
Limits
• Terminal objects
• Products
• Equalizers
• Kernels
• Pullbacks
• Inverse limit
Colimits
• Initial objects
• Coproducts
• Coequalizers
• Cokernels and quotients
• Pushout
• Direct limit
Algebraic categories
• Sets
• Relations
• Magmas
• Groups
• Abelian groups
• Rings (Fields)
• Modules (Vector spaces)
Constructions on categories
• Free category
• Functor category
• Kleisli category
• Opposite category
• Quotient category
• Product category
• Comma category
• Subcategory
Higher category theory
Key concepts
• Categorification
• Enriched category
• Higher-dimensional algebra
• Homotopy hypothesis
• Model category
• Simplex category
• String diagram
• Topos
n-categories
Weak n-categories
• Bicategory (pseudofunctor)
• Tricategory
• Tetracategory
• Kan complex
• ∞-groupoid
• ∞-topos
Strict n-categories
• 2-category (2-functor)
• 3-category
Categorified concepts
• 2-group
• 2-ring
• En-ring
• (Traced)(Symmetric) monoidal category
• n-group
• n-monoid
• Category
• Outline
• Glossary
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Wikipedia
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Stable distribution
In probability theory, a distribution is said to be stable if a linear combination of two independent random variables with this distribution has the same distribution, up to location and scale parameters. A random variable is said to be stable if its distribution is stable. The stable distribution family is also sometimes referred to as the Lévy alpha-stable distribution, after Paul Lévy, the first mathematician to have studied it.[1][2]
Stable
Probability density function
Symmetric $\alpha $-stable distributions with unit scale factor
Skewed centered stable distributions with unit scale factor
Cumulative distribution function
CDFs for symmetric $\alpha $-stable distributions
CDFs for skewed centered stable distributions
Parameters
$\alpha \in (0,2]$ — stability parameter
$\beta $ ∈ [−1, 1] — skewness parameter (note that skewness is undefined)
c ∈ (0, ∞) — scale parameter
μ ∈ (−∞, ∞) — location parameter
Support
x ∈ [μ, +∞) if $\alpha <1$ and $\beta =1$
x ∈ (-∞, μ] if $\alpha <1$ and $\beta =-1$
x ∈ R otherwise
PDF not analytically expressible, except for some parameter values
CDF not analytically expressible, except for certain parameter values
Mean μ when $\alpha >1$, otherwise undefined
Median μ when $\beta =0$, otherwise not analytically expressible
Mode μ when $\beta =0$, otherwise not analytically expressible
Variance 2c2 when $\alpha =2$, otherwise infinite
Skewness 0 when $\alpha =2$, otherwise undefined
Ex. kurtosis 0 when $\alpha =2$, otherwise undefined
Entropy not analytically expressible, except for certain parameter values
MGF $\exp \!\left(t\mu +c^{2}t^{2}\right)$ when $\alpha =2$, otherwise undefined
CF
$\exp \!{\Big [}\;it\mu -|c\,t|^{\alpha }\,(1-i\beta \operatorname {sgn}(t)\Phi )\;{\Big ]},$
where $\Phi ={\begin{cases}\tan {\tfrac {\pi \alpha }{2}}&{\text{if }}\alpha \neq 1\\-{\tfrac {2}{\pi }}\log |t|&{\text{if }}\alpha =1\end{cases}}$
Of the four parameters defining the family, most attention has been focused on the stability parameter, $\alpha $ (see panel). Stable distributions have $0<\alpha \leq 2$, with the upper bound corresponding to the normal distribution, and $\alpha =1$ to the Cauchy distribution. The distributions have undefined variance for $\alpha <2$, and undefined mean for $\alpha \leq 1$. The importance of stable probability distributions is that they are "attractors" for properly normed sums of independent and identically distributed (iid) random variables. The normal distribution defines a family of stable distributions. By the classical central limit theorem the properly normed sum of a set of random variables, each with finite variance, will tend toward a normal distribution as the number of variables increases. Without the finite variance assumption, the limit may be a stable distribution that is not normal. Mandelbrot referred to such distributions as "stable Paretian distributions",[3][4][5] after Vilfredo Pareto. In particular, he referred to those maximally skewed in the positive direction with $1<\alpha <2$ as "Pareto–Lévy distributions",[1] which he regarded as better descriptions of stock and commodity prices than normal distributions.[6]
Definition
A non-degenerate distribution is a stable distribution if it satisfies the following property:
Let X1 and X2 be independent realizations of a random variable X. Then X is said to be stable if for any constants a > 0 and b > 0 the random variable aX1 + bX2 has the same distribution as cX + d for some constants c > 0 and d. The distribution is said to be strictly stable if this holds with d = 0.[7]
Since the normal distribution, the Cauchy distribution, and the Lévy distribution all have the above property, it follows that they are special cases of stable distributions.
Such distributions form a four-parameter family of continuous probability distributions parametrized by location and scale parameters μ and c, respectively, and two shape parameters $\beta $ and $\alpha $, roughly corresponding to measures of asymmetry and concentration, respectively (see the figures).
The characteristic function $\varphi (t)$ of any probability distribution is the Fourier transform of its probability density function $f(x)$. The density function is therefore the inverse Fourier transform of the characteristic function:[8]
$f(x)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }\varphi (t)e^{-ixt}\,dt.$
Although the probability density function for a general stable distribution cannot be written analytically, the general characteristic function can be expressed analytically. A random variable X is called stable if its characteristic function can be written as[7][9]
$\varphi (t;\alpha ,\beta ,c,\mu )=\exp \left(it\mu -|ct|^{\alpha }\left(1-i\beta \operatorname {sgn}(t)\Phi \right)\right)$
where sgn(t) is just the sign of t and
$\Phi ={\begin{cases}\tan \left({\frac {\pi \alpha }{2}}\right)&\alpha \neq 1\\-{\frac {2}{\pi }}\log |t|&\alpha =1\end{cases}}$
μ ∈ R is a shift parameter, $\beta \in [-1,1]$, called the skewness parameter, is a measure of asymmetry. Notice that in this context the usual skewness is not well defined, as for $\alpha <2$ the distribution does not admit 2nd or higher moments, and the usual skewness definition is the 3rd central moment.
The reason this gives a stable distribution is that the characteristic function for the sum of two independent random variables equals the product of the two corresponding characteristic functions. Adding two random variables from a stable distribution gives something with the same values of $\alpha $ and $\beta $, but possibly different values of μ and c.
Not every function is the characteristic function of a legitimate probability distribution (that is, one whose cumulative distribution function is real and goes from 0 to 1 without decreasing), but the characteristic functions given above will be legitimate so long as the parameters are in their ranges. The value of the characteristic function at some value t is the complex conjugate of its value at −t as it should be so that the probability distribution function will be real.
In the simplest case $\beta =0$, the characteristic function is just a stretched exponential function; the distribution is symmetric about μ and is referred to as a (Lévy) symmetric alpha-stable distribution, often abbreviated SαS.
When $\alpha <1$ and $\beta =1$, the distribution is supported by [μ, ∞).
The parameter c > 0 is a scale factor which is a measure of the width of the distribution while $\alpha $ is the exponent or index of the distribution and specifies the asymptotic behavior of the distribution.
Parametrizations
The above definition is only one of the parametrizations in use for stable distributions; it is the most common but its probability density is not continuous in the parameters at $\alpha =1$.[10]
A continuous parametrization is[7]
$\varphi (t;\alpha ,\beta ,\gamma ,\delta )=\exp \left(it\delta -|\gamma t|^{\alpha }\left(1-i\beta \operatorname {sgn}(t)\Phi \right)\right)$
where:
$\Phi ={\begin{cases}\left(|\gamma t|^{1-\alpha }-1\right)\tan \left({\tfrac {\pi \alpha }{2}}\right)&\alpha \neq 1\\-{\frac {2}{\pi }}\log |\gamma t|&\alpha =1\end{cases}}$
The ranges of $\alpha $ and $\beta $ are the same as before, γ (like c) should be positive, and δ (like μ) should be real.
In either parametrization one can make a linear transformation of the random variable to get a random variable whose density is $f(y;\alpha ,\beta ,1,0)$. In the first parametrization, this is done by defining the new variable:
$y={\begin{cases}{\frac {x-\mu }{\gamma }}&\alpha \neq 1\\{\frac {x-\mu }{\gamma }}-\beta {\frac {2}{\pi }}\ln \gamma &\alpha =1\end{cases}}$
For the second parametrization, we simply use
$y={\frac {x-\delta }{\gamma }}$
no matter what $\alpha $ is. In the first parametrization, if the mean exists (that is, $\alpha >1$) then it is equal to μ, whereas in the second parametrization when the mean exists it is equal to $\delta -\beta \gamma \tan \left({\tfrac {\pi \alpha }{2}}\right).$
The distribution
A stable distribution is therefore specified by the above four parameters. It can be shown that any non-degenerate stable distribution has a smooth (infinitely differentiable) density function.[7] If $f(x;\alpha ,\beta ,c,\mu )$ denotes the density of X and Y is the sum of independent copies of X:
$Y=\sum _{i=1}^{N}k_{i}(X_{i}-\mu )$
then Y has the density ${\tfrac {1}{s}}f(y/s;\alpha ,\beta ,c,0)$ with
$s=\left(\sum _{i=1}^{N}|k_{i}|^{\alpha }\right)^{\frac {1}{\alpha }}$
The asymptotic behavior is described, for $\alpha <2$, by:[7]
$f(x)\sim {\frac {1}{|x|^{1+\alpha }}}\left(c^{\alpha }(1+\operatorname {sgn}(x)\beta )\sin \left({\frac {\pi \alpha }{2}}\right){\frac {\Gamma (\alpha +1)}{\pi }}\right)$
where Γ is the Gamma function (except that when $\alpha \geq 1$ and $\beta =\pm 1$, the tail does not vanish to the left or right, resp., of μ, although the above expression is 0). This "heavy tail" behavior causes the variance of stable distributions to be infinite for all $\alpha <2$. This property is illustrated in the log–log plots below.
When $\alpha =2$, the distribution is Gaussian (see below), with tails asymptotic to exp(−x2/4c2)/(2c√π).
One-sided stable distribution and stable count distribution
When $\alpha <1$ and $\beta =1$, the distribution is supported by [μ, ∞). This family is called one-sided stable distribution.[11] Its standard distribution (μ=0) is defined as
$L_{\alpha }(x)=f{\left(x;\alpha ,1,\cos \left({\frac {\alpha \pi }{2}}\right)^{1/\alpha },0\right)}$, where $\alpha <1$.
Let $q=\exp(-i\alpha \pi /2)$, its characteristic function is $\varphi (t;\alpha )=\exp \left(-q|t|^{\alpha }\right)$. Thus the integral form of its PDF is (note: $\operatorname {Im} (q)<0$)
${\begin{aligned}L_{\alpha }(x)&={\frac {1}{\pi }}\Re \left[\int _{-\infty }^{\infty }e^{itx}e^{-q|t|^{\alpha }}\,dt\right]\\&={\frac {2}{\pi }}\int _{0}^{\infty }e^{-\operatorname {Re} (q)\,t^{\alpha }}\sin(tx)\sin(-\operatorname {Im} (q)\,t^{\alpha })\,dt,{\text{ or }}\\&={\frac {2}{\pi }}\int _{0}^{\infty }e^{-{\text{Re}}(q)\,t^{\alpha }}\cos(tx)\cos(\operatorname {Im} (q)\,t^{\alpha })\,dt.\end{aligned}}$
The double-sine integral is more effective for very small $x$.
Consider the Lévy sum $ Y=\sum _{i=1}^{N}X_{i}$ where $ X_{i}\sim L_{\alpha }(x)$, then Y has the density $ {\frac {1}{\nu }}L_{\alpha }{\left({\frac {x}{\nu }}\right)}$ where $ \nu =N^{1/\alpha }$. Set $ x=1$, we arrive at the stable count distribution.[12] Its standard distribution is defined as
${\mathfrak {N}}_{\alpha }(\nu )={\frac {\alpha }{\Gamma \left({\frac {1}{\alpha }}\right)}}{\frac {1}{\nu }}L_{\alpha }{\left({\frac {1}{\nu }}\right)}$, where $\nu >0$ and $\alpha <1$.
The stable count distribution is the conjugate prior of the one-sided stable distribution. Its location-scale family is defined as
${\mathfrak {N}}_{\alpha }(\nu ;\nu _{0},\theta )={\frac {\alpha }{\Gamma ({\frac {1}{\alpha }})}}{\frac {1}{\nu -\nu _{0}}}L_{\alpha }{\left({\frac {\theta }{\nu -\nu _{0}}}\right)}$ ;\nu _{0},\theta )={\frac {\alpha }{\Gamma ({\frac {1}{\alpha }})}}{\frac {1}{\nu -\nu _{0}}}L_{\alpha }{\left({\frac {\theta }{\nu -\nu _{0}}}\right)}} , where $\nu >\nu _{0}$, $\theta >0$, and $\alpha <1$.
It is also a one-sided distribution supported by $[\nu _{0},\infty )$. The location parameter $\nu _{0}$ is the cut-off location, while $\theta $ defines its scale.
When $ \alpha ={\frac {1}{2}}$, $ L_{\frac {1}{2}}(x)$ is the Lévy distribution which is an inverse gamma distribution. Thus ${\mathfrak {N}}_{\frac {1}{2}}(\nu ;\nu _{0},\theta )$ ;\nu _{0},\theta )} is a shifted gamma distribution of shape 3/2 and scale $4\theta $,
${\mathfrak {N}}_{\frac {1}{2}}(\nu ;\nu _{0},\theta )={\frac {1}{4{\sqrt {\pi }}\theta ^{3/2}}}(\nu -\nu _{0})^{1/2}e^{-{\frac {\nu -\nu _{0}}{4\theta }}}$ ;\nu _{0},\theta )={\frac {1}{4{\sqrt {\pi }}\theta ^{3/2}}}(\nu -\nu _{0})^{1/2}e^{-{\frac {\nu -\nu _{0}}{4\theta }}}} , where $\nu >\nu _{0}$, $\theta >0$.
Its mean is $\nu _{0}+6\theta $ and its standard deviation is ${\sqrt {24}}\theta $. It is hypothesized that VIX is distributed like $ {\mathfrak {N}}_{\frac {1}{2}}(\nu ;\nu _{0},\theta )$ ;\nu _{0},\theta )} with $\nu _{0}=10.4$ and $\theta =1.6$ (See Section 7 of [12]). Thus the stable count distribution is the first-order marginal distribution of a volatility process. In this context, $\nu _{0}$ is called the "floor volatility".
Another approach to derive the stable count distribution is to use the Laplace transform of the one-sided stable distribution, (Section 2.4 of [12])
$\int _{0}^{\infty }e^{-zx}L_{\alpha }(x)dx=e^{-z^{\alpha }}$, where $\alpha <1$.
Let $x=1/\nu $, and one can decompose the integral on the left hand side as a product distribution of a standard Laplace distribution and a standard stable count distribution,f
$\int _{0}^{\infty }{\frac {1}{\nu }}\left({\frac {1}{2}}e^{-{\frac {|z|}{\nu }}}\right)\left({\frac {\alpha }{\Gamma ({\frac {1}{\alpha }})}}{\frac {1}{\nu }}L_{\alpha }{\left({\frac {1}{\nu }}\right)}\right)d\nu ={\frac {1}{2}}{\frac {\alpha }{\Gamma ({\frac {1}{\alpha }})}}e^{-|z|^{\alpha }}$, where $\alpha <1$.
This is called the "lambda decomposition" (See Section 4 of [12]) since the right hand side was named as "symmetric lambda distribution" in Lihn's former works. However, it has several more popular names such as "exponential power distribution", or the "generalized error/normal distribution", often referred to when $\alpha >1$.
The n-th moment of ${\mathfrak {N}}_{\alpha }(\nu )$ is the $-(n+1)$-th moment of $L_{\alpha }(x)$, All positive moments are finite. This in a way solves the thorny issue of diverging moments in the stable distribution.
Properties
• All stable distributions are infinitely divisible.
• With the exception of the normal distribution ($\alpha =2$), stable distributions are leptokurtotic and heavy-tailed distributions.
• Closure under convolution
Stable distributions are closed under convolution for a fixed value of $\alpha $. Since convolution is equivalent to multiplication of the Fourier-transformed function, it follows that the product of two stable characteristic functions with the same $\alpha $ will yield another such characteristic function. The product of two stable characteristic functions is given by:
$\exp \left(it\mu _{1}+it\mu _{2}-|c_{1}t|^{\alpha }-|c_{2}t|^{\alpha }+i\beta _{1}|c_{1}t|^{\alpha }\operatorname {sgn}(t)\Phi +i\beta _{2}|c_{2}t|^{\alpha }\operatorname {sgn}(t)\Phi \right)$
Since Φ is not a function of the μ, c or $\beta $ variables it follows that these parameters for the convolved function are given by:
${\begin{aligned}\mu &=\mu _{1}+\mu _{2}\\|c|&=\left(|c_{1}|^{\alpha }+|c_{2}|^{\alpha }\right)^{\frac {1}{\alpha }}\\[6pt]\beta &={\frac {\beta _{1}|c_{1}|^{\alpha }+\beta _{2}|c_{2}|^{\alpha }}{|c_{1}|^{\alpha }+|c_{2}|^{\alpha }}}\end{aligned}}$
In each case, it can be shown that the resulting parameters lie within the required intervals for a stable distribution.
The Generalized Central Limit Theorem
The Generalized Central Limit Theorem (GCLT) was an effort of multiple mathematicians (Berstein, Lindeberg, Lévy, Feller, Kolmogorov, and others) over the period from 1920 to 1937. [13] The first published complete proof (in French) of the GCLT was in 1937 by Paul Lévy.[14] An English language version of the complete proof of the GCLT is available in the translation of Gnedenko and Kolmogorov's 1954 book.[15]
The statement of the GLCT is as follows:[16]
A non-degenerate random variable Z is α-stable for some 0 < α ≤ 2 if and only if there is an independent, identically distributed sequence of random variables X1, X2, X3, ... and constants an > 0, bn ∈ ℝ with
an (X1 + ... + Xn) - bn → Z.
Here → means the sequence of random variable sums converges in distribution; i.e., the corresponding distributions satisfy Fn(y) → F(y) at all continuity points of F.
In other words, if sums of independent, identically distributed random variables converge in distribution to some Z, then Z must be a stable distribution.
A generalized central limit theorem
General reference: [17] by Gnedenko.
Another important property of stable distributions is the role that they play in a generalized central limit theorem. The central limit theorem states that the sum of a number of independent and identically distributed (i.i.d.) random variables with finite non-zero variances will tend to a normal distribution as the number of variables grows.
A generalization due to Gnedenko and Kolmogorov states that the sum of a number of random variables with symmetric distributions having power-law tails (Paretian tails), decreasing as $|x|^{-\alpha -1}$ where $0<\alpha \leqslant 2$ (and therefore having infinite variance), will tend to a stable distribution $f(x;\alpha ,0,c,0)$ as the number of summands grows.[18] If $\alpha >2$ then the sum converges to a stable distribution with stability parameter equal to 2, i.e. a Gaussian distribution.[19]
[20]
There are other possibilities as well. For example, if the characteristic function of the random variable is asymptotic to $1+a|t|^{\alpha }\ln |t|$ for small t (positive or negative), then we may ask how t varies with n when the value of the characteristic function for the sum of n such random variables equals a given value u:
$\varphi _{\text{sum}}=\varphi ^{n}=u$
Assuming for the moment that t → 0, we take the limit of the above as n → ∞:
$\ln u=\lim _{n\to \infty }n\ln \varphi =\lim _{n\to \infty }na|t|^{\alpha }\ln |t|.$
Therefore:
${\begin{aligned}\ln(\ln u)&=\ln \left(\lim _{n\to \infty }na|t|^{\alpha }\ln |t|\right)\\[5pt]&=\lim _{n\to \infty }\ln \left(na|t|^{\alpha }\ln |t|\right)=\lim _{n\to \infty }\left[\ln(na)+\alpha \ln |t|+\ln(\ln |t|)\right]\end{aligned}}$
This shows that $\ln |t|$ is asymptotic to ${\tfrac {-1}{\alpha }}\ln n,$ so using the previous equation we have
$|t|\sim \left({\frac {-\alpha \ln u}{na\ln n}}\right)^{1/\alpha }.$
This implies that the sum divided by
$\left({\frac {na\ln n}{\alpha }}\right)^{\frac {1}{\alpha }}$
has a characteristic function whose value at some t′ goes to u (as n increases) when $t'=(-\ln u)^{{1}/{\alpha }}.$ In other words, the characteristic function converges pointwise to $\exp(-(t')^{\alpha })$ and therefore by Lévy's continuity theorem the sum divided by
$\left({\frac {na\ln n}{\alpha }}\right)^{\frac {1}{\alpha }}$
converges in distribution to the symmetric alpha-stable distribution with stability parameter $\alpha $ and scale parameter 1.
This can be applied to a random variable whose tails decrease as $|x|^{-3}$. This random variable has a mean but the variance is infinite. Let us take the following distribution:
$f(x)={\begin{cases}{\frac {1}{3}}&|x|\leq 1\\{\frac {1}{3}}x^{-3}&|x|>1\end{cases}}$
We can write this as
$f(x)=\int _{1}^{\infty }{\frac {2}{w^{4}}}h\left({\frac {x}{w}}\right)dw$
where
$h\left({\frac {x}{w}}\right)={\begin{cases}{\frac {1}{2}}&\left|{\frac {x}{w}}\right|<1,\\0&\left|{\frac {x}{w}}\right|>1.\end{cases}}$
We want to find the leading terms of the asymptotic expansion of the characteristic function. The characteristic function of the probability distribution $ {\frac {1}{w}}h\left({\frac {x}{w}}\right)$ is $ {\tfrac {\sin(tw)}{tw}},$ so the characteristic function for f(x) is
$\varphi (t)=\int _{1}^{\infty }{\frac {2\sin(tw)}{tw^{4}}}dw$
and we can calculate:
${\begin{aligned}\varphi (t)-1&=\int _{1}^{\infty }{\frac {2}{w^{3}}}\left[{\frac {\sin(tw)}{tw}}-1\right]\,dw\\&=\int _{1}^{\frac {1}{|t|}}{\frac {2}{w^{3}}}\left[{\frac {\sin(tw)}{tw}}-1\right]\,dw+\int _{\frac {1}{|t|}}^{\infty }{\frac {2}{w^{3}}}\left[{\frac {\sin(tw)}{tw}}-1\right]\,dw\\&=\int _{1}^{\frac {1}{|t|}}{\frac {2}{w^{3}}}\left[{\frac {\sin(tw)}{tw}}-1+\left\{-{\frac {t^{2}w^{2}}{3!}}+{\frac {t^{2}w^{2}}{3!}}\right\}\right]\,dw+\int _{\frac {1}{|t|}}^{\infty }{\frac {2}{w^{3}}}\left[{\frac {\sin(tw)}{tw}}-1\right]\,dw\\&=\int _{1}^{\frac {1}{|t|}}-{\frac {t^{2}dw}{3w}}+\int _{1}^{\frac {1}{|t|}}{\frac {2}{w^{3}}}\left[{\frac {\sin(tw)}{tw}}-1+{\frac {t^{2}w^{2}}{3!}}\right]dw+\int _{\frac {1}{|t|}}^{\infty }{\frac {2}{w^{3}}}\left[{\frac {\sin(tw)}{tw}}-1\right]dw\\&=\int _{1}^{\frac {1}{|t|}}-{\frac {t^{2}dw}{3w}}+\left\{\int _{0}^{\frac {1}{|t|}}{\frac {2}{w^{3}}}\left[{\frac {\sin(tw)}{tw}}-1+{\frac {t^{2}w^{2}}{3!}}\right]dw-\int _{0}^{1}{\frac {2}{w^{3}}}\left[{\frac {\sin(tw)}{tw}}-1+{\frac {t^{2}w^{2}}{3!}}\right]dw\right\}+\int _{\frac {1}{|t|}}^{\infty }{\frac {2}{w^{3}}}\left[{\frac {\sin(tw)}{tw}}-1\right]dw\\&=\int _{1}^{\frac {1}{|t|}}-{\frac {t^{2}dw}{3w}}+t^{2}\int _{0}^{1}{\frac {2}{y^{3}}}\left[{\frac {\sin(y)}{y}}-1+{\frac {y^{2}}{6}}\right]dy-\int _{0}^{1}{\frac {2}{w^{3}}}\left[{\frac {\sin(tw)}{tw}}-1+{\frac {t^{2}w^{2}}{6}}\right]dw+t^{2}\int _{1}^{\infty }{\frac {2}{y^{3}}}\left[{\frac {\sin(y)}{y}}-1\right]dy\\&=-{\frac {t^{2}}{3}}\int _{1}^{\frac {1}{|t|}}{\frac {dw}{w}}+t^{2}C_{1}-\int _{0}^{1}{\frac {2}{w^{3}}}\left[{\frac {\sin(tw)}{tw}}-1+{\frac {t^{2}w^{2}}{6}}\right]dw+t^{2}C_{2}\\&={\frac {t^{2}}{3}}\ln |t|+t^{2}C_{3}-\int _{0}^{1}{\frac {2}{w^{3}}}\left[{\frac {\sin(tw)}{tw}}-1+{\frac {t^{2}w^{2}}{6}}\right]dw\\&={\frac {t^{2}}{3}}\ln |t|+t^{2}C_{3}-\int _{0}^{1}{\frac {2}{w^{3}}}\left[{\frac {t^{4}w^{4}}{5!}}+\cdots \right]dw\\&={\frac {t^{2}}{3}}\ln |t|+t^{2}C_{3}-{\mathcal {O}}\left(t^{4}\right)\end{aligned}}$
where $C_{1},C_{2}$ and $C_{3}$ are constants. Therefore,
$\varphi (t)\sim 1+{\frac {t^{2}}{3}}\ln |t|$
and according to what was said above (and the fact that the variance of f(x;2,0,1,0) is 2), the sum of n instances of this random variable, divided by $ {\sqrt {n(\ln n)/12}},$ will converge in distribution to a Gaussian distribution with variance 1. But the variance at any particular n will still be infinite. Note that the width of the limiting distribution grows faster than in the case where the random variable has a finite variance (in which case the width grows as the square root of n). The average, obtained by dividing the sum by n, tends toward a Gaussian whose width approaches zero as n increases, in accordance with the Law of large numbers.
Special cases
There is no general analytic solution for the form of f(x). There are, however three special cases which can be expressed in terms of elementary functions as can be seen by inspection of the characteristic function:[7][9][21]
• For $\alpha =2$ the distribution reduces to a Gaussian distribution with variance σ2 = 2c2 and mean μ; the skewness parameter $\beta $ has no effect.
• For $\alpha =1$ and $\beta =0$ the distribution reduces to a Cauchy distribution with scale parameter c and shift parameter μ.
• For $\alpha =1/2$ and $\beta =1$ the distribution reduces to a Lévy distribution with scale parameter c and shift parameter μ.
Note that the above three distributions are also connected, in the following way: A standard Cauchy random variable can be viewed as a mixture of Gaussian random variables (all with mean zero), with the variance being drawn from a standard Lévy distribution. And in fact this is a special case of a more general theorem (See p. 59 of [22]) which allows any symmetric alpha-stable distribution to be viewed in this way (with the alpha parameter of the mixture distribution equal to twice the alpha parameter of the mixing distribution—and the beta parameter of the mixing distribution always equal to one).
A general closed form expression for stable PDFs with rational values of $\alpha $ is available in terms of Meijer G-functions.[23] Fox H-Functions can also be used to express the stable probability density functions. For simple rational numbers, the closed form expression is often in terms of less complicated special functions. Several closed form expressions having rather simple expressions in terms of special functions are available. In the table below, PDFs expressible by elementary functions are indicated by an E and those that are expressible by special functions are indicated by an s.[22]
$\alpha $
1/31/22/314/33/22
$\beta $ 0sssEssE
1sEsLs
Some of the special cases are known by particular names:
• For $\alpha =1$ and $\beta =1$, the distribution is a Landau distribution (L) which has a specific usage in physics under this name.
• For $\alpha =3/2$ and $\beta =0$ the distribution reduces to a Holtsmark distribution with scale parameter c and shift parameter μ.
Also, in the limit as c approaches zero or as α approaches zero the distribution will approach a Dirac delta function δ(x − μ).
Series representation
The stable distribution can be restated as the real part of a simpler integral:[24]
$f(x;\alpha ,\beta ,c,\mu )={\frac {1}{\pi }}\Re \left[\int _{0}^{\infty }e^{it(x-\mu )}e^{-(ct)^{\alpha }(1-i\beta \Phi )}\,dt\right].$
Expressing the second exponential as a Taylor series, we have:
$f(x;\alpha ,\beta ,c,\mu )={\frac {1}{\pi }}\Re \left[\int _{0}^{\infty }e^{it(x-\mu )}\sum _{n=0}^{\infty }{\frac {(-qt^{\alpha })^{n}}{n!}}\,dt\right]$
where $q=c^{\alpha }(1-i\beta \Phi )$. Reversing the order of integration and summation, and carrying out the integration yields:
$f(x;\alpha ,\beta ,c,\mu )={\frac {1}{\pi }}\Re \left[\sum _{n=1}^{\infty }{\frac {(-q)^{n}}{n!}}\left({\frac {i}{x-\mu }}\right)^{\alpha n+1}\Gamma (\alpha n+1)\right]$
which will be valid for x ≠ μ and will converge for appropriate values of the parameters. (Note that the n = 0 term which yields a delta function in x − μ has therefore been dropped.) Expressing the first exponential as a series will yield another series in positive powers of x − μ which is generally less useful.
For one-sided stable distribution, the above series expansion needs to be modified, since $q=\exp(-i\alpha \pi /2)$ and $qi^{\alpha }=1$. There is no real part to sum. Instead, the integral of the characteristic function should be carried out on the negative axis, which yields:[25][11]
${\begin{aligned}L_{\alpha }(x)&={\frac {1}{\pi }}\Re \left[\sum _{n=1}^{\infty }{\frac {(-q)^{n}}{n!}}\left({\frac {-i}{x}}\right)^{\alpha n+1}\Gamma (\alpha n+1)\right]\\&={\frac {1}{\pi }}\sum _{n=1}^{\infty }{\frac {-\sin(n(\alpha +1)\pi )}{n!}}\left({\frac {1}{x}}\right)^{\alpha n+1}\Gamma (\alpha n+1)\end{aligned}}$
Simulation of stable variables
Simulating sequences of stable random variables is not straightforward, since there are no analytic expressions for the inverse $F^{-1}(x)$ nor the CDF $F(x)$ itself.[10][12] All standard approaches like the rejection or the inversion methods would require tedious computations. A much more elegant and efficient solution was proposed by Chambers, Mallows and Stuck (CMS),[26] who noticed that a certain integral formula[27] yielded the following algorithm:[28]
• generate a random variable $U$ uniformly distributed on $\left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)$ and an independent exponential random variable $W$ with mean 1;
• for $\alpha \neq 1$ compute:
$X=\left(1+\zeta ^{2}\right)^{\frac {1}{2\alpha }}{\frac {\sin(\alpha (U+\xi ))}{(\cos(U))^{\frac {1}{\alpha }}}}\left({\frac {\cos(U-\alpha (U+\xi ))}{W}}\right)^{\frac {1-\alpha }{\alpha }},$
• for $\alpha =1$ compute:
$X={\frac {1}{\xi }}\left\{\left({\frac {\pi }{2}}+\beta U\right)\tan U-\beta \log \left({\frac {{\frac {\pi }{2}}W\cos U}{{\frac {\pi }{2}}+\beta U}}\right)\right\},$
where
$\zeta =-\beta \tan {\frac {\pi \alpha }{2}},\qquad \xi ={\begin{cases}{\frac {1}{\alpha }}\arctan(-\zeta )&\alpha \neq 1\\{\frac {\pi }{2}}&\alpha =1\end{cases}}$
This algorithm yields a random variable $X\sim S_{\alpha }(\beta ,1,0)$. For a detailed proof see.[29]
Given the formulas for simulation of a standard stable random variable, we can easily simulate a stable random variable for all admissible values of the parameters $\alpha $, $c$, $\beta $ and $\mu $ using the following property. If $X\sim S_{\alpha }(\beta ,1,0)$ then
$Y={\begin{cases}cX+\mu &\alpha \neq 1\\cX+{\frac {2}{\pi }}\beta c\log c+\mu &\alpha =1\end{cases}}$
is $S_{\alpha }(\beta ,c,\mu )$. For $\alpha =2$ (and $\beta =0$) the CMS method reduces to the well known Box-Muller transform for generating Gaussian random variables.[30] Many other approaches have been proposed in the literature, including application of Bergström and LePage series expansions, see [31] and,[32] respectively. However, the CMS method is regarded as the fastest and the most accurate.
Applications
Stable distributions owe their importance in both theory and practice to the generalization of the central limit theorem to random variables without second (and possibly first) order moments and the accompanying self-similarity of the stable family. It was the seeming departure from normality along with the demand for a self-similar model for financial data (i.e. the shape of the distribution for yearly asset price changes should resemble that of the constituent daily or monthly price changes) that led Benoît Mandelbrot to propose that cotton prices follow an alpha-stable distribution with $\alpha $ equal to 1.7.[6] Lévy distributions are frequently found in analysis of critical behavior and financial data.[9][33]
They are also found in spectroscopy as a general expression for a quasistatically pressure broadened spectral line.[24]
The Lévy distribution of solar flare waiting time events (time between flare events) was demonstrated for CGRO BATSE hard x-ray solar flares in December 2001. Analysis of the Lévy statistical signature revealed that two different memory signatures were evident; one related to the solar cycle and the second whose origin appears to be associated with a localized or combination of localized solar active region effects.[34]
Other analytic cases
A number of cases of analytically expressible stable distributions are known. Let the stable distribution be expressed by $f(x;\alpha ,\beta ,c,\mu )$ then we know:
• The Cauchy Distribution is given by $f(x;1,0,1,0).$
• The Lévy distribution is given by $f(x;{\tfrac {1}{2}},1,1,0).$
• The Normal distribution is given by $f(x;2,0,1,0).$
• Let $S_{\mu ,\nu }(z)$ be a Lommel function, then:[35]
$f\left(x;{\tfrac {1}{3}},0,1,0\right)=\Re \left({\frac {2e^{-{\frac {i\pi }{4}}}}{3{\sqrt {3}}\pi }}{\frac {1}{\sqrt {x^{3}}}}S_{0,{\frac {1}{3}}}\left({\frac {2e^{\frac {i\pi }{4}}}{3{\sqrt {3}}}}{\frac {1}{\sqrt {x}}}\right)\right)$
• Let $S(x)$ and $C(x)$ denote the Fresnel Integrals then:[36]
$f\left(x;{\tfrac {1}{2}},0,1,0\right)={\frac {1}{\sqrt {2\pi |x|^{3}}}}\left(\sin \left({\tfrac {1}{4|x|}}\right)\left[{\frac {1}{2}}-S\left({\tfrac {1}{\sqrt {2\pi |x|}}}\right)\right]+\cos \left({\tfrac {1}{4|x|}}\right)\left[{\frac {1}{2}}-C\left({\tfrac {1}{\sqrt {2\pi |x|}}}\right)\right]\right)$
• Let $K_{v}(x)$ be the modified Bessel function of the second kind then:[36]
$f\left(x;{\tfrac {1}{3}},1,1,0\right)={\frac {1}{\pi }}{\frac {2{\sqrt {2}}}{3^{\frac {7}{4}}}}{\frac {1}{\sqrt {x^{3}}}}K_{\frac {1}{3}}\left({\frac {4{\sqrt {2}}}{3^{\frac {9}{4}}}}{\frac {1}{\sqrt {x}}}\right)$
• If the ${}_{m}F_{n}$ denote the hypergeometric functions then:[35]
${\begin{aligned}f\left(x;{\tfrac {4}{3}},0,1,0\right)&={\frac {3^{\frac {5}{4}}}{4{\sqrt {2\pi }}}}{\frac {\Gamma \left({\tfrac {7}{12}}\right)\Gamma \left({\tfrac {11}{12}}\right)}{\Gamma \left({\tfrac {6}{12}}\right)\Gamma \left({\tfrac {8}{12}}\right)}}{}_{2}F_{2}\left({\tfrac {7}{12}},{\tfrac {11}{12}};{\tfrac {6}{12}},{\tfrac {8}{12}};{\tfrac {3^{3}x^{4}}{4^{4}}}\right)-{\frac {3^{\frac {11}{4}}x^{3}}{4^{3}{\sqrt {2\pi }}}}{\frac {\Gamma \left({\tfrac {13}{12}}\right)\Gamma \left({\tfrac {17}{12}}\right)}{\Gamma \left({\tfrac {18}{12}}\right)\Gamma \left({\tfrac {15}{12}}\right)}}{}_{2}F_{2}\left({\tfrac {13}{12}},{\tfrac {17}{12}};{\tfrac {18}{12}},{\tfrac {15}{12}};{\tfrac {3^{3}x^{4}}{4^{4}}}\right)\\[6pt]f\left(x;{\tfrac {3}{2}},0,1,0\right)&={\frac {\Gamma \left({\tfrac {5}{3}}\right)}{\pi }}{}_{2}F_{3}\left({\tfrac {5}{12}},{\tfrac {11}{12}};{\tfrac {1}{3}},{\tfrac {1}{2}},{\tfrac {5}{6}};-{\tfrac {2^{2}x^{6}}{3^{6}}}\right)-{\frac {x^{2}}{3\pi }}{}_{3}F_{4}\left({\tfrac {3}{4}},1,{\tfrac {5}{4}};{\tfrac {2}{3}},{\tfrac {5}{6}},{\tfrac {7}{6}},{\tfrac {4}{3}};-{\tfrac {2^{2}x^{6}}{3^{6}}}\right)+{\frac {7x^{4}\Gamma \left({\tfrac {4}{3}}\right)}{3^{4}\pi ^{2}}}{}_{2}F_{3}\left({\tfrac {13}{12}},{\tfrac {19}{12}};{\tfrac {7}{6}},{\tfrac {3}{2}},{\tfrac {5}{3}};-{\tfrac {2^{2}x^{6}}{3^{6}}}\right)\end{aligned}}$
with the latter being the Holtsmark distribution.
• Let $W_{k,\mu }(z)$ be a Whittaker function, then:[37][38][39]
${\begin{aligned}f\left(x;{\tfrac {2}{3}},0,1,0\right)&={\frac {\sqrt {3}}{6{\sqrt {\pi }}|x|}}\exp \left({\tfrac {2}{27}}x^{-2}\right)W_{-{\frac {1}{2}},{\frac {1}{6}}}\left({\tfrac {4}{27}}x^{-2}\right)\\[8pt]f\left(x;{\tfrac {2}{3}},1,1,0\right)&={\frac {\sqrt {3}}{{\sqrt {\pi }}|x|}}\exp \left(-{\tfrac {16}{27}}x^{-2}\right)W_{{\frac {1}{2}},{\frac {1}{6}}}\left({\tfrac {32}{27}}x^{-2}\right)\\[8pt]f\left(x;{\tfrac {3}{2}},1,1,0\right)&={\begin{cases}{\frac {\sqrt {3}}{{\sqrt {\pi }}|x|}}\exp \left({\frac {1}{27}}x^{3}\right)W_{{\frac {1}{2}},{\frac {1}{6}}}\left(-{\frac {2}{27}}x^{3}\right)&x<0\\{}\\{\frac {\sqrt {3}}{6{\sqrt {\pi }}|x|}}\exp \left({\frac {1}{27}}x^{3}\right)W_{-{\frac {1}{2}},{\frac {1}{6}}}\left({\frac {2}{27}}x^{3}\right)&x\geq 0\end{cases}}\end{aligned}}$
See also
• Lévy flight
• Lévy process
• Other "power law" distributions
• Pareto distribution
• Zeta distribution
• Zipf distribution
• Zipf–Mandelbrot distribution
• Financial models with long-tailed distributions and volatility clustering
• Multivariate stable distribution
• Discrete-stable distribution
Notes
• The STABLE program for Windows is available from John Nolan's stable webpage: http://www.robustanalysis.com/public/stable.html. It calculates the density (pdf), cumulative distribution function (cdf) and quantiles for a general stable distribution, and performs maximum likelihood estimation of stable parameters and some exploratory data analysis techniques for assessing the fit of a data set.
• libstable is a C implementation for the Stable distribution pdf, cdf, random number, quantile and fitting functions (along with a benchmark replication package and an R package).
• R Package 'stabledist' by Diethelm Wuertz, Martin Maechler and Rmetrics core team members. Computes stable density, probability, quantiles, and random numbers. Updated Sept. 12, 2016.
• Python implementation is located in scipy.stats.levy_stable in the SciPy package.
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31. Mantegna, Rosario Nunzio (1994). "Fast, accurate algorithm for numerical simulation of Lévy stable stochastic processes". Physical Review E. 49 (5): 4677–4683. Bibcode:1994PhRvE..49.4677M. doi:10.1103/PhysRevE.49.4677. PMID 9961762.
32. Janicki, Aleksander; Kokoszka, Piotr (1992). "Computer investigation of the Rate of Convergence of Lepage Type Series to α-Stable Random Variables". Statistics. 23 (4): 365–373. doi:10.1080/02331889208802383. ISSN 0233-1888.
33. Rachev, Svetlozar T.; Mittnik, Stefan (2000). Stable Paretian Models in Finance. Wiley. ISBN 978-0-471-95314-2.
34. Leddon, D., A statistical Study of Hard X-Ray Solar Flares
35. Garoni, T. M.; Frankel, N. E. (2002). "Lévy flights: Exact results and asymptotics beyond all orders". Journal of Mathematical Physics. 43 (5): 2670–2689. Bibcode:2002JMP....43.2670G. doi:10.1063/1.1467095.
36. Hopcraft, K. I.; Jakeman, E.; Tanner, R. M. J. (1999). "Lévy random walks with fluctuating step number and multiscale behavior". Physical Review E. 60 (5): 5327–5343. Bibcode:1999PhRvE..60.5327H. doi:10.1103/physreve.60.5327. PMID 11970402.
37. Uchaikin, V. V.; Zolotarev, V. M. (1999). "Chance And Stability – Stable Distributions And Their Applications". VSP.
38. Zlotarev, V. M. (1961). "Expression of the density of a stable distribution with exponent alpha greater than one by means of a frequency with exponent 1/alpha". Selected Translations in Mathematical Statistics and Probability (Translated from the Russian Article: Dokl. Akad. Nauk SSSR. 98, 735–738 (1954)). 1: 163–167.
39. Zaliapin, I. V.; Kagan, Y. Y.; Schoenberg, F. P. (2005). "Approximating the Distribution of Pareto Sums". Pure and Applied Geophysics. 162 (6): 1187–1228. Bibcode:2005PApGe.162.1187Z. doi:10.1007/s00024-004-2666-3. S2CID 18754585.
Probability distributions (list)
Discrete
univariate
with finite
support
• Benford
• Bernoulli
• beta-binomial
• binomial
• categorical
• hypergeometric
• negative
• Poisson binomial
• Rademacher
• soliton
• discrete uniform
• Zipf
• Zipf–Mandelbrot
with infinite
support
• beta negative binomial
• Borel
• Conway–Maxwell–Poisson
• discrete phase-type
• Delaporte
• extended negative binomial
• Flory–Schulz
• Gauss–Kuzmin
• geometric
• logarithmic
• mixed Poisson
• negative binomial
• Panjer
• parabolic fractal
• Poisson
• Skellam
• Yule–Simon
• zeta
Continuous
univariate
supported on a
bounded interval
• arcsine
• ARGUS
• Balding–Nichols
• Bates
• beta
• beta rectangular
• continuous Bernoulli
• Irwin–Hall
• Kumaraswamy
• logit-normal
• noncentral beta
• PERT
• raised cosine
• reciprocal
• triangular
• U-quadratic
• uniform
• Wigner semicircle
supported on a
semi-infinite
interval
• Benini
• Benktander 1st kind
• Benktander 2nd kind
• beta prime
• Burr
• chi
• chi-squared
• noncentral
• inverse
• scaled
• Dagum
• Davis
• Erlang
• hyper
• exponential
• hyperexponential
• hypoexponential
• logarithmic
• F
• noncentral
• folded normal
• Fréchet
• gamma
• generalized
• inverse
• gamma/Gompertz
• Gompertz
• shifted
• half-logistic
• half-normal
• Hotelling's T-squared
• inverse Gaussian
• generalized
• Kolmogorov
• Lévy
• log-Cauchy
• log-Laplace
• log-logistic
• log-normal
• log-t
• Lomax
• matrix-exponential
• Maxwell–Boltzmann
• Maxwell–Jüttner
• Mittag-Leffler
• Nakagami
• Pareto
• phase-type
• Poly-Weibull
• Rayleigh
• relativistic Breit–Wigner
• Rice
• truncated normal
• type-2 Gumbel
• Weibull
• discrete
• Wilks's lambda
supported
on the whole
real line
• Cauchy
• exponential power
• Fisher's z
• Kaniadakis κ-Gaussian
• Gaussian q
• generalized normal
• generalized hyperbolic
• geometric stable
• Gumbel
• Holtsmark
• hyperbolic secant
• Johnson's SU
• Landau
• Laplace
• asymmetric
• logistic
• noncentral t
• normal (Gaussian)
• normal-inverse Gaussian
• skew normal
• slash
• stable
• Student's t
• Tracy–Widom
• variance-gamma
• Voigt
with support
whose type varies
• generalized chi-squared
• generalized extreme value
• generalized Pareto
• Marchenko–Pastur
• Kaniadakis κ-exponential
• Kaniadakis κ-Gamma
• Kaniadakis κ-Weibull
• Kaniadakis κ-Logistic
• Kaniadakis κ-Erlang
• q-exponential
• q-Gaussian
• q-Weibull
• shifted log-logistic
• Tukey lambda
Mixed
univariate
continuous-
discrete
• Rectified Gaussian
Multivariate
(joint)
• Discrete:
• Ewens
• multinomial
• Dirichlet
• negative
• Continuous:
• Dirichlet
• generalized
• multivariate Laplace
• multivariate normal
• multivariate stable
• multivariate t
• normal-gamma
• inverse
• Matrix-valued:
• LKJ
• matrix normal
• matrix t
• matrix gamma
• inverse
• Wishart
• normal
• inverse
• normal-inverse
• complex
Directional
Univariate (circular) directional
Circular uniform
univariate von Mises
wrapped normal
wrapped Cauchy
wrapped exponential
wrapped asymmetric Laplace
wrapped Lévy
Bivariate (spherical)
Kent
Bivariate (toroidal)
bivariate von Mises
Multivariate
von Mises–Fisher
Bingham
Degenerate
and singular
Degenerate
Dirac delta function
Singular
Cantor
Families
• Circular
• compound Poisson
• elliptical
• exponential
• natural exponential
• location–scale
• maximum entropy
• mixture
• Pearson
• Tweedie
• wrapped
• Category
• Commons
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Limit cycle
In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity or as time approaches negative infinity. Such behavior is exhibited in some nonlinear systems. Limit cycles have been used to model the behavior of many real-world oscillatory systems. The study of limit cycles was initiated by Henri Poincaré (1854–1912).
Definition
We consider a two-dimensional dynamical system of the form
$x'(t)=V(x(t))$
where
$V:\mathbb {R} ^{2}\to \mathbb {R} ^{2}$
is a smooth function. A trajectory of this system is some smooth function $x(t)$ with values in $\mathbb {R} ^{2}$ which satisfies this differential equation. Such a trajectory is called closed (or periodic) if it is not constant but returns to its starting point, i.e. if there exists some $t_{0}>0$ such that $x(t+t_{0})=x(t)$ for all $t\in \mathbb {R} $. An orbit is the image of a trajectory, a subset of $\mathbb {R} ^{2}$. A closed orbit, or cycle, is the image of a closed trajectory. A limit cycle is a cycle which is the limit set of some other trajectory.
Properties
By the Jordan curve theorem, every closed trajectory divides the plane into two regions, the interior and the exterior of the curve.
Given a limit cycle and a trajectory in its interior that approaches the limit cycle for time approaching $+\infty $, then there is a neighborhood around the limit cycle such that all trajectories in the interior that start in the neighborhood approach the limit cycle for time approaching $+\infty $. The corresponding statement holds for a trajectory in the interior that approaches the limit cycle for time approaching $-\infty $, and also for trajectories in the exterior approaching the limit cycle.
Stable, unstable and semi-stable limit cycles
In the case where all the neighboring trajectories approach the limit cycle as time approaches infinity, it is called a stable or attractive limit cycle (ω-limit cycle). If instead, all neighboring trajectories approach it as time approaches negative infinity, then it is an unstable limit cycle (α-limit cycle). If there is a neighboring trajectory which spirals into the limit cycle as time approaches infinity, and another one which spirals into it as time approaches negative infinity, then it is a semi-stable limit cycle. There are also limit cycles that are neither stable, unstable nor semi-stable: for instance, a neighboring trajectory may approach the limit cycle from the outside, but the inside of the limit cycle is approached by a family of other cycles (which wouldn't be limit cycles).
Stable limit cycles are examples of attractors. They imply self-sustained oscillations: the closed trajectory describes the perfect periodic behavior of the system, and any small perturbation from this closed trajectory causes the system to return to it, making the system stick to the limit cycle.
Finding limit cycles
Every closed trajectory contains within its interior a stationary point of the system, i.e. a point $p$ where $V'(p)=0$. The Bendixson–Dulac theorem and the Poincaré–Bendixson theorem predict the absence or existence, respectively, of limit cycles of two-dimensional nonlinear dynamical systems.
Open problems
Finding limit cycles, in general, is a very difficult problem. The number of limit cycles of a polynomial differential equation in the plane is the main object of the second part of Hilbert's sixteenth problem. It is unknown, for instance, whether there is any system $x'=V(x)$ in the plane where both components of $V$ are quadratic polynomials of the two variables, such that the system has more than 4 limit cycles.
Applications
Limit cycles are important in many scientific applications where systems with self-sustained oscillations are modelled. Some examples include:
• Aerodynamic limit-cycle oscillations[1]
• The Hodgkin–Huxley model for action potentials in neurons.
• The Sel'kov model of glycolysis.[2]
• The daily oscillations in gene expression, hormone levels and body temperature of animals, which are part of the circadian rhythm,[3][4] although this is contradicted by more recent evidence.[5]
• The migration of cancer cells in confining micro-environments follows limit cycle oscillations.[6]
• Some non-linear electrical circuits exhibit limit cycle oscillations,[7] which inspired the original Van der Pol model.
• The control of respiration and hematopoiesis, as appearing in the Mackey-Glass equations.[8]
See also
• Attractor
• Hyperbolic set
• Periodic point
• Self-oscillation
• Stable manifold
References
1. Thomas, Jeffrey P.; Dowell, Earl H.; Hall, Kenneth C. (2002), "Nonlinear Inviscid Aerodynamic Effects on Transonic Divergence, Flutter, and Limit-Cycle Oscillations" (PDF), AIAA Journal, American Institute of Aeronautics and Astronautics, 40 (4): 638, Bibcode:2002AIAAJ..40..638T, doi:10.2514/2.1720, retrieved December 9, 2019
2. Sel'kov, E. E. (1968). "Self-Oscillations in Glycolysis 1. A Simple Kinetic Model". European Journal of Biochemistry. 4 (1): 79–86. doi:10.1111/j.1432-1033.1968.tb00175.x. ISSN 1432-1033. PMID 4230812.
3. Leloup, Jean-Christophe; Gonze, Didier; Goldbeter, Albert (1999-12-01). "Limit Cycle Models for Circadian Rhythms Based on Transcriptional Regulation in Drosophila and Neurospora". Journal of Biological Rhythms. 14 (6): 433–448. doi:10.1177/074873099129000948. ISSN 0748-7304. PMID 10643740. S2CID 15074869.
4. Roenneberg, Till; Chua, Elaine Jane; Bernardo, Ric; Mendoza, Eduardo (2008-09-09). "Modelling Biological Rhythms". Current Biology. 18 (17): R826–R835. doi:10.1016/j.cub.2008.07.017. ISSN 0960-9822. PMID 18786388. S2CID 2798371.
5. Meijer, JH; Michel, S; Vanderleest, HT; Rohling, JH (December 2010). "Daily and seasonal adaptation of the circadian clock requires plasticity of the SCN neuronal network". The European Journal of Neuroscience. 32 (12): 2143–51. doi:10.1111/j.1460-9568.2010.07522.x. PMID 21143668. S2CID 12754517.
6. Brückner, David B.; Fink, Alexandra; Schreiber, Christoph; Röttgermann, Peter J. F.; Rädler, Joachim; Broedersz, Chase P. (2019). "Stochastic nonlinear dynamics of confined cell migration in two-state systems". Nature Physics. 15 (6): 595–601. Bibcode:2019NatPh..15..595B. doi:10.1038/s41567-019-0445-4. ISSN 1745-2481. S2CID 126819906.
7. Ginoux, Jean-Marc; Letellier, Christophe (2012-04-30). "Van der Pol and the history of relaxation oscillations: Toward the emergence of a concept". Chaos: An Interdisciplinary Journal of Nonlinear Science. 22 (2): 023120. arXiv:1408.4890. Bibcode:2012Chaos..22b3120G. doi:10.1063/1.3670008. ISSN 1054-1500. PMID 22757527. S2CID 293369.
8. Mackey, M.; Glass, L (1977-07-15). "Oscillation and chaos in physiological control systems". Science. 197 (4300): 287–289. Bibcode:1977Sci...197..287M. doi:10.1126/science.267326. ISSN 0036-8075. PMID 267326.
Further reading
• Steven H. Strogatz (2014). Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Avalon. ISBN 9780813349114.
• M. Vidyasagar (2002). Nonlinear Systems Analysis (Second ed.). SIAM. ISBN 9780898715262.
• Philip Hartman, "Ordinary Differential Equation", Society for Industrial and Applied Mathematics, 2002.
• Witold Hurewicz, "Lectures on Ordinary Differential Equations", Dover, 2002.
• Solomon Lefschetz, "Differential Equations: Geometric Theory", Dover, 2005.
• Lawrence Perko, "Differential Equations and Dynamical Systems", Springer-Verlag, 2006.
• Arthur Mattuck, Limit Cycles: Existence and Non-existence Criteria, MIT Open Courseware http://videolectures.net/mit1803s06_mattuck_lec32/#
External links
• "limit cycle". planetmath.org. Retrieved 2019-07-06.
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Stable manifold theorem
In mathematics, especially in the study of dynamical systems and differential equations, the stable manifold theorem is an important result about the structure of the set of orbits approaching a given hyperbolic fixed point. It roughly states that the existence of a local diffeomorphism near a fixed point implies the existence of a local stable center manifold containing that fixed point. This manifold has dimension equal to the number of eigenvalues of the Jacobian matrix of the fixed point that are less than 1.[1]
Stable manifold theorem
Let
$f:U\subset \mathbb {R} ^{n}\to \mathbb {R} ^{n}$
be a smooth map with hyperbolic fixed point at $p$. We denote by $W^{s}(p)$ the stable set and by $W^{u}(p)$ the unstable set of $p$.
The theorem[2][3][4] states that
• $W^{s}(p)$ is a smooth manifold and its tangent space has the same dimension as the stable space of the linearization of $f$ at $p$.
• $W^{u}(p)$ is a smooth manifold and its tangent space has the same dimension as the unstable space of the linearization of $f$ at $p$.
Accordingly $W^{s}(p)$ is a stable manifold and $W^{u}(p)$ is an unstable manifold.
See also
• Center manifold theorem
• Lyapunov exponent
Notes
1. Shub, Michael (1987). Global Stability of Dynamical Systems. Springer. pp. 65–66.
2. Pesin, Ya B (1977). "Characteristic Lyapunov Exponents and Smooth Ergodic Theory". Russian Mathematical Surveys. 32 (4): 55–114. Bibcode:1977RuMaS..32...55P. doi:10.1070/RM1977v032n04ABEH001639. S2CID 250877457. Retrieved 2007-03-10.
3. Ruelle, David (1979). "Ergodic theory of differentiable dynamical systems". Publications Mathématiques de l'IHÉS. 50: 27–58. doi:10.1007/bf02684768. S2CID 56389695. Retrieved 2007-03-10.
4. Teschl, Gerald (2012). Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society. ISBN 978-0-8218-8328-0.
References
• Perko, Lawrence (2001). Differential Equations and Dynamical Systems (Third ed.). New York: Springer. pp. 105–117. ISBN 0-387-95116-4.
• Sritharan, S. S. (1990). Invariant Manifold Theory for Hydrodynamic Transition. John Wiley & Sons. ISBN 0-582-06781-2.
External links
• StableManifoldTheorem at PlanetMath.
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Stable map
In mathematics, specifically in symplectic topology and algebraic geometry, one can construct the moduli space of stable maps, satisfying specified conditions, from Riemann surfaces into a given symplectic manifold. This moduli space is the essence of the Gromov–Witten invariants, which find application in enumerative geometry and type IIA string theory. The idea of stable map was proposed by Maxim Kontsevich around 1992 and published in Kontsevich (1995).
Because the construction is lengthy and difficult, it is carried out here rather than in the Gromov–Witten invariants article itself.
The moduli space of smooth pseudoholomorphic curves
Fix a closed symplectic manifold $X$ with symplectic form $\omega $. Let $g$ and $n$ be natural numbers (including zero) and $A$ a two-dimensional homology class in $X$. Then one may consider the set of pseudoholomorphic curves
$((C,j),f,(x_{1},\ldots ,x_{n}))\,$
where $(C,j)$ is a smooth, closed Riemann surface of genus $g$ with $n$ marked points $x_{1},\ldots ,x_{n}$, and
$f:C\to X\,$
is a function satisfying, for some choice of $\omega $-tame almost complex structure $J$ and inhomogeneous term $\nu $, the perturbed Cauchy–Riemann equation
${\bar {\partial }}_{j,J}f:={\frac {1}{2}}(df+J\circ df\circ j)=\nu .$
Typically one admits only those $g$ and $n$ that make the punctured Euler characteristic $2-2g-n$ of $C$ negative; then the domain is stable, meaning that there are only finitely many holomorphic automorphisms of $C$ that preserve the marked points.
The operator ${\bar {\partial }}_{j,J}$ is elliptic and thus Fredholm. After significant analytical argument (completing in a suitable Sobolev norm, applying the implicit function theorem and Sard's theorem for Banach manifolds, and using elliptic regularity to recover smoothness) one can show that, for a generic choice of $\omega $-tame $J$ and perturbation $\nu $, the set of $(j,J,\nu )$-holomorphic curves of genus $g$ with $n$ marked points that represent the class $A$ forms a smooth, oriented orbifold
$M_{g,n}^{J,\nu }(X,A)$
of dimension given by the Atiyah-Singer index theorem,
$d:=\dim _{\mathbb {R} }M_{g,n}(X,A)=2c_{1}^{X}(A)+(\dim _{\mathbb {R} }X-6)(1-g)+2n.$
The stable map compactification
This moduli space of maps is not compact, because a sequence of curves can degenerate to a singular curve, which is not in the moduli space as we've defined it. This happens, for example, when the energy of $f$ (meaning the L2-norm of the derivative) concentrates at some point on the domain. One can capture the energy by rescaling the map around the concentration point. The effect is to attach a sphere, called a bubble, to the original domain at the concentration point and to extend the map across the sphere. The rescaled map may still have energy concentrating at one or more points, so one must rescale iteratively, eventually attaching an entire bubble tree onto the original domain, with the map well-behaved on each smooth component of the new domain.
In order to make this precise, define a stable map to be a pseudoholomorphic map from a Riemann surface with at worst nodal singularities, such that there are only finitely many automorphisms of the map. Concretely, this means the following. A smooth component of a nodal Riemann surface is said to be stable if there are at most finitely many automorphisms preserving its marked and nodal points. Then a stable map is a pseudoholomorphic map with at least one stable domain component, such that for each of the other domain components
• the map is nonconstant on that component, or
• that component is stable.
It is significant that the domain of a stable map need not be a stable curve. However, one can contract its unstable components (iteratively) to produce a stable curve, called the stabilization $\mathrm {st} (C)$ of the domain $C$.
The set of all stable maps from Riemann surfaces of genus $g$ with $n$ marked points forms a moduli space
${\overline {M}}_{g,n}^{J,\nu }(X,A).$
The topology is defined by declaring that a sequence of stable maps converges if and only if
• their (stabilized) domains converge in the Deligne–Mumford moduli space of curves ${\overline {M}}_{g,n}$,
• they converge uniformly in all derivatives on compact subsets away from the nodes, and
• the energy concentrating at any point equals the energy in the bubble tree attached at that point in the limit map.
The moduli space of stable maps is compact; that is, any sequence of stable maps converges to a stable map. To show this, one iteratively rescales the sequence of maps. At each iteration there is a new limit domain, possibly singular, with less energy concentration than in the previous iteration. At this step the symplectic form $\omega $ enters in a crucial way. The energy of any smooth map representing the homology class $B$ is bounded below by the symplectic area $\omega (B)$,
$\omega (B)\leq {\frac {1}{2}}\int |df|^{2},$
with equality if and only if the map is pseudoholomorphic. This bounds the energy captured in each iteration of the rescaling and thus implies that only finitely many rescalings are needed to capture all of the energy. In the end, the limit map on the new limit domain is stable.
The compactified space is again a smooth, oriented orbifold. Maps with nontrivial automorphisms correspond to points with isotropy in the orbifold.
The Gromov–Witten pseudocycle
To construct Gromov–Witten invariants, one pushes the moduli space of stable maps forward under the evaluation map
$M_{g,n}^{J,\nu }(X,A)\to {\overline {M}}_{g,n}\times X^{n},$
$((C,j),f,(x_{1},\ldots ,x_{n}))\mapsto (\mathrm {st} (C,j),f(x_{1}),\ldots ,f(x_{n}))$
to obtain, under suitable conditions, a rational homology class
$GW_{g,n}^{X,A}\in H_{d}({\overline {M}}_{g,n}\times X^{n},\mathbb {Q} ).$
Rational coefficients are necessary because the moduli space is an orbifold. The homology class defined by the evaluation map is independent of the choice of generic $\omega $-tame $J$ and perturbation $\nu $. It is called the Gromov–Witten (GW) invariant of $X$ for the given data $g$, $n$, and $A$. A cobordism argument can be used to show that this homology class is independent of the choice of $\omega $, up to isotopy. Thus Gromov–Witten invariants are invariants of symplectic isotopy classes of symplectic manifolds.
The "suitable conditions" are rather subtle, primarily because multiply covered maps (maps that factor through a branched covering of the domain) can form moduli spaces of larger dimension than expected.
The simplest way to handle this is to assume that the target manifold $X$ is semipositive or Fano in a certain sense. This assumption is chosen exactly so that the moduli space of multiply covered maps has codimension at least two in the space of non-multiply-covered maps. Then the image of the evaluation map forms a pseudocycle, which induces a well-defined homology class of the expected dimension.
Defining Gromov–Witten invariants without assuming some kind of semipositivity requires a difficult, technical construction known as the virtual moduli cycle.
References
• Dusa McDuff and Dietmar Salamon, J-Holomorphic Curves and Symplectic Topology, American Mathematical Society colloquium publications, 2004. ISBN 0-8218-3485-1.
• Kontsevich, Maxim (1995). "Enumeration of rational curves via torus actions". Progr. Math. 129: 335–368. MR 1363062.
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Stable marriage problem
In mathematics, economics, and computer science, the stable marriage problem (also stable matching problem or SMP) is the problem of finding a stable matching between two equally sized sets of elements given an ordering of preferences for each element. A matching is a bijection from the elements of one set to the elements of the other set. A matching is not stable if:
1. There is an element A of the first matched set which prefers some given element B of the second matched set over the element to which A is already matched, and
2. B also prefers A over the element to which B is already matched.
In other words, a matching is stable when there does not exist any pair (A, B) which both prefer each other to their current partner under the matching.
The stable marriage problem has been stated as follows:
Given n men and n women, where each person has ranked all members of the opposite sex in order of preference, marry the men and women together such that there are no two people of opposite sex who would both rather have each other than their current partners. When there are no such pairs of people, the set of marriages is deemed stable.
The existence of two classes that need to be paired with each other (heterosexual men and women in this example) distinguishes this problem from the stable roommates problem.
Applications
Algorithms for finding solutions to the stable marriage problem have applications in a variety of real-world situations, perhaps the best known of these being in the assignment of graduating medical students to their first hospital appointments.[1] In 2012, the Nobel Memorial Prize in Economic Sciences was awarded to Lloyd S. Shapley and Alvin E. Roth "for the theory of stable allocations and the practice of market design."[2]
An important and large-scale application of stable marriage is in assigning users to servers in a large distributed Internet service.[3] Billions of users access web pages, videos, and other services on the Internet, requiring each user to be matched to one of (potentially) hundreds of thousands of servers around the world that offer that service. A user prefers servers that are proximal enough to provide a faster response time for the requested service, resulting in a (partial) preferential ordering of the servers for each user. Each server prefers to serve users that it can with a lower cost, resulting in a (partial) preferential ordering of users for each server. Content delivery networks that distribute much of the world's content and services solve this large and complex stable marriage problem between users and servers every tens of seconds to enable billions of users to be matched up with their respective servers that can provide the requested web pages, videos, or other services.[3]
Different stable matchings
Main article: Lattice of stable matchings
In general, there may be many different stable matchings. For example, suppose there are three men (A,B,C) and three women (X,Y,Z) which have preferences of:
A: YXZ B: ZYX C: XZY
X: BAC Y: CBA Z: ACB
There are three stable solutions to this matching arrangement:
• men get their first choice and women their third - (AY, BZ, CX);
• all participants get their second choice - (AX, BY, CZ);
• women get their first choice and men their third - (AZ, BX, CY).
All three are stable, because instability requires both of the participants to be happier with an alternative match. Giving one group their first choices ensures that the matches are stable because they would be unhappy with any other proposed match. Giving everyone their second choice ensures that any other match would be disliked by one of the parties. In general, the family of solutions to any instance of the stable marriage problem can be given the structure of a finite distributive lattice, and this structure leads to efficient algorithms for several problems on stable marriages.[4]
In a uniformly-random instance of the stable marriage problem with n men and n women, the average number of stable matchings is asymptotically $e^{-1}n\ln n$.[5] In a stable marriage instance chosen to maximize the number of different stable matchings, this number is an exponential function of n.[6] Counting the number of stable matchings in a given instance is #P-complete.[7]
Algorithmic solution
Main article: Gale–Shapley algorithm
In 1962, David Gale and Lloyd Shapley proved that, for any equal number of men and women, it is always possible to solve the SMP and make all marriages stable. They presented an algorithm to do so.[8][9]
The Gale–Shapley algorithm (also known as the deferred acceptance algorithm) involves a number of "rounds" (or "iterations"):
• In the first round, first a) each unengaged man proposes to the woman he prefers most, and then b) each woman replies "maybe" to her suitor she most prefers and "no" to all other suitors. She is then provisionally "engaged" to the suitor she most prefers so far, and that suitor is likewise provisionally engaged to her.
• In each subsequent round, first a) each unengaged man proposes to the most-preferred woman to whom he has not yet proposed (regardless of whether the woman is already engaged), and then b) each woman replies "maybe" if she is currently not engaged or if she prefers this man over her current provisional partner (in this case, she rejects her current provisional partner who becomes unengaged). The provisional nature of engagements preserves the right of an already-engaged woman to "trade up" (and, in the process, to "jilt" her until-then partner).
• This process is repeated until everyone is engaged.
This algorithm is guaranteed to produce a stable marriage for all participants in time $O(n^{2})$ where $n$ is the number of men or women.[10]
Among all possible different stable matchings, it always yields the one that is best for all men among all stable matchings, and worst for all women. It is a truthful mechanism from the point of view of men (the proposing side), i.e., no man can get a better matching for himself by misrepresenting his preferences. Moreover, the GS algorithm is even group-strategy proof for men, i.e., no coalition of men can coordinate a misrepresentation of their preferences such that all men in the coalition are strictly better-off.[11] However, it is possible for some coalition to misrepresent their preferences such that some men are better-off and the other men retain the same partner.[12] The GS algorithm is non-truthful for the women (the reviewing side): each woman may be able to misrepresent her preferences and get a better match.
Rural hospitals theorem
The rural hospitals theorem concerns a more general variant of the stable matching problem, like that applying in the problem of matching doctors to positions at hospitals, differing in the following ways from the basic n-to-n form of the stable marriage problem:
• Each participant may only be willing to be matched to a subset of the participants on the other side of the matching.
• The participants on one side of the matching (the hospitals) may have a numerical capacity, specifying the number of doctors they are willing to hire.
• The total number of participants on one side might not equal the total capacity to which they are to be matched on the other side.
• The resulting matching might not match all of the participants.
In this case, the condition of stability is that no unmatched pair prefer each other to their situation in the matching (whether that situation is another partner or being unmatched). With this condition, a stable matching will still exist, and can still be found by the Gale–Shapley algorithm.
For this kind of stable matching problem, the rural hospitals theorem states that:
• The set of assigned doctors, and the number of filled positions in each hospital, are the same in all stable matchings.
• Any hospital that has some empty positions in some stable matching, receives exactly the same set of doctors in all stable matchings.
Related problems
In stable matching with indifference, some men might be indifferent between two or more women and vice versa.
The stable roommates problem is similar to the stable marriage problem, but differs in that all participants belong to a single pool (instead of being divided into equal numbers of "men" and "women").
The hospitals/residents problem – also known as the college admissions problem – differs from the stable marriage problem in that a hospital can take multiple residents, or a college can take an incoming class of more than one student. Algorithms to solve the hospitals/residents problem can be hospital-oriented (as the NRMP was before 1995)[13] or resident-oriented. This problem was solved, with an algorithm, in the same original paper by Gale and Shapley, in which the stable marriage problem was solved.[8]
The hospitals/residents problem with couples allows the set of residents to include couples who must be assigned together, either to the same hospital or to a specific pair of hospitals chosen by the couple (e.g., a married couple want to ensure that they will stay together and not be stuck in programs that are far away from each other). The addition of couples to the hospitals/residents problem renders the problem NP-complete.[14]
The assignment problem seeks to find a matching in a weighted bipartite graph that has maximum weight. Maximum weighted matchings do not have to be stable, but in some applications a maximum weighted matching is better than a stable one.
The matching with contracts problem is a generalization of matching problem, in which participants can be matched with different terms of contracts.[15] An important special case of contracts is matching with flexible wages.[16]
See also
• Matching (graph theory) – matching between different vertices of the graph; usually unrelated to preference-ordering.
• Envy-free matching – a relaxation of stable matching for many-to-one matching problems
• Rainbow matching for edge colored graphs
• Stable matching polytope
References
1. Stable Matching Algorithms
2. "The Prize in Economic Sciences 2012". Nobelprize.org. Retrieved 2013-09-09.
3. Bruce Maggs and Ramesh Sitaraman (2015). "Algorithmic nuggets in content delivery" (PDF). ACM SIGCOMM Computer Communication Review. 45 (3).
4. Gusfield, Dan (1987). "Three fast algorithms for four problems in stable marriage". SIAM Journal on Computing. 16 (1): 111–128. doi:10.1137/0216010. MR 0873255.
5. Pittel, Boris (1989). "The average number of stable matchings". SIAM Journal on Discrete Mathematics. 2 (4): 530–549. doi:10.1137/0402048. MR 1018538.
6. Karlin, Anna R.; Gharan, Shayan Oveis; Weber, Robbie (2018). "A simply exponential upper bound on the maximum number of stable matchings". In Diakonikolas, Ilias; Kempe, David; Henzinger, Monika (eds.). Proceedings of the 50th Symposium on Theory of Computing (STOC 2018). Association for Computing Machinery. pp. 920–925. arXiv:1711.01032. doi:10.1145/3188745.3188848. MR 3826305.
7. Irving, Robert W.; Leather, Paul (1986). "The complexity of counting stable marriages". SIAM Journal on Computing. 15 (3): 655–667. doi:10.1137/0215048. MR 0850415.
8. Gale, D.; Shapley, L. S. (1962). "College Admissions and the Stability of Marriage". American Mathematical Monthly. 69 (1): 9–14. doi:10.2307/2312726. JSTOR 2312726. Archived from the original on September 25, 2017.
9. Harry Mairson: "The Stable Marriage Problem", The Brandeis Review 12, 1992 (online).
10. Iwama, Kazuo; Miyazaki, Shuichi (2008). "A Survey of the Stable Marriage Problem and Its Variants". International Conference on Informatics Education and Research for Knowledge-Circulating Society (ICKS 2008). IEEE. pp. 131–136. doi:10.1109/ICKS.2008.7. hdl:2433/226940. ISBN 978-0-7695-3128-1.
11. Dubins, L. E.; Freedman, D. A. (1981). "Machiavelli and the Gale–Shapley algorithm". American Mathematical Monthly. 88 (7): 485–494. doi:10.2307/2321753. JSTOR 2321753. MR 0628016.
12. Huang, Chien-Chung (2006). "Cheating by men in the Gale-Shapley stable matching algorithm". In Azar, Yossi; Erlebach, Thomas (eds.). Algorithms - ESA 2006, 14th Annual European Symposium, Zurich, Switzerland, September 11-13, 2006, Proceedings. Lecture Notes in Computer Science. Vol. 4168. Springer. pp. 418–431. doi:10.1007/11841036_39. MR 2347162.
13. Robinson, Sara (April 2003). "Are Medical Students Meeting Their (Best Possible) Match?" (PDF). SIAM News (3): 36. Retrieved 2 January 2018.
14. Gusfield, D.; Irving, R. W. (1989). The Stable Marriage Problem: Structure and Algorithms. MIT Press. p. 54. ISBN 0-262-07118-5.
15. Hatfield, John William; Milgrom, Paul (2005). "Matching with Contracts". American Economic Review. 95 (4): 913–935. doi:10.1257/0002828054825466. JSTOR 4132699.
16. Crawford, Vincent; Knoer, Elsie Marie (1981). "Job Matching with Heterogeneous Firms and Workers". Econometrica. 49 (2): 437–450. doi:10.2307/1913320. JSTOR 1913320.
Further reading
• Kleinberg, J., and Tardos, E. (2005) Algorithm Design, Chapter 1, pp 1–12. See companion website for the Text Archived 2011-05-14 at the Wayback Machine.
• Knuth, D. E. (1996). Stable Marriage and Its Relation to Other Combinatorial Problems: An Introduction to the Mathematical Analysis of Algorithms. CRM Proceedings and Lecture Notes. English translation. American Mathematical Society.
• Pittel, B. (1992). "On likely solutions of a stable marriage problem". The Annals of Applied Probability. 2 (2): 358–401. doi:10.1214/aoap/1177005708. JSTOR 2959755.
• Roth, A. E. (1984). "The evolution of the labor market for medical interns and residents: A case study in game theory" (PDF). Journal of Political Economy. 92 (6): 991–1016. doi:10.1086/261272. S2CID 1360205.
• Roth, A. E.; Sotomayor, M. A. O. (1990). Two-sided matching: A study in game-theoretic modeling and analysis. Cambridge University Press.
• Shoham, Yoav; Leyton-Brown, Kevin (2009). Multiagent Systems: Algorithmic, Game-Theoretic, and Logical Foundations. New York: Cambridge University Press. ISBN 978-0-521-89943-7. See Section 10.6.4; downloadable free online.
• Schummer, J.; Vohra, R. V. (2007). "Mechanism design without money" (PDF). In Nisan, Noam; Roughgarden, Tim; Tardos, Eva; Vazirani, Vijay (eds.). Algorithmic Game Theory. pp. 255–262. ISBN 978-0521872829.
External links
• Interactive Flash Demonstration of SMP
• https://web.archive.org/web/20080512150525/http://kuznets.fas.harvard.edu/~aroth/alroth.html#NRMP
• http://www.dcs.gla.ac.uk/research/algorithms/stable/EGSapplet/EGS.html
• SMP Lecture Notes
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Wikipedia
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Stable matching polytope
In mathematics, economics, and computer science, the stable matching polytope or stable marriage polytope is a convex polytope derived from the solutions to an instance of the stable matching problem.[1][2]
Description
The stable matching polytope is the convex hull of the indicator vectors of the stable matchings of the given problem. It has a dimension for each pair of elements that can be matched, and a vertex for each stable matchings. For each vertex, the Cartesian coordinates are one for pairs that are matched in the corresponding matching, and zero for pairs that are not matched.[1]
The stable matching polytope has a polynomial number of facets. These include the conventional inequalities describing matchings without the requirement of stability (each coordinate must be between 0 and 1, and for each element to be matched the sum of coordinates for the pairs involving that element must be exactly one), together with inequalities constraining the resulting matching to be stable (for each potential matched pair elements, the sum of coordinates for matches that are at least as good for one of the two elements must be at least one). The points satisfying all of these constraints can be thought of as the fractional solutions of a linear programming relaxation of the stable matching problem.
Integrality
It is a theorem of Vande Vate (1989) that the polytope described by the facet constraints listed above has only the vertices described above. In particular it is an integral polytope. This can be seen as an analogue of the theorem of Garrett Birkhoff that an analogous polytope, the Birkhoff polytope describing the set of all fractional matchings between two sets, is integral.[3]
An equivalent way of stating the same theorem is that every fractional matching can be expressed as a convex combination of integral matchings. Teo & Sethuraman (1998) prove this by constructing a probability distribution on integral matchings whose expected value can be set equal to any given fractional matching. To do so, they perform the following steps:
• Consider for each element on one side of the stable matching problem (the doctors, say, in a problem matching doctors to hospitals) the fractional values assigned to pairings with the elements on the other side (the hospitals), and sort these values in decreasing order by that doctor's preferences.
• Partition the unit interval into subintervals, of lengths equal to these fractional values, in the sorted order. Choosing a random number in the unit interval will give a random match for the selected doctor, with probability equal to the fractional weight of that match.
• Symmetrically, consider for each element on the other side of the stable matching (the hospitals), sort the fractional values for pairings involving that element in increasing order by preference, and construct a partition of the unit interval whose subintervals have these fractional values in the sorted order.
• It can be proven that, for each matched pair, the subintervals associated with that pair are the same in both the partition for the doctor and the partition for the hospital in that pair. Therefore, choosing a single random number in the unit interval, and using that choice to simultaneously select a hospital for each doctor and a doctor for each hospital, gives a matching. Moreover, this matching can be shown to be stable.
The resulting randomly chosen stable matching chooses any particular matched pair with probability equal to the fractional coordinate value of that pair. Therefore, the probability distribution over stable matchings constructed in this way provides a representation of the given fractional matching as a convex combination of integral stable matchings.[4]
Lattice of fractional matchings
The family of all stable matchings forms a distributive lattice, the lattice of stable matchings, in which the join of two matchings gives all doctors their preference among their assigned hospitals in the two matchings, and the meet gives all hospitals their preference.[5] The same is true of the family of all fractional stable matchings, the points of the stable matching polytope.[3]
In the stable matching polytope, one can define one matching to dominate another if, for every doctor and hospital, the total fractional value assigned to matches for that doctor that are at least as good (for the doctor) as that hospital are at least as large in the first matching as in the second. This defines a partial order on the fractional matchings. This partial order has a unique largest element, the integer stable matching found by a version of the Gale–Shapley algorithm in which the doctors propose matches and the hospitals respond to the proposals. It also has a unique smallest element, the integer stable matching found by a version of the Gale–Shapley algorithm in which the hospitals make the proposals.[3]
Consistently with this partial order, one can define the meet of two fractional matchings to be a fractional matching that is as low as possible in the partial order while dominating the two matchings. For each doctor and hospital, it assigns to that potential matched pair a weight that makes the total weight of that pair and all better pairs for the same doctor equal to the larger of the corresponding totals from the two given matchings. The join is defined symmetrically.[3]
Applications
By applying linear programming to the stable matching polytope, one can find the minimum or maximum weight stable matching.[1] Alternative methods for the same problem include applying the closure problem to a partially ordered set derived from the lattice of stable matchings,[6] or applying linear programming to the order polytope of this partial order.
Relation to order polytope
The property of the stable matching polytope, of defining a continuous distributive lattice is analogous to the defining property of a distributive polytope, a polytope in which coordinatewise maximization and minimization form the meet and join operations of a lattice.[7] However, the meet and join operations for the stable matching polytope are defined in a different way than coordinatewise maximization and minimization. Instead, the order polytope of the underlying partial order of the lattice of stable matchings provides a distributive polytope associated with the set of stable matchings, but one for which it is more difficult to read off the fractional value associated with each matched pair. In fact, the stable matching polytope and the order polytope of the underlying partial order are very closely related to each other: each is an affine transformation of the other.[8]
References
1. Vande Vate, John H. (1989), "Linear programming brings marital bliss", Operations Research Letters, 8 (3): 147–153, doi:10.1016/0167-6377(89)90041-2, MR 1007271
2. Ratier, Guillaume (1996), "On the stable marriage polytope" (PDF), Discrete Mathematics, 148 (1–3): 141–159, doi:10.1016/0012-365X(94)00237-D, MR 1368286
3. Roth, Alvin E.; Rothblum, Uriel G.; Vande Vate, John H. (1993), "Stable matchings, optimal assignments, and linear programming", Mathematics of Operations Research, 18 (4): 803–828, doi:10.1287/moor.18.4.803, JSTOR 3690124, MR 1251681
4. Teo, Chung-Piaw; Sethuraman, Jay (1998), "The geometry of fractional stable matchings and its applications", Mathematics of Operations Research, 23 (4): 874–891, doi:10.1287/moor.23.4.874, MR 1662426
5. Knuth, Donald E. (1976), Mariages stables et leurs relations avec d'autres problèmes combinatoires (PDF) (in French), Montréal, Quebec: Les Presses de l'Université de Montréal, ISBN 0-8405-0342-3, MR 0488980. See in particular Problem 6, pp. 87–94.
6. Irving, Robert W.; Leather, Paul; Gusfield, Dan (1987), "An efficient algorithm for the "optimal" stable marriage", Journal of the ACM, 34 (3): 532–543, doi:10.1145/28869.28871, MR 0904192
7. Felsner, Stefan; Knauer, Kolja (2011), "Distributive lattices, polyhedra, and generalized flows", European Journal of Combinatorics, 32 (1): 45–59, doi:10.1016/j.ejc.2010.07.011, MR 2727459.
8. Aprile, Manuel; Cevallos, Alfonso; Faenza, Yuri (2018), "On 2-level polytopes arising in combinatorial settings", SIAM Journal on Discrete Mathematics, 32 (3): 1857–1886, arXiv:1702.03187, doi:10.1137/17M1116684, MR 3835234
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Wikipedia
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Stable model category
In category theory, a branch of mathematics, a stable model category is a pointed model category in which the suspension functor is an equivalence of the homotopy category with itself.
The prototypical examples are the category of spectra in the stable homotopy theory and the category of chain complex of R-modules. On the other hand, the category of pointed topological spaces and the category of pointed simplicial sets are not stable model categories.
Any stable model category is equivalent to a category of presheaves of spectra.
References
• Mark Hovey: Model Categories, 1999, ISBN 0-8218-1359-5.
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Wikipedia
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Stable module category
In representation theory, the stable module category is a category in which projectives are "factored out."
Definition
Let R be a ring. For two modules M and N over R, define ${\underline {\mathrm {Hom} }}(M,N)$ to be the set of R-linear maps from M to N modulo the relation that f ~ g if f − g factors through a projective module. The stable module category is defined by setting the objects to be the R-modules, and the morphisms are the equivalence classes ${\underline {\mathrm {Hom} }}(M,N)$.
Given a module M, let P be a projective module with a surjection $p\colon P\to M$. Then set $\Omega (M)$ to be the kernel of p. Suppose we are given a morphism $f\colon M\to N$ and a surjection $q\colon Q\to N$ where Q is projective. Then one can lift f to a map $P\to Q$ which maps $\Omega (M)$ into $\Omega (N)$. This gives a well-defined functor $\Omega $ from the stable module category to itself.
For certain rings, such as Frobenius algebras, $\Omega $ is an equivalence of categories. In this case, the inverse $\Omega ^{-1}$ can be defined as follows. Given M, find an injective module I with an inclusion $i\colon M\to I$. Then $\Omega ^{-1}(M)$ is defined to be the cokernel of i. A case of particular interest is when the ring R is a group algebra.
The functor Ω−1 can even be defined on the module category of a general ring (without factoring out projectives), as the cokernel of the injective envelope. It need not be true in this case that the functor Ω−1 is actually an inverse to Ω. One important property of the stable module category is it allows defining the Ω functor for general rings. When R is perfect (or M is finitely generated and R is semiperfect), then Ω(M) can be defined as the kernel of the projective cover, giving a functor on the module category. However, in general projective covers need not exist, and so passing to the stable module category is necessary.
Connections with cohomology
Now we suppose that R = kG is a group algebra for some field k and some group G. One can show that there exist isomorphisms
${\underline {\mathrm {Hom} }}(\Omega ^{n}(M),N)\cong \mathrm {Ext} _{kG}^{n}(M,N)\cong {\underline {\mathrm {Hom} }}(M,\Omega ^{-n}(N))$
for every positive integer n. The group cohomology of a representation M is given by $\mathrm {H} ^{n}(G;M)=\mathrm {Ext} _{kG}^{n}(k,M)$ where k has a trivial G-action, so in this way the stable module category gives a natural setting in which group cohomology lives.
Furthermore, the above isomorphism suggests defining cohomology groups for negative values of n, and in this way one recovers Tate cohomology.
Triangulated structure
An exact sequence
$0\to X\to E\to Y\to 0$
in the usual module category defines an element of $\mathrm {Ext} _{kG}^{1}(Y,X)$, and hence an element of ${\underline {\mathrm {Hom} }}(Y,\Omega ^{-1}(X))$, so that we get a sequence
$X\to E\to Y\to \Omega ^{-1}(X).$
Taking $\Omega ^{-1}$ to be the translation functor and such sequences as above to be exact triangles, the stable module category becomes a triangulated category.
See also
• Stable homotopy theory
References
• J. F. Carlson, Lisa Townsley, Luis Valero-Elizondo, Mucheng Zhang, Cohomology Rings of Finite Groups, Springer-Verlag, 2003.
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Wikipedia
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Stable normal bundle
In surgery theory, a branch of mathematics, the stable normal bundle of a differentiable manifold is an invariant which encodes the stable normal (dually, tangential) data. There are analogs for generalizations of manifold, notably PL-manifolds and topological manifolds. There is also an analogue in homotopy theory for Poincaré spaces, the Spivak spherical fibration, named after Michael Spivak.[1]
Construction via embeddings
Given an embedding of a manifold in Euclidean space (provided by the theorem of Hassler Whitney), it has a normal bundle. The embedding is not unique, but for high dimension of the Euclidean space it is unique up to isotopy, thus the (class of the) bundle is unique, and called the stable normal bundle.
This construction works for any Poincaré space X: a finite CW-complex admits a stably unique (up to homotopy) embedding in Euclidean space, via general position, and this embedding yields a spherical fibration over X. For more restricted spaces (notably PL-manifolds and topological manifolds), one gets stronger data.
Details
Two embeddings $i,i'\colon X\hookrightarrow \mathbb {R} ^{m}$ are isotopic if they are homotopic through embeddings. Given a manifold or other suitable space X, with two embeddings into Euclidean space $i\colon X\hookrightarrow \mathbb {R} ^{m},$ $j\colon X\hookrightarrow \mathbb {R} ^{n},$ these will not in general be isotopic, or even maps into the same space ($m$ need not equal $n$). However, one can embed these into a larger space $\mathbf {R} ^{N}$ by letting the last $N-m$ coordinates be 0:
$i\colon X\hookrightarrow \mathbb {R} ^{m}\cong \mathbb {R} ^{m}\times \left\{(0,\dots ,0)\right\}\subset \mathbb {R} ^{m}\times \mathbb {R} ^{N-m}\cong \mathbb {R} ^{N}$.
This process of adjoining trivial copies of Euclidean space is called stabilization. One can thus arrange for any two embeddings into Euclidean space to map into the same Euclidean space (taking $N=\max(m,n)$), and, further, if $N$ is sufficiently large, these embeddings are isotopic, which is a theorem.
Thus there is a unique stable isotopy class of embedding: it is not a particular embedding (as there are many embeddings), nor an isotopy class (as the target space is not fixed: it is just "a sufficiently large Euclidean space"), but rather a stable isotopy class of maps. The normal bundle associated with this (stable class of) embeddings is then the stable normal bundle.
One can replace this stable isotopy class with an actual isotopy class by fixing the target space, either by using Hilbert space as the target space, or (for a fixed dimension of manifold $n$) using a fixed $N$ sufficiently large, as N depends only on n, not the manifold in question.
More abstractly, rather than stabilizing the embedding, one can take any embedding, and then take a vector bundle direct sum with a sufficient number of trivial line bundles; this corresponds exactly to the normal bundle of the stabilized embedding.
Construction via classifying spaces
An n-manifold M has a tangent bundle, which has a classifying map (up to homotopy)
$\tau _{M}\colon M\to B{\textrm {O}}(n).$
Composing with the inclusion $B{\textrm {O}}(n)\to B{\textrm {O}}$ yields (the homotopy class of a classifying map of) the stable tangent bundle. The normal bundle of an embedding $M\subset \mathbb {R} ^{n+k}$ ($k$ large) is an inverse $\nu _{M}\colon M\to B{\textrm {O}}(k)$ for $\tau _{M}$, such that the Whitney sum $\tau _{M}\oplus \nu _{M}\colon M\to B{\textrm {O}}(n+k)$ is trivial. The homotopy class of the composite $\nu _{M}\colon M\to B{\textrm {O}}(k)\to B{\textrm {O}}$ is independent of the choice of embedding, classifying the stable normal bundle $\nu _{M}$.
Motivation
There is no intrinsic notion of a normal vector to a manifold, unlike tangent or cotangent vectors – for instance, the normal space depends on which dimension one is embedding into – so the stable normal bundle instead provides a notion of a stable normal space: a normal space (and normal vectors) up to trivial summands.
Why stable normal, instead of stable tangent? Stable normal data is used instead of unstable tangential data because generalizations of manifolds have natural stable normal-type structures, coming from tubular neighborhoods and generalizations, but not unstable tangential ones, as the local structure is not smooth.
Spherical fibrations over a space X are classified by the homotopy classes of maps $X\to BG$ to a classifying space $BG$, with homotopy groups the stable homotopy groups of spheres
$\pi _{*}(BG)=\pi _{*-1}^{S}$.
The forgetful map $B{\textrm {O}}\to BG$ extends to a fibration sequence
$B{\textrm {O}}\to BG\to B(G/{\textrm {O}})$.
A Poincaré space X does not have a tangent bundle, but it does have a well-defined stable spherical fibration, which for a differentiable manifold is the spherical fibration associated to the stable normal bundle; thus a primary obstruction to X having the homotopy type of a differentiable manifold is that the spherical fibration lifts to a vector bundle, i.e., the Spivak spherical fibration $X\to BG$ must lift to $X\to B{\textrm {O}}$, which is equivalent to the map $X\to B(G/{\textrm {O}})$ being null homotopic Thus the bundle obstruction to the existence of a (smooth) manifold structure is the class $X\to B(G/{\textrm {O}})$. The secondary obstruction is the Wall surgery obstruction.
Applications
The stable normal bundle is fundamental in surgery theory as a primary obstruction:
• For a Poincaré space X to have the homotopy type of a smooth manifold, the map $X\to B(G/{\textrm {O}})$ must be null homotopic
• For a homotopy equivalence $f\colon M\to N$ between two manifolds to be homotopic to a diffeomorphism, it must pull back the stable normal bundle on N to the stable normal bundle on M.
More generally, its generalizations serve as replacements for the (unstable) tangent bundle.
References
1. Spivak, Michael (1967), "Spaces satisfying Poincaré duality", Topology, 6 (6): 77–101, doi:10.1016/0040-9383(67)90016-X, MR 0214071
Manifolds (Glossary)
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Wikipedia
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Stable principal bundle
In mathematics, and especially differential geometry and algebraic geometry, a stable principal bundle is a generalisation of the notion of a stable vector bundle to the setting of principal bundles. The concept of stability for principal bundles was introduced by Annamalai Ramanathan for the purpose of defining the moduli space of G-principal bundles over a Riemann surface, a generalisation of earlier work by David Mumford and others on the moduli spaces of vector bundles.[1][2][3]
Many statements about the stability of vector bundles can be translated into the language of stable principal bundles. For example, the analogue of the Kobayashi–Hitchin correspondence for principal bundles, that a holomorphic principal bundle over a compact Kähler manifold admits a Hermite–Einstein connection if and only if it is polystable, was shown to be true in the case of projective manifolds by Subramanian and Ramanathan, and for arbitrary compact Kähler manifolds by Anchouche and Biswas.[4][5]
Definition
The essential definition of stability for principal bundles was made by Ramanathan, but applies only to the case of Riemann surfaces.[2] In this section we state the definition as appearing in the work of Anchouche and Biswas which is valid over any Kähler manifold, and indeed makes sense more generally for algebraic varieties.[5] This reduces to Ramanathan's definition in the case the manifold is a Riemann surface.
Let $G$ be a connected reductive algebraic group over the complex numbers $\mathbb {C} $. Let $(X,\omega )$ be a compact Kähler manifold of complex dimension $n$. Suppose $P\to X$ is a holomorphic principal $G$-bundle over $X$. Holomorphic here means that the transition functions for $P$ vary holomorphically, which makes sense as the structure group is a complex Lie group. The principal bundle $P$ is called stable (resp. semi-stable) if for every reduction of structure group $\sigma :U\to P/Q$ for $Q\subset G$ a maximal parabolic subgroup where $U\subset X$ is some open subset with the codimension $\operatorname {codim} (X\backslash U)\geq 2$, we have
$\deg \sigma ^{*}T_{\operatorname {rel} }P/Q>0\quad ({\text{resp. }}\geq 0).$
Here $T_{\operatorname {rel} }P/Q$ is the relative tangent bundle of the fibre bundle $\left.P/Q\right|_{U}\to U$ otherwise known as the vertical bundle of $T(\left.P/Q\right|_{U})$. Recall that the degree of a vector bundle (or coherent sheaf) $F\to X$ is defined to be
$\operatorname {deg} (F):=\int _{X}c_{1}(F)\wedge \omega ^{n-1},$
where $c_{1}(F)$ is the first Chern class of $F$. In the above setting the degree is computed for a bundle defined over $U$ inside $X$, but since the codimension of the complement of $U$ is bigger than two, the value of the integral will agree with that over all of $X$.
Notice that in the case where $\dim X=1$, that is where $X$ is a Riemann surface, by assumption on the codimension of $U$ we must have that $U=X$, so it is enough to consider reductions of structure group over the entirety of $X$, $\sigma :X\to P/Q$.
Relation to stability of vector bundles
Given a principal $G$-bundle for a complex Lie group $G$ there are several natural vector bundles one may associate to it.
Firstly if $G=\operatorname {GL} (n,\mathbb {C} )$, the general linear group, then the standard representation of $\operatorname {GL} (n,\mathbb {C} )$ on $\mathbb {C} ^{n}$ allows one to construct the associated bundle $E=P\times _{\operatorname {GL} (n,\mathbb {C} )}\mathbb {C} ^{n}$. This is a holomorphic vector bundle over $X$, and the above definition of stability of the principal bundle is equivalent to slope stability of $E$. The essential point is that a maximal parabolic subgroup $Q\subset \operatorname {GL} (n,\mathbb {C} )$ corresponds to a choice of flag $0\subset W\subset \mathbb {C} ^{n}$, where $W$ is invariant under the subgroup $Q$. Since the structure group of $P$ has been reduced to $Q$, and $Q$ preserves the vector subspace $W\subset \mathbb {C} ^{n}$, one may take the associated bundle $F=P\times _{Q}W$, which is a sub-bundle of $E$ over the subset $U\subset X$ on which the reduction of structure group is defined, and therefore a subsheaf of $E$ over all of $X$. It can then be computed that
$\deg \sigma ^{*}T_{\operatorname {rel} }P/Q=\mu (E)-\mu (F)$
where $\mu $ denotes the slope of the vector bundles.
When the structure group is not $G=\operatorname {GL} (n,\mathbb {C} )$ there is still a natural associated vector bundle to $P$, the adjoint bundle $\operatorname {ad} P$, with fibre given by the Lie algebra ${\mathfrak {g}}$ of $G$. The principal bundle $P$ is semistable if and only if the adjoint bundle $\operatorname {ad} P$ is slope semistable, and furthermore if $P$ is stable, then $\operatorname {ad} P$ is slope polystable.[5] Again the key point here is that for a parabolic subgroup $Q\subset G$, one obtains a parabolic subalgebra ${\mathfrak {q}}\subset {\mathfrak {g}}$ and can take the associated subbundle. In this case more care must be taken because the adjoint representation of $G$ on ${\mathfrak {g}}$ is not always faithful or irreducible, the latter condition hinting at why stability of the principal bundle only leads to polystability of the adjoint bundle (because a representation that splits as a direct sum would lead to the associated bundle splitting as a direct sum).
Generalisations
Just as one can generalise a vector bundle to the notion of a Higgs bundle, it is possible to formulate a definition of a principal $G$-Higgs bundle. The above definition of stability for principal bundles generalises to these objects by requiring the reductions of structure group are compatible with the Higgs field of the principal Higgs bundle. It was shown by Anchouche and Biswas that the analogue of the nonabelian Hodge correspondence for Higgs vector bundles is true for principal $G$-Higgs bundles in the case where the base manifold $(X,\omega )$ is a complex projective variety.[5]
References
1. Ramanathan, A., 1975. Stable principal bundles on a compact Riemann surface. Mathematische Annalen, 213(2), pp.129-152.
2. Ramanathan, A., 1996, August. Moduli for principal bundles over algebraic curves: I. In Proceedings of the Indian Academy of Sciences-Mathematical Sciences (Vol. 106, No. 3, pp. 301-328). Springer India.
3. Ramanathan, A., 1996, November. Moduli for principal bundles over algebraic curves: II. In Proceedings of the Indian Academy of Sciences-Mathematical Sciences (Vol. 106, No. 4, pp. 421-449). Springer India.
4. Subramanian, S. and Ramanathan, A., 1988. Einstein-Hermitian connections on principal bundles and stability.
5. Anchouche, B. and Biswas, I., 2001. Einstein-Hermitian connections on polystable principal bundles over a compact Kähler manifold. American Journal of Mathematics, 123(2), pp.207-228.
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Wikipedia
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Stable range condition
In mathematics, particular in abstract algebra and algebraic K-theory, the stable range of a ring $R$ is the smallest integer $n$ such that whenever $v_{0},v_{1},...,v_{n}$ in $R$ generate the unit ideal (they form a unimodular row), there exist some $t_{1},...,t_{n}$in $R$ such that the elements $v_{i}-v_{0}t_{i}$ for $1\leq i\leq n$ also generate the unit ideal.
If $R$ is a commutative Noetherian ring of Krull dimension $d$ , then the stable range of $R$ is at most $d+1$ (a theorem of Bass).
Bass stable range
The Bass stable range condition $SR_{m}$ refers to precisely the same notion, but for historical reasons it is indexed differently: a ring $R$ satisfies$SR_{m}$ if for any $v_{1},...,v_{m}$ in $R$ generating the unit ideal there exist $t_{2},...,t_{m}$ in $R$ such that $v_{i}-v_{1}t_{i}$ for $2\leq i\leq m$ generate the unit ideal.
Comparing with the above definition, a ring with stable range $n$ satisfies $SR_{n+1}$. In particular, Bass's theorem states that a commutative Noetherian ring of Krull dimension $d$ satisfies $SR_{d+2}$. (For this reason, one often finds hypotheses phrased as "Suppose that $R$ satisfies Bass's stable range condition $SR_{d+2}$...")
Stable range relative to an ideal
Less commonly, one has the notion of the stable range of an ideal $I$ in a ring $R$. The stable range of the pair $(R,I)$ is the smallest integer $n$ such that for any elements $v_{0},...,v_{n}$ in $R$ that generate the unit ideal and satisfy $v\equiv 1$ mod $I$ and $v_{i}\equiv 0$ mod $I$ for $0\leq i\leq n-1$, there exist $t_{1},...,t_{n}$ in $R$ such that $v_{i}-v_{0}t_{i}$ for $1\leq i\leq n$ also generate the unit ideal. As above, in this case we say that $(R,I)$ satisfies the Bass stable range condition $SR_{n+1}$.
By definition, the stable range of $(R,I)$ is always less than or equal to the stable range of $R$.
References
• Charles Weibel, The K-book: An introduction to algebraic K-theory
H. Chen, Rings Related Stable Range Conditions, Series in Algebra 11, World Scientific, Hackensack, NJ, 2011.
External links
• Bass' stable range condition for principal ideal domains
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Wikipedia
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Stable roommates problem
In mathematics, economics and computer science, particularly in the fields of combinatorics, game theory and algorithms, the stable-roommate problem (SRP) is the problem of finding a stable matching for an even-sized set. A matching is a separation of the set into disjoint pairs ("roommates"). The matching is stable if there are no two elements which are not roommates and which both prefer each other to their roommate under the matching. This is distinct from the stable-marriage problem in that the stable-roommates problem allows matches between any two elements, not just between classes of "men" and "women".
It is commonly stated as:
In a given instance of the stable-roommates problem (SRP), each of 2n participants ranks the others in strict order of preference. A matching is a set of n disjoint pairs of participants. A matching M in an instance of SRP is stable if there are no two participants x and y, each of whom prefers the other to their partner in M. Such a pair is said to block M, or to be a blocking pair with respect to M.
Solution
Unlike the stable marriage problem, a stable matching may fail to exist for certain sets of participants and their preferences. For a minimal example of a stable pairing not existing, consider 4 people A, B, C, and D, whose rankings are:
A:(B,C,D), B:(C,A,D), C:(A,B,D), D:(A,B,C)
In this ranking, each of A, B, and C is the most preferable person for someone. In any solution, one of A, B, or C must be paired with D and the other two with each other (for example AD and BC), yet for anyone who is partnered with D, another member will have rated them highest, and D's partner will in turn prefer this other member over D. In this example, AC is a more favorable pairing than AD, but the necessary remaining pairing of BD then raises the same issue, illustrating the absence of a stable matching for these participants and their preferences.
Algorithm
An efficient algorithm was given in (Irving 1985).[1] The algorithm will determine, for any instance of the problem, whether a stable matching exists, and if so, will find such a matching. Irving's algorithm has O(n2) complexity, provided suitable data structures are used to implement the necessary manipulation of the preference lists and identification of rotations.
The algorithm consists of two phases. In Phase 1, participants propose to each other, in a manner similar to that of the Gale-Shapley algorithm for the stable marriage problem. Each participant orders the other members by preference, resulting in a preference list—an ordered set of the other participants. Participants then propose to each person on their list, in order, continuing to the next person if and when their current proposal is rejected. A participant will reject a proposal if they already hold a proposal from someone they prefer. A participant will also reject a previously-accepted proposal if they later receive a proposal that they prefer. In this case, the rejected participant will then propose to the next person on their list, continuing until a proposal is again accepted. If any participant is eventually rejected by all other participants, this indicates that no stable matching is possible. Otherwise, Phase 1 will end with each person holding a proposal from one of the others.
Consider two participants, q and p. If q holds a proposal from p, then we remove from q's list all participants x after p, and symmetrically, for each removed participant x, we remove q from x's list, so that q is first in p's list; and p, last in q's, since q and any x cannot be partners in any stable matching. The resulting reduced set of preference lists together is called the Phase 1 table. In this table, if any reduced list is empty, then there is no stable matching. Otherwise, the Phase 1 table is a stable table. A stable table, by definition, is the set of preference lists from the original table after members have been removed from one or more of the lists, and the following three conditions are satisfied (where reduced list means a list in the stable table):
(i) p is first on q's reduced list if and only if q is last on p's
(ii) p is not on q's reduced list if and only if q is not on p's if and only if q prefers the last person on their list to p; or p, the last person on their list to q
(iii) no reduced list is empty
Stable tables have several important properties, which are used to justify the remainder of the procedure:
1. Any stable table must be a subtable of the Phase 1 table, where subtable is a table where the preference lists of the subtable are those of the supertable with some individuals removed from each other's lists.
2. In any stable table, if every reduced list contains exactly one individual, then pairing each individual with the single person on their list gives a stable matching.
3. If the stable roommates problem instance has a stable matching, then there is a stable matching contained in any one of the stable tables.
4. Any stable subtable of a stable table, and in particular any stable subtable that specifies a stable matching as in 2, can be obtained by a sequence of rotation eliminations on the stable table.
These rotation eliminations comprise Phase 2 of Irving's algorithm.
By 2, if each reduced list of the Phase 1 table contains exactly one individual, then this gives a matching.
Otherwise, the algorithm enters Phase 2. A rotation in a stable table T is defined as a sequence (x0, y0), (x1, y1), ..., (xk-1, yk-1) such that the xi are distinct, yi is first on xi's reduced list (or xi is last on yi's reduced list) and yi+1 is second on xi's reduced list, for i = 0, ..., k-1 where the indices are taken modulo k. It follows that in any stable table with a reduced list containing at least two individuals, such a rotation always exists. To find it, start at such a p0 containing at least two individuals in their reduced list, and define recursively qi+1 to be the second on pi's list and pi+1 to be the last on qi+1's list, until this sequence repeats some pj, at which point a rotation is found: it is the sequence of pairs starting at the first occurrence of (pj, qj) and ending at the pair before the last occurrence. The sequence of pi up until the pj is called the tail of the rotation. The fact that it's a stable table in which this search occurs guarantees that each pi has at least two individuals on their list.
To eliminate the rotation, yi rejects xi so that xi proposes to yi+1, for each i. To restore the stable table properties (i) and (ii), for each i, all successors of xi-1 are removed from yi's list, and yi is removed from their lists. If a reduced list becomes empty during these removals, then there is no stable matching. Otherwise, the new table is again a stable table, and either already specifies a matching since each list contains exactly one individual or there remains another rotation to find and eliminate, so the step is repeated.
Phase 2 of the algorithm can now be summarized as follows:
T = Phase 1 table;
while (true) {
identify a rotation r in T;
eliminate r from T;
if some list in T becomes empty,
return null; (no stable matching can exist)
else if (each reduced list in T has size 1)
return the matching M = {{x, y} | x and y are on each other's lists in T}; (this is a stable matching)
}
To achieve an O(n2) running time, a ranking matrix whose entry at row i and column j is the position of the jth individual in the ith's list; this takes O(n2) time. With the ranking matrix, checking whether an individual prefers one to another can be done in constant time by comparing their ranks in the matrix. Furthermore, instead of explicitly removing elements from the preference lists, the indices of the first, second and last on each individual's reduced list are maintained. The first individual that is unmatched, i.e. has at least two in their reduced lists, is also maintained. Then, in Phase 2, the sequence of pi "traversed" to find a rotation is stored in a list, and an array is used to mark individuals as having been visited, as in a standard depth-first search graph traversal. After the elimination of a rotation, we continue to store only its tail, if any, in the list and as visited in the array, and start the search for the next rotation at the last individual on the tail, and otherwise at the next unmatched individual if there is no tail. This reduces repeated traversal of the tail, since it is largely unaffected by the elimination of the rotation.
Example
The following are the preference lists for a Stable Roommates instance involving 6 participants: 1, 2, 3, 4, 5, 6.
1 : 3 4 2 6 5
2 : 6 5 4 1 3
3 : 2 4 5 1 6
4 : 5 2 3 6 1
5 : 3 1 2 4 6
6 : 5 1 3 4 2
A possible execution of Phase 1 consists of the following sequence of proposals and rejections, where → represents proposes to and × represents rejects.
1 → 3
2 → 6
3 → 2
4 → 5
5 → 3; 3 × 1
1 → 4
6 → 5; 5 × 6
6 → 1
So Phase 1 ends with the following reduced preference lists: (for example we cross out 5 for 1: because 1: gets at least 6)
1 : 3 4 2 6 5
2 : 6 5 4 1 3
3 : 2 4 5 1 6
4 : 5 2 3 6 1
5 : 3 1 2 4 6
6 : 5 1 3 4 2
In Phase 2, the rotation r1 = (1,4), (3,2) is first identified. This is because 2 is 1's second favorite, and 4 is the second favorite of 3. Eliminating r1 gives:
1 : 3 4 2 6 5
2 : 6 5 4 1 3
3 : 2 4 5 1 6
4 : 5 2 3 6 1
5 : 3 1 2 4 6
6 : 5 1 3 4 2
Next, the rotation r2 = (1,2), (2,6), (4,5) is identified, and its elimination yields:
1 : 3 4 2 6 5
2 : 6 5 4 1 3
3 : 2 4 5 1 6
4 : 5 2 3 6 1
5 : 3 1 2 4 6
6 : 5 1 3 4 2
Hence 1 and 6 are matched. Finally, the rotation r3 = (2,5), (3,4) is identified, and its elimination gives:
1 : 3 4 2 6 5
2 : 6 5 4 1 3
3 : 2 4 5 1 6
4 : 5 2 3 6 1
5 : 3 1 2 4 6
6 : 5 1 3 4 2
Hence the matching {1, 6}, {2,4}, {3, 5} is stable.
Implementation in software packages
• Python: An implementation of Irving's algorithm is available as part of the matching library.[2]
• Java: A constraint programming model to find all stable matchings in the roommates problem with incomplete lists is available under the CRAPL licence.[3][4]
• R: The same constraint programming model is also available as part of the R matchingMarkets package.[5][6]
• API: The MatchingTools API provides a free application programming interface for the algorithm.[7]
• Web Application: The "Dyad Finder" website provides a free, web-based implementation of the algorithm, including source code for the website and solver written in JavaScript.[8]
• Matlab: The algorithm is implemented in the assignStableRoommates function that is part of the United States Naval Research Laboratory's free Tracker Component Library.[9]
References
1. Irving, Robert W. (1985), "An efficient algorithm for the "stable roommates" problem", Journal of Algorithms, 6 (4): 577–595, doi:10.1016/0196-6774(85)90033-1
2. Wilde, H.; Knight, V.; Gillard, J. (2020). "Matching: A Python library for solving matching games". Journal of Open Source Software. 5 (48): 2169. Bibcode:2020JOSS....5.2169W. doi:10.21105/joss.02169.
3. Prosser, P. (2014). "Stable Roommates and Constraint Programming" (PDF). Integration of AI and OR Techniques in Constraint Programming. Lecture Notes in Computer Science. Vol. 8451. pp. 15–28. doi:10.1007/978-3-319-07046-9_2. ISBN 978-3-319-07045-2.
4. "Constraint encoding for stable roommates problem". Java release.
5. Klein, T. (2015). "Analysis of Stable Matchings in R: Package matchingMarkets" (PDF). Vignette to R Package MatchingMarkets.
6. "matchingMarkets: Analysis of Stable Matchings". R Project. 2019-02-04.
7. "MatchingTools API".
8. "Dyad Finder". dyad-finder.web.app. Retrieved 2020-05-06.
9. "Tracker Component Library". Matlab Repository. Retrieved January 5, 2019.
Further reading
• Fleiner, Tamás; Irving, Robert W.; Manlove, David F. (2007), "An efficient algorithm for the "stable roommates" problem", Theoretical Computer Science, 381 (1–3): 162–176, doi:10.1016/j.tcs.2007.04.029
• Gusfield, Daniel M.; Irving, Robert W. (1989), The Stable Marriage Problem: Structure and Algorithms, MIT Press
• Irving, Robert W.; Manlove, David F. (2002), "The Stable Roommates Problem with Ties" (PDF), Journal of Algorithms, 43 (1): 85–105, CiteSeerX 10.1.1.108.7366, doi:10.1006/jagm.2002.1219
Topics in game theory
Definitions
• Congestion game
• Cooperative game
• Determinacy
• Escalation of commitment
• Extensive-form game
• First-player and second-player win
• Game complexity
• Graphical game
• Hierarchy of beliefs
• Information set
• Normal-form game
• Preference
• Sequential game
• Simultaneous game
• Simultaneous action selection
• Solved game
• Succinct game
Equilibrium
concepts
• Bayesian Nash equilibrium
• Berge equilibrium
• Core
• Correlated equilibrium
• Epsilon-equilibrium
• Evolutionarily stable strategy
• Gibbs equilibrium
• Mertens-stable equilibrium
• Markov perfect equilibrium
• Nash equilibrium
• Pareto efficiency
• Perfect Bayesian equilibrium
• Proper equilibrium
• Quantal response equilibrium
• Quasi-perfect equilibrium
• Risk dominance
• Satisfaction equilibrium
• Self-confirming equilibrium
• Sequential equilibrium
• Shapley value
• Strong Nash equilibrium
• Subgame perfection
• Trembling hand
Strategies
• Backward induction
• Bid shading
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• Grim trigger
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• Pure strategy
• Mixed strategy
• Strategy-stealing argument
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Classes
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• Mechanism design
• n-player game
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Games
• Go
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• Infinite chess
• Checkers
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• Optional prisoner's dilemma
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• Volunteer's dilemma
• Dollar auction
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• Stag hunt
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• Guess 2/3 of the average
• Kuhn poker
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• Trust game
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Theorems
• Arrow's impossibility theorem
• Aumann's agreement theorem
• Folk theorem
• Minimax theorem
• Nash's theorem
• Negamax theorem
• Purification theorem
• Revelation principle
• Sprague–Grundy theorem
• Zermelo's theorem
Key
figures
• Albert W. Tucker
• Amos Tversky
• Antoine Augustin Cournot
• Ariel Rubinstein
• Claude Shannon
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• David M. Kreps
• Donald B. Gillies
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• William Vickrey
Miscellaneous
• All-pay auction
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• Game Description Language
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• No-win situation
• Solving chess
• Topological game
• Tragedy of the commons
• Tyranny of small decisions
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Wikipedia
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Fiber product of schemes
In mathematics, specifically in algebraic geometry, the fiber product of schemes is a fundamental construction. It has many interpretations and special cases. For example, the fiber product describes how an algebraic variety over one field determines a variety over a bigger field, or the pullback of a family of varieties, or a fiber of a family of varieties. Base change is a closely related notion.
Definition
The category of schemes is a broad setting for algebraic geometry. A fruitful philosophy (known as Grothendieck's relative point of view) is that much of algebraic geometry should be developed for a morphism of schemes X → Y (called a scheme X over Y), rather than for a single scheme X. For example, rather than simply studying algebraic curves, one can study families of curves over any base scheme Y. Indeed, the two approaches enrich each other.
In particular, a scheme over a commutative ring R means a scheme X together with a morphism X → Spec(R). The older notion of an algebraic variety over a field k is equivalent to a scheme over k with certain properties. (There are different conventions for exactly which schemes should be called "varieties". One standard choice is that a variety over a field k means an integral separated scheme of finite type over k.[1])
In general, a morphism of schemes X → Y can be imagined as a family of schemes parametrized by the points of Y. Given a morphism from some other scheme Z to Y, there should be a "pullback" family of schemes over Z. This is exactly the fiber product X ×Y Z → Z.
Formally: it is a useful property of the category of schemes that the fiber product always exists.[2] That is, for any morphisms of schemes X → Y and Z → Y, there is a scheme X ×Y Z with morphisms to X and Z, making the diagram
commutative, and which is universal with that property. That is, for any scheme W with morphisms to X and Z whose compositions to Y are equal, there is a unique morphism from W to X ×Y Z that makes the diagram commute. As always with universal properties, this condition determines the scheme X ×Y Z up to a unique isomorphism, if it exists. The proof that fiber products of schemes always do exist reduces the problem to the tensor product of commutative rings (cf. gluing schemes). In particular, when X, Y, and Z are all affine schemes, so X = Spec(A), Y = Spec(B), and Z = Spec(C) for some commutative rings A,B,C, the fiber product is the affine scheme
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle X\times_Y Z = \operatorname{Spec}(A\otimes_B C).}
The morphism X ×Y Z → Z is called the base change or pullback of the morphism X → Y via the morphism Z → Y.
In some cases, the fiber product of schemes has a right adjoint, the restriction of scalars.
Interpretations and special cases
• In the category of schemes over a field k, the product X × Y means the fiber product X ×k Y (which is shorthand for the fiber product over Spec(k)). For example, the product of affine spaces Am and An over a field k is the affine space Am+n over k.
• For a scheme X over a field k and any field extension E of k, the base change XE means the fiber product X ×Spec(k) Spec(E). Here XE is a scheme over E. For example, if X is the curve in the projective plane P2
R
over the real numbers R defined by the equation xy2 = 7z3, then XC is the complex curve in P2
C
defined by the same equation. Many properties of an algebraic variety over a field k can be defined in terms of its base change to the algebraic closure of k, which makes the situation simpler.
• Let f: X → Y be a morphism of schemes, and let y be a point in Y. Then there is a morphism Spec(k(y)) → Y with image y, where k(y) is the residue field of y. The fiber of f over y is defined as the fiber product X ×Y Spec(k(y)); this is a scheme over the field k(y).[3] This concept helps to justify the rough idea of a morphism of schemes X → Y as a family of schemes parametrized by Y.
• Let X, Y, and Z be schemes over a field k, with morphisms X → Y and Z → Y over k. Then the set of k-rational points of the fiber product X xY Z is easy to describe:
$(X\times _{Y}Z)(k)=X(k)\times _{Y(k)}Z(k).$
That is, a k-point of X xY Z can be identified with a pair of k-points of X and Z that have the same image in Y. This is immediate from the universal property of the fiber product of schemes.
• If X and Z are closed subschemes of a scheme Y, then the fiber product X xY Z is exactly the intersection X ∩ Z, with its natural scheme structure.[4] The same goes for open subschemes.
Base change and descent
Some important properties P of morphisms of schemes are preserved under arbitrary base change. That is, if X → Y has property P and Z → Y is any morphism of schemes, then the base change X xY Z → Z has property P. For example, flat morphisms, smooth morphisms, proper morphisms, and many other classes of morphisms are preserved under arbitrary base change.[5]
The word descent refers to the reverse question: if the pulled-back morphism X xY Z → Z has some property P, must the original morphism X → Y have property P? Clearly this is impossible in general: for example, Z might be the empty scheme, in which case the pulled-back morphism loses all information about the original morphism. But if the morphism Z → Y is flat and surjective (also called faithfully flat) and quasi-compact, then many properties do descend from Z to Y. Properties that descend include flatness, smoothness, properness, and many other classes of morphisms.[6] These results form part of Grothendieck's theory of faithfully flat descent.
Example: for any field extension k ⊂ E, the morphism Spec(E) → Spec(k) is faithfully flat and quasi-compact. So the descent results mentioned imply that a scheme X over k is smooth over k if and only if the base change XE is smooth over E. The same goes for properness and many other properties.
Notes
1. Stacks Project, Tag 020D.
2. Grothendieck, EGA I, Théorème 3.2.6; Hartshorne (1977), Theorem II.3.3.
3. Hartshorne (1977), section II.3.
4. Stacks Project, Tag 0C4I.
5. Stacks Project, Tag 02WE.
6. Stacks Project, Tag 02YJ.
References
• Grothendieck, Alexandre; Dieudonné, Jean (1960). "Éléments de géométrie algébrique: I. Le langage des schémas". Publications Mathématiques de l'IHÉS. 4. doi:10.1007/bf02684778. MR 0217083.
• Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157
External links
• The Stacks Project Authors, The Stacks Project
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Wikipedia
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Algorithms for calculating variance
Algorithms for calculating variance play a major role in computational statistics. A key difficulty in the design of good algorithms for this problem is that formulas for the variance may involve sums of squares, which can lead to numerical instability as well as to arithmetic overflow when dealing with large values.
Naïve algorithm
A formula for calculating the variance of an entire population of size N is:
$\sigma ^{2}={\overline {(x^{2})}}-{\bar {x}}^{2}={\frac {\sum _{i=1}^{N}x_{i}^{2}-(\sum _{i=1}^{N}x_{i})^{2}/N}{N}}.$
Using Bessel's correction to calculate an unbiased estimate of the population variance from a finite sample of n observations, the formula is:
$s^{2}=\left({\frac {\sum _{i=1}^{n}x_{i}^{2}}{n}}-\left({\frac {\sum _{i=1}^{n}x_{i}}{n}}\right)^{2}\right)\cdot {\frac {n}{n-1}}.$
Therefore, a naïve algorithm to calculate the estimated variance is given by the following:
• Let n ← 0, Sum ← 0, SumSq ← 0
• For each datum x:
• n ← n + 1
• Sum ← Sum + x
• SumSq ← SumSq + x × x
• Var = (SumSq − (Sum × Sum) / n) / (n − 1)
This algorithm can easily be adapted to compute the variance of a finite population: simply divide by n instead of n − 1 on the last line.
Because SumSq and (Sum×Sum)/n can be very similar numbers, cancellation can lead to the precision of the result to be much less than the inherent precision of the floating-point arithmetic used to perform the computation. Thus this algorithm should not be used in practice,[1][2] and several alternate, numerically stable, algorithms have been proposed.[3] This is particularly bad if the standard deviation is small relative to the mean.
Computing shifted data
The variance is invariant with respect to changes in a location parameter, a property which can be used to avoid the catastrophic cancellation in this formula.
$\operatorname {Var} (X-K)=\operatorname {Var} (X).$
with $K$ any constant, which leads to the new formula
$\sigma ^{2}={\frac {\sum _{i=1}^{n}(x_{i}-K)^{2}-(\sum _{i=1}^{n}(x_{i}-K))^{2}/n}{n-1}}.$
the closer $K$ is to the mean value the more accurate the result will be, but just choosing a value inside the samples range will guarantee the desired stability. If the values $(x_{i}-K)$ are small then there are no problems with the sum of its squares, on the contrary, if they are large it necessarily means that the variance is large as well. In any case the second term in the formula is always smaller than the first one therefore no cancellation may occur.[2]
If just the first sample is taken as $K$ the algorithm can be written in Python programming language as
def shifted_data_variance(data):
if len(data) < 2:
return 0.0
K = data[0]
n = Ex = Ex2 = 0.0
for x in data:
n += 1
Ex += x - K
Ex2 += (x - K) ** 2
variance = (Ex2 - Ex**2 / n) / (n - 1)
# use n instead of (n-1) if want to compute the exact variance of the given data
# use (n-1) if data are samples of a larger population
return variance
This formula also facilitates the incremental computation that can be expressed as
K = Ex = Ex2 = 0.0
n = 0
def add_variable(x):
global K, n, Ex, Ex2
if n == 0:
K = x
n += 1
Ex += x - K
Ex2 += (x - K) ** 2
def remove_variable(x):
global K, n, Ex, Ex2
n -= 1
Ex -= x - K
Ex2 -= (x - K) ** 2
def get_mean():
global K, n, Ex
return K + Ex / n
def get_variance():
global n, Ex, Ex2
return (Ex2 - Ex**2 / n) / (n - 1)
Two-pass algorithm
An alternative approach, using a different formula for the variance, first computes the sample mean,
${\bar {x}}={\frac {\sum _{j=1}^{n}x_{j}}{n}},$
and then computes the sum of the squares of the differences from the mean,
${\text{sample variance}}=s^{2}={\dfrac {\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}}{n-1}},$
where s is the standard deviation. This is given by the following code:
def two_pass_variance(data):
n = len(data)
mean = sum(data) / n
variance = sum([(x - mean) ** 2 for x in data]) / (n - 1)
return variance
This algorithm is numerically stable if n is small.[1][4] However, the results of both of these simple algorithms ("naïve" and "two-pass") can depend inordinately on the ordering of the data and can give poor results for very large data sets due to repeated roundoff error in the accumulation of the sums. Techniques such as compensated summation can be used to combat this error to a degree.
Welford's online algorithm
It is often useful to be able to compute the variance in a single pass, inspecting each value $x_{i}$ only once; for example, when the data is being collected without enough storage to keep all the values, or when costs of memory access dominate those of computation. For such an online algorithm, a recurrence relation is required between quantities from which the required statistics can be calculated in a numerically stable fashion.
The following formulas can be used to update the mean and (estimated) variance of the sequence, for an additional element xn. Here, $ {\overline {x}}_{n}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}$ denotes the sample mean of the first n samples $(x_{1},\dots ,x_{n})$, $ \sigma _{n}^{2}={\frac {1}{n}}\sum _{i=1}^{n}\left(x_{i}-{\overline {x}}_{n}\right)^{2}$ their biased sample variance, and $ s_{n}^{2}={\frac {1}{n-1}}\sum _{i=1}^{n}\left(x_{i}-{\overline {x}}_{n}\right)^{2}$ their unbiased sample variance.
${\bar {x}}_{n}={\frac {(n-1)\,{\bar {x}}_{n-1}+x_{n}}{n}}={\bar {x}}_{n-1}+{\frac {x_{n}-{\bar {x}}_{n-1}}{n}}$
$\sigma _{n}^{2}={\frac {(n-1)\,\sigma _{n-1}^{2}+(x_{n}-{\bar {x}}_{n-1})(x_{n}-{\bar {x}}_{n})}{n}}=\sigma _{n-1}^{2}+{\frac {(x_{n}-{\bar {x}}_{n-1})(x_{n}-{\bar {x}}_{n})-\sigma _{n-1}^{2}}{n}}.$
$s_{n}^{2}={\frac {n-2}{n-1}}\,s_{n-1}^{2}+{\frac {(x_{n}-{\bar {x}}_{n-1})^{2}}{n}}=s_{n-1}^{2}+{\frac {(x_{n}-{\bar {x}}_{n-1})^{2}}{n}}-{\frac {s_{n-1}^{2}}{n-1}},\quad n>1$
These formulas suffer from numerical instability , as they repeatedly subtract a small number from a big number which scales with n. A better quantity for updating is the sum of squares of differences from the current mean, $ \sum _{i=1}^{n}(x_{i}-{\bar {x}}_{n})^{2}$, here denoted $M_{2,n}$:
${\begin{aligned}M_{2,n}&=M_{2,n-1}+(x_{n}-{\bar {x}}_{n-1})(x_{n}-{\bar {x}}_{n})\\[4pt]\sigma _{n}^{2}&={\frac {M_{2,n}}{n}}\\[4pt]s_{n}^{2}&={\frac {M_{2,n}}{n-1}}\end{aligned}}$
This algorithm was found by Welford,[5][6] and it has been thoroughly analyzed.[2][7] It is also common to denote $M_{k}={\bar {x}}_{k}$ and $S_{k}=M_{2,k}$.[8]
An example Python implementation for Welford's algorithm is given below.
# For a new value new_value, compute the new count, new mean, the new M2.
# mean accumulates the mean of the entire dataset
# M2 aggregates the squared distance from the mean
# count aggregates the number of samples seen so far
def update(existing_aggregate, new_value):
(count, mean, M2) = existing_aggregate
count += 1
delta = new_value - mean
mean += delta / count
delta2 = new_value - mean
M2 += delta * delta2
return (count, mean, M2)
# Retrieve the mean, variance and sample variance from an aggregate
def finalize(existing_aggregate):
(count, mean, M2) = existing_aggregate
if count < 2:
return float("nan")
else:
(mean, variance, sample_variance) = (mean, M2 / count, M2 / (count - 1))
return (mean, variance, sample_variance)
This algorithm is much less prone to loss of precision due to catastrophic cancellation, but might not be as efficient because of the division operation inside the loop. For a particularly robust two-pass algorithm for computing the variance, one can first compute and subtract an estimate of the mean, and then use this algorithm on the residuals.
The parallel algorithm below illustrates how to merge multiple sets of statistics calculated online.
Weighted incremental algorithm
The algorithm can be extended to handle unequal sample weights, replacing the simple counter n with the sum of weights seen so far. West (1979)[9] suggests this incremental algorithm:
def weighted_incremental_variance(data_weight_pairs):
w_sum = w_sum2 = mean = S = 0
for x, w in data_weight_pairs:
w_sum = w_sum + w
w_sum2 = w_sum2 + w**2
mean_old = mean
mean = mean_old + (w / w_sum) * (x - mean_old)
S = S + w * (x - mean_old) * (x - mean)
population_variance = S / w_sum
# Bessel's correction for weighted samples
# Frequency weights
sample_frequency_variance = S / (w_sum - 1)
# Reliability weights
sample_reliability_variance = S / (w_sum - w_sum2 / w_sum)
Parallel algorithm
Chan et al.[10] note that Welford's online algorithm detailed above is a special case of an algorithm that works for combining arbitrary sets $A$ and $B$:
${\begin{aligned}n_{AB}&=n_{A}+n_{B}\\\delta &={\bar {x}}_{B}-{\bar {x}}_{A}\\{\bar {x}}_{AB}&={\bar {x}}_{A}+\delta \cdot {\frac {n_{B}}{n_{AB}}}\\M_{2,AB}&=M_{2,A}+M_{2,B}+\delta ^{2}\cdot {\frac {n_{A}n_{B}}{n_{AB}}}\\\end{aligned}}$.
This may be useful when, for example, multiple processing units may be assigned to discrete parts of the input.
Chan's method for estimating the mean is numerically unstable when $n_{A}\approx n_{B}$ and both are large, because the numerical error in $\delta ={\bar {x}}_{B}-{\bar {x}}_{A}$ is not scaled down in the way that it is in the $n_{B}=1$ case. In such cases, prefer $ {\bar {x}}_{AB}={\frac {n_{A}{\bar {x}}_{A}+n_{B}{\bar {x}}_{B}}{n_{AB}}}$.
def parallel_variance(n_a, avg_a, M2_a, n_b, avg_b, M2_b):
n = n_a + n_b
delta = avg_b - avg_a
M2 = M2_a + M2_b + delta**2 * n_a * n_b / n
var_ab = M2 / (n - 1)
return var_ab
This can be generalized to allow parallelization with AVX, with GPUs, and computer clusters, and to covariance.[3]
Example
Assume that all floating point operations use standard IEEE 754 double-precision arithmetic. Consider the sample (4, 7, 13, 16) from an infinite population. Based on this sample, the estimated population mean is 10, and the unbiased estimate of population variance is 30. Both the naïve algorithm and two-pass algorithm compute these values correctly.
Next consider the sample (108 + 4, 108 + 7, 108 + 13, 108 + 16), which gives rise to the same estimated variance as the first sample. The two-pass algorithm computes this variance estimate correctly, but the naïve algorithm returns 29.333333333333332 instead of 30.
While this loss of precision may be tolerable and viewed as a minor flaw of the naïve algorithm, further increasing the offset makes the error catastrophic. Consider the sample (109 + 4, 109 + 7, 109 + 13, 109 + 16). Again the estimated population variance of 30 is computed correctly by the two-pass algorithm, but the naïve algorithm now computes it as −170.66666666666666. This is a serious problem with naïve algorithm and is due to catastrophic cancellation in the subtraction of two similar numbers at the final stage of the algorithm.
Higher-order statistics
Terriberry[11] extends Chan's formulae to calculating the third and fourth central moments, needed for example when estimating skewness and kurtosis:
${\begin{aligned}M_{3,X}=M_{3,A}+M_{3,B}&{}+\delta ^{3}{\frac {n_{A}n_{B}(n_{A}-n_{B})}{n_{X}^{2}}}+3\delta {\frac {n_{A}M_{2,B}-n_{B}M_{2,A}}{n_{X}}}\\[6pt]M_{4,X}=M_{4,A}+M_{4,B}&{}+\delta ^{4}{\frac {n_{A}n_{B}\left(n_{A}^{2}-n_{A}n_{B}+n_{B}^{2}\right)}{n_{X}^{3}}}\\[6pt]&{}+6\delta ^{2}{\frac {n_{A}^{2}M_{2,B}+n_{B}^{2}M_{2,A}}{n_{X}^{2}}}+4\delta {\frac {n_{A}M_{3,B}-n_{B}M_{3,A}}{n_{X}}}\end{aligned}}$
Here the $M_{k}$ are again the sums of powers of differences from the mean $ \sum (x-{\overline {x}})^{k}$, giving
${\begin{aligned}&{\text{skewness}}=g_{1}={\frac {{\sqrt {n}}M_{3}}{M_{2}^{3/2}}},\\[4pt]&{\text{kurtosis}}=g_{2}={\frac {nM_{4}}{M_{2}^{2}}}-3.\end{aligned}}$
For the incremental case (i.e., $B=\{x\}$), this simplifies to:
${\begin{aligned}\delta &=x-m\\[5pt]m'&=m+{\frac {\delta }{n}}\\[5pt]M_{2}'&=M_{2}+\delta ^{2}{\frac {n-1}{n}}\\[5pt]M_{3}'&=M_{3}+\delta ^{3}{\frac {(n-1)(n-2)}{n^{2}}}-{\frac {3\delta M_{2}}{n}}\\[5pt]M_{4}'&=M_{4}+{\frac {\delta ^{4}(n-1)(n^{2}-3n+3)}{n^{3}}}+{\frac {6\delta ^{2}M_{2}}{n^{2}}}-{\frac {4\delta M_{3}}{n}}\end{aligned}}$
By preserving the value $\delta /n$, only one division operation is needed and the higher-order statistics can thus be calculated for little incremental cost.
An example of the online algorithm for kurtosis implemented as described is:
def online_kurtosis(data):
n = mean = M2 = M3 = M4 = 0
for x in data:
n1 = n
n = n + 1
delta = x - mean
delta_n = delta / n
delta_n2 = delta_n**2
term1 = delta * delta_n * n1
mean = mean + delta_n
M4 = M4 + term1 * delta_n2 * (n**2 - 3*n + 3) + 6 * delta_n2 * M2 - 4 * delta_n * M3
M3 = M3 + term1 * delta_n * (n - 2) - 3 * delta_n * M2
M2 = M2 + term1
# Note, you may also calculate variance using M2, and skewness using M3
# Caution: If all the inputs are the same, M2 will be 0, resulting in a division by 0.
kurtosis = (n * M4) / (M2**2) - 3
return kurtosis
Pébaÿ[12] further extends these results to arbitrary-order central moments, for the incremental and the pairwise cases, and subsequently Pébaÿ et al.[13] for weighted and compound moments. One can also find there similar formulas for covariance.
Choi and Sweetman[14] offer two alternative methods to compute the skewness and kurtosis, each of which can save substantial computer memory requirements and CPU time in certain applications. The first approach is to compute the statistical moments by separating the data into bins and then computing the moments from the geometry of the resulting histogram, which effectively becomes a one-pass algorithm for higher moments. One benefit is that the statistical moment calculations can be carried out to arbitrary accuracy such that the computations can be tuned to the precision of, e.g., the data storage format or the original measurement hardware. A relative histogram of a random variable can be constructed in the conventional way: the range of potential values is divided into bins and the number of occurrences within each bin are counted and plotted such that the area of each rectangle equals the portion of the sample values within that bin:
$H(x_{k})={\frac {h(x_{k})}{A}}$
where $h(x_{k})$ and $H(x_{k})$ represent the frequency and the relative frequency at bin $x_{k}$ and $ A=\sum _{k=1}^{K}h(x_{k})\,\Delta x_{k}$ is the total area of the histogram. After this normalization, the $n$ raw moments and central moments of $x(t)$ can be calculated from the relative histogram:
$m_{n}^{(h)}=\sum _{k=1}^{K}x_{k}^{n}H(x_{k})\,\Delta x_{k}={\frac {1}{A}}\sum _{k=1}^{K}x_{k}^{n}h(x_{k})\,\Delta x_{k}$
$\theta _{n}^{(h)}=\sum _{k=1}^{K}{\Big (}x_{k}-m_{1}^{(h)}{\Big )}^{n}\,H(x_{k})\,\Delta x_{k}={\frac {1}{A}}\sum _{k=1}^{K}{\Big (}x_{k}-m_{1}^{(h)}{\Big )}^{n}h(x_{k})\,\Delta x_{k}$
where the superscript $^{(h)}$ indicates the moments are calculated from the histogram. For constant bin width $\Delta x_{k}=\Delta x$ these two expressions can be simplified using $I=A/\Delta x$:
$m_{n}^{(h)}={\frac {1}{I}}\sum _{k=1}^{K}x_{k}^{n}\,h(x_{k})$
$\theta _{n}^{(h)}={\frac {1}{I}}\sum _{k=1}^{K}{\Big (}x_{k}-m_{1}^{(h)}{\Big )}^{n}h(x_{k})$
The second approach from Choi and Sweetman[14] is an analytical methodology to combine statistical moments from individual segments of a time-history such that the resulting overall moments are those of the complete time-history. This methodology could be used for parallel computation of statistical moments with subsequent combination of those moments, or for combination of statistical moments computed at sequential times.
If $Q$ sets of statistical moments are known: $(\gamma _{0,q},\mu _{q},\sigma _{q}^{2},\alpha _{3,q},\alpha _{4,q})\quad $ for $q=1,2,\ldots ,Q$, then each $\gamma _{n}$ can be expressed in terms of the equivalent $n$ raw moments:
$\gamma _{n,q}=m_{n,q}\gamma _{0,q}\qquad \quad {\textrm {for}}\quad n=1,2,3,4\quad {\text{ and }}\quad q=1,2,\dots ,Q$
where $\gamma _{0,q}$ is generally taken to be the duration of the $q^{th}$ time-history, or the number of points if $\Delta t$ is constant.
The benefit of expressing the statistical moments in terms of $\gamma $ is that the $Q$ sets can be combined by addition, and there is no upper limit on the value of $Q$.
$\gamma _{n,c}=\sum _{q=1}^{Q}\gamma _{n,q}\quad \quad {\text{for }}n=0,1,2,3,4$
where the subscript $_{c}$ represents the concatenated time-history or combined $\gamma $. These combined values of $\gamma $ can then be inversely transformed into raw moments representing the complete concatenated time-history
$m_{n,c}={\frac {\gamma _{n,c}}{\gamma _{0,c}}}\quad {\text{for }}n=1,2,3,4$
Known relationships between the raw moments ($m_{n}$) and the central moments ($\theta _{n}=\operatorname {E} [(x-\mu )^{n}])$) are then used to compute the central moments of the concatenated time-history. Finally, the statistical moments of the concatenated history are computed from the central moments:
$\mu _{c}=m_{1,c}\qquad \sigma _{c}^{2}=\theta _{2,c}\qquad \alpha _{3,c}={\frac {\theta _{3,c}}{\sigma _{c}^{3}}}\qquad \alpha _{4,c}={\frac {\theta _{4,c}}{\sigma _{c}^{4}}}-3$
Covariance
Very similar algorithms can be used to compute the covariance.
Naïve algorithm
The naïve algorithm is
$\operatorname {Cov} (X,Y)={\frac {\sum _{i=1}^{n}x_{i}y_{i}-(\sum _{i=1}^{n}x_{i})(\sum _{i=1}^{n}y_{i})/n}{n}}.$
For the algorithm above, one could use the following Python code:
def naive_covariance(data1, data2):
n = len(data1)
sum1 = sum(data1)
sum2 = sum(data2)
sum12 = sum([i1 * i2 for i1, i2 in zip(data1, data2)])
covariance = (sum12 - sum1 * sum2 / n) / n
return covariance
With estimate of the mean
As for the variance, the covariance of two random variables is also shift-invariant, so given any two constant values $k_{x}$ and $k_{y},$ it can be written:
$\operatorname {Cov} (X,Y)=\operatorname {Cov} (X-k_{x},Y-k_{y})={\dfrac {\sum _{i=1}^{n}(x_{i}-k_{x})(y_{i}-k_{y})-(\sum _{i=1}^{n}(x_{i}-k_{x}))(\sum _{i=1}^{n}(y_{i}-k_{y}))/n}{n}}.$
and again choosing a value inside the range of values will stabilize the formula against catastrophic cancellation as well as make it more robust against big sums. Taking the first value of each data set, the algorithm can be written as:
def shifted_data_covariance(data_x, data_y):
n = len(data_x)
if n < 2:
return 0
kx = data_x[0]
ky = data_y[0]
Ex = Ey = Exy = 0
for ix, iy in zip(data_x, data_y):
Ex += ix - kx
Ey += iy - ky
Exy += (ix - kx) * (iy - ky)
return (Exy - Ex * Ey / n) / n
Two-pass
The two-pass algorithm first computes the sample means, and then the covariance:
${\bar {x}}=\sum _{i=1}^{n}x_{i}/n$
${\bar {y}}=\sum _{i=1}^{n}y_{i}/n$
$\operatorname {Cov} (X,Y)={\frac {\sum _{i=1}^{n}(x_{i}-{\bar {x}})(y_{i}-{\bar {y}})}{n}}.$
The two-pass algorithm may be written as:
def two_pass_covariance(data1, data2):
n = len(data1)
mean1 = sum(data1) / n
mean2 = sum(data2) / n
covariance = 0
for i1, i2 in zip(data1, data2):
a = i1 - mean1
b = i2 - mean2
covariance += a * b / n
return covariance
A slightly more accurate compensated version performs the full naive algorithm on the residuals. The final sums $ \sum _{i}x_{i}$ and $ \sum _{i}y_{i}$ should be zero, but the second pass compensates for any small error.
Online
A stable one-pass algorithm exists, similar to the online algorithm for computing the variance, that computes co-moment $ C_{n}=\sum _{i=1}^{n}(x_{i}-{\bar {x}}_{n})(y_{i}-{\bar {y}}_{n})$:
${\begin{alignedat}{2}{\bar {x}}_{n}&={\bar {x}}_{n-1}&\,+\,&{\frac {x_{n}-{\bar {x}}_{n-1}}{n}}\\[5pt]{\bar {y}}_{n}&={\bar {y}}_{n-1}&\,+\,&{\frac {y_{n}-{\bar {y}}_{n-1}}{n}}\\[5pt]C_{n}&=C_{n-1}&\,+\,&(x_{n}-{\bar {x}}_{n})(y_{n}-{\bar {y}}_{n-1})\\[5pt]&=C_{n-1}&\,+\,&(x_{n}-{\bar {x}}_{n-1})(y_{n}-{\bar {y}}_{n})\end{alignedat}}$
The apparent asymmetry in that last equation is due to the fact that $ (x_{n}-{\bar {x}}_{n})={\frac {n-1}{n}}(x_{n}-{\bar {x}}_{n-1})$, so both update terms are equal to $ {\frac {n-1}{n}}(x_{n}-{\bar {x}}_{n-1})(y_{n}-{\bar {y}}_{n-1})$. Even greater accuracy can be achieved by first computing the means, then using the stable one-pass algorithm on the residuals.
Thus the covariance can be computed as
${\begin{aligned}\operatorname {Cov} _{N}(X,Y)={\frac {C_{N}}{N}}&={\frac {\operatorname {Cov} _{N-1}(X,Y)\cdot (N-1)+(x_{n}-{\bar {x}}_{n})(y_{n}-{\bar {y}}_{n-1})}{N}}\\&={\frac {\operatorname {Cov} _{N-1}(X,Y)\cdot (N-1)+(x_{n}-{\bar {x}}_{n-1})(y_{n}-{\bar {y}}_{n})}{N}}\\&={\frac {\operatorname {Cov} _{N-1}(X,Y)\cdot (N-1)+{\frac {N-1}{N}}(x_{n}-{\bar {x}}_{n-1})(y_{n}-{\bar {y}}_{n-1})}{N}}\\&={\frac {\operatorname {Cov} _{N-1}(X,Y)\cdot (N-1)+{\frac {N}{N-1}}(x_{n}-{\bar {x}}_{n})(y_{n}-{\bar {y}}_{n})}{N}}.\end{aligned}}$
def online_covariance(data1, data2):
meanx = meany = C = n = 0
for x, y in zip(data1, data2):
n += 1
dx = x - meanx
meanx += dx / n
meany += (y - meany) / n
C += dx * (y - meany)
population_covar = C / n
# Bessel's correction for sample variance
sample_covar = C / (n - 1)
A small modification can also be made to compute the weighted covariance:
def online_weighted_covariance(data1, data2, data3):
meanx = meany = 0
wsum = wsum2 = 0
C = 0
for x, y, w in zip(data1, data2, data3):
wsum += w
wsum2 += w * w
dx = x - meanx
meanx += (w / wsum) * dx
meany += (w / wsum) * (y - meany)
C += w * dx * (y - meany)
population_covar = C / wsum
# Bessel's correction for sample variance
# Frequency weights
sample_frequency_covar = C / (wsum - 1)
# Reliability weights
sample_reliability_covar = C / (wsum - wsum2 / wsum)
Likewise, there is a formula for combining the covariances of two sets that can be used to parallelize the computation:[3]
$C_{X}=C_{A}+C_{B}+({\bar {x}}_{A}-{\bar {x}}_{B})({\bar {y}}_{A}-{\bar {y}}_{B})\cdot {\frac {n_{A}n_{B}}{n_{X}}}.$
Weighted batched version
A version of the weighted online algorithm that does batched updated also exists: let $w_{1},\dots w_{N}$ denote the weights, and write
${\begin{alignedat}{2}{\bar {x}}_{n+k}&={\bar {x}}_{n}&\,+\,&{\frac {\sum _{i=n+1}^{n+k}w_{i}(x_{i}-{\bar {x}}_{n})}{\sum _{i=1}^{n+k}w_{i}}}\\{\bar {y}}_{n+k}&={\bar {y}}_{n}&\,+\,&{\frac {\sum _{i=n+1}^{n+k}w_{i}(y_{i}-{\bar {y}}_{n})}{\sum _{i=1}^{n+k}w_{i}}}\\C_{n+k}&=C_{n}&\,+\,&\sum _{i=n+1}^{n+k}w_{i}(x_{i}-{\bar {x}}_{n+k})(y_{i}-{\bar {y}}_{n})\\&=C_{n}&\,+\,&\sum _{i=n+1}^{n+k}w_{i}(x_{i}-{\bar {x}}_{n})(y_{i}-{\bar {y}}_{n+k})\\\end{alignedat}}$
The covariance can then be computed as
$\operatorname {Cov} _{N}(X,Y)={\frac {C_{N}}{\sum _{i=1}^{N}w_{i}}}$
See also
• Kahan summation algorithm
• Squared deviations from the mean
• Yamartino method
References
1. Einarsson, Bo (2005). Accuracy and Reliability in Scientific Computing. SIAM. p. 47. ISBN 978-0-89871-584-2.
2. Chan, Tony F.; Golub, Gene H.; LeVeque, Randall J. (1983). "Algorithms for computing the sample variance: Analysis and recommendations" (PDF). The American Statistician. 37 (3): 242–247. doi:10.1080/00031305.1983.10483115. JSTOR 2683386. Archived (PDF) from the original on 9 October 2022.
3. Schubert, Erich; Gertz, Michael (9 July 2018). Numerically stable parallel computation of (co-)variance. ACM. p. 10. doi:10.1145/3221269.3223036. ISBN 9781450365055. S2CID 49665540.
4. Higham, Nicholas (2002). Accuracy and Stability of Numerical Algorithms (2 ed) (Problem 1.10). SIAM.
5. Welford, B. P. (1962). "Note on a method for calculating corrected sums of squares and products". Technometrics. 4 (3): 419–420. doi:10.2307/1266577. JSTOR 1266577.
6. Donald E. Knuth (1998). The Art of Computer Programming, volume 2: Seminumerical Algorithms, 3rd edn., p. 232. Boston: Addison-Wesley.
7. Ling, Robert F. (1974). "Comparison of Several Algorithms for Computing Sample Means and Variances". Journal of the American Statistical Association. 69 (348): 859–866. doi:10.2307/2286154. JSTOR 2286154.
8. "Accurately computing sample variance online".
9. West, D. H. D. (1979). "Updating Mean and Variance Estimates: An Improved Method". Communications of the ACM. 22 (9): 532–535. doi:10.1145/359146.359153. S2CID 30671293.
10. Chan, Tony F.; Golub, Gene H.; LeVeque, Randall J. (1979), "Updating Formulae and a Pairwise Algorithm for Computing Sample Variances." (PDF), Technical Report STAN-CS-79-773, Department of Computer Science, Stanford University.
11. Terriberry, Timothy B. (2007), Computing Higher-Order Moments Online, archived from the original on 23 April 2014, retrieved 5 May 2008
12. Pébaÿ, Philippe (2008), "Formulas for Robust, One-Pass Parallel Computation of Covariances and Arbitrary-Order Statistical Moments" (PDF), Technical Report SAND2008-6212, Sandia National Laboratories, archived (PDF) from the original on 9 October 2022
13. Pébaÿ, Philippe; Terriberry, Timothy; Kolla, Hemanth; Bennett, Janine (2016), "Numerically Stable, Scalable Formulas for Parallel and Online Computation of Higher-Order Multivariate Central Moments with Arbitrary Weights", Computational Statistics, Springer, 31 (4): 1305–1325, doi:10.1007/s00180-015-0637-z, S2CID 124570169
14. Choi, Myoungkeun; Sweetman, Bert (2010), "Efficient Calculation of Statistical Moments for Structural Health Monitoring", Journal of Structural Health Monitoring, 9 (1): 13–24, doi:10.1177/1475921709341014, S2CID 17534100
External links
• Weisstein, Eric W. "Sample Variance Computation". MathWorld.
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Stably finite ring
In mathematics, particularly in abstract algebra, a ring R is said to be stably finite (or weakly finite) if, for all square matrices A and B of the same size with entries in R, AB = 1 implies BA = 1. This is a stronger property for a ring than having the invariant basis number (IBN) property. Namely, any nontrivial[1] stably finite ring has IBN. Commutative rings, noetherian rings and artinian rings are stably finite. Subrings of stably finite rings and matrix rings over stably finite rings are stably finite. A ring satisfying Klein's nilpotence condition is stably finite.
References
1. A trivial ring is stably finite but doesn't have IBN.
• P.M. Cohn (2003). Basic Algebra, Springer.
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Stably free module
In mathematics, a stably free module is a module which is close to being free.
Definition
A finitely generated module M over a ring R is stably free if there exist free finitely generated modules F and G over R such that
$M\oplus F=G.\,$
Properties
• A projective module is stably free if and only if it possesses a finite free resolution.[1]
• An infinitely generated module is stably free if and only if it is free.[2]
See also
• Free object
• Eilenberg–Mazur swindle
• Hermite ring
References
1. Lang, Serge (1993), Algebra (Third ed.), Reading, Mass.: Addison-Wesley, ISBN 978-0-201-55540-0, Zbl 0848.13001
2. Lam, T. Y. (1978). Serre's Conjecture. p. 23.
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Stack-sortable permutation
In mathematics and computer science, a stack-sortable permutation (also called a tree permutation)[1] is a permutation whose elements may be sorted by an algorithm whose internal storage is limited to a single stack data structure. The stack-sortable permutations are exactly the permutations that do not contain the permutation pattern 231; they are counted by the Catalan numbers, and may be placed in bijection with many other combinatorial objects with the same counting function including Dyck paths and binary trees.
Sorting with a stack
The problem of sorting an input sequence using a stack was first posed by Knuth (1968), who gave the following linear time algorithm (closely related to algorithms for the later all nearest smaller values problem):
• Initialize an empty stack
• For each input value x:
• While the stack is nonempty and x is larger than the top item on the stack, pop the stack to the output
• Push x onto the stack
• While the stack is nonempty, pop it to the output
Knuth observed that this algorithm correctly sorts some input sequences, and fails to sort others. For instance, the sequence 3,2,1 is correctly sorted: the three elements are all pushed onto the stack, and then popped in the order 1,2,3. However, the sequence 2,3,1 is not correctly sorted: the algorithm first pushes 2, and pops it when it sees the larger input value 3, causing 2 to be output before 1 rather than after it.
Because this algorithm is a comparison sort, its success or failure does not depend on the numerical values of the input sequence, but only on their relative order; that is, an input may be described by the permutation needed to form that input from a sorted sequence of the same length. Knuth characterized the permutations that this algorithm correctly sorts as being exactly the permutations that do not contain the permutation pattern 231: three elements x, y, and z, appearing in the input in that respective order, with z < x < y. Moreover, he observed that, if the algorithm fails to sort an input, then that input cannot be sorted with a single stack.
As well as inspiring much subsequent work on sorting using more complicated systems of stacks and related data structures,[2] Knuth's research kicked off the study of permutation patterns and of permutation classes defined by forbidden patterns.
Bijections and enumeration
The sequence of pushes and pops performed by Knuth's sorting algorithm as it sorts a stack-sortable permutation form a Dyck language: reinterpreting a push as a left parenthesis and a pop as a right parenthesis produces a string of balanced parentheses. Moreover, every Dyck string comes from a stack-sortable permutation in this way, and every two different stack-sortable permutations produce different Dyck strings. For this reason, the number of stack-sortable permutations of length n is the same as the number of Dyck strings of length 2n, the Catalan number
$C_{n}={\frac {1}{n+1}}{\binom {2n}{n}}.$[3]
Stack-sortable permutations may also be translated directly to and from (unlabeled) binary trees, another combinatorial class whose counting function is the sequence of Catalan numbers. A binary tree may be transformed into a stack-sortable permutation by numbering its nodes in left-to-right order, and then listing these numbers in the order they would be visited by a preorder traversal of the tree: the root first, then the left subtree, then the right subtree, continuing recursively within each subtree. In the reverse direction, a stack-sortable permutation may be decoded into a tree in which the first value x of the permutation corresponds to the root of the tree, the next x − 1 values are decoded recursively to give the left child of the root, and the remaining values are again decoded recursively to give the right child.[1]
Several other classes of permutations may also be placed in bijection with the stack-sortable permutations. For instance, the permutations that avoid the patterns 132, 213, and 312 may be formed respectively from the stack-sortable (231-avoiding) permutations by reversing the permutation, replacing each value x in the permutation by n + 1 − x, or both operations combined. The 312-avoiding permutations are also the inverses of the 231-avoiding permutations, and have been called the stack-realizable permutations as they are the permutations that can be formed from the identity permutation by a sequence of push-from-input and pop-to-output operations on a stack.[4] As Knuth (1968) noted, the 123-avoiding and 321-avoiding permutations also have the same counting function despite being less directly related to the stack-sortable permutations.
Random stack-sortable permutations
Rotem (1981) investigates the properties of stack-sortable permutations chosen uniformly at random among all such permutations of a given length. The expected length of the longest descending subsequence in such a permutation is ${\sqrt {\pi n}}-O(1)$, differing by a constant factor from unconstrained random permutations (for which the expected length is approximately $2{\sqrt {n}}$). The expected length of the longest ascending sequence differs even more strongly from unconstrained permutations: it is $(n+1)/2$. The expected number of values within the permutation that are larger than all previous values is only $3-6/(n+2)$, smaller than its logarithmic value for unconstrained permutations. And the expected number of inversions is $\Theta (n^{3/2})$, in contrast to its value of $\Theta (n^{2})$ for unconstrained permutations.
Additional properties
Every permutation defines a permutation graph, a graph whose vertices are the elements of the permutation and whose edges connect pairs of elements that are inverted by the permutation. The permutation graphs of stack-sortable permutations are trivially perfect.[4]
For each element i of a permutation p, define bi to be the number of other elements that are to the left of and greater than i. Then p is stack-sortable if and only if, for all i, bi − bi + 1 ≤ 1.[1]
Algorithms
Knott (1977) uses the bijection between stack-sortable permutations and binary trees to define a numerical rank for each binary tree, and to construct efficient algorithms for computing the rank of a tree ("ranking") and for computing the tree with a given rank ("unranking").
Micheli & Rossin (2006) defined two edit operations on permutations: deletion (making a permutation pattern) and its inverse. Using the same correspondence between trees and permutations, they observed that these operations correspond to edge contraction in a tree and its inverse. By applying a polynomial time dynamic programming algorithm for edit distance in trees, they showed that the edit distance between two stack-sortable permutations (and hence also the longest common pattern) can be found in polynomial time. This technique was later generalized to algorithms for finding longest common patterns of separable permutations;[5] however, the longest common pattern problem is NP-complete for arbitrary permutations.[6]
Notes
1. Knott (1977).
2. Tarjan (1972); Avis & Newborn (1981); Rosenstiehl & Tarjan (1984); Bóna (2002); Felsner & Pergel (2008). See also the many additional references given by Bóna.
3. Knuth (1968); Rotem (1981).
4. Rotem (1981).
5. Bouvel, Rossin & Vialette (2007).
6. Micheli & Rossin (2006).
References
• Avis, David; Newborn, Monroe (1981), "On pop-stacks in series", Utilitas Mathematica, 19: 129–140, MR 0624050.
• Bóna, Miklós (2002), "A survey of stack-sorting disciplines", Electronic Journal of Combinatorics, 9 (2): A1, MR 2028290.
• Bouvel, Mathilde; Rossin, Dominique; Vialette, Stéphane (2007), "Longest common separable pattern among permutations", Combinatorial Pattern Matching (CPM 2007), Lecture Notes in Computer Science, vol. 4580, Springer, pp. 316–327, doi:10.1007/978-3-540-73437-6_32.
• Felsner, Stefan; Pergel, Martin (2008), "The complexity of sorting with networks of stacks and queues", Proc. 16th Eur. Symp. Algorithms, Karlsruhe, Germany, pp. 417–429, doi:10.1007/978-3-540-87744-8_35, ISBN 978-3-540-87743-1{{citation}}: CS1 maint: location missing publisher (link).
• Knott, Gary D. (February 1977), "A numbering system for binary trees", Communications of the ACM, 20 (2): 113–115, doi:10.1145/359423.359434.
• Knuth, Donald (1968), "Vol. 1: Fundamental Algorithms", The Art of Computer Programming, Reading, Mass.: Addison-Wesley.
• Micheli, Anne; Rossin, Dominique (2006), "Edit distance between unlabeled ordered trees", Theoretical Informatics and Applications, 40 (4): 593–609, arXiv:math/0506538, doi:10.1051/ita:2006043, MR 2277052.
• Rosenstiehl, Pierre; Tarjan, Robert E. (1984), "Gauss codes, planar Hamiltonian graphs, and stack-sortable permutations", Journal of Algorithms, 5 (3): 375–390, doi:10.1016/0196-6774(84)90018-X, MR 0756164
• Rotem, D. (1981), "Stack sortable permutations", Discrete Mathematics, 33 (2): 185–196, doi:10.1016/0012-365X(81)90165-5, MR 0599081.
• Tarjan, Robert (April 1972), "Sorting Using Networks of Queues and Stacks", Journal of the ACM, 19 (2): 341–346, doi:10.1145/321694.321704.
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Quotient stack
In algebraic geometry, a quotient stack is a stack that parametrizes equivariant objects. Geometrically, it generalizes a quotient of a scheme or a variety by a group: a quotient variety, say, would be a coarse approximation of a quotient stack.
The notion is of fundamental importance in the study of stacks: a stack that arises in nature is often either a quotient stack itself or admits a stratification by quotient stacks (e.g., a Deligne–Mumford stack.) A quotient stack is also used to construct other stacks like classifying stacks.
Definition
A quotient stack is defined as follows. Let G be an affine smooth group scheme over a scheme S and X an S-scheme on which G acts. Let the quotient stack $[X/G]$ be the category over the category of S-schemes:
• an object over T is a principal G-bundle $P\to T$ together with equivariant map $P\to X$;
• an arrow from $P\to T$ to $P'\to T'$ is a bundle map (i.e., forms a commutative diagram) that is compatible with the equivariant maps $P\to X$ and $P'\to X$.
Suppose the quotient $X/G$ exists as an algebraic space (for example, by the Keel–Mori theorem). The canonical map
$[X/G]\to X/G$,
that sends a bundle P over T to a corresponding T-point,[1] need not be an isomorphism of stacks; that is, the space "X/G" is usually coarser. The canonical map is an isomorphism if and only if the stabilizers are trivial (in which case $X/G$ exists.)
In general, $[X/G]$ is an Artin stack (also called algebraic stack). If the stabilizers of the geometric points are finite and reduced, then it is a Deligne–Mumford stack.
Burt Totaro (2004) has shown: let X be a normal Noetherian algebraic stack whose stabilizer groups at closed points are affine. Then X is a quotient stack if and only if it has the resolution property; i.e., every coherent sheaf is a quotient of a vector bundle. Earlier, Robert Wayne Thomason proved that a quotient stack has the resolution property.
Examples
An effective quotient orbifold, e.g., $[M/G]$ where the $G$ action has only finite stabilizers on the smooth space $M$, is an example of a quotient stack.[2]
If $X=S$ with trivial action of $G$ (often $S$ is a point), then $[S/G]$ is called the classifying stack of $G$ (in analogy with the classifying space of $G$) and is usually denoted by $BG$. Borel's theorem describes the cohomology ring of the classifying stack.
Moduli of line bundles
One of the basic examples of quotient stacks comes from the moduli stack $B\mathbb {G} _{m}$ of line bundles $[*/\mathbb {G} _{m}]$ over ${\text{Sch}}$, or $[S/\mathbb {G} _{m}]$ over ${\text{Sch}}/S$ for the trivial $\mathbb {G} _{m}$-action on $S$. For any scheme (or $S$-scheme) $X$, the $X$-points of the moduli stack are the groupoid of principal $\mathbb {G} _{m}$-bundles $P\to X$.
Moduli of line bundles with n-sections
There is another closely related moduli stack given by $[\mathbb {A} ^{n}/\mathbb {G} _{m}]$ which is the moduli stack of line bundles with $n$-sections. This follows directly from the definition of quotient stacks evaluated on points. For a scheme $X$, the $X$-points are the groupoid whose objects are given by the set
$[\mathbb {A} ^{n}/\mathbb {G} _{m}](X)=\left\{{\begin{matrix}P&\to &\mathbb {A} ^{n}\\\downarrow &&\\X\end{matrix}}:{\begin{aligned}&P\to \mathbb {A} ^{n}{\text{ is }}\mathbb {G} _{m}{\text{ equivariant and}}\\&P\to X{\text{ is a principal }}\mathbb {G} _{m}{\text{-bundle}}\end{aligned}}\right\}$
The morphism in the top row corresponds to the $n$-sections of the associated line bundle over $X$. This can be found by noting giving a $\mathbb {G} _{m}$-equivariant map $\phi :P\to \mathbb {A} ^{1}$ and restricting it to the fiber $P|_{x}$ gives the same data as a section $\sigma $ of the bundle. This can be checked by looking at a chart and sending a point $x\in X$ to the map $\phi _{x}$, noting the set of $\mathbb {G} _{m}$-equivariant maps $P|_{x}\to \mathbb {A} ^{1}$ is isomorphic to $\mathbb {G} _{m}$. This construction then globalizes by gluing affine charts together, giving a global section of the bundle. Since $\mathbb {G} _{m}$-equivariant maps to $\mathbb {A} ^{n}$ is equivalently an $n$-tuple of $\mathbb {G} _{m}$-equivariant maps to $\mathbb {A} ^{1}$, the result holds.
Moduli of formal group laws
Example:[3] Let L be the Lazard ring; i.e., $L=\pi _{*}\operatorname {MU} $. Then the quotient stack $[\operatorname {Spec} L/G]$ by $G$,
$G(R)=\{g\in R[\![t]\!]|g(t)=b_{0}t+b_{1}t^{2}+\cdots ,b_{0}\in R^{\times }\}$,
is called the moduli stack of formal group laws, denoted by ${\mathcal {M}}_{\text{FG}}$.
See also
• Homotopy quotient
• Moduli stack of principal bundles (which, roughly, is an infinite product of classifying stacks.)
• Group-scheme action
• Moduli of algebraic curves
References
1. The T-point is obtained by completing the diagram $T\leftarrow P\to X\to X/G$.
2. Orbifolds and Stringy Topology. Definition 1.7: Cambridge Tracts in Mathematics. p. 4.{{cite book}}: CS1 maint: location (link)
3. Taken from http://www.math.harvard.edu/~lurie/252xnotes/Lecture11.pdf
• Deligne, Pierre; Mumford, David (1969), "The irreducibility of the space of curves of given genus", Publications Mathématiques de l'IHÉS, 36 (36): 75–109, CiteSeerX 10.1.1.589.288, doi:10.1007/BF02684599, MR 0262240
• Totaro, Burt (2004). "The resolution property for schemes and stacks". Journal für die reine und angewandte Mathematik. 577: 1–22. arXiv:math/0207210. doi:10.1515/crll.2004.2004.577.1. MR 2108211.
Some other references are
• Behrend, Kai (1991). The Lefschetz trace formula for the moduli stack of principal bundles (PDF) (Thesis). University of California, Berkeley.
• Edidin, Dan. "Notes on the construction of the moduli space of curves" (PDF).
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Stacked polytope
In polyhedral combinatorics (a branch of mathematics), a stacked polytope is a polytope formed from a simplex by repeatedly gluing another simplex onto one of its facets.[1][2]
Examples
Every simplex is itself a stacked polytope.
In three dimensions, every stacked polytope is a polyhedron with triangular faces, and several of the deltahedra (polyhedra with equilateral triangle faces) are stacked polytopes
The quadaugmented tetrahedron on the left is a stacked polytope, but the pentagonal bipyramid on the right is not
In a stacked polytope, each newly added simplex is only allowed to touch one of the facets of the previous ones. Thus, for instance, the quadaugmented tetrahedron, a shape formed by gluing together five regular tetrahedra around a common line segment is a stacked polytope (it has a small gap between the first and last tetrahedron). However, the similar-looking pentagonal bipyramid is not a stacked polytope, because if it is formed by gluing tetrahedra together, the last tetrahedron will be glued to two triangular faces of previous tetrahedra instead of only one.
Other non-convex stacked deltahedra include:
Three tetrahedra Four tetrahedra Five tetrahedra
Combinatorial structure
The undirected graph formed by the vertices and edges of a stacked polytope in d dimensions is a (d + 1)-tree. More precisely, the graphs of stacked polytopes are exactly the (d + 1)-trees in which every d-vertex clique (complete subgraph) is contained in at most two (d + 1)-vertex cliques.[3] For instance, the graphs of three-dimensional stacked polyhedra are exactly the Apollonian networks, the graphs formed from a triangle by repeatedly subdividing a triangular face of the graph into three smaller triangles.
One reason for the significance of stacked polytopes is that, among all d-dimensional simplicial polytopes with a given number of vertices, the stacked polytopes have the fewest possible higher-dimensional faces. For three-dimensional simplicial polyhedra the numbers of edges and two-dimensional faces are determined from the number of vertices by Euler's formula, regardless of whether the polyhedron is stacked, but this is not true in higher dimensions. Analogously, the simplicial polytopes that maximize the number of higher-dimensional faces for their number of vertices are the cyclic polytopes.[2]
References
1. Grünbaum, Branko (2001), "A convex polyhedron which is not equifacettable" (PDF), Geombinatorics, 10 (4): 165–171, MR 1825338
2. Miller, Ezra; Reiner, Victor; Sturmfels, Bernd, Geometric Combinatorics, IAS/Park City mathematics series, vol. 13, American Mathematical Society, p. 621, ISBN 9780821886953.
3. Koch, Etan; Perles, Micha A. (1976), "Covering efficiency of trees and k-trees", Proceedings of the Seventh Southeastern Conference on Combinatorics, Graph Theory, and Computing (Louisiana State Univ., Baton Rouge, La., 1976), Congressus Numerantium, Winnipeg, Manitoba, Canada: Utilitas Mathematica, 17: 391–420, MR 0457265. See in particular p. 420.
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Stadium (geometry)
A stadium is a two-dimensional geometric shape constructed of a rectangle with semicircles at a pair of opposite sides.[1] The same shape is known also as a pill shape, discorectangle,[2] squectangle,[3] obround,[4][5] or sausage body.[6]
The shape is based on a stadium, a place used for athletics and horse racing tracks.
A stadium may be constructed as the Minkowski sum of a disk and a line segment.[6] Alternatively, it is the neighborhood of points within a given distance from a line segment. A stadium is a type of oval. However, unlike some other ovals such as the ellipses, it is not an algebraic curve because different parts of its boundary are defined by different equations.
Formulas
The perimeter of a stadium is calculated by the formula $P=2(\pi r+a)$ where a is the length of the straight sides and r is the radius of the semicircles. With the same parameters, the area of the stadium is $A=\pi r^{2}+2ra=r(\pi r+2a)$.[7]
Bunimovich stadium
When this shape is used in the study of dynamical billiards, it is called the Bunimovich stadium. Leonid Bunimovich used this shape to show that it is possible for billiard tracks to exhibit chaotic behavior (positive Lyapunov exponent and exponential divergence of paths) even within a convex billiard table.[8]
Related shapes
A capsule is produced by revolving a stadium around the line of symmetry that bisects the semicircles.
References
1. "Stadium - from Wolfram MathWorld". Mathworld.wolfram.com. 2013-01-19. Retrieved 2013-01-31.
2. Dzubiella, Joachim; Matthias Schmidt; Hartmut Löwen (2000). "Topological defects in nematic droplets of hard spherocylinders". Physical Review E. 62 (4): 5081–5091. arXiv:cond-mat/9906388. Bibcode:2000PhRvE..62.5081D. doi:10.1103/PhysRevE.62.5081. PMID 11089056. S2CID 31381033.
3. Cha, Ju-Hwan; Young-Jae Kim (2021). "Rethinking the Proportional Design Principles of Timber-Framed Buddhist Buildings in the Goryeo Era". Religions. 12 (11): 985. doi:10.3390/rel12110985.
4. Ackermann, Kurt. "Obround - Punching Tools - VIP, Inc". www.vista-industrial.com. Retrieved 2016-04-29.
5. "Obround Level Gauge Glass : L.J. Star Incorporated". L.J.Star Incorporated. Archived from the original on 2016-04-22. Retrieved 2016-04-29.
6. Huang, Pingliang; Pan, Shengliang; Yang, Yunlong (2015). "Positive center sets of convex curves". Discrete & Computational Geometry. 54 (3): 728–740. doi:10.1007/s00454-015-9715-9. MR 3392976.
7. "Stadium Calculator". Calculatorsoup.com. Retrieved 2013-01-31.
8. Bunimovič, L. A. (1974). "The ergodic properties of certain billiards". Funkcional. Anal. I Priložen. 8 (3): 73–74. MR 0357736.
External links
• Weisstein, Eric W. "Stadium". MathWorld.
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Gigliola Staffilani
Gigliola Staffilani (born March 24, 1966)[1] is an Italian-American[2][3] mathematician who works as the Abby Rockefeller Mauze Professor of Mathematics at the Massachusetts Institute of Technology.[4][5][2] Her research concerns harmonic analysis and partial differential equations, including the Korteweg–de Vries equation and Schrödinger equation.
Gigliola Staffilani
Gigliola Staffilani (2013)
BornMarch 24, 1966 (1966-03-24) (age 57)
Nationality
• Italy
• United States
Alma materUniversity of Chicago
Awards
Fellow, American Mathematical Society (2012)
Fellow, American Academy of Arts and Sciences (2014) Member, National Academy of Sciences (2021)
Scientific career
FieldsMathematics
Institutions
• Stanford University
• Brown University
• MIT
ThesisThe initial value problem for some dispersive differential equations (1995)
Doctoral advisorCarlos Kenig
Websitemath.mit.edu/~gigliola/
Education and career
Staffilani grew up on a farm in Martinsicuro in central Italy, speaking only the local dialect, and with no books until her older brother brought some back from his school. Her father died when she was 10, and her mother decided that she did not need to continue on to high school, but her brother helped her change her mother's mind. She came to love mathematics at her school, and was encouraged by her teachers and brother to continue her studies, with the idea that she could return to Martinsicuro as a mathematics teacher. She earned a fellowship to study at the University of Bologna, where she earned a laurea in mathematics in 1989 with an undergraduate thesis on Green's functions for elliptic partial differential equations.[4][5][2]
At the suggestion of one of her professors at Bologna, she moved to the University of Chicago for her graduate studies, to study with Carlos Kenig. This was a big change in her previous plans, because it would mean that she could not return to Martinsicuro. When she arrived at Chicago, still knowing very little English and not having taken the Test of English as a Foreign Language, she had the wrong type of visa to obtain the teaching fellowship she had been promised. She almost returned home, but remained after Paul Sally intervened and loaned her enough money to get by until the issue could be resolved.[5][2] At Chicago, she studied dispersive partial differential equations with Kenig,[5] earning a master's degree in 1991 and a Ph.D. in 1995.[4][6]
After postdoctoral studies at the Institute for Advanced Study, Stanford University, and Princeton University, Staffilani took a tenure-track faculty position at Stanford in 1999, and earned tenure there in 2001. While at Stanford, she met her husband, Tomasz Mrowka, a mathematics professor at MIT, and after a year and a half found a faculty position closer to him at Brown University. She moved to MIT in 2002,[4][5] where, in 2006 she became the second female full professor of mathematics.[2] She served as an American Mathematical Society Council member at large from 2018 to 2020.[7]
Collaboration
Staffilani is a frequent collaborator with James Colliander, Markus Keel, Hideo Takaoka, and Terence Tao, forming a group known as the "I-team".[5][8] The name of this group has been said to come from the notation for a mollification operator used in the team's method of almost conserved quantities,[9] or as an abbreviation for "interaction", referring both to the teamwork of the group and to the interactions of light waves with each other.[10] The group's work was featured prominently in Fefferman's 2006 Fields Medal citations for group member Tao.[8][10]
Awards and honors
Staffilani was a Sloan Fellow from 2000 to 2002.[4] In 2009-2010 she was a member of the Radcliffe Institute for Advanced Study. In 2012 she became one of the inaugural fellows of the American Mathematical Society.[11] In 2014 she was inducted into the American Academy of Arts and Sciences.[12] In 2021, she was elected to the National Academy of Sciences.[13]
Major publications
• Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T. Global well-posedness for Schrödinger equations with derivative. SIAM J. Math. Anal. 33 (2001), no. 3, 649–669.
• Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T. A refined global well-posedness result for Schrödinger equations with derivative. SIAM J. Math. Anal. 34 (2002), no. 1, 64–86.
• Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T. Almost conservation laws and global rough solutions to a nonlinear Schrödinger equation. Math. Res. Lett. 9 (2002), no. 5-6, 659–682.
• Staffilani, Gigliola; Tataru, Daniel. Strichartz estimates for a Schrödinger operator with nonsmooth coefficients. Comm. Partial Differential Equations 27 (2002), no. 7-8, 1337–1372. doi:10.1081/PDE-120005841 MR1924470
• Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T. Sharp global well-posedness for KdV and modified KdV on $\mathbb {R} $ and $\mathbb {T} $. J. Amer. Math. Soc. 16 (2003), no. 3, 705–749. doi:10.1090/S0894-0347-03-00421-1 MR1969209 arXiv:math/0110045
• Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T. Multilinear estimates for periodic KdV equations, and applications. J. Funct. Anal. 211 (2004), no. 1, 173–218.
• Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T. Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $\mathbb {R} ^{3}$. Comm. Pure Appl. Math. 57 (2004), no. 8, 987–1014.
• Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T. Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $\mathbb {R} ^{3}$. Ann. of Math. (2) 167 (2008), no. 3, 767–865. doi:10.4007/annals.2008.167.767 MR2415387
References
1. Birth year from Library of Congress catalog entry, retrieved 2018-12-02.
2. Baker, Billy (April 28, 2008), "A life of unexpected twists takes her from farm to math department", Boston Globe. Archived by the Indian Academy of Sciences, Women in Science initiative.
3. Talthia Williams (2018). Power in Numbers:The rebel women of mathematics. Race Point Publishing. pp. 219–221. ISBN 978-1631064852.
4. Curriculum vitae, retrieved 2015-01-01.
5. Staffilani, Gigliola (March 18, 2012), Quello Che Si Far per Amore? Della Matematica, Careers in the Math Sciences, archived from the original on April 1, 2018, retrieved January 1, 2015. An autobiographical retrospective of Staffilani's life and career.
6. Gigliola Staffilani at the Mathematics Genealogy Project
7. "AMS Committees". American Mathematical Society. Retrieved 2023-03-29.
8. Fefferman, Charles (2006), "The work of Terence Tao" (PDF), International Congress of Mathematicians, archived from the original (PDF) on 2011-08-09, retrieved 2015-01-01.
9. I-method Archived 2012-09-25 at the Wayback Machine, Dispersive Wiki, retrieved 2015-01-02.
10. Fields Medal announcement for Terry Tao Archived 2016-03-03 at the Wayback Machine, International Congress of Mathematicians, 2006, retrieved 2015-01-02.
11. List of Fellows of the American Mathematical Society, retrieved 2015-01-01.
12. Member listing, American Academy of Arts and Sciences, retrieved 2015-06-13.
13. "2021 NAS Election". National Academy of Sciences. Retrieved 26 April 2021.
External links
• Home page
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Wikipedia
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Stagnation point flow
In fluid dynamics, a stagnation point flow refers to a fluid flow in the neighbourhood of a stagnation point (in three-dimensional flows) or a stagnation line (in two-dimensional flows) with which the stagnation point/line refers to a point/line where the velocity is zero in the inviscid approximation. The flow specifically considers a class of stagnation points known as saddle points wherein incoming streamlines gets deflected and directed outwards in a different direction; the streamline deflections are guided by separatrices. The flow in the neighborhood of the stagnation point or line can generally be described using potential flow theory, although viscous effects cannot be neglected if the stagnation point lies on a solid surface.
Stagnation point flow without solid surfaces
When two streams either of two-dimensional or axisymmetric nature impinge on each other, a stagnation plane is created, where the incoming streams are diverted tangentially outwards; thus on the stagnation plane, the velocity component normal to that plane is zero, whereas the tangential component is non-zero. In the neighborhood of the stagnation point, a local description for the velocity field can be described.
General three-dimensional velocity field
The stagnation point flow corresponds to a linear dependence on the coordinates, that can be described in the Cartesian coordinates $(x,y,z)$ with velocity components $(v_{x},v_{y},v_{z})$ as follows
$v_{x}=\alpha x,\quad v_{y}=\beta y,\quad v_{z}=\gamma z$
where $(\alpha ,\beta ,\gamma )$ are constants referred as the strain rates; these constants are not completely arbitrary since the continuity equation requires $\alpha +\beta +\gamma =0$, that is to say, only two of the three constants are independent. We shall assume $\gamma <0\leq \alpha $ so that flow is towards the stagnation point in the $z$ direction and away from the stagnation point in the $x$ direction. Without loss of generality, one can assume that $\beta \geq \alpha $. The flow field can be categorized into different types based on a single parameter[1]
$\lambda ={\frac {\alpha -\beta }{\alpha +\beta }}$
Planar stagnation-point flow
The two-dimensional stagnation-point flow belongs to the case $\beta =0\,(\lambda =1)$. The flow field is described as follows
$v_{x}=kx,\quad v_{z}=-kz$
where we let $k=\alpha =-\gamma >0$. This flow field is investigated as early as 1934 by G. I. Taylor.[2] In the laboratory, this flow field is created using a four-mill apparatus, although these flow fields are ubiquitous in turbulent flows.
Axisymmetric stagnation-point flow
The axisymmetric stagnation point flow corresponds to $\alpha =\beta \,(\lambda =0)$. The flow field can be simply described in cylindrical coordinate system $(r,\theta ,z)$ with velocity components $(v_{r},0,v_{z})$ as follows
$v_{r}=kr,\quad v_{z}=-2kz$
where we let $k=\alpha =\beta =-\gamma /2>0$.
Radial stagnation flows
In radial stagnation flows, instead of a stagnation point, we have a stagnation circle and the stagnation plane is replaced by a stagnation cylinder. The radial stagnation flow is described using the cylindrical coordinate system $(r,z)$ with velocity components $(v_{r},v_{z})$ as follows[3][4][5]
$v_{r}=-k\left(r-{\frac {r_{s}^{2}}{r}}\right),\quad v_{z}=2kz$
where $r_{s}$ is the location of the stagnation cylinder.
Hiemenz flow[6][7]
The flow due to the presence of a solid surface at $z=0$ in planar stagnation-point flow was described first by Karl Hiemenz in 1911,[8] whose numerical computations for the solutions were improved later by Leslie Howarth.[9] A familiar example where Hiemenz flow is applicable is the forward stagnation line that occurs in the flow over a circular cylinder.
The solid surface lies on the $xy$. According to potential flow theory, the fluid motion described in terms of the stream function $\psi $ and the velocity components $(v_{x},0,v_{z})$ are given by
$\psi =kxz,\quad v_{x}=kx,\quad v_{z}=-kz.$
The stagnation line for this flow is $(x,y,z)=(0,y,0)$. The velocity component $v_{x}$ is non-zero on the solid surface indicating that the above velocity field do not satisfy no-slip boundary condition on the wall. To find the velocity components that satisfy the no-slip boundary condition, one assumes the following form
$\psi ={\sqrt {\nu k}}xF(\eta ),\quad \eta ={\frac {z}{\sqrt {\nu /k}}}$
where $\nu $ is the Kinematic viscosity and ${\sqrt {\nu /k}}$ is the characteristic thickness where viscous effects are significant. The existence of constant value for the viscous effects thickness is due to the competing balance between the fluid convection that is directed towards the solid surface and viscous diffusion that is directed away from the surface. Thus the vorticity produced at the solid surface is able to diffuse only to distances of order ${\sqrt {\nu /k}}$; analogous situations that resembles this behavior occurs in asymptotic suction profile and von Kármán swirling flow. The velocity components, pressure and Navier–Stokes equations then become
$v_{x}=kxF',\quad v_{z}=-{\sqrt {\nu k}}F,\quad {\frac {p_{o}-p}{\rho }}={\frac {1}{2}}k^{2}x^{2}+k\nu F'+{\frac {1}{2}}k\nu F^{2}$
$F'''+FF''-F'^{2}+1=0$
The requirements that $(v_{x},v_{z})=(0,0)$ at $z=0$ and that $v_{x}\rightarrow kx$ as $z\rightarrow \infty $ translate to
$F(0)=0,\ F'(0)=0,F'(\infty )=1.$
The condition for $v_{z}$ as $z\rightarrow \infty $ cannot be prescribed and is obtained as a part of the solution. The problem formulated here is a special case of Falkner-Skan boundary layer. The solution can be obtained from numerical integrations and is shown in the figure. The asymptotic behaviors for large $\eta \rightarrow \infty $ are
$F\sim \eta -0.6479,\quad v_{x}\sim kx,\quad v_{z}\sim -k(z-\delta ^{*}),\quad \delta ^{*}=0.6479\delta $
where $\delta ^{*}$ is the displacement thickness.
Stagnation point flow with a translating wall[10]
Hiemenz flow when the solid wall translates with a constant velocity $U$ along the $x$ was solved by Rott (1956).[11] This problem describes the flow in the neighbourhood of the forward stagnation line occurring in a flow over a rotating cylinder. The required stream function is
$\psi ={\sqrt {\nu k}}xF(\eta )+U\delta \int _{0}^{\eta }G(\eta )d\eta $
where the function $G(\eta )$ satisfies
$G''+FG'-F'G=0,\quad G(0)=1,\quad G(\infty )=0$
The solution to the above equation is given by $G(\eta )=F''(\eta )/F''(0).$
Oblique stagnation point flow
If the incoming stream is perpendicular to the stagnation line, but approaches obliquely, the outer flow is not potential, but has a constant vorticity $-\zeta _{o}$. The appropriate stream function for oblique stagnation point flow is given by
$\psi =kxz+{\frac {1}{2}}\zeta _{o}z^{2}$
Viscous effects due to the presence of a solid wall was studied by Stuart (1959),[12] Tamada (1979)[13] and Dorrepaal (1986).[14] In their approach, the streamfunction takes the form
$\psi ={\sqrt {\nu k}}xF(\eta )+\zeta _{o}\delta ^{2}\int _{0}^{\eta }H(\eta )d\eta $
where the function $H(\eta )$
$H''+FH'-F'H=0,\quad H(0)=0,\quad H'(\infty )=1$.
Homann flow
The solution for axisymmetric stagnation point flow in the presence of a solid wall was first obtained by Homann (1936).[15] A typical example of this flow is the forward stagnation point appearing in a flow past a sphere. Paul A. Libby (1974)[16](1976)[17] extended Homann's work by allowing the solid wall to translate along its own plane with a constant speed and allowing constant suction or injection at the solid surface.
The solution for this problem is obtained in the cylindrical coorindate system $(r,\theta ,z)$ by introducing
$\eta ={\frac {z}{\sqrt {\nu /k}}},\quad \gamma =-{\frac {V}{2{\sqrt {k\nu }}}},\quad v_{r}=krF'(\eta )+U\cos \theta G(\eta ),\quad v_{\theta }=-U\sin \theta G(\eta ),\quad v_{z}=-2{\sqrt {k\nu }}F(\eta )$
where $U$ is the translational speed of the wall and $V$ is the injection (or, suction) velocity at the wall. The problem is axisymmetric only when $U=0$. The pressure is given by
${\frac {p-p_{o}}{\rho }}=-{\frac {1}{2}}k^{2}r^{2}-2k\nu (F^{2}+F')$
The Navier–Stokes equations then reduce to
${\begin{aligned}F'''+2FF''-F'^{2}+1&=0,\\G''+2FG'-F'G&=0\end{aligned}}$
along with boundary conditions,
$F(0)=\gamma ,\quad F'(0)=0,\quad F'(\infty )=1,\quad G(0)=1,\quad G(\infty )=0.$
When $U=V=0$, the classical Homann problem is recovered.
Plane counterflows
Jets emerging from a slot-jets creates stagnation point in between according to potential theory. The flow near the stagnation point can by studied using self-similar solution. This setup is widely used in combustion experiments. The initial study of impinging stagnation flows are due to C.Y. Wang.[18][19] Let two fluids with constant properties denoted with suffix $1({\text{top}}),\ 2({\text{bottom}})$ flowing from opposite direction impinge, and assume the two fluids are immiscible and the interface (located at $y=0$) is planar. The velocity is given by
$u_{1}=k_{1}x,\quad v_{1}=-k_{1}y,\quad u_{2}=k_{2}x,\quad v_{2}=-k_{2}y$
where $k_{1},\ k_{2}$ are strain rates of the fluids. At the interface, velocities, tangential stress and pressure must be continuous. Introducing the self-similar transformation,
$\eta _{1}={\sqrt {\frac {\nu _{1}}{k_{1}}}}y,\quad u_{1}=k_{1}xF_{1}',\quad v_{1}=-{\sqrt {\nu _{1}k_{1}}}F_{1}$
$\eta _{2}={\sqrt {\frac {\nu _{2}}{k_{2}}}}y,\quad u_{2}=k_{2}xF_{2}',\quad v_{2}=-{\sqrt {\nu _{2}k_{2}}}F_{2}$
results equations,
$F_{1}'''+F_{1}F_{1}''-F_{1}'^{2}+1=0,\quad {\frac {p_{o1}-p_{1}}{\rho _{1}}}={\frac {1}{2}}k_{1}^{2}x^{2}+k_{1}\nu _{1}F_{1}'+{\frac {1}{2}}k_{1}\nu _{1}F_{1}^{2}$
$F_{2}'''+F_{2}F_{2}''-F_{2}'^{2}+1=0,\quad {\frac {p_{o2}-p_{2}}{\rho _{2}}}={\frac {1}{2}}k_{2}^{2}x^{2}+k_{2}\nu _{2}F_{2}'+{\frac {1}{2}}k_{2}\nu _{2}F_{2}^{2}.$
The no-penetration condition at the interface and free stream condition far away from the stagnation plane become
$F_{1}(0)=0,\quad F_{1}'(\infty )=1,\quad F_{2}(0)=0,\quad F_{2}'(-\infty )=1.$
But the equations require two more boundary conditions. At $\eta =0$, the tangential velocities $u_{1}=u_{2}$, the tangential stress $\rho _{1}\nu _{1}\partial u_{1}/\partial y=\rho _{2}\nu _{2}\partial u_{2}/\partial y$ and the pressure $p_{1}=p_{2}$ are continuous. Therefore,
${\begin{aligned}k_{1}F_{1}'(0)&=k_{2}F_{2}'(0),\\\rho _{1}{\sqrt {\nu _{1}k_{1}^{3}}}F_{1}''(0)&=\rho _{2}{\sqrt {\nu _{2}k_{2}^{3}}}F_{2}''(0),\\p_{o1}-\rho _{1}\nu _{1}k_{1}F_{1}'(0)&=p_{o2}-\rho _{2}\nu _{2}k_{2}F_{2}'(0).\end{aligned}}$
where $\rho _{1}k_{1}^{2}=\rho _{2}k_{2}^{2}$ (from outer inviscid problem) is used. Both $F_{i}'(0),F_{i}''(0)$ are not known apriori, but derived from matching conditions. The third equation is determine variation of outer pressure $p_{o1}-p_{o2}$ due to the effect of viscosity. So there are only two parameters, which governs the flow, which are
$\Lambda ={\frac {k_{1}}{k_{2}}}=\left({\frac {\rho _{2}}{\rho _{1}}}\right)^{1/2},\quad \Gamma ={\frac {\nu _{2}}{\nu _{1}}}$
then the boundary conditions become
$F_{1}'(0)=\Lambda F_{2}'(0),\quad F_{1}''(0)={\sqrt {\frac {\Gamma }{\Lambda }}}F_{2}''(0)$.
References
1. Moffatt, H. K., Kida, S., & Ohkitani, K. (1994). Stretched vortices–the sinews of turbulence; large-Reynolds-number asymptotics. Journal of Fluid Mechanics, 259, 241-264.
2. Taylor, G. I. (1934). The formation of emulsions in definable fields of flow. Proceedings of the Royal Society of London. Series A, containing papers of a mathematical and physical character, 146(858), 501-523.
3. Wang, C. Y. (1974). Axisymmetric stagnation flow on a cylinder. Quarterly of Applied Mathematics, 32(2), 207-213.
4. Craik, A. D. (2009). Exact vortex solutions of the Navier–Stokes equations with axisymmetric strain and suction or injection. Journal of fluid mechanics, 626, 291-306.
5. Rajamanickam, P., & Weiss, A. D. (2021). Steady axisymmetric vortices in radial stagnation flows. The Quarterly Journal of Mechanics and Applied Mathematics, 74(3), 367-378.
6. Rosenhead, Louis, ed. Laminar boundary layers. Clarendon Press, 1963.
7. Batchelor, George Keith. An introduction to fluid dynamics. Cambridge University Press, 2000.
8. Hiemenz, Karl. Die Grenzschicht an einem in den gleichförmigen Flüssigkeitsstrom eingetauchten geraden Kreiszylinder... Diss. 1911.
9. Howarth, Leslie. On the calculation of steady flow in the boundary layer near the surface of a cylinder in a stream. No. ARC-R/M-1632. AERONAUTICAL RESEARCH COUNCIL LONDON (UNITED KINGDOM), 1934.
10. Drazin, Philip G., and Norman Riley. The Navier–Stokes equations: a classification of flows and exact solutions. No. 334. Cambridge University Press, 2006.
11. Rott, Nicholas. "Unsteady viscous flow in the vicinity of a stagnation point." Quarterly of Applied Mathematics 13.4 (1956): 444–451.
12. Stuart, J. T. "The viscous flow near a stagnation point when the external flow has uniform vorticity." Journal of the Aerospace Sciences (2012).
13. Tamada, Ko. "Two-dimensional stagnation-point flow impinging obliquely on a plane wall." Journal of the Physical Society of Japan 46 (1979): 310.
14. Dorrepaal, J. M. "An exact solution of the Navier–Stokes equation which describes non-orthogonal stagnation-point flow in two dimensions." Journal of Fluid Mechanics 163 (1986): 141–147.
15. Homann, Fritz. "Der Einfluss grosser Zähigkeit bei der Strömung um den Zylinder und um die Kugel." ZAMM‐Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 16.3 (1936): 153–164.
16. Libby, Paul A. "Wall shear at a three-dimensional stagnation point with a moving wall." AIAA Journal 12.3 (1974): 408–409.
17. Libby, Paul A. "Laminar flow at a three-dimensional stagnation point with large rates of injection." AIAA Journal 14.9 (1976): 1273–1279.
18. Wang, C. Y. "Stagnation flow on the surface of a quiescent fluid—an exact solution of the Navier–Stokes equations." Quarterly of applied mathematics 43.2 (1985): 215–223.
19. Wang, C. Y. "Impinging stagnation flows." The Physics of fluids 30.3 (1987): 915–917.
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Wikipedia
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Stahl's theorem
In matrix analysis Stahl's theorem is a theorem proved in 2011 by Herbert Stahl concerning Laplace transforms for special matrix functions.[1] It originated in 1975 as the Bessis-Moussa-Villani (BMV) conjecture by Daniel Bessis, Pierre Moussa, and Marcel Villani.[2] In 2004 Elliott H. Lieb and Robert Seiringer gave two important reformulations of the BMV conjecture.[3] In 2015 Alexandre Eremenko gave a simplified proof of Stahl's theorem.[4]
Statement of the theorem
Let $\operatorname {tr} $ denote the trace of a matrix. If $A$ and $B$ are $n\times n$ Hermitian matrices and $B$ is positive semidefinite, define $\mathbf {f} (t)=\operatorname {tr} (\exp(A-tB))$, for all real $t\geq 0$. Then $\mathbf {f} $ can be represented as the Laplace transform of a non-negative Borel measure $\mu $ on $[0,\infty )$. In other words, for all real $t\geq 0$,
$\mathbf {f} $(t) = $\int _{[0,\infty )}e^{-ts}\,d\mu (s)$,
for some non-negative measure $\mu $ depending upon $A$ and $B$.[5]
References
1. Stahl, Herbert R. (2013). "Proof of the BMV conjecture". Acta Mathematica. 211 (2): 255–290. arXiv:1107.4875. doi:10.1007/s11511-013-0104-z.
2. Bessis, D.; Moussa, P.; Villani, M. (1975). "Monotonic converging variational approximations to the functional integrals in quantum statistical mechanics". Journal of Mathematical Physics. 16 (11): 2318–2325. Bibcode:1975JMP....16.2318B. doi:10.1063/1.522463.
3. Lieb, Elliott; Seiringer, Robert (2004). "Equivalent forms of the Bessis-Moussa-Villani conjecture". Journal of Statistical Physics. 115 (1–2): 185–190. arXiv:math-ph/0210027. Bibcode:2004JSP...115..185L. doi:10.1023/B:JOSS.0000019811.15510.27.
4. Eremenko, A. È. (2015). "Herbert Stahl's proof of the BMV conjecture". Sbornik: Mathematics. 206 (1): 87–92. arXiv:1312.6003. Bibcode:2015SbMat.206...87E. doi:10.1070/SM2015v206n01ABEH004447.
5. Clivaz, Fabien (2016). Stahl's Theorem (aka BMV Conjecture): Insights and Intuition on its Proof. Operator Theory: Advances and Applications. Vol. 254. pp. 107–117. arXiv:1702.06403. doi:10.1007/978-3-319-29992-1_6. ISBN 978-3-319-29990-7. ISSN 0255-0156.
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Wikipedia
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Polite number
In number theory, a polite number is a positive integer that can be written as the sum of two or more consecutive positive integers. A positive integer which is not polite is called impolite.[1][2] The impolite numbers are exactly the powers of two, and the polite numbers are the natural numbers that are not powers of two.
Polite numbers have also been called staircase numbers because the Young diagrams which represent graphically the partitions of a polite number into consecutive integers (in the French notation of drawing these diagrams) resemble staircases.[3][4][5] If all numbers in the sum are strictly greater than one, the numbers so formed are also called trapezoidal numbers because they represent patterns of points arranged in a trapezoid.[6][7][8][9][10][11][12]
The problem of representing numbers as sums of consecutive integers and of counting the number of representations of this type has been studied by Sylvester,[13] Mason,[14][15] Leveque,[16] and many other more recent authors.[1][2][17][18][19][20][21][22][23] The polite numbers describe the possible numbers of sides of the Reinhardt polygons.[24]
Examples and characterization
The first few polite numbers are
3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, ... (sequence A138591 in the OEIS).
The impolite numbers are exactly the powers of two.[13] It follows from the Lambek–Moser theorem that the nth polite number is f(n + 1), where
$f(n)=n+\left\lfloor \log _{2}\left(n+\log _{2}n\right)\right\rfloor .$
Politeness
The politeness of a positive number is defined as the number of ways it can be expressed as the sum of consecutive integers. For every x, the politeness of x equals the number of odd divisors of x that are greater than one.[13] The politeness of the numbers 1, 2, 3, ... is
0, 0, 1, 0, 1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 3, 0, 1, 2, 1, 1, 3, ... (sequence A069283 in the OEIS).
For instance, the politeness of 9 is 2 because it has two odd divisors, 3 and 9, and two polite representations
9 = 2 + 3 + 4 = 4 + 5;
the politeness of 15 is 3 because it has three odd divisors, 3, 5, and 15, and (as is familiar to cribbage players)[25] three polite representations
15 = 4 + 5 + 6 = 1 + 2 + 3 + 4 + 5 = 7 + 8.
An easy way of calculating the politeness of a positive number by decomposing the number into its prime factors, taking the powers of all prime factors greater than 2, adding 1 to all of them, multiplying the numbers thus obtained with each other and subtracting 1. For instance 90 has politeness 5 because $90=2\times 3^{2}\times 5^{1}$; the powers of 3 and 5 are respectively 2 and 1, and applying this method $(2+1)\times (1+1)-1=5$.
Construction of polite representations from odd divisors
To see the connection between odd divisors and polite representations, suppose a number x has the odd divisor y > 1. Then y consecutive integers centered on x/y (so that their average value is x/y) have x as their sum:
$x=\sum _{i={\frac {x}{y}}-{\frac {y-1}{2}}}^{{\frac {x}{y}}+{\frac {y-1}{2}}}i.$
Some of the terms in this sum may be zero or negative. However, if a term is zero it can be omitted and any negative terms may be used to cancel positive ones, leading to a polite representation for x. (The requirement that y > 1 corresponds to the requirement that a polite representation have more than one term; applying the same construction for y = 1 would just lead to the trivial one-term representation x = x.) For instance, the polite number x = 14 has a single nontrivial odd divisor, 7. It is therefore the sum of 7 consecutive numbers centered at 14/7 = 2:
14 = (2 − 3) + (2 − 2) + (2 − 1) + 2 + (2 + 1) + (2 + 2) + (2 + 3).
The first term, −1, cancels a later +1, and the second term, zero, can be omitted, leading to the polite representation
14 = 2 + (2 + 1) + (2 + 2) + (2 + 3) = 2 + 3 + 4 + 5.
Conversely, every polite representation of x can be formed from this construction. If a representation has an odd number of terms, x/y is the middle term, while if it has an even number of terms and its minimum value is m it may be extended in a unique way to a longer sequence with the same sum and an odd number of terms, by including the 2m − 1 numbers −(m − 1), −(m − 2), ..., −1, 0, 1, ..., m − 2, m − 1. After this extension, again, x/y is the middle term. By this construction, the polite representations of a number and its odd divisors greater than one may be placed into a one-to-one correspondence, giving a bijective proof of the characterization of polite numbers and politeness.[13][26] More generally, the same idea gives a two-to-one correspondence between, on the one hand, representations as a sum of consecutive integers (allowing zero, negative numbers, and single-term representations) and on the other hand odd divisors (including 1).[15]
Another generalization of this result states that, for any n, the number of partitions of n into odd numbers having k distinct values equals the number of partitions of n into distinct numbers having k maximal runs of consecutive numbers.[13][27][28] Here a run is one or more consecutive values such that the next larger and the next smaller consecutive values are not part of the partition; for instance the partition 10 = 1 + 4 + 5 has two runs, 1 and 4 + 5. A polite representation has a single run, and a partition with one value d is equivalent to a factorization of n as the product d ⋅ (n/d), so the special case k = 1 of this result states again the equivalence between polite representations and odd factors (including in this case the trivial representation n = n and the trivial odd factor 1).
Trapezoidal numbers
If a polite representation starts with 1, the number so represented is a triangular number
$T_{n}={\frac {n(n+1)}{2}}=1+2+\cdots +n.$
Otherwise, it is the difference of two nonconsecutive triangular numbers
$i+(i+1)+(i+2)+\cdots +j=T_{j}-T_{i-1}\quad (j>i\geq 2).$
This second case is called a trapezoidal number.[12] One can also consider polite numbers that aren't trapezoidal. The only such numbers are the triangular numbers with only one nontrivial odd divisor, because for those numbers, according to the bijection described earlier, the odd divisor corresponds to the triangular representation and there can be no other polite representations. Thus, non-trapezoidal polite number must have the form of a power of two multiplied by an odd prime. As Jones and Lord observe,[12] there are exactly two types of triangular numbers with this form:
1. the even perfect numbers 2n − 1(2n − 1) formed by the product of a Mersenne prime 2n − 1 with half the nearest power of two, and
2. the products 2n − 1(2n + 1) of a Fermat prime 2n + 1 with half the nearest power of two.
(sequence A068195 in the OEIS). For instance, the perfect number 28 = 23 − 1(23 − 1) and the number 136 = 24 − 1(24 + 1) are both this type of polite number. It is conjectured that there are infinitely many Mersenne primes, in which case there are also infinitely many polite numbers of this type.
References
1. Adams, Ken (March 1993), "How polite is x?", The Mathematical Gazette, 77 (478): 79–80, doi:10.2307/3619263, JSTOR 3619263, S2CID 171530924.
2. Griggs, Terry S. (December 1991), "Impolite Numbers", The Mathematical Gazette, 75 (474): 442–443, doi:10.2307/3618630, JSTOR 3618630, S2CID 171681914.
3. Mason, John; Burton, Leone; Stacey, Kaye (1982), Thinking Mathematically, Addison-Wesley, ISBN 978-0-201-10238-3.
4. Stacey, K.; Groves, S. (1985), Strategies for Problem Solving, Melbourne: Latitude.
5. Stacey, K.; Scott, N. (2000), "Orientation to deep structure when trying examples: a key to successful problem solving", in Carillo, J.; Contreras, L. C. (eds.), Resolucion de Problemas en los Albores del Siglo XXI: Una vision Internacional desde Multiples Perspectivas y Niveles Educativos (PDF), Huelva, Spain: Hergue, pp. 119–147, archived from the original (PDF) on 2008-07-26.
6. Gamer, Carlton; Roeder, David W.; Watkins, John J. (1985), "Trapezoidal numbers", Mathematics Magazine, 58 (2): 108–110, doi:10.2307/2689901, JSTOR 2689901.
7. Jean, Charles-É. (March 1991), "Les nombres trapézoïdaux" (French), Bulletin de l'AMQ: 6–11.
8. Haggard, Paul W.; Morales, Kelly L. (1993), "Discovering relationships and patterns by exploring trapezoidal numbers", International Journal of Mathematical Education in Science and Technology, 24 (1): 85–90, doi:10.1080/0020739930240111.
9. Feinberg-McBrian, Carol (1996), "The case of trapezoidal numbers", Mathematics Teacher, 89 (1): 16–24, doi:10.5951/MT.89.1.0016.
10. Smith, Jim (1997), "Trapezoidal numbers", Mathematics in School, 5: 42.
11. Verhoeff, T. (1999), "Rectangular and trapezoidal arrangements", Journal of Integer Sequences, 2: 16, Bibcode:1999JIntS...2...16V, Article 99.1.6.
12. Jones, Chris; Lord, Nick (1999), "Characterising non-trapezoidal numbers", The Mathematical Gazette, 83 (497): 262–263, doi:10.2307/3619053, JSTOR 3619053, S2CID 125545112.
13. Sylvester, J. J.; Franklin, F (1882), "A constructive theory of partitions, arranged in three acts, an interact and an exodion", American Journal of Mathematics, 5 (1): 251–330, doi:10.2307/2369545, JSTOR 2369545. In The collected mathematical papers of James Joseph Sylvester (December 1904), H. F. Baker, ed. Sylvester defines the class of a partition into distinct integers as the number of blocks of consecutive integers in the partition, so in his notation a polite partition is of first class.
14. Mason, T. E. (1911), "On the representations of a number as a sum of consecutive integers", Proceedings of the Indiana Academy of Science: 273–274.
15. Mason, Thomas E. (1912), "On the representation of an integer as the sum of consecutive integers", American Mathematical Monthly, 19 (3): 46–50, doi:10.2307/2972423, JSTOR 2972423, MR 1517654.
16. Leveque, W. J. (1950), "On representations as a sum of consecutive integers", Canadian Journal of Mathematics, 2: 399–405, doi:10.4153/CJM-1950-036-3, MR 0038368, S2CID 124093945,
17. Pong, Wai Yan (2007), "Sums of consecutive integers", College Math. J., 38 (2): 119–123, arXiv:math/0701149, Bibcode:2007math......1149P, doi:10.1080/07468342.2007.11922226, MR 2293915, S2CID 14169613.
18. Britt, Michael J. C.; Fradin, Lillie; Philips, Kathy; Feldman, Dima; Cooper, Leon N. (2005), "On sums of consecutive integers", Quart. Appl. Math., 63 (4): 791–792, doi:10.1090/S0033-569X-05-00991-1, MR 2187932.
19. Frenzen, C. L. (1997), "Proof without words: sums of consecutive positive integers", Math. Mag., 70 (4): 294, doi:10.1080/0025570X.1997.11996560, JSTOR 2690871, MR 1573264.
20. Guy, Robert (1982), "Sums of consecutive integers" (PDF), Fibonacci Quarterly, 20 (1): 36–38, Zbl 0475.10014.
21. Apostol, Tom M. (2003), "Sums of consecutive positive integers", The Mathematical Gazette, 87 (508): 98–101, doi:10.1017/S002555720017216X, JSTOR 3620570, S2CID 125202845.
22. Prielipp, Robert W.; Kuenzi, Norbert J. (1975), "Sums of consecutive positive integers", Mathematics Teacher, 68 (1): 18–21, doi:10.5951/MT.68.1.0018.
23. Parker, John (1998), "Sums of consecutive integers", Mathematics in School, 27 (2): 8–11.
24. Mossinghoff, Michael J. (2011), "Enumerating isodiametric and isoperimetric polygons", Journal of Combinatorial Theory, Series A, 118 (6): 1801–1815, doi:10.1016/j.jcta.2011.03.004, MR 2793611
25. Graham, Ronald; Knuth, Donald; Patashnik, Oren (1988), "Problem 2.30", Concrete Mathematics, Addison-Wesley, p. 65, ISBN 978-0-201-14236-5.
26. Vaderlind, Paul; Guy, Richard K.; Larson, Loren C. (2002), The inquisitive problem solver, Mathematical Association of America, pp. 205–206, ISBN 978-0-88385-806-6.
27. Andrews, G. E. (1966), "On generalizations of Euler's partition theorem", Michigan Mathematical Journal, 13 (4): 491–498, doi:10.1307/mmj/1028999609, MR 0202617.
28. Ramamani, V.; Venkatachaliengar, K. (1972), "On a partition theorem of Sylvester", The Michigan Mathematical Journal, 19 (2): 137–140, doi:10.1307/mmj/1029000844, MR 0304323.
External links
• Polite Numbers, NRICH, University of Cambridge, December 2002
• An Introduction to Runsums, R. Knott.
• Is there any pattern to the set of trapezoidal numbers? Intellectualism.org question of the day, October 2, 2003. With a diagram showing trapezoidal numbers color-coded by the number of terms in their expansions.
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Wikipedia
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Ivar Stakgold
Ivar Stakgold (December 13, 1925 – May 29, 2018)[1] was a Norwegian-born American academic mathematician and bridge player from Newark, Delaware.[2] As the sole author of two books he specialized in boundary value problems (LCSH).[3][4]
Life
Stakgold was born in Oslo, Norway to parents with Russian-Jewish heritage.[5][6] He studied applied mathematics at Harvard University and earned the Ph.D. in 1949 with a dissertation, The Cauchy Relations In A Molecular Theory of Elasticity, under Léon Nicolas Brillouin.[7] He was professor emeritus of mathematical sciences at the University of Delaware and a researcher at the University of California, San Diego.[8] He was a former president of the Society for Industrial and Applied Mathematics (SIAM).[3]
Books
By Stakgold
• Boundary Value Problems of Mathematical Physics, 2 vols. (Macmillan, 1967), Macmillan series in advanced mathematics and theoretical physics, LCCN 67-10304; reprint 2000, SIAM Classics in applied mathematics, no. 29[4]
• Nonlinear Problems in the Physical Sciences and Biology: proceedings of a Battelle Summer Institute, Seattle, July 3–28, 1972, eds. Stakgold and others (Springer-Verlag, 1973), LCCN 73-78428
• Green's Functions and Boundary Value Problems (Wiley, 1979); 2nd ed. 1998; 3rd ed. 2011, Stakgold and Michael J. Holst[3]
• Analytical and Computational Methods in Scattering and Applied Mathematics, eds. Fadil Santosa and Stakgold (Chapman & Hall/CRC, 2000) – "A volume to the memory of Ralph Ellis Kleinman", LCCN 99-87683
Other
• Nonlinear Problems in Applied Mathematics: in honor of Ivar Stakgold on his 70th birthday,eds. T.S. Angell and others (Philadelphia:SIAM, 1996), LCCN 96-112982
Bridge accomplishments
Awards
• Mott-Smith Trophy (1) 1958
Wins
• North American Bridge Championships (5)
• Silodor Open Pairs (1) 1958 [9]
• Vanderbilt (1) 1958 [10]
• Chicago Mixed Board-a-Match (1) 1969 [11]
• Reisinger (1) 1958 [12]
• Spingold (1) 1962 [13]
Runners-up
• Bermuda Bowl (1) 1959
• North American Bridge Championships
• Silodor Open Pairs (1) 1963 [9]
• Mitchell Board-a-Match Teams (1) 1957 [14]
• Spingold (1) 1958 [13]
References
1. "Obituary: Ivar Stakgold". The San Diego Union Tribune. San Diego , California. 12 June 2018. Retrieved 14 June 2018.
2. Francis, Henry G.; Truscott, Alan F.; Francis, Dorthy A., eds. (1994). The Official Encyclopedia of Bridge (5th ed.). Memphis, TN: American Contract Bridge League. p. 746. ISBN 0-943855-48-9. LCCN 96188639.
3. "Contributor biographical information for Green's functions and boundary value problems / Ivar Stakgold". Library of Congress. [1998?]. Retrieved 2014-11-04.
With more information linked.
4. "Publisher description for Boundary value problems of mathematical physics / Ivar Stakgold". Library of Congress. [2000]. Retrieved 2014-11-04.
With more information linked.
5. "Ivar Stakgold Obituary (1925 - 2018) - San Diego, Ca, CA - San Diego Union-Tribune". Legacy.com.
6. American Men & Women of Science: Q-S. Thomson Gale. 2003. p. 943. ISBN 0787665290.
7. "Ivar Stakgold". Mathematics Genealogy Project. North Dakota State University. Retrieved 2014-11-04.
8. UCSD faculty bio.
9. "Silodor Open Pairs Previous Winners" (PDF). American Contract Bridge League. 2014-07-27. p. 11. Archived from the original (PDF) on 2014-10-21. Retrieved 2014-10-17.
10. "Vanderbilt Previous Winners" (PDF). American Contract Bridge League. 2014-03-24. p. 6. Archived from the original (PDF) on 2014-10-21. Retrieved 2014-10-17.
11. "Mixed BAM Previous Winners" (PDF). American Contract Bridge League. 2014-07-24. p. 14. Archived from the original (PDF) on 2014-10-21. Retrieved 2014-10-17.
12. "Reisinger Winners" (PDF). American Contract Bridge League. 2013-12-06. p. 6. Archived from the original (PDF) on 2014-10-21. Retrieved 2014-10-17.
13. "Spingold Previous Winners" (PDF). American Contract Bridge League. 2014-07-21. p. 12. Archived from the original (PDF) on 2014-10-21. Retrieved 2014-10-17.
14. "Mitchell BAM Winners" (PDF). American Contract Bridge League. 2013-12-01. p. 8. Archived from the original (PDF) on 2014-10-21. Retrieved 2014-10-17.
External links
• "International record for Ivar Stakgold". World Bridge Federation.
• Ivar Stakgold at Library of Congress, with 8 library catalog records
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Wikipedia
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Stallings–Zeeman theorem
In mathematics, the Stallings–Zeeman theorem is a result in algebraic topology, used in the proof of the Poincaré conjecture for dimension greater than or equal to five. It is named after the mathematicians John R. Stallings and Christopher Zeeman.
Statement of the theorem
Let M be a finite simplicial complex of dimension dim(M) = m ≥ 5. Suppose that M has the homotopy type of the m-dimensional sphere Sm and that M is locally piecewise linearly homeomorphic to m-dimensional Euclidean space Rm. Then M is homeomorphic to Sm under a map that is piecewise linear except possibly at a single point x. That is, M \ {x} is piecewise linearly homeomorphic to Rm.
References
• Stallings, John (1962). "The piecewise-linear structure of Euclidean space". Proc. Cambridge Philos. Soc. 58 (3): 481–488. Bibcode:1962PCPS...58..481S. doi:10.1017/s0305004100036756. S2CID 120418488. MR0149457
• Zeeman, Christopher (1961). "The generalised Poincaré conjecture". Bull. Amer. Math. Soc. 67 (3): 270. doi:10.1090/S0002-9904-1961-10578-8. MR0124906
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Wikipedia
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Stallings theorem about ends of groups
In the mathematical subject of group theory, the Stallings theorem about ends of groups states that a finitely generated group $G$ has more than one end if and only if the group $G$ admits a nontrivial decomposition as an amalgamated free product or an HNN extension over a finite subgroup. In the modern language of Bass–Serre theory the theorem says that a finitely generated group $G$ has more than one end if and only if $G$ admits a nontrivial (that is, without a global fixed point) action on a simplicial tree with finite edge-stabilizers and without edge-inversions.
The theorem was proved by John R. Stallings, first in the torsion-free case (1968)[1] and then in the general case (1971).[2]
Ends of graphs
Main article: End (graph theory)
Let $\Gamma $ be a connected graph where the degree of every vertex is finite. One can view $\Gamma $ as a topological space by giving it the natural structure of a one-dimensional cell complex. Then the ends of $\Gamma $ are the ends of this topological space. A more explicit definition of the number of ends of a graph is presented below for completeness.
Let $n\geqslant 0$ be a non-negative integer. The graph $\Gamma $ is said to satisfy $e(\Gamma )\leqslant n$ if for every finite collection $F$ of edges of $\Gamma $ the graph $\Gamma -F$ has at most $n$ infinite connected components. By definition, $e(\Gamma )=m$ if $e(\Gamma )\leqslant m$ and if for every $0\leqslant n<m$ the statement $e(\Gamma )\leqslant n$ is false. Thus $e(\Gamma )=m$ if $m$ is the smallest nonnegative integer $n$ such that $e(\Gamma )\leqslant n$. If there does not exist an integer $n\geqslant 0$ such that $e(\Gamma )\leqslant n$, put $e(\Gamma )=\infty $. The number $e(\Gamma )$ is called the number of ends of $\Gamma $.
Informally, $e(\Gamma )$ is the number of "connected components at infinity" of $\Gamma $. If $e(\Gamma )=m<\infty $, then for any finite set $F$ of edges of $\Gamma $ there exists a finite set $K$ of edges of $\Gamma $ with $F\subseteq K$ such that $\Gamma -F$ has exactly $m$ infinite connected components. If $e(\Gamma )=\infty $, then for any finite set $F$ of edges of $\Gamma $ and for any integer $n\geqslant 0$ there exists a finite set $K$ of edges of $\Gamma $ with $F\subseteq K$ such that $\Gamma -K$ has at least $n$ infinite connected components.
Ends of groups
Let $G$ be a finitely generated group. Let $S\subseteq G$ be a finite generating set of $G$ and let $\Gamma (G,S)$ be the Cayley graph of $G$ with respect to $S$. The number of ends of $G$ is defined as $e(G)=e(\Gamma (G,S))$. A basic fact in the theory of ends of groups says that $e(\Gamma (G,S))$ does not depend on the choice of a finite generating set $S$ of $G$, so that $e(G)$ is well-defined.
Basic facts and examples
• For a finitely generated group $G$ we have $e(G)=0$ if and only if $G$ is finite.
• For the infinite cyclic group $\mathbb {Z} $ we have $e(\mathbb {Z} )=2.$
• For the free abelian group of rank two $\mathbb {Z} ^{2}$ we have $e(\mathbb {Z} ^{2})=1.$
• For a free group $F(X)$ where $1<|X|<\infty $ we have $e(F(X))=\infty $.
Freudenthal-Hopf theorems
Hans Freudenthal[3] and independently Heinz Hopf[4] established in the 1940s the following two facts:
• For any finitely generated group $G$ we have $e(G)\in \{0,1,2,\infty \}$.
• For any finitely generated group $G$ we have $e(G)=2$ if and only if $G$ is virtually infinite cyclic (that is, $G$ contains an infinite cyclic subgroup of finite index).
Charles T. C. Wall proved in 1967 the following complementary fact:[5]
• A group $G$ is virtually infinite cyclic if and only if it has a finite normal subgroup $W$ such that $G/W$ is either infinite cyclic or infinite dihedral.
Cuts and almost invariant sets
Let $G$ be a finitely generated group, $S\subseteq G$ be a finite generating set of $G$ and let $\Gamma =\Gamma (G,S)$ be the Cayley graph of $G$ with respect to $S$. For a subset $A\subseteq G$ denote by $A^{*}$ the complement $G-A$ of $A$ in $G$.
For a subset $A\subseteq G$, the edge boundary or the co-boundary $\delta A$ of $A$ consists of all (topological) edges of $\Gamma $ connecting a vertex from $A$ with a vertex from $A^{*}$. Note that by definition $\delta A=\delta A^{*}$.
An ordered pair $(A,A^{*})$ is called a cut in $\Gamma $ if $\delta A$ is finite. A cut $(A,A^{*})$ is called essential if both the sets $A$ and $A^{*}$ are infinite.
A subset $A\subseteq G$ is called almost invariant if for every $g\in G$ the symmetric difference between $A$ and $Ag$ is finite. It is easy to see that $(A,A^{*})$ is a cut if and only if the sets $A$ and $A^{*}$ are almost invariant (equivalently, if and only if the set $A$ is almost invariant).
Cuts and ends
A simple but important observation states:
$e(G)>1$ if and only if there exists at least one essential cut $(A,A^{*})$ in Γ.
Cuts and splittings over finite groups
If $G=H*K$ where $H$ and $K$ are nontrivial finitely generated groups then the Cayley graph of $G$ has at least one essential cut and hence $e(G)>1$. Indeed, let $X$ and $Y$ be finite generating sets for $H$ and $K$ accordingly so that $S=X\cup Y$ is a finite generating set for $G$ and let $\Gamma =\Gamma (G,S)$ be the Cayley graph of $G$ with respect to $S$. Let $A$ consist of the trivial element and all the elements of $G$ whose normal form expressions for $G=H*K$ starts with a nontrivial element of $H$. Thus $A^{*}$ consists of all elements of $G$ whose normal form expressions for $G=H*K$ starts with a nontrivial element of $K$. It is not hard to see that $(A,A^{*})$ is an essential cut in Γ so that $e(G)>1$.
A more precise version of this argument shows that for a finitely generated group $G$:
• If $G=H*_{C}K$ is a free product with amalgamation where $C$ is a finite group such that $C\neq H$ and $C\neq K$ then $H$ and $K$ are finitely generated and $e(G)>1$ .
• If $G=\langle H,t|t^{-1}C_{1}t=C_{2}\rangle $ is an HNN-extension where $C_{1}$, $C_{2}$ are isomorphic finite subgroups of $H$ then $G$ is a finitely generated group and $e(G)>1$.
Stallings' theorem shows that the converse is also true.
Formal statement of Stallings' theorem
Let $G$ be a finitely generated group.
Then $e(G)>1$ if and only if one of the following holds:
• The group $G$ admits a splitting $G=H*_{C}K$ as a free product with amalgamation where $C$ is a finite group such that $C\neq H$ and $C\neq K$.
• The group $G$ is an HNN extension $G=\langle H,t|t^{-1}C_{1}t=C_{2}\rangle $ where and $C_{1}$, $C_{2}$ are isomorphic finite subgroups of $H$.
In the language of Bass–Serre theory this result can be restated as follows: For a finitely generated group $G$ we have $e(G)>1$ if and only if $G$ admits a nontrivial (that is, without a global fixed vertex) action on a simplicial tree with finite edge-stabilizers and without edge-inversions.
For the case where $G$ is a torsion-free finitely generated group, Stallings' theorem implies that $e(G)=\infty $ if and only if $G$ admits a proper free product decomposition $G=A*B$ with both $A$ and $B$ nontrivial.
Applications and generalizations
• Among the immediate applications of Stallings' theorem was a proof by Stallings[6] of a long-standing conjecture that every finitely generated group of cohomological dimension one is free and that every torsion-free virtually free group is free.
• Stallings' theorem also implies that the property of having a nontrivial splitting over a finite subgroup is a quasi-isometry invariant of a finitely generated group since the number of ends of a finitely generated group is easily seen to be a quasi-isometry invariant. For this reason Stallings' theorem is considered to be one of the first results in geometric group theory.
• Stallings' theorem was a starting point for Dunwoody's accessibility theory. A finitely generated group $G$ is said to be accessible if the process of iterated nontrivial splitting of $G$ over finite subgroups always terminates in a finite number of steps. In Bass–Serre theory terms that the number of edges in a reduced splitting of $G$ as the fundamental group of a graph of groups with finite edge groups is bounded by some constant depending on $G$. Dunwoody proved[7] that every finitely presented group is accessible but that there do exist finitely generated groups that are not accessible.[8] Linnell[9] showed that if one bounds the size of finite subgroups over which the splittings are taken then every finitely generated group is accessible in this sense as well. These results in turn gave rise to other versions of accessibility such as Bestvina-Feighn accessibility[10] of finitely presented groups (where the so-called "small" splittings are considered), acylindrical accessibility,[11][12] strong accessibility,[13] and others.
• Stallings' theorem is a key tool in proving that a finitely generated group $G$ is virtually free if and only if $G$ can be represented as the fundamental group of a finite graph of groups where all vertex and edge groups are finite (see, for example,[14]).
• Using Dunwoody's accessibility result, Stallings' theorem about ends of groups and the fact that if $G$ is a finitely presented group with asymptotic dimension 1 then $G$ is virtually free[15] one can show [16] that for a finitely presented word-hyperbolic group $G$ the hyperbolic boundary of $G$ has topological dimension zero if and only if $G$ is virtually free.
• Relative versions of Stallings' theorem and relative ends of finitely generated groups with respect to subgroups have also been considered. For a subgroup $H\leqslant G$ of a finitely generated group $G$ one defines the number of relative ends $e(G,H)$ as the number of ends of the relative Cayley graph (the Schreier coset graph) of $G$ with respect to $H$. The case where $e(G,H)>1$ is called a semi-splitting of $G$ over $H$. Early work on semi-splittings, inspired by Stallings' theorem, was done in the 1970s and 1980s by Scott,[17] Swarup,[18] and others.[19][20] The work of Sageev[21] and Gerasimov[22] in the 1990s showed that for a subgroup $H\leqslant G$ the condition $e(G,H)>1$ corresponds to the group $G$ admitting an essential isometric action on a CAT(0)-cubing where a subgroup commensurable with $H$ stabilizes an essential "hyperplane" (a simplicial tree is an example of a CAT(0)-cubing where the hyperplanes are the midpoints of edges). In certain situations such a semi-splitting can be promoted to an actual algebraic splitting, typically over a subgroup commensurable with $H$, such as for the case where $H$ is finite (Stallings' theorem). Another situation where an actual splitting can be obtained (modulo a few exceptions) is for semi-splittings over virtually polycyclic subgroups. Here the case of semi-splittings of word-hyperbolic groups over two-ended (virtually infinite cyclic) subgroups was treated by Scott-Swarup[23] and by Bowditch.[24] The case of semi-splittings of finitely generated groups with respect to virtually polycyclic subgroups is dealt with by the algebraic torus theorem of Dunwoody-Swenson.[25]
• A number of new proofs of Stallings' theorem have been obtained by others after Stallings' original proof. Dunwoody gave a proof[26] based on the ideas of edge-cuts. Later Dunwoody also gave a proof of Stallings' theorem for finitely presented groups using the method of "tracks" on finite 2-complexes.[7] Niblo obtained a proof[27] of Stallings' theorem as a consequence of Sageev's CAT(0)-cubing relative version, where the CAT(0)-cubing is eventually promoted to being a tree. Niblo's paper also defines an abstract group-theoretic obstruction (which is a union of double cosets of $H$ in $G$) for obtaining an actual splitting from a semi-splitting. It is also possible to prove Stallings' theorem for finitely presented groups using Riemannian geometry techniques of minimal surfaces, where one first realizes a finitely presented group as the fundamental group of a compact $4$-manifold (see, for example, a sketch of this argument in the survey article of Wall[28]). Gromov outlined a proof (see pp. 228–230 in [16]) where the minimal surfaces argument is replaced by an easier harmonic analysis argument and this approach was pushed further by Kapovich to cover the original case of finitely generated groups.[15][29]
See also
• Free product with amalgamation
• HNN extension
• Bass–Serre theory
• Graph of groups
• Geometric group theory
Notes
1. John R. Stallings. On torsion-free groups with infinitely many ends. Annals of Mathematics (2), vol. 88 (1968), pp. 312–334
2. John Stallings. Group theory and three-dimensional manifolds. A James K. Whittemore Lecture in Mathematics given at Yale University, 1969. Yale Mathematical Monographs, 4. Yale University Press, New Haven, Conn.-London, 1971.
3. H. Freudenthal. Über die Enden diskreter Räume und Gruppen. Comment. Math. Helv. 17, (1945). 1-38.
4. H. Hopf. Enden offener Räume und unendliche diskontinuierliche Gruppen. Comment. Math. Helv. 16, (1944). 81-100
5. Lemma 4.1 in C. T. C. Wall, Poincaré Complexes: I. Annals of Mathematics, Second Series, Vol. 86, No. 2 (Sep., 1967), pp. 213-245
6. John R. Stallings. Groups of dimension 1 are locally free. Bulletin of the American Mathematical Society, vol. 74 (1968), pp. 361–364
7. M. J. Dunwoody. The accessibility of finitely presented groups. Inventiones Mathematicae, vol. 81 (1985), no. 3, pp. 449-457
8. M. J. Dunwoody. An inaccessible group. Geometric group theory, Vol. 1 (Sussex, 1991), pp. 75–78, London Mathematical Society Lecture Note Series, vol. 181, Cambridge University Press, Cambridge, 1993; ISBN 0-521-43529-3
9. P. A. Linnell. On accessibility of groups. Journal of Pure and Applied Algebra, vol. 30 (1983), no. 1, pp. 39–46.
10. M. Bestvina and M. Feighn. Bounding the complexity of simplicial group actions on trees. Inventiones Mathematicae, vol. 103 (1991), no. 3, pp. 449–469
11. Z. Sela. Acylindrical accessibility for groups. Inventiones Mathematicae, vol. 129 (1997), no. 3, pp. 527–565
12. T. Delzant. Sur l'accessibilité acylindrique des groupes de présentation finie. Archived 2011-06-05 at the Wayback Machine Université de Grenoble. Annales de l'Institut Fourier, vol. 49 (1999), no. 4, pp. 1215–1224
13. T. Delzant, and L. Potyagailo. Accessibilité hiérarchique des groupes de présentation finie. Topology, vol. 40 (2001), no. 3, pp. 617–629
14. H. Bass. Covering theory for graphs of groups. Journal of Pure and Applied Algebra, vol. 89 (1993), no. 1-2, pp. 3–47
15. Gentimis Thanos, Asymptotic dimension of finitely presented groups, http://www.ams.org/journals/proc/2008-136-12/S0002-9939-08-08973-9/home.html
16. M. Gromov, Hyperbolic Groups, in "Essays in Group Theory" (G. M. Gersten, ed.), MSRI Publ. 8, 1987, pp. 75-263
17. Peter Scott. Ends of pairs of groups. Journal of Pure and Applied Algebra, vol. 11 (1977/78), no. 1–3, pp. 179–198
18. G. A. Swarup. Relative version of a theorem of Stallings. Journal of Pure and Applied Algebra, vol. 11 (1977/78), no. 1–3, pp. 75–82
19. H. Müller. Decomposition theorems for group pairs. Mathematische Zeitschrift, vol. 176 (1981), no. 2, pp. 223–246
20. P. H. Kropholler, and M. A. Roller. Relative ends and duality groups. Journal of Pure and Applied Algebra, vol. 61 (1989), no. 2, pp. 197–210
21. Michah Sageev. Ends of group pairs and non-positively curved cube complexes. Proceedings of the London Mathematical Society (3), vol. 71 (1995), no. 3, pp. 585–617
22. V. N. Gerasimov. Semi-splittings of groups and actions on cubings. (in Russian) Algebra, geometry, analysis and mathematical physics (Novosibirsk, 1996), pp. 91–109, 190, Izdat. Ross. Akad. Nauk Sib. Otd. Inst. Mat., Novosibirsk, 1997
23. G. P. Scott, and G. A. Swarup. An algebraic annulus theorem. Archived 2007-07-15 at the Wayback Machine Pacific Journal of Mathematics, vol. 196 (2000), no. 2, pp. 461–506
24. B. H. Bowditch. Cut points and canonical splittings of hyperbolic groups. Acta Mathematica, vol. 180 (1998), no. 2, pp. 145–186
25. M. J. Dunwoody, and E. L. Swenson. The algebraic torus theorem. Inventiones Mathematicae, vol. 140 (2000), no. 3, pp. 605–637
26. M. J. Dunwoody. Cutting up graphs. Combinatorica, vol. 2 (1982), no. 1, pp. 15–23
27. Graham A. Niblo. A geometric proof of Stallings' theorem on groups with more than one end. Geometriae Dedicata, vol. 105 (2004), pp. 61–76
28. C. T. C. Wall. The geometry of abstract groups and their splittings. Revista Matemática Complutense vol. 16(2003), no. 1, pp. 5–101
29. M. Kapovich. Energy of harmonic functions and Gromov's proof of Stallings' theorem, preprint, 2007, arXiv:0707.4231
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Wikipedia
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Coupon collector's problem
In probability theory, the coupon collector's problem describes "collect all coupons and win" contests. It asks the following question: If each box of a brand of cereals contains a coupon, and there are n different types of coupons, what is the probability that more than t boxes need to be bought to collect all n coupons? An alternative statement is: Given n coupons, how many coupons do you expect you need to draw with replacement before having drawn each coupon at least once? The mathematical analysis of the problem reveals that the expected number of trials needed grows as $\Theta (n\log(n))$.[lower-alpha 1] For example, when n = 50 it takes about 225[lower-alpha 2] trials on average to collect all 50 coupons.
Solution
Calculating the expectation
Let time T be the number of draws needed to collect all n coupons, and let ti be the time to collect the i-th coupon after i − 1 coupons have been collected. Then $T=t_{1}+\cdots +t_{n}$. Think of T and ti as random variables. Observe that the probability of collecting a new coupon is $p_{i}={\frac {n-(i-1)}{n}}={\frac {n-i+1}{n}}$. Therefore, $t_{i}$ has geometric distribution with expectation ${\frac {1}{p_{i}}}={\frac {n}{n-i+1}}$. By the linearity of expectations we have:
${\begin{aligned}\operatorname {E} (T)&{}=\operatorname {E} (t_{1}+t_{2}+\cdots +t_{n})\\&{}=\operatorname {E} (t_{1})+\operatorname {E} (t_{2})+\cdots +\operatorname {E} (t_{n})\\&{}={\frac {1}{p_{1}}}+{\frac {1}{p_{2}}}+\cdots +{\frac {1}{p_{n}}}\\&{}={\frac {n}{n}}+{\frac {n}{n-1}}+\cdots +{\frac {n}{1}}\\&{}=n\cdot \left({\frac {1}{1}}+{\frac {1}{2}}+\cdots +{\frac {1}{n}}\right)\\&{}=n\cdot H_{n}.\end{aligned}}$
Here Hn is the n-th harmonic number. Using the asymptotics of the harmonic numbers, we obtain:
$\operatorname {E} (T)=n\cdot H_{n}=n\log n+\gamma n+{\frac {1}{2}}+O(1/n),$
where $\gamma \approx 0.5772156649$ is the Euler–Mascheroni constant.
Using the Markov inequality to bound the desired probability:
$\operatorname {P} (T\geq cnH_{n})\leq {\frac {1}{c}}.$
The above can be modified slightly to handle the case when we've already collected some of the coupons. Let k be the number of coupons already collected, then:
${\begin{aligned}\operatorname {E} (T_{k})&{}=\operatorname {E} (t_{k+1}+t_{k+2}+\cdots +t_{n})\\&{}=n\cdot \left({\frac {1}{1}}+{\frac {1}{2}}+\cdots +{\frac {1}{n-k}}\right)\\&{}=n\cdot H_{n-k}\end{aligned}}$
And when $k=0$ then we get the original result.
Calculating the variance
Using the independence of random variables ti, we obtain:
${\begin{aligned}\operatorname {Var} (T)&{}=\operatorname {Var} (t_{1}+\cdots +t_{n})\\&{}=\operatorname {Var} (t_{1})+\operatorname {Var} (t_{2})+\cdots +\operatorname {Var} (t_{n})\\&{}={\frac {1-p_{1}}{p_{1}^{2}}}+{\frac {1-p_{2}}{p_{2}^{2}}}+\cdots +{\frac {1-p_{n}}{p_{n}^{2}}}\\&{}<\left({\frac {n^{2}}{n^{2}}}+{\frac {n^{2}}{(n-1)^{2}}}+\cdots +{\frac {n^{2}}{1^{2}}}\right)\\&{}=n^{2}\cdot \left({\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+\cdots +{\frac {1}{n^{2}}}\right)\\&{}<{\frac {\pi ^{2}}{6}}n^{2}\end{aligned}}$
since ${\frac {\pi ^{2}}{6}}={\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+\cdots +{\frac {1}{n^{2}}}+\cdots $ (see Basel problem).
Bound the desired probability using the Chebyshev inequality:
$\operatorname {P} \left(|T-nH_{n}|\geq cn\right)\leq {\frac {\pi ^{2}}{6c^{2}}}.$
Tail estimates
A stronger tail estimate for the upper tail be obtained as follows. Let ${Z}_{i}^{r}$ denote the event that the $i$-th coupon was not picked in the first $r$ trials. Then
${\begin{aligned}P\left[{Z}_{i}^{r}\right]=\left(1-{\frac {1}{n}}\right)^{r}\leq e^{-r/n}.\end{aligned}}$
Thus, for $r=\beta n\log n$, we have $P\left[{Z}_{i}^{r}\right]\leq e^{(-\beta n\log n)/n}=n^{-\beta }$. Via a union bound over the $n$ coupons, we obtain
${\begin{aligned}P\left[T>\beta n\log n\right]=P\left[\bigcup _{i}{Z}_{i}^{\beta n\log n}\right]\leq n\cdot P[{Z}_{1}^{\beta n\log n}]\leq n^{-\beta +1}.\end{aligned}}$
Extensions and generalizations
• Pierre-Simon Laplace, but also Paul Erdős and Alfréd Rényi, proved the limit theorem for the distribution of T. This result is a further extension of previous bounds. A proof is found in.[1]
$\operatorname {P} (T<n\log n+cn)\to e^{-e^{-c}},{\text{ as }}n\to \infty .$
• Donald J. Newman and Lawrence Shepp gave a generalization of the coupon collector's problem when m copies of each coupon need to be collected. Let Tm be the first time m copies of each coupon are collected. They showed that the expectation in this case satisfies:
$\operatorname {E} (T_{m})=n\log n+(m-1)n\log \log n+O(n),{\text{ as }}n\to \infty .$
Here m is fixed. When m = 1 we get the earlier formula for the expectation.
• Common generalization, also due to Erdős and Rényi:
$\operatorname {P} \left(T_{m}<n\log n+(m-1)n\log \log n+cn\right)\to e^{-e^{-c}/(m-1)!},{\text{ as }}n\to \infty .$
• In the general case of a nonuniform probability distribution, according to Philippe Flajolet et al.[2]
$\operatorname {E} (T)=\int _{0}^{\infty }\left(1-\prod _{i=1}^{m}\left(1-e^{-p_{i}t}\right)\right)dt.$
This is equal to
$\operatorname {E} (T)=\sum _{q=0}^{m-1}(-1)^{m-1-q}\sum _{|J|=q}{\frac {1}{1-P_{J}}},$
where m denotes the number of coupons to be collected and PJ denotes the probability of getting any coupon in the set of coupons J.
See also
• McDonald's Monopoly – an example of the coupon collector's problem that further increases the challenge by making some coupons of the set rarer
• Watterson estimator
• Birthday problem
Notes
1. Here and throughout this article, "log" refers to the natural logarithm rather than a logarithm to some other base. The use of Θ here invokes big O notation.
2. E(50) = 50(1 + 1/2 + 1/3 + ... + 1/50) = 224.9603, the expected number of trials to collect all 50 coupons. The approximation $n\log n+\gamma n+1/2$ for this expected number gives in this case $50\log 50+50\gamma +1/2\approx 195.6011+28.8608+0.5\approx 224.9619$.
References
1. Mitzenmacher, Michael (2017). Probability and computing : randomization and probabilistic techniques in algorithms and data analysis. Eli Upfal (2nd ed.). Cambridge, United Kingdom. Theorem 5.13. ISBN 978-1-107-15488-9. OCLC 960841613.{{cite book}}: CS1 maint: location missing publisher (link)
2. Flajolet, Philippe; Gardy, Danièle; Thimonier, Loÿs (1992), "Birthday paradox, coupon collectors, caching algorithms and self-organizing search", Discrete Applied Mathematics, 39 (3): 207–229, CiteSeerX 10.1.1.217.5965, doi:10.1016/0166-218x(92)90177-c
• Blom, Gunnar; Holst, Lars; Sandell, Dennis (1994), "7.5 Coupon collecting I, 7.6 Coupon collecting II, and 15.4 Coupon collecting III", Problems and Snapshots from the World of Probability, New York: Springer-Verlag, pp. 85–87, 191, ISBN 0-387-94161-4, MR 1265713.
• Dawkins, Brian (1991), "Siobhan's problem: the coupon collector revisited", The American Statistician, 45 (1): 76–82, doi:10.2307/2685247, JSTOR 2685247.
• Erdős, Paul; Rényi, Alfréd (1961), "On a classical problem of probability theory" (PDF), Magyar Tudományos Akadémia Matematikai Kutató Intézetének Közleményei, 6: 215–220, MR 0150807.
• Laplace, Pierre-Simon (1812), Théorie analytique des probabilités, pp. 194–195.
• Newman, Donald J.; Shepp, Lawrence (1960), "The double dixie cup problem", American Mathematical Monthly, 67 (1): 58–61, doi:10.2307/2308930, JSTOR 2308930, MR 0120672
• Flajolet, Philippe; Gardy, Danièle; Thimonier, Loÿs (1992), "Birthday paradox, coupon collectors, caching algorithms and self-organizing search", Discrete Applied Mathematics, 39 (3): 207–229, doi:10.1016/0166-218X(92)90177-C, MR 1189469.
• Isaac, Richard (1995), "8.4 The coupon collector's problem solved", The Pleasures of Probability, Undergraduate Texts in Mathematics, New York: Springer-Verlag, pp. 80–82, ISBN 0-387-94415-X, MR 1329545.
• Motwani, Rajeev; Raghavan, Prabhakar (1995), "3.6. The Coupon Collector's Problem", Randomized algorithms, Cambridge: Cambridge University Press, pp. 57–63, ISBN 9780521474658, MR 1344451.
External links
• "Coupon Collector Problem" by Ed Pegg, Jr., the Wolfram Demonstrations Project. Mathematica package.
• How Many Singles, Doubles, Triples, Etc., Should The Coupon Collector Expect?, a short note by Doron Zeilberger.
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Wikipedia
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Map folding
In the mathematics of paper folding, map folding and stamp folding are two problems of counting the number of ways that a piece of paper can be folded. In the stamp folding problem, the paper is a strip of stamps with creases between them, and the folds must lie on the creases. In the map folding problem, the paper is a map, divided by creases into rectangles, and the folds must again lie only along these creases.
Lucas (1891) credits the invention of the stamp folding problem to Émile Lemoine.[1] Touchard (1950) provides several other early references.[2]
Labeled stamps
In the stamp folding problem, the paper to be folded is a strip of square or rectangular stamps, separated by creases, and the stamps can only be folded along those creases. In one commonly considered version of the problem, each stamp is considered to be distinguishable from each other stamp, so two foldings of a strip of stamps are considered equivalent only when they have the same vertical sequence of stamps.[3] For example, there are six ways to fold a strip of three different stamps:
These include all six permutations of the stamps, but for more than three stamps not all permutations are possible. If, for a permutation p, there are two numbers i and j with the same parity such that the four numbers i, j, i + 1, and j + 1 appear in p in that cyclic order, then p cannot be folded. The parity condition implies that the creases between stamps i and i + 1, and between stamps j and j + 1, appear on the same side of the stack of folded stamps, but the cyclic ordering condition implies that these two creases cross each other, a physical impossibility. For instance, the four-element permutation 1324 cannot be folded, because it has this forbidden pattern with i = 1 and j = 3. All remaining permutations, without this pattern, can be folded.[3] The number of different ways to fold a strip of n stamps is given by the sequence
1, 2, 6, 16, 50, 144, 462, 1392, 4536, 14060, 46310, 146376, 485914, 1557892, 5202690, ... (sequence A000136 in the OEIS).
These numbers are always divisible by n (because a cyclic permutation of a foldable stamp sequence is always itself foldable),[3][4] and the quotients of this division are
1, 1, 2, 4, 10, 24, 66, 174, 504, 1406, 4210, 12198, 37378, 111278, 346846, 1053874, ... (sequence A000682 in the OEIS),
the number of topologically distinct ways that a half-infinite curve can make n crossings with a line, called "semimeanders". These are closely related to meanders, ways for a closed curve to make the same number of crossings with a line. Meanders correspond to solutions of the stamp-folding problem in which the first and last stamp can be connected to each other to form a continuous loop of stamps.
Unsolved problem in mathematics:
Is there a formula or polynomial-time algorithm for counting solutions to the stamp-folding problem?
(more unsolved problems in mathematics)
In the 1960s, John E. Koehler and W. F. Lunnon implemented algorithms that, at that time, could calculate these numbers for up to 28 stamps.[5][6][7] Despite additional research, the known methods for calculating these numbers take exponential time as a function of n.[8][9] Thus, there is no formula or efficient algorithm known that could extend this sequence to very large values of n. Nevertheless, heuristic methods from physics can be used to predict the rate of exponential growth of this sequence.[10]
The stamp folding problem usually considers only the number of possible folded states of the strip of stamps, without considering whether it is possible to physically construct the fold by a sequence of moves starting from an unfolded strip of stamps. However, according to the solution of the carpenter's rule problem, every folded state can be constructed (or equivalently, can be unfolded).[11]
Unlabeled stamps
In another variation of the stamp folding problem, the strip of stamps is considered to be blank, so that it is not possible to tell one of its ends from the other, and two foldings are considered distinct only when they have different shapes. Turning a folded strip upside-down or back-to-front is not considered to change its shape, so three stamps have only two foldings, an S-curve and a spiral.[3] More generally, the numbers of foldings with this definition are
1, 1, 2, 5, 14, 38, 120, 353, 1148, 3527, 11622, 36627, 121622, 389560, 1301140, 4215748, ... (sequence A001011 in the OEIS)
Maps
Map folding is the question of how many ways there are to fold a rectangular map along its creases, allowing each crease to form either a mountain or a valley fold. It differs from stamp folding in that it includes both vertical and horizontal creases, rather than only creases in a single direction.[12]
There are eight ways to fold a 2 × 2 map along its creases, counting each different vertical sequence of folded squares as a distinct way of folding the map:[5]
However, the general problem of counting the number of ways to fold a map remains unsolved. The numbers of ways of folding an n × n map are known only for n ≤ 7. They are:
1, 8, 1368, 300608, 186086600, 123912532224, 129950723279272 (sequence A001418 in the OEIS).
Complexity
Unsolved problem in mathematics:
Given a mountain-valley assignment for the creases of a map, is it possible to test efficiently whether it can be folded flat?
(more unsolved problems in mathematics)
The map folding and stamp folding problems are related to a problem in the mathematics of origami of whether a square with a crease pattern can be folded to a flat figure. If a folding direction (either a mountain fold or a valley fold) is assigned to each crease of a strip of stamps, it is possible to test whether the result can be folded flat in polynomial time.[13]
For the same problem on a map (divided into rectangles by creases with assigned directions) it is unknown whether a polynomial time folding algorithm exists in general, although a polynomial algorithm is known for 2 × n maps.[14] In a restricted case where the map is to be folded by a sequence of "simple" folds, which fold the paper along a single line, the problem is polynomial. Some extensions of the problem, for instance to non-rectangular sheets of paper, are NP-complete.[13]
Even for a one-dimensional strip of stamps, with its creases already labeled as mountain or valley folds, it is NP-hard to find a way to fold it that minimizes the maximum number of stamps that lie between the two stamps of any crease.[15]
See also
• Regular paperfolding sequence, an infinite sequence of 0s and 1s that describes one way of folding strips of stamps
References
1. Lucas, Édouard (1891), Théorie des nombres (in French), vol. I, Paris: Gauthier-Villars, p. 120.
2. Touchard, Jacques (1950), "Contribution à l'étude du problème des timbres poste", Canadian Journal of Mathematics (in French), 2: 385–398, doi:10.4153/CJM-1950-035-6, MR 0037815, S2CID 124708270.
3. Legendre, Stéphane (2014), "Foldings and meanders", The Australasian Journal of Combinatorics, 58: 275–291, arXiv:1302.2025, Bibcode:2013arXiv1302.2025L, MR 3211783
4. Sainte-Laguë, André (1937), Avec des nombres et des lignes (in French), Paris: Vuibert, pp. 147–162. As cited by Legendre (2014)
5. Gardner, Martin (1983), "The combinatorics of paper folding", Wheels, Life and Other Mathematical Amusements, New York: W. H. Freeman, pp. 60–73, Bibcode:1983wlom.book.....G. See in particular pp. 60–62.
6. Koehler, John E. (1968), "Folding a strip of stamps", Journal of Combinatorial Theory, 5 (2): 135–152, doi:10.1016/S0021-9800(68)80048-1, MR 0228364
7. Lunnon, W. F. (1968), "A map-folding problem", Mathematics of Computation, 22 (101): 193–199, doi:10.2307/2004779, JSTOR 2004779, MR 0221957
8. Jensen, Iwan (2000), "A transfer matrix approach to the enumeration of plane meanders", Journal of Physics A: Mathematical and General, 33 (34): 5953, arXiv:cond-mat/0008178, Bibcode:2000JPhA...33.5953J, doi:10.1088/0305-4470/33/34/301, S2CID 14259684
9. Sawada, Joe; Li, Roy (2012), "Stamp foldings, semi-meanders, and open meanders: fast generation algorithms", Electronic Journal of Combinatorics, 19 (2): Paper 43, 16pp, doi:10.37236/2404, MR 2946101
10. Di Francesco, P. (2000), "Exact asymptotics of meander numbers", Formal power series and algebraic combinatorics (Moscow, 2000), Springer, Berlin, pp. 3–14, doi:10.1007/978-3-662-04166-6_1, ISBN 978-3-642-08662-5, MR 1798197
11. Connelly, Robert; Demaine, Erik D.; Rote, Günter (2003), "Straightening polygonal arcs and convexifying polygonal cycles" (PDF), Discrete and Computational Geometry, 30 (2): 205–239, doi:10.1007/s00454-003-0006-7, MR 1931840
12. Lunnon, W. F. (1971), "Multi-dimensional map-folding", The Computer Journal, 14: 75–80, doi:10.1093/comjnl/14.1.75, MR 0285409
13. Arkin, Esther M.; Bender, Michael A.; Demaine, Erik D.; Demaine, Martin L.; Mitchell, Joseph S. B.; Sethia, Saurabh; Skiena, Steven S. (September 2004), "When can you fold a map?" (PDF), Computational Geometry: Theory and Applications, 29 (1): 23–46, doi:10.1016/j.comgeo.2004.03.012.
14. Morgan, Thomas D. (May 21, 2012), Map folding (Thesis), Master's thesis, Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, hdl:1721.1/77030
15. Umesato, Takuya; Saitoh, Toshiki; Uehara, Ryuhei; Ito, Hiro; Okamoto, Yoshio (2013), "The complexity of the stamp folding problem", Theoretical Computer Science, 497: 13–19, doi:10.1016/j.tcs.2012.08.006, MR 3084129
External links
• Eric W. Weisstein, Map Folding (Stamp Folding) at MathWorld.
• "Folding a Strip of Labeled Stamps" from The Wolfram Demonstrations Project: http://demonstrations.wolfram.com/FoldingAStripOfLabeledStamps/
Mathematics of paper folding
Flat folding
• Big-little-big lemma
• Crease pattern
• Huzita–Hatori axioms
• Kawasaki's theorem
• Maekawa's theorem
• Map folding
• Napkin folding problem
• Pureland origami
• Yoshizawa–Randlett system
Strip folding
• Dragon curve
• Flexagon
• Möbius strip
• Regular paperfolding sequence
3d structures
• Miura fold
• Modular origami
• Paper bag problem
• Rigid origami
• Schwarz lantern
• Sonobe
• Yoshimura buckling
Polyhedra
• Alexandrov's uniqueness theorem
• Blooming
• Flexible polyhedron (Bricard octahedron, Steffen's polyhedron)
• Net
• Source unfolding
• Star unfolding
Miscellaneous
• Fold-and-cut theorem
• Lill's method
Publications
• Geometric Exercises in Paper Folding
• Geometric Folding Algorithms
• Geometric Origami
• A History of Folding in Mathematics
• Origami Polyhedra Design
• Origamics
People
• Roger C. Alperin
• Margherita Piazzola Beloch
• Robert Connelly
• Erik Demaine
• Martin Demaine
• Rona Gurkewitz
• David A. Huffman
• Tom Hull
• Kôdi Husimi
• Humiaki Huzita
• Toshikazu Kawasaki
• Robert J. Lang
• Anna Lubiw
• Jun Maekawa
• Kōryō Miura
• Joseph O'Rourke
• Tomohiro Tachi
• Eve Torrence
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Stanley Osher
Stanley Osher (born April 24, 1942) is an American mathematician, known for his many contributions in shock capturing, level-set methods, and PDE-based methods in computer vision and image processing. Osher is a professor at the University of California, Los Angeles (UCLA), Director of Special Projects in the Institute for Pure and Applied Mathematics (IPAM) and member of the California NanoSystems Institute (CNSI) at UCLA.
Stanley Joel Osher
Osher in 1968
Born (1942-04-24) April 24, 1942
Brooklyn, New York, U.S.
Known for
• Level-set method
• Shock-capturing methods
• image processing
• L1/TV methods
• Bregman method
Scientific career
FieldsApplied mathematics
Institutions
• UCLA
• SUNY, Stony Brook
• UC Berkeley
Doctoral advisorJacob Schwartz
Doctoral students
• Rosa Donat
• Ron Fedkiw
• Chiu-Yen Kao
• Chi-Wang Shu
He has a daughter, Kathryn, and a son, Joel.
Education
• BS, Brooklyn College, 1962
• MS, New York University, 1964
• PhD, New York University, 1966
Research interests
• Level-set methods for computing moving fronts
• Approximation methods for hyperbolic conservation laws and Hamilton–Jacobi equations
• Total variation (TV) and other PDE-based image processing techniques
• Scientific computing
• Applied partial differential equations
• L1/TV-based convex optimization
Osher is listed as an ISI highly cited researcher.[1]
Research contributions
Osher was the inventor (or co-inventor) and developer of many highly successful numerical methods for computational physics, image processing and other fields, including:
• High resolution numerical schemes to compute flows having shocks and steep gradients, including ENO (essentially non-oscillatory) schemes (with Harten, Chakravarthy, Engquist, Shu), WENO (weighted ENO) schemes (with Liu and Chan), the Osher scheme, the Engquist-Osher scheme, and the Hamilton–Jacobi versions of these methods. These methods have been widely used in computational fluid dynamics (CFD) and related fields.
• Total variation (TV)-based image restoration (with Rudin and Fatemi) and shock filters (with Rudin). These are pioneering - and widely used - methods for PDE based image processing and have also been used for inverse problems.
• Level-set method (with Sethian) for capturing moving interfaces, which has been phenomenally successful as a key tool in PDE based image processing and computer vision, as well as applications in differential geometry, image segmentation, inverse problems, optimal design, Two-phase flow, crystal growth, deposition and etching.
• Bregman iteration and augmented Lagrangian type methods for L1 and L1-related optimization problems which are fundamental to the fields of compressed sensing, matrix completion, robust principal component analysis, etc.
• Overcoming the curse of dimensionality for Hamilton–Jacobi equations arising in control theory and differential games.
Osher has founded (or co-founded) three successful companies:
• Cognitech (with Rudin)
• Level Set Systems
• Luminescent Technologies (with Yablonovitch)
Osher has been a thesis advisor for at least 53 PhD students, with 188 descendants, as well as postdoctoral adviser and collaborator for many applied mathematicians. His PhD students have been evenly distributed among academia and industry and labs, most of them are involved in applying mathematical and computational tools to industrial or scientific application areas.
Honors
• National Academy of Engineering (NAE), 2018
• William Benter Prize in Applied Mathematics, 2016.[2]
• Carl Friedrich Gauss Prize, 2014.
• John von Neumann Lecture prize from SIAM, 2013.[3]
• Fellow of the American Mathematical Society, 2013.[4]
• Plenary speaker, International Congress of Mathematicians, 2010[5]
• American Academy of Arts and Sciences, 2009
• Fellow, Society for Industrial and Applied Mathematics (SIAM), 2009 [6]
• Honorary Doctoral Degree, Hong Kong Baptist University, 2009
• International Cooperation Award, International Congress of Chinese Mathematicians, 2007
• Computational and Applied Sciences Award, United States Association for Computational Mechanics, 2007
• Docteur Honoris Causa, ENS Cachan, France 2006
• National Academy of Sciences (NAS), 2005
• SIAM Kleinman Prize, 2005[7]
• ICIAM Pioneer Prize, 2003[5]
• Computational Mechanics Award, Japan Society of Mechanical Engineering, 2002
• NASA Public Service Group Achievement Award, 1992
• US-Israel BSF Fellow, 1986
• SERC Fellowship (England), 1982
• Alfred P. Sloan Fellow, 1972–1974[8]
• Fulbright Fellow, 1971[9]
Books authored
• Osher, Stanley (2003). Level set methods and dynamic implicit surfaces. New York: Springer. ISBN 978-0-387-22746-7. OCLC 53224633.
• Osher, Stanley (2003). Geometric level set methods in imaging, vision, and graphics. New York: Springer. ISBN 978-0-387-21810-6. OCLC 56066930.
• Glowinski, R (2016). Splitting methods in communication, imaging, science, and engineering. Cham, Switzerland: Springer. ISBN 978-3-319-41589-5. OCLC 967938355.
See also
• James Sethian, co-developer of level-set methods.
References
1. Thomson ISI, Osher, Stanley, ISI Highly Cited Researchers, archived from the original on May 18, 2006, retrieved June 20, 2009
2. "William Benter Prize in Applied Mathematics". www.cityu.edu.hk. Retrieved December 17, 2021.
3. "John von Neumann Prize". SIAM. Retrieved December 17, 2021.
4. List of Fellows of the American Mathematical Society, retrieved March 20, 2013.
5. "ICM Plenary and Invited Speakers since 1897". International Congress of Mathematicians.
6. "Fellows Program | SIAM". www.siam.org. Retrieved December 17, 2021.
7. "Ralph E. Kleinman Prize". SIAM. Retrieved December 17, 2021.
8. "Sloan Research Fellowships | Alfred P. Sloan Foundation". sloan.org. Retrieved December 17, 2021.
9. "Fulbright Scholar Directories". www.cies.org. Archived from the original on March 5, 2005.
External links
• Osher's home page at UCLA
• Stanley Osher at the Mathematics Genealogy Project
• Stanley Osher publications indexed by Google Scholar
John von Neumann Lecturers
• Lars Ahlfors (1960)
• Mark Kac (1961)
• Jean Leray (1962)
• Stanislaw Ulam (1963)
• Solomon Lefschetz (1964)
• Freeman Dyson (1965)
• Eugene Wigner (1966)
• Chia-Chiao Lin (1967)
• Peter Lax (1968)
• George F. Carrier (1969)
• James H. Wilkinson (1970)
• Paul Samuelson (1971)
• Jule Charney (1974)
• James Lighthill (1975)
• René Thom (1976)
• Kenneth Arrow (1977)
• Peter Henrici (1978)
• Kurt O. Friedrichs (1979)
• Keith Stewartson (1980)
• Garrett Birkhoff (1981)
• David Slepian (1982)
• Joseph B. Keller (1983)
• Jürgen Moser (1984)
• John W. Tukey (1985)
• Jacques-Louis Lions (1986)
• Richard M. Karp (1987)
• Germund Dahlquist (1988)
• Stephen Smale (1989)
• Andrew Majda (1990)
• R. Tyrrell Rockafellar (1992)
• Martin D. Kruskal (1994)
• Carl de Boor (1996)
• William Kahan (1997)
• Olga Ladyzhenskaya (1998)
• Charles S. Peskin (1999)
• Persi Diaconis (2000)
• David Donoho (2001)
• Eric Lander (2002)
• Heinz-Otto Kreiss (2003)
• Alan C. Newell (2004)
• Jerrold E. Marsden (2005)
• George C. Papanicolaou (2006)
• Nancy Kopell (2007)
• David Gottlieb (2008)
• Franco Brezzi (2009)
• Bernd Sturmfels (2010)
• Ingrid Daubechies (2011)
• John M. Ball (2012)
• Stanley Osher (2013)
• Leslie Greengard (2014)
• Jennifer Tour Chayes (2015)
• Donald Knuth (2016)
• Bernard J. Matkowsky (2017)
• Charles F. Van Loan (2018)
• Margaret H. Wright (2019)
• Nick Trefethen (2020)
• Chi-Wang Shu (2021)
• Leah Keshet (2022)
Authority control
International
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National
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• BnF data
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Wikipedia
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Stanisław Ulam
Stanisław Marcin Ulam ([sta'ɲiswaf 'mart͡ɕin 'ulam]; 13 April 1909 – 13 May 1984) was a Polish-American mathematician and nuclear physicist. He participated in the Manhattan Project, originated the Teller–Ulam design of thermonuclear weapons, discovered the concept of the cellular automaton, invented the Monte Carlo method of computation, and suggested nuclear pulse propulsion. In pure and applied mathematics, he proved some theorems and proposed several conjectures.
Stanisław Ulam
Ulam at Los Alamos
Born
Stanisław Marcin Ulam
(1909-04-13)13 April 1909
Lemberg, Austria-Hungary
(now Lviv, Ukraine)
Died13 May 1984(1984-05-13) (aged 75)
Santa Fe, New Mexico, U.S.
CitizenshipPoland, United States (naturalized in 1941)
EducationLwów Polytechnic Institute, Second Polish Republic
Known forMathematical formulations in the fields of Physics, Computer Science, and Biology
Teller–Ulam design
Monte Carlo method
Fermi–Pasta–Ulam–Tsingou problem
Nuclear pulse propulsion
Scientific career
FieldsMathematics
InstitutionsInstitute for Advanced Study
Harvard University
University of Wisconsin
Los Alamos National Laboratory
University of Colorado
University of Florida
Doctoral advisorKazimierz Kuratowski
Włodzimierz Stożek
Doctoral studentsPaul Kelly
Born into a wealthy Polish Jewish family, Ulam studied mathematics at the Lwów Polytechnic Institute, where he earned his PhD in 1933 under the supervision of Kazimierz Kuratowski and Włodzimierz Stożek.[1] In 1935, John von Neumann, whom Ulam had met in Warsaw, invited him to come to the Institute for Advanced Study in Princeton, New Jersey, for a few months. From 1936 to 1939, he spent summers in Poland and academic years at Harvard University in Cambridge, Massachusetts, where he worked to establish important results regarding ergodic theory. On 20 August 1939, he sailed for the United States for the last time with his 17-year-old brother Adam Ulam. He became an assistant professor at the University of Wisconsin–Madison in 1940, and a United States citizen in 1941.
In October 1943, he received an invitation from Hans Bethe to join the Manhattan Project at the secret Los Alamos Laboratory in New Mexico. There, he worked on the hydrodynamic calculations to predict the behavior of the explosive lenses that were needed by an implosion-type weapon. He was assigned to Edward Teller's group, where he worked on Teller's "Super" bomb for Teller and Enrico Fermi. After the war he left to become an associate professor at the University of Southern California, but returned to Los Alamos in 1946 to work on thermonuclear weapons. With the aid of a cadre of female "computers" he found that Teller's "Super" design was unworkable. In January 1951, Ulam and Teller came up with the Teller–Ulam design, which became the basis for all thermonuclear weapons.
Ulam considered the problem of nuclear propulsion of rockets, which was pursued by Project Rover, and proposed, as an alternative to Rover's nuclear thermal rocket, to harness small nuclear explosions for propulsion, which became Project Orion. With Fermi, John Pasta, and Mary Tsingou, Ulam studied the Fermi–Pasta–Ulam–Tsingou problem, which became the inspiration for the field of non-linear science. He is probably best known for realising that electronic computers made it practical to apply statistical methods to functions without known solutions, and as computers have developed, the Monte Carlo method has become a common and standard approach to many problems.
Poland
Ulam was born in Lemberg, Galicia, on 13 April 1909.[2][3][4] At this time, Galicia was in the Kingdom of Galicia and Lodomeria of the Austro-Hungarian Empire, which was known to Poles as the Austrian partition. In 1918, it became part of the newly restored Poland, the Second Polish Republic, and the city took its Polish name again, Lwów.[5]
The Ulams were a wealthy Polish Jewish family of bankers, industrialists, and other professionals. Ulam's immediate family was "well-to-do but hardly rich".[6] His father, Józef Ulam, was born in Lwów and was a lawyer,[5] and his mother, Anna (née Auerbach), was born in Stryj.[7] His uncle, Michał Ulam, was an architect, building contractor, and lumber industrialist.[8] From 1916 until 1918, Józef's family lived temporarily in Vienna.[9] After they returned, Lwów became the epicenter of the Polish–Ukrainian War, during which the city experienced a Ukrainian siege.[5]
In 1919, Ulam entered Lwów Gymnasium Nr. VII, from which he graduated in 1927.[10] He then studied mathematics at the Lwów Polytechnic Institute. Under the supervision of Kazimierz Kuratowski, he received his Master of Arts degree in 1932, and became a Doctor of Science in 1933.[9][11] At the age of 20, in 1929, he published his first paper Concerning Function of Sets in the journal Fundamenta Mathematicae.[11] From 1931 until 1935, he traveled to and studied in Wilno (Vilnius), Vienna, Zurich, Paris, and Cambridge, England, where he met G. H. Hardy and Subrahmanyan Chandrasekhar.[12]
Along with Stanisław Mazur, Mark Kac, Włodzimierz Stożek, Kuratowski, and others, Ulam was a member of the Lwów School of Mathematics. Its founders were Hugo Steinhaus and Stefan Banach, who were professors at the Jan Kazimierz University. Mathematicians of this "school" met for long hours at the Scottish Café, where the problems they discussed were collected in the Scottish Book, a thick notebook provided by Banach's wife. Ulam was a major contributor to the book. Of the 193 problems recorded between 1935 and 1941, he contributed 40 problems as a single author, another 11 with Banach and Mazur, and an additional 15 with others. In 1957, he received from Steinhaus a copy of the book, which had survived the war, and translated it into English.[13] In 1981, Ulam's friend R. Daniel Mauldin published an expanded and annotated version.[14]
Move to the United States
In 1935, John von Neumann, whom Ulam had met in Warsaw, invited him to come to the Institute for Advanced Study in Princeton, New Jersey, for a few months. In December of that year, Ulam sailed to the US. At Princeton, he went to lectures and seminars, where he heard Oswald Veblen, James Alexander, and Albert Einstein. During a tea party at von Neumann's house, he encountered G. D. Birkhoff, who suggested that he apply for a position with the Harvard Society of Fellows.[9] Following up on Birkhoff's suggestion, Ulam spent summers in Poland and academic years at Harvard University in Cambridge, Massachusetts from 1936 to 1939, where he worked with John C. Oxtoby to establish results regarding ergodic theory. These appeared in Annals of Mathematics in 1941.[10][15] In 1938, Stanislaw's mother Anna hanna Ulam (maiden name Auerbach) died of cancer.
On 20 August 1939, in Gdynia, Józef Ulam, along with his brother Szymon, put his two sons, Stanislaw and 17 year old Adam, on a ship headed for the US.[9] Eleven days later, the Germans invaded Poland. Within two months, the Germans completed their occupation of western Poland, and the Soviets invaded and occupied eastern Poland. Within two years, Józef Ulam and the rest of his family, including Stanislaw's sister Stefania Ulam, were victims of the Holocaust, Hugo Steinhaus was in hiding, Kazimierz Kuratowski was lecturing at the underground university in Warsaw, Włodzimierz Stożek and his two sons had been killed in the massacre of Lwów professors, and the last problem had been recorded in the Scottish Book. Stefan Banach survived the Nazi occupation by feeding lice at Rudolf Weigl's typhus research institute. In 1963, Adam Ulam, who had become an eminent kremlinologist at Harvard,[16] received a letter from George Volsky,[17] who hid in Józef Ulam's house after deserting from the Polish army. This reminiscence gave a chilling account of Lwów's chaotic scenes in late 1939.[18] In later life Ulam described himself as "an agnostic. Sometimes I muse deeply on the forces that are for me invisible. When I am almost close to the idea of God, I feel immediately estranged by the horrors of this world, which he seems to tolerate".[19]
In 1940, after being recommended by Birkhoff, Ulam became an assistant professor at the University of Wisconsin–Madison. Here, he became a United States citizen in 1941.[9] That year, he married Françoise Aron.[10] She had been a French exchange student at Mount Holyoke College, whom he met in Cambridge. They had one daughter, Claire. In Madison, Ulam met his friend and colleague C. J. Everett, with whom he collaborated on a number of papers.[20]
Manhattan Project
In early 1943, Ulam asked von Neumann to find him a war job. In October, he received an invitation to join an unidentified project near Santa Fe, New Mexico.[9] The letter was signed by Hans Bethe, who had been appointed as leader of the theoretical division of Los Alamos National Laboratory by Robert Oppenheimer, its scientific director.[21] Knowing nothing of the area, he borrowed a New Mexico guide book. On the checkout card, he found the names of his Wisconsin colleagues, Joan Hinton, David Frisch, and Joseph McKibben, all of whom had mysteriously disappeared.[9] This was Ulam's introduction to the Manhattan Project, which was the US's wartime effort to create the atomic bomb.[22]
Hydrodynamical calculations of implosion
A few weeks after Ulam reached Los Alamos in February 1944, the project experienced a crisis. In April, Emilio Segrè discovered that plutonium made in reactors would not work in a gun-type plutonium weapon like the "Thin Man", which was being developed in parallel with a uranium weapon, the "Little Boy" that was dropped on Hiroshima. This problem threatened to waste an enormous investment in new reactors at the Hanford site and to make slow uranium isotope separation the only way to prepare fissile material suitable for use in bombs. To respond, Oppenheimer implemented, in August, a sweeping reorganization of the laboratory to focus on development of an implosion-type weapon and appointed George Kistiakowsky head of the implosion department. He was a professor at Harvard and an expert on precise use of explosives.[23]
The basic concept of implosion is to use chemical explosives to crush a chunk of fissile material into a critical mass, where neutron multiplication leads to a nuclear chain reaction, releasing a large amount of energy. Cylindrical implosive configurations had been studied by Seth Neddermeyer, but von Neumann, who had experience with shaped charges used in armor-piercing ammunition, was a vocal advocate of spherical implosion driven by explosive lenses. He realized that the symmetry and speed with which implosion compressed the plutonium were critical issues,[23] and enlisted Ulam to help design lens configurations that would provide nearly spherical implosion. Within an implosion, because of enormous pressures and high temperatures, solid materials behave much like fluids. This meant that hydrodynamical calculations were needed to predict and minimize asymmetries that would spoil a nuclear detonation. Of these calculations, Ulam said:
The hydrodynamical problem was simply stated, but very difficult to calculate – not only in detail, but even in order of magnitude. In this discussion, I stressed pure pragmatism and the necessity to get a heuristic survey of the problem by simple-minded brute force, rather than by massive numerical work.[9]
Nevertheless, with the primitive facilities available at the time, Ulam and von Neumann did carry out numerical computations that led to a satisfactory design. This motivated their advocacy of a powerful computational capability at Los Alamos, which began during the war years,[24] continued through the cold war, and still exists.[25] Otto Frisch remembered Ulam as "a brilliant Polish topologist with a charming French wife. At once he told me that he was a pure mathematician who had sunk so low that his latest paper actually contained numbers with decimal points!"[26]
Statistics of branching and multiplicative processes
Even the inherent statistical fluctuations of neutron multiplication within a chain reaction have implications with regard to implosion speed and symmetry. In November 1944, David Hawkins[27] and Ulam addressed this problem in a report entitled "Theory of Multiplicative Processes".[28] This report, which invokes probability-generating functions, is also an early entry in the extensive literature on statistics of branching and multiplicative processes. In 1948, its scope was extended by Ulam and Everett.[29]
Early in the Manhattan project, Enrico Fermi's attention was focused on the use of reactors to produce plutonium. In September 1944, he arrived at Los Alamos, shortly after breathing life into the first Hanford reactor, which had been poisoned by a xenon isotope.[30] Soon after Fermi's arrival, Teller's "Super" bomb group, of which Ulam was a part, was transferred to a new division headed by Fermi.[31] Fermi and Ulam formed a relationship that became very fruitful after the war.[32]
Post war Los Alamos
In September 1945, Ulam left Los Alamos to become an associate professor at the University of Southern California in Los Angeles. In January 1946, he suffered an acute attack of encephalitis, which put his life in danger, but which was alleviated by emergency brain surgery. During his recuperation, many friends visited, including Nicholas Metropolis from Los Alamos and the famous mathematician Paul Erdős,[33] who remarked: "Stan, you are just like before."[9] This was encouraging, because Ulam was concerned about the state of his mental faculties, for he had lost the ability to speak during the crisis. Another friend, Gian-Carlo Rota, asserted in a 1987 article that the attack changed Ulam's personality: afterwards, he turned from rigorous pure mathematics to more speculative conjectures concerning the application of mathematics to physics and biology; Rota also cites Ulam's former collaborator Paul Stein as noting that Ulam was sloppier in his clothing afterwards, and John Oxtoby as noting that Ulam before the encephalitis could work for hours on end doing calculations, while when Rota worked with him, was reluctant to solve even a quadratic equation.[34] This assertion was not accepted by Françoise Aron Ulam.[35]
By late April 1946, Ulam had recovered enough to attend a secret conference at Los Alamos to discuss thermonuclear weapons. Those in attendance included Ulam, von Neumann, Metropolis, Teller, Stan Frankel, and others. Throughout his participation in the Manhattan Project, Teller's efforts had been directed toward the development of a "super" weapon based on nuclear fusion, rather than toward development of a practical fission bomb. After extensive discussion, the participants reached a consensus that his ideas were worthy of further exploration. A few weeks later, Ulam received an offer of a position at Los Alamos from Metropolis and Robert D. Richtmyer, the new head of its theoretical division, at a higher salary, and the Ulams returned to Los Alamos.[36]
Monte Carlo method
Late in the war, under the sponsorship of von Neumann, Frankel and Metropolis began to carry out calculations on the first general-purpose electronic computer, the ENIAC at the Aberdeen Proving Ground in Maryland. Shortly after returning to Los Alamos, Ulam participated in a review of results from these calculations.[37] Earlier, while playing solitaire during his recovery from surgery, Ulam had thought about playing hundreds of games to estimate statistically the probability of a successful outcome.[38] With ENIAC in mind, he realized that the availability of computers made such statistical methods very practical. John von Neumann immediately saw the significance of this insight. In March 1947 he proposed a statistical approach to the problem of neutron diffusion in fissionable material.[39] Because Ulam had often mentioned his uncle, Michał Ulam, "who just had to go to Monte Carlo" to gamble, Metropolis dubbed the statistical approach "The Monte Carlo method".[37] Metropolis and Ulam published the first unclassified paper on the Monte Carlo method in 1949.[40]
Fermi, learning of Ulam's breakthrough, devised an analog computer known as the Monte Carlo trolley, later dubbed the FERMIAC. The device performed a mechanical simulation of random diffusion of neutrons. As computers improved in speed and programmability, these methods became more useful. In particular, many Monte Carlo calculations carried out on modern massively parallel supercomputers are embarrassingly parallel applications, whose results can be very accurate.[25]
Teller–Ulam design
On 29 August 1949, the Soviet Union tested its first fission bomb, the RDS-1. Created under the supervision of Lavrentiy Beria, who sought to duplicate the US effort, this weapon was nearly identical to Fat Man, for its design was based on information provided by spies Klaus Fuchs, Theodore Hall, and David Greenglass. In response, on 31 January 1950, President Harry S. Truman announced a crash program to develop a fusion bomb.[41]
To advocate an aggressive development program, Ernest Lawrence and Luis Alvarez came to Los Alamos, where they conferred with Norris Bradbury, the laboratory director, and with George Gamow, Edward Teller, and Ulam. Soon, these three became members of a short-lived committee appointed by Bradbury to study the problem, with Teller as chairman.[9] At this time, research on the use of a fission weapon to create a fusion reaction had been ongoing since 1942, but the design was still essentially the one originally proposed by Teller. His concept was to put tritium and/or deuterium in close proximity to a fission bomb, with the hope that the heat and intense flux of neutrons released when the bomb exploded, would ignite a self-sustaining fusion reaction. Reactions of these isotopes of hydrogen are of interest because the energy per unit mass of fuel released by their fusion is much larger than that from fission of heavy nuclei.[42]
Because the results of calculations based on Teller's concept were discouraging, many scientists believed it could not lead to a successful weapon, while others had moral and economic grounds for not proceeding. Consequently, several senior people of the Manhattan Project opposed development, including Bethe and Oppenheimer.[43] To clarify the situation, Ulam and von Neumann resolved to do new calculations to determine whether Teller's approach was feasible. To carry out these studies, von Neumann decided to use electronic computers: ENIAC at Aberdeen, a new computer, MANIAC, at Princeton, and its twin, which was under construction at Los Alamos. Ulam enlisted Everett to follow a completely different approach, one guided by physical intuition. Françoise Ulam was one of a cadre of women "computers" who carried out laborious and extensive computations of thermonuclear scenarios on mechanical calculators, supplemented and confirmed by Everett's slide rule. Ulam and Fermi collaborated on further analysis of these scenarios. The results showed that, in workable configurations, a thermonuclear reaction would not ignite, and if ignited, it would not be self-sustaining. Ulam had used his expertise in combinatorics to analyze the chain reaction in deuterium, which was much more complicated than the ones in uranium and plutonium, and he concluded that no self-sustaining chain reaction would take place at the (low) densities that Teller was considering.[44] In late 1950, these conclusions were confirmed by von Neumann's results.[35][45]
In January 1951, Ulam had another idea: to channel the mechanical shock of a nuclear explosion so as to compress the fusion fuel. On the recommendation of his wife,[35] Ulam discussed this idea with Bradbury and Mark before he told Teller about it.[46] Almost immediately, Teller saw its merit, but noted that soft X-rays from the fission bomb would compress the thermonuclear fuel more strongly than mechanical shock and suggested ways to enhance this effect. On 9 March 1951, Teller and Ulam submitted a joint report describing these innovations.[47] A few weeks later, Teller suggested placing a fissile rod or cylinder at the center of the fusion fuel. The detonation of this "spark plug"[48] would help to initiate and enhance the fusion reaction. The design based on these ideas, called staged radiation implosion, has become the standard way to build thermonuclear weapons. It is often described as the "Teller–Ulam design".[49]
In September 1951, after a series of differences with Bradbury and other scientists, Teller resigned from Los Alamos, and returned to the University of Chicago.[50] At about the same time, Ulam went on leave as a visiting professor at Harvard for a semester.[51] Although Teller and Ulam submitted a joint report on their design[47] and jointly applied for a patent on it,[22] they soon became involved in a dispute over who deserved credit.[46] After the war, Bethe returned to Cornell University, but he was deeply involved in the development of thermonuclear weapons as a consultant. In 1954, he wrote an article on the history of the H-bomb,[52] which presents his opinion that both men contributed very significantly to the breakthrough. This balanced view is shared by others who were involved, including Mark and Fermi, but Teller persistently attempted to downplay Ulam's role.[53] "After the H-bomb was made," Bethe recalled, "reporters started to call Teller the father of the H-bomb. For the sake of history, I think it is more precise to say that Ulam is the father, because he provided the seed, and Teller is the mother, because he remained with the child. As for me, I guess I am the midwife."[54]
With the basic fusion reactions confirmed, and with a feasible design in hand, there was nothing to prevent Los Alamos from testing a thermonuclear device. On 1 November 1952, the first thermonuclear explosion occurred when Ivy Mike was detonated on Enewetak Atoll, within the US Pacific Proving Grounds. This device, which used liquid deuterium as its fusion fuel, was immense and utterly unusable as a weapon. Nevertheless, its success validated the Teller–Ulam design, and stimulated intensive development of practical weapons.[51]
Fermi–Pasta–Ulam–Tsingou problem
When Ulam returned to Los Alamos, his attention turned away from weapon design and toward the use of computers to investigate problems in physics and mathematics. With John Pasta, who helped Metropolis to bring MANIAC on line in March 1952, he explored these ideas in a report "Heuristic Studies in Problems of Mathematical Physics on High Speed Computing Machines", which was submitted on 9 June 1953. It treated several problems that cannot be addressed within the framework of traditional analytic methods: billowing of fluids, rotational motion in gravitating systems, magnetic lines of force, and hydrodynamic instabilities.[55]
Soon, Pasta and Ulam became experienced with electronic computation on MANIAC, and by this time, Enrico Fermi had settled into a routine of spending academic years at the University of Chicago and summers at Los Alamos. During these summer visits, Pasta, Ulam, and Mary Tsingou, a programmer in the MANIAC group, joined him to study a variation of the classic problem of a string of masses held together by springs that exert forces linearly proportional to their displacement from equilibrium.[56] Fermi proposed to add to this force a nonlinear component, which could be chosen to be proportional to either the square or cube of the displacement, or to a more complicated "broken linear" function. This addition is the key element of the Fermi–Pasta–Ulam–Tsingou problem, which is often designated by the abbreviation FPUT.[57][58]
A classical spring system can be described in terms of vibrational modes, which are analogous to the harmonics that occur on a stretched violin string. If the system starts in a particular mode, vibrations in other modes do not develop. With the nonlinear component, Fermi expected energy in one mode to transfer gradually to other modes, and eventually, to be distributed equally among all modes. This is roughly what began to happen shortly after the system was initialized with all its energy in the lowest mode, but much later, essentially all the energy periodically reappeared in the lowest mode.[58] This behavior is very different from the expected equipartition of energy. It remained mysterious until 1965, when Kruskal and Zabusky showed that, after appropriate mathematical transformations, the system can be described by the Korteweg–de Vries equation, which is the prototype of nonlinear partial differential equations that have soliton solutions. This means that FPUT behavior can be understood in terms of solitons.[59]
Nuclear propulsion
Starting in 1955, Ulam and Frederick Reines considered nuclear propulsion of aircraft and rockets.[60] This is an attractive possibility, because the nuclear energy per unit mass of fuel is a million times greater than that available from chemicals. From 1955 to 1972, their ideas were pursued during Project Rover, which explored the use of nuclear reactors to power rockets.[61] In response to a question by Senator John O. Pastore at a congressional committee hearing on "Outer Space Propulsion by Nuclear Energy", on January 22, 1958, Ulam replied that "the future as a whole of mankind is to some extent involved inexorably now with going outside the globe."[62]
Ulam and C. J. Everett also proposed, in contrast to Rover's continuous heating of rocket exhaust, to harness small nuclear explosions for propulsion.[63] Project Orion was a study of this idea. It began in 1958 and ended in 1965, after the Partial Nuclear Test Ban Treaty of 1963 banned nuclear weapons tests in the atmosphere and in space.[64] Work on this project was spearheaded by physicist Freeman Dyson, who commented on the decision to end Orion in his article, "Death of a Project".[65]
Bradbury appointed Ulam and John H. Manley as research advisors to the laboratory director in 1957. These newly created positions were on the same administrative level as division leaders, and Ulam held his until he retired from Los Alamos. In this capacity, he was able to influence and guide programs in many divisions: theoretical, physics, chemistry, metallurgy, weapons, health, Rover, and others.[61]
In addition to these activities, Ulam continued to publish technical reports and research papers. One of these introduced the Fermi–Ulam model, an extension of Fermi's theory of the acceleration of cosmic rays.[66] Another, with Paul Stein and Mary Tsingou, titled "Quadratic Transformations", was an early investigation of chaos theory and is considered the first published use of the phrase "chaotic behavior".[67][68]
Return to academia
During his years at Los Alamos, Ulam was a visiting professor at Harvard from 1951 to 1952, MIT from 1956 to 1957, the University of California, San Diego, in 1963, and the University of Colorado at Boulder from 1961 to 1962 and 1965 to 1967. In 1967, the last of these positions became permanent, when Ulam was appointed Professor and Chairman of the Department of Mathematics at the University of Colorado. He kept a residence in Santa Fe, which made it convenient to spend summers at Los Alamos as a consultant.[69] He was an elected member of the American Academy of Arts and Sciences, the United States National Academy of Sciences, and the American Philosophical Society.[70][71][72]
In Colorado, where he rejoined his friends Gamow, Richtmyer, and Hawkins, Ulam's research interests turned toward biology. In 1968, recognizing this emphasis, the University of Colorado School of Medicine appointed Ulam as Professor of Biomathematics, and he held this position until his death. With his Los Alamos colleague Robert Schrandt he published a report, "Some Elementary Attempts at Numerical Modeling of Problems Concerning Rates of Evolutionary Processes", which applied his earlier ideas on branching processes to evolution.[73] Another, report, with William Beyer, Temple F. Smith, and M. L. Stein, titled "Metrics in Biology", introduced new ideas about numerical taxonomy and evolutionary distances.[74]
When he retired from Colorado in 1975, Ulam began to spend winter semesters at the University of Florida, where he was a graduate research professor. In 1976, he was awarded the Commander's Cross with the Star of the Order of Polonia Restituta by the Polish government-in-exile in London.[75] Except for sabbaticals at the University of California, Davis from 1982 to 1983, and at Rockefeller University from 1980 to 1984,[69] this pattern of spending summers in Colorado and Los Alamos and winters in Florida continued until Ulam died of an apparent heart attack in Santa Fe on 13 May 1984.[3] Paul Erdős noted that "he died suddenly of heart failure, without fear or pain, while he could still prove and conjecture."[33] In 1987, Françoise Ulam deposited his papers with the American Philosophical Society Library in Philadelphia.[76] She continued to live in Santa Fe until she died in 2011, at the age of 93. Both Françoise and her husband were buried with her family in Montparnasse Cemetery in Paris.[77][78]
Challenge to economics
Alfred Marshall and his disciples dominated economic theory until the end of WWII. With the Cold War, the theory changed, emphasizing that a market economy was superior and the only sensible way. In Paul Samuelson's "Economics: An Introductory Analysis", 1948, Adam Smith's "invisible hand" was only a footnote. In later editions, it became the central theme. As Samuelson remembers, all this was challenged by Stanislaw Ulam: "[Y]ears ago... I was in the Society of Fellows at Harvard along with the mathematician Stanislaw Ulam. Ulam, who was to become an originator of the Monte Carlo method and co-discoverer of the hydrogen-bomb,... used to tease me by saying, 'Name me one proposition in all of the social sciences which is both true and non-trivial.' This was the test that I always failed. But now, some thirty years later ... an appropriate answer occurs to me: The Ricardian theory of comparative advantage ... That it is logically true need not be argued before a mathematician; that it is not trivial is attested by the thousands of important and intelligent men who have never been able to grasp the doctrine for themselves or to believe it after it was explained to them."[79][80]
Impact and legacy
Ulam participated in the creation of a hydrogen bomb as part of the Los Alamos Laboratory nuclear project. From the publication of his first paper as a student in 1929 until his death, Ulam was constantly writing on mathematics. The list of Ulam's publications includes more than 150 papers.[10] Topics represented by a significant number of papers are: set theory (including measurable cardinals and abstract measures), topology, functional analysis, transformation theory, ergodic theory, group theory, projective algebra, number theory, combinatorics, and graph theory.[81]
Notable results of this work are:
• Borsuk–Ulam theorem
• Mazur–Ulam theorem
• Kuratowski–Ulam theorem
• Hyers–Ulam–Rassias stability
• Lucky number
• Ulam spiral
• Ulam conjecture (in Number Theory)
• Ulam conjecture (in Graph theory)
• Ulam's packing conjecture
• Ulam's game
• Ulam matrix
• Ulam numbers
Ulam played pivotal role in the development of thermonuclear weapons. According to Françoise Ulam: "Stan would reassure me that, barring accidents, the H-bomb rendered nuclear war impossible."[35] In 1980, Ulam and his wife appeared in the television documentary The Day After Trinity.[82]
The Monte Carlo method has become a ubiquitous and standard approach to computation, and the method has been applied to a vast number of scientific problems.[83] In addition to problems in physics and mathematics, the method has been applied to finance, social science,[84] environmental risk assessment,[85] linguistics,[86] radiation therapy,[87] and sports.[88]
The Fermi–Pasta–Ulam–Tsingou problem is credited not only as "the birth of experimental mathematics",[58] but also as inspiration for the vast field of Nonlinear Science. In his Lilienfeld Prize lecture, David K. Campbell noted this relationship and described how FPUT gave rise to ideas in chaos, solitons, and dynamical systems.[89] In 1980, Donald Kerr, laboratory director at Los Alamos, with the strong support of Ulam and Mark Kac,[90] founded the Center for Nonlinear Studies (CNLS).[91] In 1985, CNLS initiated the Stanislaw M. Ulam Distinguished Scholar program, which provides an annual award that enables a noted scientist to spend a year carrying out research at Los Alamos.[92]
The fiftieth anniversary of the original FPUT paper was the subject of the March 2005 issue of the journal Chaos,[93] and the topic of the 25th Annual International Conference of CNLS.[94] The University of Southern Mississippi and the University of Florida supported the Ulam Quarterly,[95] which was active from 1992 to 1996, and which was one of the first online mathematical journals.[96] Florida's Department of Mathematics has sponsored, since 1998, the annual Ulam Colloquium Lecture,[97] and in March 2009, the Ulam Centennial Conference.[98]
Ulam's work on non-Euclidean distance metrics in the context of molecular biology made a significant contribution to sequence analysis[99] and his contributions in theoretical biology are considered watersheds in the development of cellular automata theory, population biology, pattern recognition, and biometrics generally (David Sankoff, however, challenged conclusions of Walter by writing that Ulam had only modest influence on early development of sequence alignment methods.[100]). Colleagues noted that some of his greatest contributions were in clearly identifying problems to be solved and general techniques for solving them.[101]
In 1987, Los Alamos issued a special issue of its Science publication, which summarized his accomplishments,[102] and which appeared, in 1989, as the book From Cardinals to Chaos. Similarly, in 1990, the University of California Press issued a compilation of mathematical reports by Ulam and his Los Alamos collaborators: Analogies Between Analogies.[103] During his career, Ulam was awarded honorary degrees by the Universities of New Mexico, Wisconsin, and Pittsburgh.[9]
In 2021, German film director Thorsten Klein made a film adaptation of the book Adventures of a Mathematician about Ulam's life.
Ulam is the grandfather of Rebecca Weiner, the New York Police Department’s deputy commissioner of intelligence and counterterrorism.[104][105]
Bibliography
• Kac, Mark; Ulam, Stanisław (1968). Mathematics and Logic: Retrospect and Prospects. New York: Praeger. ISBN 978-0-486-67085-0. OCLC 24847821.
• Ulam, Stanisław (1974). Beyer, W. A.; Mycielski and, J.; Rota, G.-C. (eds.). Sets, Numbers, and Universes: selected works. Mathematicians of Our Time. Vol. 9. The MIT Press, Cambridge, Mass.-London. ISBN 978-0-262-02108-1. MR 0441664.
• Ulam, Stanisław (1960). A Collection of Mathematical Problems. New York: Interscience Publishers. OCLC 526673.
• Ulam, Stanisław (1983). Adventures of a Mathematician. New York: Charles Scribner's Sons. ISBN 978-0-684-14391-0. OCLC 1528346. (autobiography).
• Ulam, Stanisław (1986). Science, Computers, and People: From the Tree of Mathematics. Boston: Birkhauser. ISBN 978-3-7643-3276-1. OCLC 11260216.
• Ulam, Stanisław; Ulam, Françoise (1990). Analogies Between Analogies: The Mathematical Reports of S.M. Ulam and his Los Alamos Collaborators. Berkeley: University of California Press. ISBN 978-0-520-05290-1. OCLC 20318499.
See also
• List of Polish mathematicians
• List of Polish physicists
• List of things named after Stanislaw Ulam
• Timeline of Polish science and technology
• Biopic about Stanislaw Ulam, based on his autobiography, starring Jakub Gierszal[106]
References
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External links
• 1979 Audio Interview with Stanislaus Ulam by Martin Sherwin Voices of the Manhattan Project
• 1965 Audio Interview with Stanislaus Ulam by Richard Rhodes Voices of the Manhattan Project
• "Publications of Stanislaw M. Ulam" (PDF). Los Alamos Science (Special Issue): 313. 1987. ISSN 0273-7116. Archived (PDF) from the original on 2022-10-09.
• Von Neumann: The Interaction of Mathematics and Computing on YouTube – 1976 lecture on The First International Research Conference on the History of Computing.
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Wiener process
In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion.[1] It is often also called Brownian motion due to its historical connection with the physical process of the same name originally observed by Scottish botanist Robert Brown. It is one of the best known Lévy processes (càdlàg stochastic processes with stationary independent increments) and occurs frequently in pure and applied mathematics, economics, quantitative finance, evolutionary biology, and physics.
The Wiener process plays an important role in both pure and applied mathematics. In pure mathematics, the Wiener process gave rise to the study of continuous time martingales. It is a key process in terms of which more complicated stochastic processes can be described. As such, it plays a vital role in stochastic calculus, diffusion processes and even potential theory. It is the driving process of Schramm–Loewner evolution. In applied mathematics, the Wiener process is used to represent the integral of a white noise Gaussian process, and so is useful as a model of noise in electronics engineering (see Brownian noise), instrument errors in filtering theory and disturbances in control theory.
The Wiener process has applications throughout the mathematical sciences. In physics it is used to study Brownian motion, the diffusion of minute particles suspended in fluid, and other types of diffusion via the Fokker–Planck and Langevin equations. It also forms the basis for the rigorous path integral formulation of quantum mechanics (by the Feynman–Kac formula, a solution to the Schrödinger equation can be represented in terms of the Wiener process) and the study of eternal inflation in physical cosmology. It is also prominent in the mathematical theory of finance, in particular the Black–Scholes option pricing model.
Characterisations of the Wiener process
The Wiener process $W_{t}$ is characterised by the following properties:[2]
1. $W_{0}=0$ almost surely
2. $W$ has independent increments: for every $t>0,$ the future increments $W_{t+u}-W_{t},$ $u\geq 0,$ are independent of the past values $W_{s}$, $s<t.$
3. $W$ has Gaussian increments: $W_{t+u}-W_{t}$ is normally distributed with mean $0$ and variance $u$, $W_{t+u}-W_{t}\sim {\mathcal {N}}(0,u).$
4. $W$ has almost surely continuous paths: $W_{t}$ is almost surely continuous in $t$.
That the process has independent increments means that if 0 ≤ s1 < t1 ≤ s2 < t2 then Wt1 − Ws1 and Wt2 − Ws2 are independent random variables, and the similar condition holds for n increments.
An alternative characterisation of the Wiener process is the so-called Lévy characterisation that says that the Wiener process is an almost surely continuous martingale with W0 = 0 and quadratic variation [Wt, Wt] = t (which means that Wt2 − t is also a martingale).
A third characterisation is that the Wiener process has a spectral representation as a sine series whose coefficients are independent N(0, 1) random variables. This representation can be obtained using the Karhunen–Loève theorem.
Another characterisation of a Wiener process is the definite integral (from time zero to time t) of a zero mean, unit variance, delta correlated ("white") Gaussian process.[3]
The Wiener process can be constructed as the scaling limit of a random walk, or other discrete-time stochastic processes with stationary independent increments. This is known as Donsker's theorem. Like the random walk, the Wiener process is recurrent in one or two dimensions (meaning that it returns almost surely to any fixed neighborhood of the origin infinitely often) whereas it is not recurrent in dimensions three and higher.[4] Unlike the random walk, it is scale invariant, meaning that
$\alpha ^{-1}W_{\alpha ^{2}t}$
is a Wiener process for any nonzero constant α. The Wiener measure is the probability law on the space of continuous functions g, with g(0) = 0, induced by the Wiener process. An integral based on Wiener measure may be called a Wiener integral.
Wiener process as a limit of random walk
Let $\xi _{1},\xi _{2},\ldots $ be i.i.d. random variables with mean 0 and variance 1. For each n, define a continuous time stochastic process
$W_{n}(t)={\frac {1}{\sqrt {n}}}\sum \limits _{1\leq k\leq \lfloor nt\rfloor }\xi _{k},\qquad t\in [0,1].$
This is a random step function. Increments of $W_{n}$ are independent because the $\xi _{k}$ are independent. For large n, $W_{n}(t)-W_{n}(s)$ is close to $N(0,t-s)$ by the central limit theorem. Donsker's theorem asserts that as $n\to \infty $, $W_{n}$ approaches a Wiener process, which explains the ubiquity of Brownian motion.[5]
Properties of a one-dimensional Wiener process
Basic properties
The unconditional probability density function follows a normal distribution with mean = 0 and variance = t, at a fixed time t:
$f_{W_{t}}(x)={\frac {1}{\sqrt {2\pi t}}}e^{-x^{2}/(2t)}.$
The expectation is zero:
$\operatorname {E} [W_{t}]=0.$
The variance, using the computational formula, is t:
$\operatorname {Var} (W_{t})=t.$
These results follow immediately from the definition that increments have a normal distribution, centered at zero. Thus
$W_{t}=W_{t}-W_{0}\sim N(0,t).$
Covariance and correlation
The covariance and correlation (where $s\leq t$):
${\begin{aligned}\operatorname {cov} (W_{s},W_{t})&=s,\\\operatorname {corr} (W_{s},W_{t})&={\frac {\operatorname {cov} (W_{s},W_{t})}{\sigma _{W_{s}}\sigma _{W_{t}}}}={\frac {s}{\sqrt {st}}}={\sqrt {\frac {s}{t}}}.\end{aligned}}$
These results follow from the definition that non-overlapping increments are independent, of which only the property that they are uncorrelated is used. Suppose that $t_{1}\leq t_{2}$.
$\operatorname {cov} (W_{t_{1}},W_{t_{2}})=\operatorname {E} \left[(W_{t_{1}}-\operatorname {E} [W_{t_{1}}])\cdot (W_{t_{2}}-\operatorname {E} [W_{t_{2}}])\right]=\operatorname {E} \left[W_{t_{1}}\cdot W_{t_{2}}\right].$
Substituting
$W_{t_{2}}=(W_{t_{2}}-W_{t_{1}})+W_{t_{1}}$
we arrive at:
${\begin{aligned}\operatorname {E} [W_{t_{1}}\cdot W_{t_{2}}]&=\operatorname {E} \left[W_{t_{1}}\cdot ((W_{t_{2}}-W_{t_{1}})+W_{t_{1}})\right]\\&=\operatorname {E} \left[W_{t_{1}}\cdot (W_{t_{2}}-W_{t_{1}})\right]+\operatorname {E} \left[W_{t_{1}}^{2}\right].\end{aligned}}$
Since $W_{t_{1}}=W_{t_{1}}-W_{t_{0}}$ and $W_{t_{2}}-W_{t_{1}}$ are independent,
$\operatorname {E} \left[W_{t_{1}}\cdot (W_{t_{2}}-W_{t_{1}})\right]=\operatorname {E} [W_{t_{1}}]\cdot \operatorname {E} [W_{t_{2}}-W_{t_{1}}]=0.$
Thus
$\operatorname {cov} (W_{t_{1}},W_{t_{2}})=\operatorname {E} \left[W_{t_{1}}^{2}\right]=t_{1}.$
A corollary useful for simulation is that we can write, for t1 < t2:
$W_{t_{2}}=W_{t_{1}}+{\sqrt {t_{2}-t_{1}}}\cdot Z$
where Z is an independent standard normal variable.
Wiener representation
Wiener (1923) also gave a representation of a Brownian path in terms of a random Fourier series. If $\xi _{n}$ are independent Gaussian variables with mean zero and variance one, then
$W_{t}=\xi _{0}t+{\sqrt {2}}\sum _{n=1}^{\infty }\xi _{n}{\frac {\sin \pi nt}{\pi n}}$
and
$W_{t}={\sqrt {2}}\sum _{n=1}^{\infty }\xi _{n}{\frac {\sin \left(\left(n-{\frac {1}{2}}\right)\pi t\right)}{\left(n-{\frac {1}{2}}\right)\pi }}$
represent a Brownian motion on $[0,1]$. The scaled process
${\sqrt {c}}\,W\left({\frac {t}{c}}\right)$
is a Brownian motion on $[0,c]$ (cf. Karhunen–Loève theorem).
Running maximum
The joint distribution of the running maximum
$M_{t}=\max _{0\leq s\leq t}W_{s}$
and Wt is
$f_{M_{t},W_{t}}(m,w)={\frac {2(2m-w)}{t{\sqrt {2\pi t}}}}e^{-{\frac {(2m-w)^{2}}{2t}}},\qquad m\geq 0,w\leq m.$
To get the unconditional distribution of $f_{M_{t}}$, integrate over −∞ < w ≤ m:
${\begin{aligned}f_{M_{t}}(m)&=\int _{-\infty }^{m}f_{M_{t},W_{t}}(m,w)\,dw=\int _{-\infty }^{m}{\frac {2(2m-w)}{t{\sqrt {2\pi t}}}}e^{-{\frac {(2m-w)^{2}}{2t}}}\,dw\\[5pt]&={\sqrt {\frac {2}{\pi t}}}e^{-{\frac {m^{2}}{2t}}},\qquad m\geq 0,\end{aligned}}$
the probability density function of a Half-normal distribution. The expectation[6] is
$\operatorname {E} [M_{t}]=\int _{0}^{\infty }mf_{M_{t}}(m)\,dm=\int _{0}^{\infty }m{\sqrt {\frac {2}{\pi t}}}e^{-{\frac {m^{2}}{2t}}}\,dm={\sqrt {\frac {2t}{\pi }}}$
If at time $t$ the Wiener process has a known value $W_{t}$, it is possible to calculate the conditional probability distribution of the maximum in interval $[0,t]$ (cf. Probability distribution of extreme points of a Wiener stochastic process). The cumulative probability distribution function of the maximum value, conditioned by the known value $W_{t}$, is:
$\,F_{M_{W_{t}}}(m)=\Pr \left(M_{W_{t}}=\max _{0\leq s\leq t}W(s)\leq m\mid W(t)=W_{t}\right)=\ 1-\ e^{-2{\frac {m(m-W_{t})}{t}}}\ \,,\,\ \ m>\max(0,W_{t})$
Self-similarity
Brownian scaling
For every c > 0 the process $V_{t}=(1/{\sqrt {c}})W_{ct}$ is another Wiener process.
Time reversal
The process $V_{t}=W_{1}-W_{1-t}$ for 0 ≤ t ≤ 1 is distributed like Wt for 0 ≤ t ≤ 1.
Time inversion
The process $V_{t}=tW_{1/t}$ is another Wiener process.
A class of Brownian martingales
If a polynomial p(x, t) satisfies the partial differential equation
$\left({\frac {\partial }{\partial t}}-{\frac {1}{2}}{\frac {\partial ^{2}}{\partial x^{2}}}\right)p(x,t)=0$
then the stochastic process
$M_{t}=p(W_{t},t)$
is a martingale.
Example: $W_{t}^{2}-t$ is a martingale, which shows that the quadratic variation of W on [0, t] is equal to t. It follows that the expected time of first exit of W from (−c, c) is equal to c2.
More generally, for every polynomial p(x, t) the following stochastic process is a martingale:
$M_{t}=p(W_{t},t)-\int _{0}^{t}a(W_{s},s)\,\mathrm {d} s,$
where a is the polynomial
$a(x,t)=\left({\frac {\partial }{\partial t}}+{\frac {1}{2}}{\frac {\partial ^{2}}{\partial x^{2}}}\right)p(x,t).$
Example: $p(x,t)=\left(x^{2}-t\right)^{2},$ $a(x,t)=4x^{2};$ the process
$\left(W_{t}^{2}-t\right)^{2}-4\int _{0}^{t}W_{s}^{2}\,\mathrm {d} s$
is a martingale, which shows that the quadratic variation of the martingale $W_{t}^{2}-t$ on [0, t] is equal to
$4\int _{0}^{t}W_{s}^{2}\,\mathrm {d} s.$
About functions p(xa, t) more general than polynomials, see local martingales.
Some properties of sample paths
The set of all functions w with these properties is of full Wiener measure. That is, a path (sample function) of the Wiener process has all these properties almost surely.
Qualitative properties
• For every ε > 0, the function w takes both (strictly) positive and (strictly) negative values on (0, ε).
• The function w is continuous everywhere but differentiable nowhere (like the Weierstrass function).
• For any $\epsilon >0$, $w(t)$ is almost surely not $({\tfrac {1}{2}}+\epsilon )$-Hölder continuous, and almost surely $({\tfrac {1}{2}}-\epsilon )$-Hölder continuous.[7]
• Points of local maximum of the function w are a dense countable set; the maximum values are pairwise different; each local maximum is sharp in the following sense: if w has a local maximum at t then
$\lim _{s\to t}{\frac {|w(s)-w(t)|}{|s-t|}}\to \infty .$
The same holds for local minima.
• The function w has no points of local increase, that is, no t > 0 satisfies the following for some ε in (0, t): first, w(s) ≤ w(t) for all s in (t − ε, t), and second, w(s) ≥ w(t) for all s in (t, t + ε). (Local increase is a weaker condition than that w is increasing on (t − ε, t + ε).) The same holds for local decrease.
• The function w is of unbounded variation on every interval.
• The quadratic variation of w over [0,t] is t.
• Zeros of the function w are a nowhere dense perfect set of Lebesgue measure 0 and Hausdorff dimension 1/2 (therefore, uncountable).
Law of the iterated logarithm
$\limsup _{t\to +\infty }{\frac {|w(t)|}{\sqrt {2t\log \log t}}}=1,\quad {\text{almost surely}}.$
Modulus of continuity
Local modulus of continuity:
$\limsup _{\varepsilon \to 0+}{\frac {|w(\varepsilon )|}{\sqrt {2\varepsilon \log \log(1/\varepsilon )}}}=1,\qquad {\text{almost surely}}.$
Global modulus of continuity (Lévy):
$\limsup _{\varepsilon \to 0+}\sup _{0\leq s<t\leq 1,t-s\leq \varepsilon }{\frac {|w(s)-w(t)|}{\sqrt {2\varepsilon \log(1/\varepsilon )}}}=1,\qquad {\text{almost surely}}.$
Dimension doubling theorem
The dimension doubling theorems say that the Hausdorff dimension of a set under a Brownian motion doubles almost surely.
Local time
The image of the Lebesgue measure on [0, t] under the map w (the pushforward measure) has a density Lt. Thus,
$\int _{0}^{t}f(w(s))\,\mathrm {d} s=\int _{-\infty }^{+\infty }f(x)L_{t}(x)\,\mathrm {d} x$
for a wide class of functions f (namely: all continuous functions; all locally integrable functions; all non-negative measurable functions). The density Lt is (more exactly, can and will be chosen to be) continuous. The number Lt(x) is called the local time at x of w on [0, t]. It is strictly positive for all x of the interval (a, b) where a and b are the least and the greatest value of w on [0, t], respectively. (For x outside this interval the local time evidently vanishes.) Treated as a function of two variables x and t, the local time is still continuous. Treated as a function of t (while x is fixed), the local time is a singular function corresponding to a nonatomic measure on the set of zeros of w.
These continuity properties are fairly non-trivial. Consider that the local time can also be defined (as the density of the pushforward measure) for a smooth function. Then, however, the density is discontinuous, unless the given function is monotone. In other words, there is a conflict between good behavior of a function and good behavior of its local time. In this sense, the continuity of the local time of the Wiener process is another manifestation of non-smoothness of the trajectory.
Information rate
The information rate of the Wiener process with respect to the squared error distance, i.e. its quadratic rate-distortion function, is given by [8]
$R(D)={\frac {2}{\pi ^{2}\ln 2D}}\approx 0.29D^{-1}.$
Therefore, it is impossible to encode $\{w_{t}\}_{t\in [0,T]}$ using a binary code of less than $TR(D)$ bits and recover it with expected mean squared error less than $D$. On the other hand, for any $\varepsilon >0$, there exists $T$ large enough and a binary code of no more than $2^{TR(D)}$ distinct elements such that the expected mean squared error in recovering $\{w_{t}\}_{t\in [0,T]}$ from this code is at most $D-\varepsilon $.
In many cases, it is impossible to encode the Wiener process without sampling it first. When the Wiener process is sampled at intervals $T_{s}$ before applying a binary code to represent these samples, the optimal trade-off between code rate $R(T_{s},D)$ and expected mean square error $D$ (in estimating the continuous-time Wiener process) follows the parametric representation [9]
$R(T_{s},D_{\theta })={\frac {T_{s}}{2}}\int _{0}^{1}\log _{2}^{+}\left[{\frac {S(\varphi )-{\frac {1}{6}}}{\theta }}\right]d\varphi ,$
$D_{\theta }={\frac {T_{s}}{6}}+T_{s}\int _{0}^{1}\min \left\{S(\varphi )-{\frac {1}{6}},\theta \right\}d\varphi ,$
where $S(\varphi )=(2\sin(\pi \varphi /2))^{-2}$ and $\log ^{+}[x]=\max\{0,\log(x)\}$. In particular, $T_{s}/6$ is the mean squared error associated only with the sampling operation (without encoding).
Related processes
The stochastic process defined by
$X_{t}=\mu t+\sigma W_{t}$
is called a Wiener process with drift μ and infinitesimal variance σ2. These processes exhaust continuous Lévy processes.
Two random processes on the time interval [0, 1] appear, roughly speaking, when conditioning the Wiener process to vanish on both ends of [0,1]. With no further conditioning, the process takes both positive and negative values on [0, 1] and is called Brownian bridge. Conditioned also to stay positive on (0, 1), the process is called Brownian excursion.[10] In both cases a rigorous treatment involves a limiting procedure, since the formula P(A|B) = P(A ∩ B)/P(B) does not apply when P(B) = 0.
A geometric Brownian motion can be written
$e^{\mu t-{\frac {\sigma ^{2}t}{2}}+\sigma W_{t}}.$
It is a stochastic process which is used to model processes that can never take on negative values, such as the value of stocks.
The stochastic process
$X_{t}=e^{-t}W_{e^{2t}}$
is distributed like the Ornstein–Uhlenbeck process with parameters $\theta =1$, $\mu =0$, and $\sigma ^{2}=2$.
The time of hitting a single point x > 0 by the Wiener process is a random variable with the Lévy distribution. The family of these random variables (indexed by all positive numbers x) is a left-continuous modification of a Lévy process. The right-continuous modification of this process is given by times of first exit from closed intervals [0, x].
The local time L = (Lxt)x ∈ R, t ≥ 0 of a Brownian motion describes the time that the process spends at the point x. Formally
$L^{x}(t)=\int _{0}^{t}\delta (x-B_{t})\,ds$
where δ is the Dirac delta function. The behaviour of the local time is characterised by Ray–Knight theorems.
Brownian martingales
Let A be an event related to the Wiener process (more formally: a set, measurable with respect to the Wiener measure, in the space of functions), and Xt the conditional probability of A given the Wiener process on the time interval [0, t] (more formally: the Wiener measure of the set of trajectories whose concatenation with the given partial trajectory on [0, t] belongs to A). Then the process Xt is a continuous martingale. Its martingale property follows immediately from the definitions, but its continuity is a very special fact – a special case of a general theorem stating that all Brownian martingales are continuous. A Brownian martingale is, by definition, a martingale adapted to the Brownian filtration; and the Brownian filtration is, by definition, the filtration generated by the Wiener process.
Integrated Brownian motion
The time-integral of the Wiener process
$W^{(-1)}(t):=\int _{0}^{t}W(s)\,ds$
is called integrated Brownian motion or integrated Wiener process. It arises in many applications and can be shown to have the distribution N(0, t3/3),[11] calculated using the fact that the covariance of the Wiener process is $t\wedge s=\min(t,s)$.[12]
For the general case of the process defined by
$V_{f}(t)=\int _{0}^{t}f'(s)W(s)\,ds=\int _{0}^{t}(f(t)-f(s))\,dW_{s}$
Then, for $a>0$,
$\operatorname {Var} (V_{f}(t))=\int _{0}^{t}(f(t)-f(s))^{2}\,ds$
$\operatorname {cov} (V_{f}(t+a),V_{f}(t))=\int _{0}^{t}(f(t+a)-f(s))(f(t)-f(s))\,ds$
In fact, $V_{f}(t)$ is always a zero mean normal random variable. This allows for simulation of $V_{f}(t+a)$ given $V_{f}(t)$ by taking
$V_{f}(t+a)=A\cdot V_{f}(t)+B\cdot Z$
where Z is a standard normal variable and
$A={\frac {\operatorname {cov} (V_{f}(t+a),V_{f}(t))}{\operatorname {Var} (V_{f}(t))}}$
$B^{2}=\operatorname {Var} (V_{f}(t+a))-A^{2}\operatorname {Var} (V_{f}(t))$
The case of $V_{f}(t)=W^{(-1)}(t)$ corresponds to $f(t)=t$. All these results can be seen as direct consequences of Itô isometry. The n-times-integrated Wiener process is a zero-mean normal variable with variance ${\frac {t}{2n+1}}\left({\frac {t^{n}}{n!}}\right)^{2}$. This is given by the Cauchy formula for repeated integration.
Time change
Every continuous martingale (starting at the origin) is a time changed Wiener process.
Example: 2Wt = V(4t) where V is another Wiener process (different from W but distributed like W).
Example. $W_{t}^{2}-t=V_{A(t)}$ where $A(t)=4\int _{0}^{t}W_{s}^{2}\,\mathrm {d} s$ and V is another Wiener process.
In general, if M is a continuous martingale then $M_{t}-M_{0}=V_{A(t)}$ where A(t) is the quadratic variation of M on [0, t], and V is a Wiener process.
Corollary. (See also Doob's martingale convergence theorems) Let Mt be a continuous martingale, and
$M_{\infty }^{-}=\liminf _{t\to \infty }M_{t},$
$M_{\infty }^{+}=\limsup _{t\to \infty }M_{t}.$
Then only the following two cases are possible:
$-\infty <M_{\infty }^{-}=M_{\infty }^{+}<+\infty ,$
$-\infty =M_{\infty }^{-}<M_{\infty }^{+}=+\infty ;$ ;}
other cases (such as $M_{\infty }^{-}=M_{\infty }^{+}=+\infty ,$ $M_{\infty }^{-}<M_{\infty }^{+}<+\infty $ etc.) are of probability 0.
Especially, a nonnegative continuous martingale has a finite limit (as t → ∞) almost surely.
All stated (in this subsection) for martingales holds also for local martingales.
Change of measure
A wide class of continuous semimartingales (especially, of diffusion processes) is related to the Wiener process via a combination of time change and change of measure.
Using this fact, the qualitative properties stated above for the Wiener process can be generalized to a wide class of continuous semimartingales.[13][14]
Complex-valued Wiener process
The complex-valued Wiener process may be defined as a complex-valued random process of the form $Z_{t}=X_{t}+iY_{t}$ where $X_{t}$ and $Y_{t}$ are independent Wiener processes (real-valued).[15]
Self-similarity
Brownian scaling, time reversal, time inversion: the same as in the real-valued case.
Rotation invariance: for every complex number $c$ such that $|c|=1$ the process $c\cdot Z_{t}$ is another complex-valued Wiener process.
Time change
If $f$ is an entire function then the process $f(Z_{t})-f(0)$ is a time-changed complex-valued Wiener process.
Example: $Z_{t}^{2}=\left(X_{t}^{2}-Y_{t}^{2}\right)+2X_{t}Y_{t}i=U_{A(t)}$ where
$A(t)=4\int _{0}^{t}|Z_{s}|^{2}\,\mathrm {d} s$
and $U$ is another complex-valued Wiener process.
In contrast to the real-valued case, a complex-valued martingale is generally not a time-changed complex-valued Wiener process. For example, the martingale $2X_{t}+iY_{t}$ is not (here $X_{t}$ and $Y_{t}$ are independent Wiener processes, as before).
Brownian sheet
Main article: Brownian sheet
The Brownian sheet is a multiparamateric generalization. The definition varies from authors, some define the Brownian sheet to have specifically a two-dimensional time parameter $t$ while others define it for general dimensions.
See also
Generalities:
• Abstract Wiener space
• Classical Wiener space
• Chernoff's distribution
• Fractal
• Brownian web
• Probability distribution of extreme points of a Wiener stochastic process
Numerical path sampling:
• Euler–Maruyama method
• Walk-on-spheres method
Notes
1. N.Wiener Collected Works vol.1
2. Durrett, Rick (2019). "Brownian Motion". Probability: Theory and Examples (5th ed.). ISBN 9781108591034.
3. Huang, Steel T.; Cambanis, Stamatis (1978). "Stochastic and Multiple Wiener Integrals for Gaussian Processes". The Annals of Probability. 6 (4): 585–614. doi:10.1214/aop/1176995480. ISSN 0091-1798. JSTOR 2243125.
4. "Pólya's Random Walk Constants". Wolfram Mathworld.
5. Steven Lalley, Mathematical Finance 345 Lecture 5: Brownian Motion (2001)
6. Shreve, Steven E (2008). Stochastic Calculus for Finance II: Continuous Time Models. Springer. p. 114. ISBN 978-0-387-40101-0.
7. Mörters, Peter; Peres, Yuval; Schramm, Oded; Werner, Wendelin (2010). Brownian motion. Cambridge series in statistical and probabilistic mathematics. Cambridge: Cambridge University Press. p. 18. ISBN 978-0-521-76018-8.
8. T. Berger, "Information rates of Wiener processes," in IEEE Transactions on Information Theory, vol. 16, no. 2, pp. 134-139, March 1970. doi: 10.1109/TIT.1970.1054423
9. Kipnis, A., Goldsmith, A.J. and Eldar, Y.C., 2019. The distortion-rate function of sampled Wiener processes. IEEE Transactions on Information Theory, 65(1), pp.482-499.
10. Vervaat, W. (1979). "A relation between Brownian bridge and Brownian excursion". Annals of Probability. 7 (1): 143–149. doi:10.1214/aop/1176995155. JSTOR 2242845.
11. "Interview Questions VII: Integrated Brownian Motion – Quantopia". www.quantopia.net. Retrieved 2017-05-14.
12. Forum, "Variance of integrated Wiener process", 2009.
13. Revuz, D., & Yor, M. (1999). Continuous martingales and Brownian motion (Vol. 293). Springer.
14. Doob, J. L. (1953). Stochastic processes (Vol. 101). Wiley: New York.
15. Navarro-moreno, J.; Estudillo-martinez, M.D; Fernandez-alcala, R.M.; Ruiz-molina, J.C. (2009), "Estimation of Improper Complex-Valued Random Signals in Colored Noise by Using the Hilbert Space Theory", IEEE Transactions on Information Theory, 55 (6): 2859–2867, doi:10.1109/TIT.2009.2018329, S2CID 5911584
References
• Kleinert, Hagen (2004). Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets (4th ed.). Singapore: World Scientific. ISBN 981-238-107-4. (also available online: PDF-files)
• Stark, Henry; Woods, John (2002). Probability and Random Processes with Applications to Signal Processing (3rd ed.). New Jersey: Prentice Hall. ISBN 0-13-020071-9.
• Revuz, Daniel; Yor, Marc (1994). Continuous martingales and Brownian motion (Second ed.). Springer-Verlag.
External links
• Article for the school-going child
• Brownian Motion, "Diverse and Undulating"
• Discusses history, botany and physics of Brown's original observations, with videos
• "Einstein's prediction finally witnessed one century later" : a test to observe the velocity of Brownian motion
• "Interactive Web Application: Stochastic Processes used in Quantitative Finance".
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Wikipedia
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Standard L-function
In mathematics, the term standard L-function refers to a particular type of automorphic L-function described by Robert P. Langlands.[1][2] Here, standard refers to the finite-dimensional representation r being the standard representation of the L-group as a matrix group.
Relations to other L-functions
Standard L-functions are thought to be the most general type of L-function. Conjecturally, they include all examples of L-functions, and in particular are expected to coincide with the Selberg class. Furthermore, all L-functions over arbitrary number fields are widely thought to be instances of standard L-functions for the general linear group GL(n) over the rational numbers Q. This makes them a useful testing ground for statements about L-functions, since it sometimes affords structure from the theory of automorphic forms.
Analytic properties
These L-functions were proven to always be entire by Roger Godement and Hervé Jacquet,[3] with the sole exception of Riemann ζ-function, which arises for n = 1. Another proof was later given by Freydoon Shahidi using the Langlands–Shahidi method. For a broader discussion, see Gelbart & Shahidi (1988).[4]
See also
• Zeta function
References
1. Langlands, R.P. (1978), L-Functions and Automorphic Representations (ICM report at Helsinki) (PDF).
2. Borel, A. (1979), "Automorphic L-functions", Automorphic forms, representations and L-functions (Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., vol. XXXIII, Providence, R.I.: Amer. Math. Soc., pp. 27–61, MR 0546608.
3. Godement, Roger; Jacquet, Hervé (1972), Zeta functions of simple algebras, Lecture Notes in Mathematics, vol. 260, Berlin-New York: Springer-Verlag, MR 0342495.
4. Gelbart, Stephen; Shahidi, Freydoon (1988), Analytic properties of automorphic L-functions, Perspectives in Mathematics, vol. 6, Boston, MA: Academic Press, Inc., ISBN 0-12-279175-4, MR 0951897.
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Wikipedia
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Standard basis
In mathematics, the standard basis (also called natural basis or canonical basis) of a coordinate vector space (such as $\mathbb {R} ^{n}$ or $\mathbb {C} ^{n}$) is the set of vectors, each of whose components are all zero, except one that equals 1.[1] For example, in the case of the Euclidean plane $\mathbb {R} ^{2}$ formed by the pairs (x, y) of real numbers, the standard basis is formed by the vectors
$\mathbf {e} _{x}=(1,0),\quad \mathbf {e} _{y}=(0,1).$
For broader coverage of this topic, see Canonical basis.
Not to be confused with another name for a Gröbner basis.
Similarly, the standard basis for the three-dimensional space $\mathbb {R} ^{3}$ is formed by vectors
$\mathbf {e} _{x}=(1,0,0),\quad \mathbf {e} _{y}=(0,1,0),\quad \mathbf {e} _{z}=(0,0,1).$
Here the vector ex points in the x direction, the vector ey points in the y direction, and the vector ez points in the z direction. There are several common notations for standard-basis vectors, including {ex, ey, ez}, {e1, e2, e3}, {i, j, k}, and {x, y, z}. These vectors are sometimes written with a hat to emphasize their status as unit vectors (standard unit vectors).
These vectors are a basis in the sense that any other vector can be expressed uniquely as a linear combination of these.[2] For example, every vector v in three-dimensional space can be written uniquely as
$v_{x}\,\mathbf {e} _{x}+v_{y}\,\mathbf {e} _{y}+v_{z}\,\mathbf {e} _{z},$
the scalars $v_{x}$, $v_{y}$, $v_{z}$ being the scalar components of the vector v.
In the n-dimensional Euclidean space $\mathbb {R} ^{n}$, the standard basis consists of n distinct vectors
$\{\mathbf {e} _{i}:1\leq i\leq n\},$
where ei denotes the vector with a 1 in the ith coordinate and 0's elsewhere.
Standard bases can be defined for other vector spaces, whose definition involves coefficients, such as polynomials and matrices. In both cases, the standard basis consists of the elements of the space such that all coefficients but one are 0 and the non-zero one is 1. For polynomials, the standard basis thus consists of the monomials and is commonly called monomial basis. For matrices ${\mathcal {M}}_{m\times n}$, the standard basis consists of the m×n-matrices with exactly one non-zero entry, which is 1. For example, the standard basis for 2×2 matrices is formed by the 4 matrices
$\mathbf {e} _{11}={\begin{pmatrix}1&0\\0&0\end{pmatrix}},\quad \mathbf {e} _{12}={\begin{pmatrix}0&1\\0&0\end{pmatrix}},\quad \mathbf {e} _{21}={\begin{pmatrix}0&0\\1&0\end{pmatrix}},\quad \mathbf {e} _{22}={\begin{pmatrix}0&0\\0&1\end{pmatrix}}.$
Properties
By definition, the standard basis is a sequence of orthogonal unit vectors. In other words, it is an ordered and orthonormal basis.
However, an ordered orthonormal basis is not necessarily a standard basis. For instance the two vectors representing a 30° rotation of the 2D standard basis described above, i.e.
$v_{1}=\left({{\sqrt {3}} \over 2},{1 \over 2}\right)\,$
$v_{2}=\left({1 \over 2},{-{\sqrt {3}} \over 2}\right)\,$
are also orthogonal unit vectors, but they are not aligned with the axes of the Cartesian coordinate system, so the basis with these vectors does not meet the definition of standard basis.
Generalizations
There is a standard basis also for the ring of polynomials in n indeterminates over a field, namely the monomials.
All of the preceding are special cases of the family
${(e_{i})}_{i\in I}=((\delta _{ij})_{j\in I})_{i\in I}$
where $I$ is any set and $\delta _{ij}$ is the Kronecker delta, equal to zero whenever i ≠ j and equal to 1 if i = j. This family is the canonical basis of the R-module (free module)
$R^{(I)}$
of all families
$f=(f_{i})$
from I into a ring R, which are zero except for a finite number of indices, if we interpret 1 as 1R, the unit in R.[3]
Other usages
The existence of other 'standard' bases has become a topic of interest in algebraic geometry, beginning with work of Hodge from 1943 on Grassmannians. It is now a part of representation theory called standard monomial theory. The idea of standard basis in the universal enveloping algebra of a Lie algebra is established by the Poincaré–Birkhoff–Witt theorem.
Gröbner bases are also sometimes called standard bases.
In physics, the standard basis vectors for a given Euclidean space are sometimes referred to as the versors of the axes of the corresponding Cartesian coordinate system.
See also
• Canonical units
• Examples of vector spaces § Generalized coordinate space
Citations
1. Roman 2008, p. 47, ch. 1.
2. Axler (2015) harvp error: no target: CITEREFAxler2015 (help) p. 39-40, §2.29
3. Roman 2008, p. 131, ch. 5.
References
• Axler, Sheldon (18 December 2014). Linear Algebra Done Right. Undergraduate Texts in Mathematics (3rd ed.). Springer Publishing (published 2015). ISBN 978-3-319-11079-0.
• Roman, Stephen (2008). Advanced Linear Algebra. Graduate Texts in Mathematics (Third ed.). Springer. ISBN 978-0-387-72828-5. (page 47)
• Ryan, Patrick J. (2000). Euclidean and non-Euclidean geometry: an analytical approach. Cambridge; New York: Cambridge University Press. ISBN 0-521-27635-7. (page 198)
• Schneider, Philip J.; Eberly, David H. (2003). Geometric tools for computer graphics. Amsterdam; Boston: Morgan Kaufmann Publishers. ISBN 1-55860-594-0. (page 112)
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Wikipedia
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Standard conjectures on algebraic cycles
In mathematics, the standard conjectures about algebraic cycles are several conjectures describing the relationship of algebraic cycles and Weil cohomology theories. One of the original applications of these conjectures, envisaged by Alexander Grothendieck, was to prove that his construction of pure motives gave an abelian category that is semisimple. Moreover, as he pointed out, the standard conjectures also imply the hardest part of the Weil conjectures, namely the "Riemann hypothesis" conjecture that remained open at the end of the 1960s and was proved later by Pierre Deligne; for details on the link between Weil and standard conjectures, see Kleiman (1968). The standard conjectures remain open problems, so that their application gives only conditional proofs of results. In quite a few cases, including that of the Weil conjectures, other methods have been found to prove such results unconditionally.
The classical formulations of the standard conjectures involve a fixed Weil cohomology theory H. All of the conjectures deal with "algebraic" cohomology classes, which means a morphism on the cohomology of a smooth projective variety
H ∗(X) → H ∗(X)
induced by an algebraic cycle with rational coefficients on the product X × X via the cycle class map, which is part of the structure of a Weil cohomology theory.
Conjecture A is equivalent to Conjecture B (see Grothendieck (1969), p. 196), and so is not listed.
Lefschetz type Standard Conjecture (Conjecture B)
One of the axioms of a Weil theory is the so-called hard Lefschetz theorem (or axiom):
Begin with a fixed smooth hyperplane section
W = H ∩ X,
where X is a given smooth projective variety in the ambient projective space P N and H is a hyperplane. Then for i ≤ n = dim(X), the Lefschetz operator
L : H i(X) → H i+2(X),
which is defined by intersecting cohomology classes with W, gives an isomorphism
Ln−i : H i(X) → H 2n−i(X).
Now, for i ≤ n define:
Λ = (Ln−i+2)−1 ∘ L ∘ (Ln−i) : H i(X) → H i−2(X)
Λ = (Ln−i) ∘ L ∘ (Ln−i+2)−1 : H 2n−i+2(X) → H 2n−i(X)
The conjecture states that the Lefschetz operator (Λ) is induced by an algebraic cycle.
Künneth type Standard Conjecture (Conjecture C)
It is conjectured that the projectors
H ∗(X) ↠ Hi(X) ↣ H ∗(X)
are algebraic, i.e. induced by a cycle π i ⊂ X × X with rational coefficients. This implies that the motive of any smooth projective variety (and more generally, every pure motive) decomposes as
$h(X)=\bigoplus _{i=0}^{2dim(X)}h^{i}(X).$
The motives $h^{0}(X)$ and $h^{2dim(X)}$ can always be split off as direct summands. The conjecture therefore immediately holds for curves. It was proved for surfaces by Murre (1990). Katz & Messing (1974) have used the Weil conjectures to show the conjecture for algebraic varieties defined over finite fields, in arbitrary dimension.
Šermenev (1974) proved the Künneth decomposition for abelian varieties A. Deninger & Murre (1991) refined this result by exhibiting a functorial Künneth decomposition of the Chow motive of A such that the n-multiplication on the abelian variety acts as $n^{i}$ on the i-th summand $h^{i}(A)$. de Cataldo & Migliorini (2002) proved the Künneth decomposition for the Hilbert scheme of points in a smooth surface.
Conjecture D (numerical equivalence vs. homological equivalence)
Conjecture D states that numerical and homological equivalence agree. (It implies in particular the latter does not depend on the choice of the Weil cohomology theory). This conjecture implies the Lefschetz conjecture. If the Hodge standard conjecture holds, then the Lefschetz conjecture and Conjecture D are equivalent.
This conjecture was shown by Lieberman for varieties of dimension at most 4, and for abelian varieties.[1]
The Hodge Standard Conjecture
The Hodge standard conjecture is modelled on the Hodge index theorem. It states the definiteness (positive or negative, according to the dimension) of the cup product pairing on primitive algebraic cohomology classes. If it holds, then the Lefschetz conjecture implies Conjecture D. In characteristic zero the Hodge standard conjecture holds, being a consequence of Hodge theory. In positive characteristic the Hodge standard conjecture is known for surfaces (Grothendieck (1958)) and for abelian varieties of dimension 4 (Ancona (2020)).
The Hodge standard conjecture is not to be confused with the Hodge conjecture which states that for smooth projective varieties over C, every rational (p, p)-class is algebraic. The Hodge conjecture implies the Lefschetz and Künneth conjectures and conjecture D for varieties over fields of characteristic zero. The Tate conjecture implies Lefschetz, Künneth, and conjecture D for ℓ-adic cohomology over all fields.
Permanence properties of the standard conjectures
For two algebraic varieties X and Y, Arapura (2006) has introduced a condition that Y is motivated by X. The precise condition is that the motive of Y is (in André's category of motives) expressible starting from the motive of X by means of sums, summands, and products. For example, Y is motivated if there is a surjective morphism $X^{n}\to Y$.[2] If Y is not found in the category, it is unmotivated in that context. For smooth projective complex algebraic varieties X and Y, such that Y is motivated by X, the standard conjectures D (homological equivalence equals numerical), B (Lefschetz), the Hodge conjecture and also the generalized Hodge conjecture hold for Y if they hold for all powers of X.[3] This fact can be applied to show, for example, the Lefschetz conjecture for the Hilbert scheme of points on an algebraic surface.
Relation to other conjectures
Beilinson (2012) has shown that the (conjectural) existence of the so-called motivic t-structure on the triangulated category of motives implies the Lefschetz and Künneth standard conjectures B and C.
References
1. Lieberman, David I. (1968), "Numerical and homological equivalence of algebraic cycles on Hodge manifolds", Amer. J. Math., 90 (2): 366–374, doi:10.2307/2373533, JSTOR 2373533
2. Arapura (2006, Cor. 1.2)
3. Arapura (2006, Lemma 4.2)
• Ancona, Giuseppe (2020), "Standard conjectures for abelian fourfolds", Invent. Math., arXiv:1806.03216, doi:10.1007/s00222-020-00990-7, S2CID 119579196
• Arapura, Donu (2006), "Motivation for Hodge cycles", Advances in Mathematics, 207 (2): 762–781, arXiv:math/0501348, doi:10.1016/j.aim.2006.01.005, MR 2271985, S2CID 13897239
• Beilinson, A. (2012), "Remarks on Grothendieck's standard conjectures", Regulators, Contemp. Math., vol. 571, Amer. Math. Soc., Providence, RI, pp. 25–32, arXiv:1006.1116, doi:10.1090/conm/571/11319, ISBN 9780821853221, MR 2953406, S2CID 119687821
• de Cataldo, Mark Andrea A.; Migliorini, Luca (2002), "The Chow groups and the motive of the Hilbert scheme of points on a surface", Journal of Algebra, 251 (2): 824–848, arXiv:math/0005249, doi:10.1006/jabr.2001.9105, MR 1919155, S2CID 16431761
• Deninger, Christopher; Murre, Jacob (1991), "Motivic decomposition of abelian schemes and the Fourier transform", J. Reine Angew. Math., 422: 201–219, MR 1133323
• Grothendieck, A. (1969), "Standard Conjectures on Algebraic Cycles", Algebraic Geometry (Internat. Colloq., Tata Inst. Fund. Res., Bombay, 1968) (PDF), Oxford University Press, pp. 193–199, MR 0268189.
• Grothendieck, A. (1958), "Sur une note de Mattuck-Tate", J. Reine Angew. Math., 1958 (200): 208–215, doi:10.1515/crll.1958.200.208, MR 0136607, S2CID 115548848
• Katz, Nicholas M.; Messing, William (1974), "Some consequences of the Riemann hypothesis for varieties over finite fields", Inventiones Mathematicae, 23: 73–77, Bibcode:1974InMat..23...73K, doi:10.1007/BF01405203, MR 0332791, S2CID 121989640
• Kleiman, Steven L. (1968), "Algebraic cycles and the Weil conjectures", Dix exposés sur la cohomologie des schémas, Amsterdam: North-Holland, pp. 359–386, MR 0292838.
• Murre, J. P. (1990), "On the motive of an algebraic surface", J. Reine Angew. Math., 1990 (409): 190–204, doi:10.1515/crll.1990.409.190, MR 1061525, S2CID 117483201
• Kleiman, Steven L. (1994), "The standard conjectures", Motives (Seattle, WA, 1991), Proceedings of Symposia in Pure Mathematics, vol. 55, American Mathematical Society, pp. 3–20, MR 1265519.
• Šermenev, A. M. (1974), "Motif of an Abelian variety", Funckcional. Anal. I Priložen, 8 (1): 55–61, MR 0335523
External links
• Progress on the standard conjectures on algebraic cycles
• Analogues Kähleriens de certaines conjectures de Weil. J.-P Serre (extrait d'une lettre a A. Weil, 9 Nov. 1959) scan
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Wikipedia
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Standard deviation line
In statistics, the standard deviation line (or SD line) marks points on a scatter plot that are an equal number of standard deviations away from the average in each dimension. For example, in a 2-dimensional scatter diagram with variables $x$ and $y$, points that are 1 standard deviation away from the mean of $x$ and also 1 standard deviation away from the mean of $y$ are on the SD line.[1] The SD line is a useful visual tool since points in a scatter diagram tend to cluster around it,[1] more or less tightly depending on their correlation.
Properties
Relation to regression line
The SD line goes through the point of averages and has a slope of ${\frac {\sigma _{y}}{\sigma _{x}}}$ when the correlation between $x$ and $y$ is positive, and $-{\frac {\sigma _{y}}{\sigma _{x}}}$ when the correlation is negative.[1][2] Unlike the regression line, the SD line does not take into account the relationship between $x$ and $y$.[3] The slope of the SD line is related to that of the regression line by $a=r{\frac {\sigma _{y}}{\sigma _{x}}}$ where $a$ is the slope of the regression line, $r$ is the correlation coefficient, and ${\frac {\sigma _{y}}{\sigma _{x}}}$ is the magnitude of the slope of the SD line.[2]
Typical distance of points to SD line
The root mean square vertical distance of points from the SD line is ${\sqrt {2(1-|r|)}}\times \sigma _{y}$.[1] This gives an idea of the spread of points around the SD line.
1. Freedman, David (1998). Statistics. Robert Pisani, Roger Purves (3rd ed.). New York: W.W. Norton. ISBN 0-393-97083-3. OCLC 36922529.
2. Stark. "Regression". www.stat.berkeley.edu. Retrieved 2022-11-12.
3. Cochran. "Regression". www.stat.ucla.edu. Retrieved 2022-11-12.
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Wikipedia
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Standard error
The standard error (SE)[1] of a statistic (usually an estimate of a parameter) is the standard deviation of its sampling distribution[2] or an estimate of that standard deviation. If the statistic is the sample mean, it is called the standard error of the mean (SEM).[1]
The sampling distribution of a mean is generated by repeated sampling from the same population and recording of the sample means obtained. This forms a distribution of different means, and this distribution has its own mean and variance. Mathematically, the variance of the sampling mean distribution obtained is equal to the variance of the population divided by the sample size. This is because as the sample size increases, sample means cluster more closely around the population mean.
Therefore, the relationship between the standard error of the mean and the standard deviation is such that, for a given sample size, the standard error of the mean equals the standard deviation divided by the square root of the sample size.[1] In other words, the standard error of the mean is a measure of the dispersion of sample means around the population mean.
In regression analysis, the term "standard error" refers either to the square root of the reduced chi-squared statistic or the standard error for a particular regression coefficient (as used in, say, confidence intervals).
Standard error of the sample mean
Exact value
Suppose a statistically independent sample of $n$ observations $x_{1},x_{2},\ldots ,x_{n}$ is taken from a statistical population with a standard deviation of $\sigma $. The mean value calculated from the sample, ${\bar {x}}$, will have an associated standard error on the mean, ${\sigma }_{\bar {x}}$, given by:[1]
${\sigma }_{\bar {x}}\ ={\frac {\sigma }{\sqrt {n}}}$.
Practically this tells us that when trying to estimate the value of a population mean, due to the factor $1/{\sqrt {n}}$, reducing the error on the estimate by a factor of two requires acquiring four times as many observations in the sample; reducing it by a factor of ten requires a hundred times as many observations.
Estimate
The standard deviation $\sigma $ of the population being sampled is seldom known. Therefore, the standard error of the mean is usually estimated by replacing $\sigma $ with the sample standard deviation $\sigma _{x}$ instead:
${\sigma }_{\bar {x}}\ \approx {\frac {\sigma _{x}}{\sqrt {n}}}$.
As this is only an estimator for the true "standard error", it is common to see other notations here such as:
${\widehat {\sigma }}_{\bar {x}}:={\frac {\sigma _{x}}{\sqrt {n}}}$ or alternately ${s}_{\bar {x}}\ :={\frac {s}{\sqrt {n}}}$ :={\frac {s}{\sqrt {n}}}} .
A common source of confusion occurs when failing to distinguish clearly between:
• the standard deviation of the population ($\sigma $),
• the standard deviation of the sample ($\sigma _{x}$),
• the standard deviation of the mean itself ($\sigma _{\bar {x}}$, which is the standard error), and
• the estimator of the standard deviation of the mean (${\widehat {\sigma }}_{\bar {x}}$, which is the most often calculated quantity, and is also often colloquially called the standard error).
Accuracy of the estimator
When the sample size is small, using the standard deviation of the sample instead of the true standard deviation of the population will tend to systematically underestimate the population standard deviation, and therefore also the standard error. With n = 2, the underestimate is about 25%, but for n = 6, the underestimate is only 5%. Gurland and Tripathi (1971) provide a correction and equation for this effect.[3] Sokal and Rohlf (1981) give an equation of the correction factor for small samples of n < 20.[4] See unbiased estimation of standard deviation for further discussion.
Derivation
The standard error on the mean may be derived from the variance of a sum of independent random variables,[5] given the definition of variance and some simple properties thereof. If $x_{1},x_{2},\ldots ,x_{n}$ is a sample of $n$ independent observations from a population with mean ${\bar {x}}$ and standard deviation $\sigma $, then we can define the total
$T=(x_{1}+x_{2}+\cdots +x_{n})$
which due to the Bienaymé formula, will have variance
$\operatorname {Var} (T)={\big (}\operatorname {Var} (x_{1})+\operatorname {Var} (x_{2})+\cdots +\operatorname {Var} (x_{n}){\big )}=n\sigma ^{2}.$
where we've approximated the standard deviations, i.e., the uncertainties, of the measurements themselves with the best value for the standard deviation of the population. The mean of these measurements ${\bar {x}}$ is simply given by
${\bar {x}}=T/n$.
The variance of the mean is then
$\operatorname {Var} ({\bar {x}})=\operatorname {Var} \left({\frac {T}{n}}\right)={\frac {1}{n^{2}}}\operatorname {Var} (T)={\frac {1}{n^{2}}}n\sigma ^{2}={\frac {\sigma ^{2}}{n}}.$
The standard error is, by definition, the standard deviation of ${\bar {x}}$ which is simply the square root of the variance:
$\sigma _{\bar {x}}={\sqrt {\frac {\sigma ^{2}}{n}}}={\frac {\sigma }{\sqrt {n}}}$.
For correlated random variables the sample variance needs to be computed according to the Markov chain central limit theorem.
Independent and identically distributed random variables with random sample size
There are cases when a sample is taken without knowing, in advance, how many observations will be acceptable according to some criterion. In such cases, the sample size $N$ is a random variable whose variation adds to the variation of $X$ such that,
$\operatorname {Var} (T)=\operatorname {E} (N)\operatorname {Var} (X)+\operatorname {Var} (N){\big (}\operatorname {E} (X){\big )}^{2}$[6]
which follows from the law of total variance.
If $N$ has a Poisson distribution, then $\operatorname {E} (N)=\operatorname {Var} (N)$ with estimator $n=N$. Hence the estimator of $\operatorname {Var} (T)$ becomes $nS_{X}^{2}+n{\bar {X}}^{2}$, leading the following formula for standard error:
$\operatorname {Standard~Error} ({\bar {X}})={\sqrt {\frac {S_{X}^{2}+{\bar {X}}^{2}}{n}}}$
(since the standard deviation is the square root of the variance).
Student approximation when σ value is unknown
Further information: Student's t-distribution § Confidence intervals, and Normal distribution § Confidence intervals
In many practical applications, the true value of σ is unknown. As a result, we need to use a distribution that takes into account that spread of possible σ's. When the true underlying distribution is known to be Gaussian, although with unknown σ, then the resulting estimated distribution follows the Student t-distribution. The standard error is the standard deviation of the Student t-distribution. T-distributions are slightly different from Gaussian, and vary depending on the size of the sample. Small samples are somewhat more likely to underestimate the population standard deviation and have a mean that differs from the true population mean, and the Student t-distribution accounts for the probability of these events with somewhat heavier tails compared to a Gaussian. To estimate the standard error of a Student t-distribution it is sufficient to use the sample standard deviation "s" instead of σ, and we could use this value to calculate confidence intervals.
Note: The Student's probability distribution is approximated well by the Gaussian distribution when the sample size is over 100. For such samples one can use the latter distribution, which is much simpler.
Assumptions and usage
Further information: Confidence interval
An example of how $\operatorname {SE} $ is used is to make confidence intervals of the unknown population mean. If the sampling distribution is normally distributed, the sample mean, the standard error, and the quantiles of the normal distribution can be used to calculate confidence intervals for the true population mean. The following expressions can be used to calculate the upper and lower 95% confidence limits, where ${\bar {x}}$ is equal to the sample mean, $\operatorname {SE} $ is equal to the standard error for the sample mean, and 1.96 is the approximate value of the 97.5 percentile point of the normal distribution:
Upper 95% limit $={\bar {x}}+(\operatorname {SE} \times 1.96),$ and
Lower 95% limit $={\bar {x}}-(\operatorname {SE} \times 1.96).$
In particular, the standard error of a sample statistic (such as sample mean) is the actual or estimated standard deviation of the sample mean in the process by which it was generated. In other words, it is the actual or estimated standard deviation of the sampling distribution of the sample statistic. The notation for standard error can be any one of SE, SEM (for standard error of measurement or mean), or SE.
Standard errors provide simple measures of uncertainty in a value and are often used because:
• in many cases, if the standard error of several individual quantities is known then the standard error of some function of the quantities can be easily calculated;
• when the probability distribution of the value is known, it can be used to calculate an exact confidence interval;
• when the probability distribution is unknown, Chebyshev's or the Vysochanskiï–Petunin inequalities can be used to calculate a conservative confidence interval; and
• as the sample size tends to infinity the central limit theorem guarantees that the sampling distribution of the mean is asymptotically normal.
Standard error of mean versus standard deviation
In scientific and technical literature, experimental data are often summarized either using the mean and standard deviation of the sample data or the mean with the standard error. This often leads to confusion about their interchangeability. However, the mean and standard deviation are descriptive statistics, whereas the standard error of the mean is descriptive of the random sampling process. The standard deviation of the sample data is a description of the variation in measurements, while the standard error of the mean is a probabilistic statement about how the sample size will provide a better bound on estimates of the population mean, in light of the central limit theorem.[7]
Put simply, the standard error of the sample mean is an estimate of how far the sample mean is likely to be from the population mean, whereas the standard deviation of the sample is the degree to which individuals within the sample differ from the sample mean.[8] If the population standard deviation is finite, the standard error of the mean of the sample will tend to zero with increasing sample size, because the estimate of the population mean will improve, while the standard deviation of the sample will tend to approximate the population standard deviation as the sample size increases.
Extensions
Finite population correction (FPC)
The formula given above for the standard error assumes that the population is infinite. Nonetheless, it is often used for finite populations when people are interested in measuring the process that created the existing finite population (this is called an analytic study). Though the above formula is not exactly correct when the population is finite, the difference between the finite- and infinite-population versions will be small when sampling fraction is small (e.g. a small proportion of a finite population is studied). In this case people often do not correct for the finite population, essentially treating it as an "approximately infinite" population.
If one is interested in measuring an existing finite population that will not change over time, then it is necessary to adjust for the population size (called an enumerative study). When the sampling fraction (often termed f) is large (approximately at 5% or more) in an enumerative study, the estimate of the standard error must be corrected by multiplying by a ''finite population correction'' (a.k.a.: FPC):[9] [10]
$\operatorname {FPC} ={\sqrt {\frac {N-n}{N-1}}}$
which, for large N:
$\operatorname {FPC} \approx {\sqrt {1-{\frac {n}{N}}}}={\sqrt {1-f}}$
to account for the added precision gained by sampling close to a larger percentage of the population. The effect of the FPC is that the error becomes zero when the sample size n is equal to the population size N.
This happens in survey methodology when sampling without replacement. If sampling with replacement, then FPC does not come into play.
Correction for correlation in the sample
If values of the measured quantity A are not statistically independent but have been obtained from known locations in parameter space x, an unbiased estimate of the true standard error of the mean (actually a correction on the standard deviation part) may be obtained by multiplying the calculated standard error of the sample by the factor f:
$f={\sqrt {\frac {1+\rho }{1-\rho }}},$
where the sample bias coefficient ρ is the widely used Prais–Winsten estimate of the autocorrelation-coefficient (a quantity between −1 and +1) for all sample point pairs. This approximate formula is for moderate to large sample sizes; the reference gives the exact formulas for any sample size, and can be applied to heavily autocorrelated time series like Wall Street stock quotes. Moreover, this formula works for positive and negative ρ alike.[11] See also unbiased estimation of standard deviation for more discussion.
See also
• Illustration of the central limit theorem
• Margin of error
• Probable error
• Standard error of the weighted mean
• Sample mean and sample covariance
• Standard error of the median
• Variance
• Variance of the mean and predicted responses
References
1. Altman, Douglas G; Bland, J Martin (2005-10-15). "Standard deviations and standard errors". BMJ: British Medical Journal. 331 (7521): 903. doi:10.1136/bmj.331.7521.903. ISSN 0959-8138. PMC 1255808. PMID 16223828.
2. Everitt, B. S. (2003). The Cambridge Dictionary of Statistics. CUP. ISBN 978-0-521-81099-9.
3. Gurland, J; Tripathi RC (1971). "A simple approximation for unbiased estimation of the standard deviation". American Statistician. 25 (4): 30–32. doi:10.2307/2682923. JSTOR 2682923.
4. Sokal; Rohlf (1981). Biometry: Principles and Practice of Statistics in Biological Research (2nd ed.). p. 53. ISBN 978-0-7167-1254-1.
5. Hutchinson, T. P. (1993). Essentials of Statistical Methods, in 41 pages. Adelaide: Rumsby. ISBN 978-0-646-12621-0.
6. Cornell, J R, and Benjamin, C A, Probability, Statistics, and Decisions for Civil Engineers, McGraw-Hill, NY, 1970, ISBN 0486796094, pp. 178–9.
7. Barde, M. (2012). "What to use to express the variability of data: Standard deviation or standard error of mean?". Perspect. Clin. Res. 3 (3): 113–116. doi:10.4103/2229-3485.100662. PMC 3487226. PMID 23125963.
8. Wassertheil-Smoller, Sylvia (1995). Biostatistics and Epidemiology : A Primer for Health Professionals (Second ed.). New York: Springer. pp. 40–43. ISBN 0-387-94388-9.
9. Isserlis, L. (1918). "On the value of a mean as calculated from a sample". Journal of the Royal Statistical Society. 81 (1): 75–81. doi:10.2307/2340569. JSTOR 2340569. (Equation 1)
10. Bondy, Warren; Zlot, William (1976). "The Standard Error of the Mean and the Difference Between Means for Finite Populations". The American Statistician. 30 (2): 96–97. doi:10.1080/00031305.1976.10479149. JSTOR 2683803. (Equation 2)
11. Bence, James R. (1995). "Analysis of Short Time Series: Correcting for Autocorrelation". Ecology. 76 (2): 628–639. doi:10.2307/1941218. JSTOR 1941218.
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Ordinary least squares
In statistics, ordinary least squares (OLS) is a type of linear least squares method for choosing the unknown parameters in a linear regression model (with fixed level-one effects of a linear function of a set of explanatory variables) by the principle of least squares: minimizing the sum of the squares of the differences between the observed dependent variable (values of the variable being observed) in the input dataset and the output of the (linear) function of the independent variable.
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Geometrically, this is seen as the sum of the squared distances, parallel to the axis of the dependent variable, between each data point in the set and the corresponding point on the regression surface—the smaller the differences, the better the model fits the data. The resulting estimator can be expressed by a simple formula, especially in the case of a simple linear regression, in which there is a single regressor on the right side of the regression equation.
The OLS estimator is consistent for the level-one fixed effects when the regressors are exogenous and forms perfect colinearity (rank condition), consistent for the variance estimate of the residuals when regressors have finite fourth moments [1] and—by the Gauss–Markov theorem—optimal in the class of linear unbiased estimators when the errors are homoscedastic and serially uncorrelated. Under these conditions, the method of OLS provides minimum-variance mean-unbiased estimation when the errors have finite variances. Under the additional assumption that the errors are normally distributed with zero mean, OLS is the maximum likelihood estimator that outperforms any non-linear unbiased estimator.
Linear model
Main article: Linear regression model
Suppose the data consists of $n$ observations $\left\{\mathbf {x} _{i},y_{i}\right\}_{i=1}^{n}$. Each observation $i$ includes a scalar response $y_{i}$ and a column vector $\mathbf {x} _{i}$ of $p$ parameters (regressors), i.e., $\mathbf {x} _{i}=\left[x_{i1},x_{i2},\dots ,x_{ip}\right]^{\mathsf {T}}$. In a linear regression model, the response variable, $y_{i}$, is a linear function of the regressors:
$y_{i}=\beta _{1}\ x_{i1}+\beta _{2}\ x_{i2}+\cdots +\beta _{p}\ x_{ip}+\varepsilon _{i},$
or in vector form,
$y_{i}=\mathbf {x} _{i}^{\mathsf {T}}{\boldsymbol {\beta }}+\varepsilon _{i},\,$
where $\mathbf {x} _{i}$, as introduced previously, is a column vector of the $i$-th observation of all the explanatory variables; ${\boldsymbol {\beta }}$ is a $p\times 1$ vector of unknown parameters; and the scalar $\varepsilon _{i}$ represents unobserved random variables (errors) of the $i$-th observation. $\varepsilon _{i}$ accounts for the influences upon the responses $y_{i}$ from sources other than the explanatory variables $\mathbf {x} _{i}$. This model can also be written in matrix notation as
$\mathbf {y} =\mathbf {X} {\boldsymbol {\beta }}+{\boldsymbol {\varepsilon }},\,$
where $\mathbf {y} $ and ${\boldsymbol {\varepsilon }}$ are $n\times 1$ vectors of the response variables and the errors of the $n$ observations, and $\mathbf {X} $ is an $n\times p$ matrix of regressors, also sometimes called the design matrix, whose row $i$ is $\mathbf {x} _{i}^{\mathsf {T}}$ and contains the $i$-th observations on all the explanatory variables.
Typically, a constant term is included in the set of regressors $\mathbf {X} $, say, by taking $x_{i1}=1$ for all $i=1,\dots ,n$. The coefficient $\beta _{1}$ corresponding to this regressor is called the intercept. Without the intercept, the fitted line is forced to cross the origin when $x_{i}={\vec {0}}$.
Regressors do not have to be independent: there can be any desired relationship between the regressors (so long as it is not a linear relationship). For instance, we might suspect the response depends linearly both on a value and its square; in which case we would include one regressor whose value is just the square of another regressor. In that case, the model would be quadratic in the second regressor, but none-the-less is still considered a linear model because the model is still linear in the parameters (${\boldsymbol {\beta }}$).
Matrix/vector formulation
Consider an overdetermined system
$\sum _{j=1}^{p}x_{ij}\beta _{j}=y_{i},\ (i=1,2,\dots ,n),$
of $n$ linear equations in $p$ unknown coefficients, $\beta _{1},\beta _{2},\dots ,\beta _{p}$, with $n>p$. This can be written in matrix form as
$\mathbf {X} {\boldsymbol {\beta }}=\mathbf {y} ,$
where
$\mathbf {X} ={\begin{bmatrix}X_{11}&X_{12}&\cdots &X_{1p}\\X_{21}&X_{22}&\cdots &X_{2p}\\\vdots &\vdots &\ddots &\vdots \\X_{n1}&X_{n2}&\cdots &X_{np}\end{bmatrix}},\qquad {\boldsymbol {\beta }}={\begin{bmatrix}\beta _{1}\\\beta _{2}\\\vdots \\\beta _{p}\end{bmatrix}},\qquad \mathbf {y} ={\begin{bmatrix}y_{1}\\y_{2}\\\vdots \\y_{n}\end{bmatrix}}.$
(Note: for a linear model as above, not all elements in $\mathbf {X} $ contains information on the data points. The first column is populated with ones, $X_{i1}=1$. Only the other columns contain actual data. So here $p$ is equal to the number of regressors plus one).
Such a system usually has no exact solution, so the goal is instead to find the coefficients ${\boldsymbol {\beta }}$ which fit the equations "best", in the sense of solving the quadratic minimization problem
${\hat {\boldsymbol {\beta }}}={\underset {\boldsymbol {\beta }}{\operatorname {arg\,min} }}\,S({\boldsymbol {\beta }}),$
where the objective function $S$ is given by
$S({\boldsymbol {\beta }})=\sum _{i=1}^{n}\left|y_{i}-\sum _{j=1}^{p}X_{ij}\beta _{j}\right|^{2}=\left\|\mathbf {y} -\mathbf {X} {\boldsymbol {\beta }}\right\|^{2}.$
A justification for choosing this criterion is given in Properties below. This minimization problem has a unique solution, provided that the $p$ columns of the matrix $\mathbf {X} $ are linearly independent, given by solving the so-called normal equations:
$\left(\mathbf {X} ^{\mathsf {T}}\mathbf {X} \right){\hat {\boldsymbol {\beta }}}=\mathbf {X} ^{\mathsf {T}}\mathbf {y} \ .$
The matrix $\mathbf {X} ^{\mathsf {T}}\mathbf {X} $ is known as the normal matrix or Gram matrix and the matrix $\mathbf {X} ^{\mathsf {T}}\mathbf {y} $ is known as the moment matrix of regressand by regressors.[2] Finally, ${\hat {\boldsymbol {\beta }}}$ is the coefficient vector of the least-squares hyperplane, expressed as
${\hat {\boldsymbol {\beta }}}=\left(\mathbf {X} ^{\mathsf {T}}\mathbf {X} \right)^{-1}\mathbf {X} ^{\mathsf {T}}\mathbf {y} .$
or
${\hat {\boldsymbol {\beta }}}={\boldsymbol {\beta }}+\left(\mathbf {X} ^{\mathsf {T}}\mathbf {X} \right)^{-1}\mathbf {X} ^{\mathsf {T}}{\boldsymbol {\varepsilon }}.$
Estimation
Suppose b is a "candidate" value for the parameter vector β. The quantity yi − xiTb, called the residual for the i-th observation, measures the vertical distance between the data point (xi, yi) and the hyperplane y = xTb, and thus assesses the degree of fit between the actual data and the model. The sum of squared residuals (SSR) (also called the error sum of squares (ESS) or residual sum of squares (RSS))[3] is a measure of the overall model fit:
$S(b)=\sum _{i=1}^{n}(y_{i}-x_{i}^{\mathrm {T} }b)^{2}=(y-Xb)^{\mathrm {T} }(y-Xb),$
where T denotes the matrix transpose, and the rows of X, denoting the values of all the independent variables associated with a particular value of the dependent variable, are Xi = xiT. The value of b which minimizes this sum is called the OLS estimator for β. The function S(b) is quadratic in b with positive-definite Hessian, and therefore this function possesses a unique global minimum at $b={\hat {\beta }}$, which can be given by the explicit formula:[4][proof]
${\hat {\beta }}=\operatorname {argmin} _{b\in \mathbb {R} ^{p}}S(b)=(X^{\mathrm {T} }X)^{-1}X^{\mathrm {T} }y\ .$
The product N=XT X is a Gram matrix and its inverse, Q=N–1, is the cofactor matrix of β,[5][6][7] closely related to its covariance matrix, Cβ. The matrix (XT X)–1 XT=Q XT is called the Moore–Penrose pseudoinverse matrix of X. This formulation highlights the point that estimation can be carried out if, and only if, there is no perfect multicollinearity between the explanatory variables (which would cause the gram matrix to have no inverse).
After we have estimated β, the fitted values (or predicted values) from the regression will be
${\hat {y}}=X{\hat {\beta }}=Py,$
where P = X(XTX)−1XT is the projection matrix onto the space V spanned by the columns of X. This matrix P is also sometimes called the hat matrix because it "puts a hat" onto the variable y. Another matrix, closely related to P is the annihilator matrix M = In − P; this is a projection matrix onto the space orthogonal to V. Both matrices P and M are symmetric and idempotent (meaning that P2 = P and M2 = M), and relate to the data matrix X via identities PX = X and MX = 0.[8] Matrix M creates the residuals from the regression:
${\hat {\varepsilon }}=y-{\hat {y}}=y-X{\hat {\beta }}=My=M(X\beta +\varepsilon )=(MX)\beta +M\varepsilon =M\varepsilon .$
Using these residuals we can estimate the value of σ 2 using the reduced chi-squared statistic:
$s^{2}={\frac {{\hat {\varepsilon }}^{\mathrm {T} }{\hat {\varepsilon }}}{n-p}}={\frac {(My)^{\mathrm {T} }My}{n-p}}={\frac {y^{\mathrm {T} }M^{\mathrm {T} }My}{n-p}}={\frac {y^{\mathrm {T} }My}{n-p}}={\frac {S({\hat {\beta }})}{n-p}},\qquad {\hat {\sigma }}^{2}={\frac {n-p}{n}}\;s^{2}$
The denominator, n−p, is the statistical degrees of freedom. The first quantity, s2, is the OLS estimate for σ2, whereas the second, $\scriptstyle {\hat {\sigma }}^{2}$, is the MLE estimate for σ2. The two estimators are quite similar in large samples; the first estimator is always unbiased, while the second estimator is biased but has a smaller mean squared error. In practice s2 is used more often, since it is more convenient for the hypothesis testing. The square root of s2 is called the regression standard error,[9] standard error of the regression,[10][11] or standard error of the equation.[8]
It is common to assess the goodness-of-fit of the OLS regression by comparing how much the initial variation in the sample can be reduced by regressing onto X. The coefficient of determination R2 is defined as a ratio of "explained" variance to the "total" variance of the dependent variable y, in the cases where the regression sum of squares equals the sum of squares of residuals:[12]
$R^{2}={\frac {\sum ({\hat {y}}_{i}-{\overline {y}})^{2}}{\sum (y_{i}-{\overline {y}})^{2}}}={\frac {y^{\mathrm {T} }P^{\mathrm {T} }LPy}{y^{\mathrm {T} }Ly}}=1-{\frac {y^{\mathrm {T} }My}{y^{\mathrm {T} }Ly}}=1-{\frac {\rm {RSS}}{\rm {TSS}}}$
where TSS is the total sum of squares for the dependent variable, $ L=I_{n}-{\frac {1}{n}}J_{n}$, and $ J_{n}$ is an n×n matrix of ones. ($L$ is a centering matrix which is equivalent to regression on a constant; it simply subtracts the mean from a variable.) In order for R2 to be meaningful, the matrix X of data on regressors must contain a column vector of ones to represent the constant whose coefficient is the regression intercept. In that case, R2 will always be a number between 0 and 1, with values close to 1 indicating a good degree of fit.
The variance in the prediction of the independent variable as a function of the dependent variable is given in the article Polynomial least squares.
Simple linear regression model
Main article: Simple linear regression
If the data matrix X contains only two variables, a constant and a scalar regressor xi, then this is called the "simple regression model". This case is often considered in the beginner statistics classes, as it provides much simpler formulas even suitable for manual calculation. The parameters are commonly denoted as (α, β):
$y_{i}=\alpha +\beta x_{i}+\varepsilon _{i}.$
The least squares estimates in this case are given by simple formulas
${\begin{aligned}{\widehat {\beta }}&={\frac {\sum _{i=1}^{n}{(x_{i}-{\bar {x}})(y_{i}-{\bar {y}})}}{\sum _{i=1}^{n}{(x_{i}-{\bar {x}})^{2}}}}\\[2pt]{\widehat {\alpha }}&={\bar {y}}-{\widehat {\beta }}\,{\bar {x}}\ ,\end{aligned}}$
Alternative derivations
In the previous section the least squares estimator ${\hat {\beta }}$ was obtained as a value that minimizes the sum of squared residuals of the model. However it is also possible to derive the same estimator from other approaches. In all cases the formula for OLS estimator remains the same: ^β = (XTX)−1XTy; the only difference is in how we interpret this result.
Projection
For mathematicians, OLS is an approximate solution to an overdetermined system of linear equations Xβ ≈ y, where β is the unknown. Assuming the system cannot be solved exactly (the number of equations n is much larger than the number of unknowns p), we are looking for a solution that could provide the smallest discrepancy between the right- and left- hand sides. In other words, we are looking for the solution that satisfies
${\hat {\beta }}={\rm {arg}}\min _{\beta }\,\lVert \mathbf {y} -\mathbf {X} {\boldsymbol {\beta }}\rVert ,$
where ‖·‖ is the standard L2 norm in the n-dimensional Euclidean space Rn. The predicted quantity Xβ is just a certain linear combination of the vectors of regressors. Thus, the residual vector y − Xβ will have the smallest length when y is projected orthogonally onto the linear subspace spanned by the columns of X. The OLS estimator ${\hat {\beta }}$ in this case can be interpreted as the coefficients of vector decomposition of ^y = Py along the basis of X.
In other words, the gradient equations at the minimum can be written as:
$(\mathbf {y} -\mathbf {X} {\hat {\boldsymbol {\beta }}})^{\top }\mathbf {X} =0.$
A geometrical interpretation of these equations is that the vector of residuals, $\mathbf {y} -X{\hat {\boldsymbol {\beta }}}$ is orthogonal to the column space of X, since the dot product $(\mathbf {y} -\mathbf {X} {\hat {\boldsymbol {\beta }}})\cdot \mathbf {X} \mathbf {v} $ is equal to zero for any conformal vector, v. This means that $\mathbf {y} -\mathbf {X} {\boldsymbol {\hat {\beta }}}$ is the shortest of all possible vectors $\mathbf {y} -\mathbf {X} {\boldsymbol {\beta }}$, that is, the variance of the residuals is the minimum possible. This is illustrated at the right.
Introducing ${\hat {\boldsymbol {\gamma }}}$ and a matrix K with the assumption that a matrix $[\mathbf {X} \ \mathbf {K} ]$ is non-singular and KT X = 0 (cf. Orthogonal projections), the residual vector should satisfy the following equation:
${\hat {\mathbf {r} }}:=\mathbf {y} -\mathbf {X} {\hat {\boldsymbol {\beta }}}=\mathbf {K} {\hat {\boldsymbol {\gamma }}}.$
The equation and solution of linear least squares are thus described as follows:
${\begin{aligned}\mathbf {y} &={\begin{bmatrix}\mathbf {X} &\mathbf {K} \end{bmatrix}}{\begin{bmatrix}{\hat {\boldsymbol {\beta }}}\\{\hat {\boldsymbol {\gamma }}}\end{bmatrix}},\\{}\Rightarrow {\begin{bmatrix}{\hat {\boldsymbol {\beta }}}\\{\hat {\boldsymbol {\gamma }}}\end{bmatrix}}&={\begin{bmatrix}\mathbf {X} &\mathbf {K} \end{bmatrix}}^{-1}\mathbf {y} ={\begin{bmatrix}\left(\mathbf {X} ^{\top }\mathbf {X} \right)^{-1}\mathbf {X} ^{\top }\\\left(\mathbf {K} ^{\top }\mathbf {K} \right)^{-1}\mathbf {K} ^{\top }\end{bmatrix}}\mathbf {y} .\end{aligned}}$
Another way of looking at it is to consider the regression line to be a weighted average of the lines passing through the combination of any two points in the dataset.[13] Although this way of calculation is more computationally expensive, it provides a better intuition on OLS.
Maximum likelihood
The OLS estimator is identical to the maximum likelihood estimator (MLE) under the normality assumption for the error terms.[14][proof] This normality assumption has historical importance, as it provided the basis for the early work in linear regression analysis by Yule and Pearson. From the properties of MLE, we can infer that the OLS estimator is asymptotically efficient (in the sense of attaining the Cramér–Rao bound for variance) if the normality assumption is satisfied.[15]
Generalized method of moments
In iid case the OLS estimator can also be viewed as a GMM estimator arising from the moment conditions
$\mathrm {E} {\big [}\,x_{i}\left(y_{i}-x_{i}^{\mathsf {T}}\beta \right)\,{\big ]}=0.$
These moment conditions state that the regressors should be uncorrelated with the errors. Since xi is a p-vector, the number of moment conditions is equal to the dimension of the parameter vector β, and thus the system is exactly identified. This is the so-called classical GMM case, when the estimator does not depend on the choice of the weighting matrix.
Note that the original strict exogeneity assumption E[εi | xi] = 0 implies a far richer set of moment conditions than stated above. In particular, this assumption implies that for any vector-function ƒ, the moment condition E[ƒ(xi)·εi] = 0 will hold. However it can be shown using the Gauss–Markov theorem that the optimal choice of function ƒ is to take ƒ(x) = x, which results in the moment equation posted above.
Properties
Assumptions
See also: Linear regression § Assumptions
There are several different frameworks in which the linear regression model can be cast in order to make the OLS technique applicable. Each of these settings produces the same formulas and same results. The only difference is the interpretation and the assumptions which have to be imposed in order for the method to give meaningful results. The choice of the applicable framework depends mostly on the nature of data in hand, and on the inference task which has to be performed.
One of the lines of difference in interpretation is whether to treat the regressors as random variables, or as predefined constants. In the first case (random design) the regressors xi are random and sampled together with the yi's from some population, as in an observational study. This approach allows for more natural study of the asymptotic properties of the estimators. In the other interpretation (fixed design), the regressors X are treated as known constants set by a design, and y is sampled conditionally on the values of X as in an experiment. For practical purposes, this distinction is often unimportant, since estimation and inference is carried out while conditioning on X. All results stated in this article are within the random design framework.
Classical linear regression model
The classical model focuses on the "finite sample" estimation and inference, meaning that the number of observations n is fixed. This contrasts with the other approaches, which study the asymptotic behavior of OLS, and in which the number of observations is allowed to grow to infinity.
• Correct specification. The linear functional form must coincide with the form of the actual data-generating process.
• Strict exogeneity. The errors in the regression should have conditional mean zero:[16]
$\operatorname {E} [\,\varepsilon \mid X\,]=0.$
The immediate consequence of the exogeneity assumption is that the errors have mean zero: E[ε] = 0 (for the law of total expectation), and that the regressors are uncorrelated with the errors: E[XTε] = 0.
The exogeneity assumption is critical for the OLS theory. If it holds then the regressor variables are called exogenous. If it doesn't, then those regressors that are correlated with the error term are called endogenous,[17] and the OLS estimator becomes biased. In such case the method of instrumental variables may be used to carry out inference.
• No linear dependence. The regressors in X must all be linearly independent. Mathematically, this means that the matrix X must have full column rank almost surely:[18]
$\Pr \!{\big [}\,\operatorname {rank} (X)=p\,{\big ]}=1.$
Usually, it is also assumed that the regressors have finite moments up to at least the second moment. Then the matrix Qxx = E[XTX / n] is finite and positive semi-definite.
When this assumption is violated the regressors are called linearly dependent or perfectly multicollinear. In such case the value of the regression coefficient β cannot be learned, although prediction of y values is still possible for new values of the regressors that lie in the same linearly dependent subspace.
• Spherical errors:[18]
$\operatorname {Var} [\,\varepsilon \mid X\,]=\sigma ^{2}I_{n},$
where In is the identity matrix in dimension n, and σ2 is a parameter which determines the variance of each observation. This σ2 is considered a nuisance parameter in the model, although usually it is also estimated. If this assumption is violated then the OLS estimates are still valid, but no longer efficient.
It is customary to split this assumption into two parts:
• Homoscedasticity: E[ εi2 | X ] = σ2, which means that the error term has the same variance σ2 in each observation. When this requirement is violated this is called heteroscedasticity, in such case a more efficient estimator would be weighted least squares. If the errors have infinite variance then the OLS estimates will also have infinite variance (although by the law of large numbers they will nonetheless tend toward the true values so long as the errors have zero mean). In this case, robust estimation techniques are recommended.
• No autocorrelation: the errors are uncorrelated between observations: E[ εiεj | X ] = 0 for i ≠ j. This assumption may be violated in the context of time series data, panel data, cluster samples, hierarchical data, repeated measures data, longitudinal data, and other data with dependencies. In such cases generalized least squares provides a better alternative than the OLS. Another expression for autocorrelation is serial correlation.
• Normality. It is sometimes additionally assumed that the errors have normal distribution conditional on the regressors:[19]
$\varepsilon \mid X\sim {\mathcal {N}}(0,\sigma ^{2}I_{n}).$
This assumption is not needed for the validity of the OLS method, although certain additional finite-sample properties can be established in case when it does (especially in the area of hypotheses testing). Also when the errors are normal, the OLS estimator is equivalent to the maximum likelihood estimator (MLE), and therefore it is asymptotically efficient in the class of all regular estimators. Importantly, the normality assumption applies only to the error terms; contrary to a popular misconception, the response (dependent) variable is not required to be normally distributed.[20]
Independent and identically distributed (iid)
In some applications, especially with cross-sectional data, an additional assumption is imposed — that all observations are independent and identically distributed. This means that all observations are taken from a random sample which makes all the assumptions listed earlier simpler and easier to interpret. Also this framework allows one to state asymptotic results (as the sample size n → ∞), which are understood as a theoretical possibility of fetching new independent observations from the data generating process. The list of assumptions in this case is:
• iid observations: (xi, yi) is independent from, and has the same distribution as, (xj, yj) for all i ≠ j;
• no perfect multicollinearity: Qxx = E[ xi xiT ] is a positive-definite matrix;
• exogeneity: E[ εi | xi ] = 0;
• homoscedasticity: Var[ εi | xi ] = σ2.
Time series model
• The stochastic process {xi, yi} is stationary and ergodic; if {xi, yi} is nonstationary, OLS results are often spurious unless {xi, yi} is co-integrating.[21]
• The regressors are predetermined: E[xiεi] = 0 for all i = 1, ..., n;
• The p×p matrix Qxx = E[ xi xiT ] is of full rank, and hence positive-definite;
• {xiεi} is a martingale difference sequence, with a finite matrix of second moments Qxxε² = E[ εi2xi xiT ].
Finite sample properties
First of all, under the strict exogeneity assumption the OLS estimators $\scriptstyle {\hat {\beta }}$ and s2 are unbiased, meaning that their expected values coincide with the true values of the parameters:[22][proof]
$\operatorname {E} [\,{\hat {\beta }}\mid X\,]=\beta ,\quad \operatorname {E} [\,s^{2}\mid X\,]=\sigma ^{2}.$
If the strict exogeneity does not hold (as is the case with many time series models, where exogeneity is assumed only with respect to the past shocks but not the future ones), then these estimators will be biased in finite samples.
The variance-covariance matrix (or simply covariance matrix) of $\scriptstyle {\hat {\beta }}$ is equal to[23]
$\operatorname {Var} [\,{\hat {\beta }}\mid X\,]=\sigma ^{2}\left(X^{\mathsf {T}}X\right)^{-1}=\sigma ^{2}Q.$
In particular, the standard error of each coefficient $\scriptstyle {\hat {\beta }}_{j}$ is equal to square root of the j-th diagonal element of this matrix. The estimate of this standard error is obtained by replacing the unknown quantity σ2 with its estimate s2. Thus,
${\widehat {\operatorname {s.\!e.} }}({\hat {\beta }}_{j})={\sqrt {s^{2}\left(X^{\mathsf {T}}X\right)_{jj}^{-1}}}$
It can also be easily shown that the estimator $\scriptstyle {\hat {\beta }}$ is uncorrelated with the residuals from the model:[23]
$\operatorname {Cov} [\,{\hat {\beta }},{\hat {\varepsilon }}\mid X\,]=0.$
The Gauss–Markov theorem states that under the spherical errors assumption (that is, the errors should be uncorrelated and homoscedastic) the estimator $\scriptstyle {\hat {\beta }}$ is efficient in the class of linear unbiased estimators. This is called the best linear unbiased estimator (BLUE). Efficiency should be understood as if we were to find some other estimator $\scriptstyle {\tilde {\beta }}$ which would be linear in y and unbiased, then [23]
$\operatorname {Var} [\,{\tilde {\beta }}\mid X\,]-\operatorname {Var} [\,{\hat {\beta }}\mid X\,]\geq 0$
in the sense that this is a nonnegative-definite matrix. This theorem establishes optimality only in the class of linear unbiased estimators, which is quite restrictive. Depending on the distribution of the error terms ε, other, non-linear estimators may provide better results than OLS.
Assuming normality
The properties listed so far are all valid regardless of the underlying distribution of the error terms. However, if you are willing to assume that the normality assumption holds (that is, that ε ~ N(0, σ2In)), then additional properties of the OLS estimators can be stated.
The estimator $\scriptstyle {\hat {\beta }}$ is normally distributed, with mean and variance as given before:[24]
${\hat {\beta }}\ \sim \ {\mathcal {N}}{\big (}\beta ,\ \sigma ^{2}(X^{\mathrm {T} }X)^{-1}{\big )}.$
This estimator reaches the Cramér–Rao bound for the model, and thus is optimal in the class of all unbiased estimators.[15] Note that unlike the Gauss–Markov theorem, this result establishes optimality among both linear and non-linear estimators, but only in the case of normally distributed error terms.
The estimator s2 will be proportional to the chi-squared distribution:[25]
$s^{2}\ \sim \ {\frac {\sigma ^{2}}{n-p}}\cdot \chi _{n-p}^{2}$
The variance of this estimator is equal to 2σ4/(n − p), which does not attain the Cramér–Rao bound of 2σ4/n. However it was shown that there are no unbiased estimators of σ2 with variance smaller than that of the estimator s2.[26] If we are willing to allow biased estimators, and consider the class of estimators that are proportional to the sum of squared residuals (SSR) of the model, then the best (in the sense of the mean squared error) estimator in this class will be ~σ2 = SSR / (n − p + 2), which even beats the Cramér–Rao bound in case when there is only one regressor (p = 1).[27]
Moreover, the estimators $\scriptstyle {\hat {\beta }}$ and s2 are independent,[28] the fact which comes in useful when constructing the t- and F-tests for the regression.
Influential observations
Main article: Influential observation
See also: Leverage (statistics)
As was mentioned before, the estimator ${\hat {\beta }}$ is linear in y, meaning that it represents a linear combination of the dependent variables yi. The weights in this linear combination are functions of the regressors X, and generally are unequal. The observations with high weights are called influential because they have a more pronounced effect on the value of the estimator.
To analyze which observations are influential we remove a specific j-th observation and consider how much the estimated quantities are going to change (similarly to the jackknife method). It can be shown that the change in the OLS estimator for β will be equal to [29]
${\hat {\beta }}^{(j)}-{\hat {\beta }}=-{\frac {1}{1-h_{j}}}(X^{\mathrm {T} }X)^{-1}x_{j}^{\mathrm {T} }{\hat {\varepsilon }}_{j}\,,$
where hj = xjT (XTX)−1xj is the j-th diagonal element of the hat matrix P, and xj is the vector of regressors corresponding to the j-th observation. Similarly, the change in the predicted value for j-th observation resulting from omitting that observation from the dataset will be equal to [29]
${\hat {y}}_{j}^{(j)}-{\hat {y}}_{j}=x_{j}^{\mathrm {T} }{\hat {\beta }}^{(j)}-x_{j}^{T}{\hat {\beta }}=-{\frac {h_{j}}{1-h_{j}}}\,{\hat {\varepsilon }}_{j}$
From the properties of the hat matrix, 0 ≤ hj ≤ 1, and they sum up to p, so that on average hj ≈ p/n. These quantities hj are called the leverages, and observations with high hj are called leverage points.[30] Usually the observations with high leverage ought to be scrutinized more carefully, in case they are erroneous, or outliers, or in some other way atypical of the rest of the dataset.
Partitioned regression
Sometimes the variables and corresponding parameters in the regression can be logically split into two groups, so that the regression takes form
$y=X_{1}\beta _{1}+X_{2}\beta _{2}+\varepsilon ,$
where X1 and X2 have dimensions n×p1, n×p2, and β1, β2 are p1×1 and p2×1 vectors, with p1 + p2 = p.
The Frisch–Waugh–Lovell theorem states that in this regression the residuals ${\hat {\varepsilon }}$ and the OLS estimate $\scriptstyle {\hat {\beta }}_{2}$ will be numerically identical to the residuals and the OLS estimate for β2 in the following regression:[31]
$M_{1}y=M_{1}X_{2}\beta _{2}+\eta \,,$
where M1 is the annihilator matrix for regressors X1.
The theorem can be used to establish a number of theoretical results. For example, having a regression with a constant and another regressor is equivalent to subtracting the means from the dependent variable and the regressor and then running the regression for the de-meaned variables but without the constant term.
Constrained estimation
Suppose it is known that the coefficients in the regression satisfy a system of linear equations
$A\colon \quad Q^{T}\beta =c,\,$
where Q is a p×q matrix of full rank, and c is a q×1 vector of known constants, where q < p. In this case least squares estimation is equivalent to minimizing the sum of squared residuals of the model subject to the constraint A. The constrained least squares (CLS) estimator can be given by an explicit formula:[32]
${\hat {\beta }}^{c}={\hat {\beta }}-(X^{T}X)^{-1}Q{\Big (}Q^{T}(X^{T}X)^{-1}Q{\Big )}^{-1}(Q^{T}{\hat {\beta }}-c).$
This expression for the constrained estimator is valid as long as the matrix XTX is invertible. It was assumed from the beginning of this article that this matrix is of full rank, and it was noted that when the rank condition fails, β will not be identifiable. However it may happen that adding the restriction A makes β identifiable, in which case one would like to find the formula for the estimator. The estimator is equal to [33]
${\hat {\beta }}^{c}=R(R^{T}X^{T}XR)^{-1}R^{T}X^{T}y+{\Big (}I_{p}-R(R^{T}X^{T}XR)^{-1}R^{T}X^{T}X{\Big )}Q(Q^{T}Q)^{-1}c,$
where R is a p×(p − q) matrix such that the matrix [Q R] is non-singular, and RTQ = 0. Such a matrix can always be found, although generally it is not unique. The second formula coincides with the first in case when XTX is invertible.[33]
Large sample properties
The least squares estimators are point estimates of the linear regression model parameters β. However, generally we also want to know how close those estimates might be to the true values of parameters. In other words, we want to construct the interval estimates.
Since we haven't made any assumption about the distribution of error term εi, it is impossible to infer the distribution of the estimators ${\hat {\beta }}$ and ${\hat {\sigma }}^{2}$. Nevertheless, we can apply the central limit theorem to derive their asymptotic properties as sample size n goes to infinity. While the sample size is necessarily finite, it is customary to assume that n is "large enough" so that the true distribution of the OLS estimator is close to its asymptotic limit.
We can show that under the model assumptions, the least squares estimator for β is consistent (that is ${\hat {\beta }}$ converges in probability to β) and asymptotically normal:[proof]
$({\hat {\beta }}-\beta )\ {\xrightarrow {d}}\ {\mathcal {N}}{\big (}0,\;\sigma ^{2}Q_{xx}^{-1}{\big )},$
where $Q_{xx}=X^{T}X.$
Intervals
Main articles: Confidence interval and Prediction interval
Using this asymptotic distribution, approximate two-sided confidence intervals for the j-th component of the vector ${\hat {\beta }}$ can be constructed as
$\beta _{j}\in {\bigg [}\ {\hat {\beta }}_{j}\pm q_{1-{\frac {\alpha }{2}}}^{{\mathcal {N}}(0,1)}\!{\sqrt {{\hat {\sigma }}^{2}\left[Q_{xx}^{-1}\right]_{jj}}}\ {\bigg ]}$ at the 1 − α confidence level,
where q denotes the quantile function of standard normal distribution, and [·]jj is the j-th diagonal element of a matrix.
Similarly, the least squares estimator for σ2 is also consistent and asymptotically normal (provided that the fourth moment of εi exists) with limiting distribution
$({\hat {\sigma }}^{2}-\sigma ^{2})\ {\xrightarrow {d}}\ {\mathcal {N}}\left(0,\;\operatorname {E} \left[\varepsilon _{i}^{4}\right]-\sigma ^{4}\right).$
These asymptotic distributions can be used for prediction, testing hypotheses, constructing other estimators, etc.. As an example consider the problem of prediction. Suppose $x_{0}$ is some point within the domain of distribution of the regressors, and one wants to know what the response variable would have been at that point. The mean response is the quantity $y_{0}=x_{0}^{\mathrm {T} }\beta $, whereas the predicted response is ${\hat {y}}_{0}=x_{0}^{\mathrm {T} }{\hat {\beta }}$. Clearly the predicted response is a random variable, its distribution can be derived from that of ${\hat {\beta }}$:
$\left({\hat {y}}_{0}-y_{0}\right)\ {\xrightarrow {d}}\ {\mathcal {N}}\left(0,\;\sigma ^{2}x_{0}^{\mathrm {T} }Q_{xx}^{-1}x_{0}\right),$
which allows construct confidence intervals for mean response $y_{0}$ to be constructed:
$y_{0}\in \left[\ x_{0}^{\mathrm {T} }{\hat {\beta }}\pm q_{1-{\frac {\alpha }{2}}}^{{\mathcal {N}}(0,1)}\!{\sqrt {{\hat {\sigma }}^{2}x_{0}^{\mathrm {T} }Q_{xx}^{-1}x_{0}}}\ \right]$ at the 1 − α confidence level.
Hypothesis testing
Main article: Hypothesis testing
Two hypothesis tests are particularly widely used. First, one wants to know if the estimated regression equation is any better than simply predicting that all values of the response variable equal its sample mean (if not, it is said to have no explanatory power). The null hypothesis of no explanatory value of the estimated regression is tested using an F-test. If the calculated F-value is found to be large enough to exceed its critical value for the pre-chosen level of significance, the null hypothesis is rejected and the alternative hypothesis, that the regression has explanatory power, is accepted. Otherwise, the null hypothesis of no explanatory power is accepted.
Second, for each explanatory variable of interest, one wants to know whether its estimated coefficient differs significantly from zero—that is, whether this particular explanatory variable in fact has explanatory power in predicting the response variable. Here the null hypothesis is that the true coefficient is zero. This hypothesis is tested by computing the coefficient's t-statistic, as the ratio of the coefficient estimate to its standard error. If the t-statistic is larger than a predetermined value, the null hypothesis is rejected and the variable is found to have explanatory power, with its coefficient significantly different from zero. Otherwise, the null hypothesis of a zero value of the true coefficient is accepted.
In addition, the Chow test is used to test whether two subsamples both have the same underlying true coefficient values. The sum of squared residuals of regressions on each of the subsets and on the combined data set are compared by computing an F-statistic; if this exceeds a critical value, the null hypothesis of no difference between the two subsets is rejected; otherwise, it is accepted.
Example with real data
See also: Simple linear regression § Example, and Linear least squares § Example
The following data set gives average heights and weights for American women aged 30–39 (source: The World Almanac and Book of Facts, 1975).
Height (m) 1.471.501.521.551.57
Weight (kg) 52.2153.1254.4855.8457.20
Height (m) 1.601.631.651.681.70
Weight (kg) 58.5759.9361.2963.1164.47
Height (m) 1.731.751.781.801.83
Weight (kg) 66.2868.1069.9272.1974.46
When only one dependent variable is being modeled, a scatterplot will suggest the form and strength of the relationship between the dependent variable and regressors. It might also reveal outliers, heteroscedasticity, and other aspects of the data that may complicate the interpretation of a fitted regression model. The scatterplot suggests that the relationship is strong and can be approximated as a quadratic function. OLS can handle non-linear relationships by introducing the regressor HEIGHT2. The regression model then becomes a multiple linear model:
$w_{i}=\beta _{1}+\beta _{2}h_{i}+\beta _{3}h_{i}^{2}+\varepsilon _{i}.$
The output from most popular statistical packages will look similar to this:
MethodLeast squares
Dependent variableWEIGHT
Observations15
Parameter Value Std error t-statistic p-value
$\beta _{1}$ 128.812816.30837.89860.0000
$\beta _{2}$ –143.162019.8332–7.21830.0000
$\beta _{3}$ 61.96036.008410.31220.0000
R20.9989 S.E. of regression0.2516
Adjusted R20.9987 Model sum-of-sq.692.61
Log-likelihood1.0890 Residual sum-of-sq.0.7595
Durbin–Watson stat.2.1013 Total sum-of-sq.693.37
Akaike criterion0.2548 F-statistic5471.2
Schwarz criterion0.3964 p-value (F-stat)0.0000
In this table:
• The Value column gives the least squares estimates of parameters βj
• The Std error column shows standard errors of each coefficient estimate: ${\hat {\sigma }}_{j}=\left({\hat {\sigma }}^{2}\left[Q_{xx}^{-1}\right]_{jj}\right)^{\frac {1}{2}}$
• The t-statistic and p-value columns are testing whether any of the coefficients might be equal to zero. The t-statistic is calculated simply as $t={\hat {\beta }}_{j}/{\hat {\sigma }}_{j}$. If the errors ε follow a normal distribution, t follows a Student-t distribution. Under weaker conditions, t is asymptotically normal. Large values of t indicate that the null hypothesis can be rejected and that the corresponding coefficient is not zero. The second column, p-value, expresses the results of the hypothesis test as a significance level. Conventionally, p-values smaller than 0.05 are taken as evidence that the population coefficient is nonzero.
• R-squared is the coefficient of determination indicating goodness-of-fit of the regression. This statistic will be equal to one if fit is perfect, and to zero when regressors X have no explanatory power whatsoever. This is a biased estimate of the population R-squared, and will never decrease if additional regressors are added, even if they are irrelevant.
• Adjusted R-squared is a slightly modified version of $R^{2}$, designed to penalize for the excess number of regressors which do not add to the explanatory power of the regression. This statistic is always smaller than $R^{2}$, can decrease as new regressors are added, and even be negative for poorly fitting models:
${\overline {R}}^{2}=1-{\frac {n-1}{n-p}}(1-R^{2})$
• Log-likelihood is calculated under the assumption that errors follow normal distribution. Even though the assumption is not very reasonable, this statistic may still find its use in conducting LR tests.
• Durbin–Watson statistic tests whether there is any evidence of serial correlation between the residuals. As a rule of thumb, the value smaller than 2 will be an evidence of positive correlation.
• Akaike information criterion and Schwarz criterion are both used for model selection. Generally when comparing two alternative models, smaller values of one of these criteria will indicate a better model.[34]
• Standard error of regression is an estimate of σ, standard error of the error term.
• Total sum of squares, model sum of squared, and residual sum of squares tell us how much of the initial variation in the sample were explained by the regression.
• F-statistic tries to test the hypothesis that all coefficients (except the intercept) are equal to zero. This statistic has F(p–1,n–p) distribution under the null hypothesis and normality assumption, and its p-value indicates probability that the hypothesis is indeed true. Note that when errors are not normal this statistic becomes invalid, and other tests such as Wald test or LR test should be used.
Ordinary least squares analysis often includes the use of diagnostic plots designed to detect departures of the data from the assumed form of the model. These are some of the common diagnostic plots:
• Residuals against the explanatory variables in the model. A non-linear relation between these variables suggests that the linearity of the conditional mean function may not hold. Different levels of variability in the residuals for different levels of the explanatory variables suggests possible heteroscedasticity.
• Residuals against explanatory variables not in the model. Any relation of the residuals to these variables would suggest considering these variables for inclusion in the model.
• Residuals against the fitted values, ${\hat {y}}$.
• Residuals against the preceding residual. This plot may identify serial correlations in the residuals.
An important consideration when carrying out statistical inference using regression models is how the data were sampled. In this example, the data are averages rather than measurements on individual women. The fit of the model is very good, but this does not imply that the weight of an individual woman can be predicted with high accuracy based only on her height.
Sensitivity to rounding
This example also demonstrates that coefficients determined by these calculations are sensitive to how the data is prepared. The heights were originally given rounded to the nearest inch and have been converted and rounded to the nearest centimetre. Since the conversion factor is one inch to 2.54 cm this is not an exact conversion. The original inches can be recovered by Round(x/0.0254) and then re-converted to metric without rounding. If this is done the results become:
ConstHeightHeight2
Converted to metric with rounding. 128.8128−143.16261.96033
Converted to metric without rounding. 119.0205−131.507658.5046
Using either of these equations to predict the weight of a 5' 6" (1.6764 m) woman gives similar values: 62.94 kg with rounding vs. 62.98 kg without rounding. Thus a seemingly small variation in the data has a real effect on the coefficients but a small effect on the results of the equation.
While this may look innocuous in the middle of the data range it could become significant at the extremes or in the case where the fitted model is used to project outside the data range (extrapolation).
This highlights a common error: this example is an abuse of OLS which inherently requires that the errors in the independent variable (in this case height) are zero or at least negligible. The initial rounding to nearest inch plus any actual measurement errors constitute a finite and non-negligible error. As a result, the fitted parameters are not the best estimates they are presumed to be. Though not totally spurious the error in the estimation will depend upon relative size of the x and y errors.
Another example with less real data
Problem statement
We can use the least square mechanism to figure out the equation of a two body orbit in polar base co-ordinates. The equation typically used is $r(\theta )={\frac {p}{1-e\cos(\theta )}}$ where $r(\theta )$ is the radius of how far the object is from one of the bodies. In the equation the parameters $p$ and $e$ are used to determine the path of the orbit. We have measured the following data.
$\theta $ (in degrees) 43 45 52 93 108 116
$r(\theta )$ 4.7126 4.5542 4.0419 2.2187 1.8910 1.7599
We need to find the least-squares approximation of $e$ and $p$ for the given data.
Solution
First we need to represent e and p in a linear form. So we are going to rewrite the equation $r(\theta )$ as ${\frac {1}{r(\theta )}}={\frac {1}{p}}-{\frac {e}{p}}\cos(\theta )$. Now we can use this form to represent our observational data as:
$A^{T}A{\binom {x}{y}}=A^{T}b$ where $x$ is ${\frac {1}{p}}$ and $y$ is ${\frac {e}{p}}$ and $A$ is constructed by the first column being the coefficient of ${\frac {1}{p}}$ and the second column being the coefficient of ${\frac {e}{p}}$ and $b$ is the values for the respective ${\frac {1}{r(\theta )}}$ so $A={\begin{bmatrix}1&-0.731354\\1&-0.707107\\1&-0.615661\\1&\ 0.052336\\1&0.309017\\1&0.438371\end{bmatrix}}$ and $b={\begin{bmatrix}0.21220\\0.21958\\0.24741\\0.45071\\0.52883\\0.56820\end{bmatrix}}.$
On solving we get ${\binom {x}{y}}={\binom {0.43478}{0.30435}}$
so $p={\frac {1}{x}}=2.3000$ and $e=p\cdot y=0.70001$
See also
• Bayesian least squares
• Fama–MacBeth regression
• Nonlinear least squares
• Numerical methods for linear least squares
• Nonlinear system identification
References
1. "What is a complete list of the usual assumptions for linear regression?". Cross Validated. Retrieved 2022-09-28.
2. Goldberger, Arthur S. (1964). "Classical Linear Regression". Econometric Theory. New York: John Wiley & Sons. pp. 158. ISBN 0-471-31101-4.
3. Hayashi, Fumio (2000). Econometrics. Princeton University Press. p. 15.
4. Hayashi (2000, page 18)
5. Ghilani, Charles D.; Paul r. Wolf, Ph. D. (12 June 2006). Adjustment Computations: Spatial Data Analysis. ISBN 9780471697282.
6. Hofmann-Wellenhof, Bernhard; Lichtenegger, Herbert; Wasle, Elmar (20 November 2007). GNSS – Global Navigation Satellite Systems: GPS, GLONASS, Galileo, and more. ISBN 9783211730171.
7. Xu, Guochang (5 October 2007). GPS: Theory, Algorithms and Applications. ISBN 9783540727156.
8. Hayashi (2000, page 19)
9. Julian Faraway (2000), Practical Regression and Anova using R
10. Kenney, J.; Keeping, E. S. (1963). Mathematics of Statistics. van Nostrand. p. 187.
11. Zwillinger, D. (1995). Standard Mathematical Tables and Formulae. Chapman&Hall/CRC. p. 626. ISBN 0-8493-2479-3.
12. Hayashi (2000, page 20)
13. Akbarzadeh, Vahab (7 May 2014). "Line Estimation".
14. Hayashi (2000, page 49)
15. Hayashi (2000, page 52)
16. Hayashi (2000, page 7)
17. Hayashi (2000, page 187)
18. Hayashi (2000, page 10)
19. Hayashi (2000, page 34)
20. Williams, M. N; Grajales, C. A. G; Kurkiewicz, D (2013). "Assumptions of multiple regression: Correcting two misconceptions". Practical Assessment, Research & Evaluation. 18 (11).
21. "Memento on EViews Output" (PDF). Retrieved 28 December 2020.
22. Hayashi (2000, pages 27, 30)
23. Hayashi (2000, page 27)
24. Amemiya, Takeshi (1985). Advanced Econometrics. Harvard University Press. p. 13. ISBN 9780674005600.
25. Amemiya (1985, page 14)
26. Rao, C. R. (1973). Linear Statistical Inference and its Applications (Second ed.). New York: J. Wiley & Sons. p. 319. ISBN 0-471-70823-2.
27. Amemiya (1985, page 20)
28. Amemiya (1985, page 27)
29. Davidson, Russell; MacKinnon, James G. (1993). Estimation and Inference in Econometrics. New York: Oxford University Press. p. 33. ISBN 0-19-506011-3.
30. Davidson & MacKinnon (1993, page 36)
31. Davidson & MacKinnon (1993, page 20)
32. Amemiya (1985, page 21)
33. Amemiya (1985, page 22)
34. Burnham, Kenneth P.; David Anderson (2002). Model Selection and Multi-Model Inference (2nd ed.). Springer. ISBN 0-387-95364-7.
Further reading
• Dougherty, Christopher (2002). Introduction to Econometrics (2nd ed.). New York: Oxford University Press. pp. 48–113. ISBN 0-19-877643-8.
• Gujarati, Damodar N.; Porter, Dawn C. (2009). Basic Econometics (Fifth ed.). Boston: McGraw-Hill Irwin. pp. 55–96. ISBN 978-0-07-337577-9.
• Heij, Christiaan; Boer, Paul; Franses, Philip H.; Kloek, Teun; van Dijk, Herman K. (2004). Econometric Methods with Applications in Business and Economics (1st ed.). Oxford: Oxford University Press. pp. 76–115. ISBN 978-0-19-926801-6.
• Hill, R. Carter; Griffiths, William E.; Lim, Guay C. (2008). Principles of Econometrics (3rd ed.). Hoboken, NJ: John Wiley & Sons. pp. 8–47. ISBN 978-0-471-72360-8.
• Wooldridge, Jeffrey (2008). "The Simple Regression Model". Introductory Econometrics: A Modern Approach (4th ed.). Mason, OH: Cengage Learning. pp. 22–67. ISBN 978-0-324-58162-1.
Least squares and regression analysis
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Wikipedia
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Standard normal deviate
A standard normal deviate is a normally distributed deviate. It is a realization of a standard normal random variable, defined as a random variable with expected value 0 and variance 1.[1] Where collections of such random variables are used, there is often an associated (possibly unstated) assumption that members of such collections are statistically independent.
Standard normal variables play a major role in theoretical statistics in the description of many types of models, particularly in regression analysis, the analysis of variance and time series analysis.
When the term "deviate" is used, rather than "variable", there is a connotation that the value concerned is treated as the no-longer-random outcome of a standard normal random variable. The terminology here is the same as that for random variable and random variate. Standard normal deviates arise in practical statistics in two ways.
• Given a model for a set of observed data, a set of manipulations of the data can result in a derived quantity which, assuming that the model is a true representation of reality, is a standard normal deviate (perhaps in an approximate sense). This enables a significance test to be made for the validity of the model.
• In the computer generation of a pseudorandom number sequence, the aim may be to generate random numbers having a normal distribution: these can be obtained from standard normal deviates (themselves the output of a pseudorandom number sequence) by multiplying by the scale parameter and adding the location parameter. More generally, the generation of pseudorandom number sequence having other marginal distributions may involve manipulating sequences of standard normal deviates: an example here is the chi-squared distribution, random values of which can be obtained by adding the squares of standard normal deviates (although this would seldom be the fastest method of generating such values).
See also
• Standard normal table
References
1. Dodge, Y. (2003) The Oxford Dictionary of Statistical Terms. OUP. ISBN 0-19-920613-9
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Orientation (vector space)
The orientation of a real vector space or simply orientation of a vector space is the arbitrary choice of which ordered bases are "positively" oriented and which are "negatively" oriented. In the three-dimensional Euclidean space, right-handed bases are typically declared to be positively oriented, but the choice is arbitrary, as they may also be assigned a negative orientation. A vector space with an orientation selected is called an oriented vector space, while one not having an orientation selected, is called unoriented.
Not to be confused with Orientation (geometry).
In mathematics, orientability is a broader notion that, in two dimensions, allows one to say when a cycle goes around clockwise or counterclockwise, and in three dimensions when a figure is left-handed or right-handed. In linear algebra over the real numbers, the notion of orientation makes sense in arbitrary finite dimension, and is a kind of asymmetry that makes a reflection impossible to replicate by means of a simple displacement. Thus, in three dimensions, it is impossible to make the left hand of a human figure into the right hand of the figure by applying a displacement alone, but it is possible to do so by reflecting the figure in a mirror. As a result, in the three-dimensional Euclidean space, the two possible basis orientations are called right-handed and left-handed (or right-chiral and left-chiral).
Definition
Let V be a finite-dimensional real vector space and let b1 and b2 be two ordered bases for V. It is a standard result in linear algebra that there exists a unique linear transformation A : V → V that takes b1 to b2. The bases b1 and b2 are said to have the same orientation (or be consistently oriented) if A has positive determinant; otherwise they have opposite orientations. The property of having the same orientation defines an equivalence relation on the set of all ordered bases for V. If V is non-zero, there are precisely two equivalence classes determined by this relation. An orientation on V is an assignment of +1 to one equivalence class and −1 to the other.[1]
Every ordered basis lives in one equivalence class or another. Thus any choice of a privileged ordered basis for V determines an orientation: the orientation class of the privileged basis is declared to be positive.
For example, the standard basis on Rn provides a standard orientation on Rn (in turn, the orientation of the standard basis depends on the orientation of the Cartesian coordinate system on which it is built). Any choice of a linear isomorphism between V and Rn will then provide an orientation on V.
The ordering of elements in a basis is crucial. Two bases with a different ordering will differ by some permutation. They will have the same/opposite orientations according to whether the signature of this permutation is ±1. This is because the determinant of a permutation matrix is equal to the signature of the associated permutation.
Similarly, let A be a nonsingular linear mapping of vector space Rn to Rn. This mapping is orientation-preserving if its determinant is positive.[2] For instance, in R3 a rotation around the Z Cartesian axis by an angle α is orientation-preserving:
$\mathbf {A} _{1}={\begin{pmatrix}\cos \alpha &-\sin \alpha &0\\\sin \alpha &\cos \alpha &0\\0&0&1\end{pmatrix}}$
while a reflection by the XY Cartesian plane is not orientation-preserving:
$\mathbf {A} _{2}={\begin{pmatrix}1&0&0\\0&1&0\\0&0&-1\end{pmatrix}}$
Zero-dimensional case
The concept of orientation degenerates in the zero-dimensional case. A zero-dimensional vector space has only a single point, the zero vector. Consequently, the only basis of a zero-dimensional vector space is the empty set $\emptyset $. Therefore, there is a single equivalence class of ordered bases, namely, the class $\{\emptyset \}$ whose sole member is the empty set. This means that an orientation of a zero-dimensional space is a function
$\{\{\emptyset \}\}\to \{\pm 1\}.$
It is therefore possible to orient a point in two different ways, positive and negative.
Because there is only a single ordered basis $\emptyset $, a zero-dimensional vector space is the same as a zero-dimensional vector space with ordered basis. Choosing $\{\emptyset \}\mapsto +1$ or $\{\emptyset \}\mapsto -1$ therefore chooses an orientation of every basis of every zero-dimensional vector space. If all zero-dimensional vector spaces are assigned this orientation, then, because all isomorphisms among zero-dimensional vector spaces preserve the ordered basis, they also preserve the orientation. This is unlike the case of higher-dimensional vector spaces where there is no way to choose an orientation so that it is preserved under all isomorphisms.
However, there are situations where it is desirable to give different orientations to different points. For example, consider the fundamental theorem of calculus as an instance of Stokes' theorem. A closed interval [a, b] is a one-dimensional manifold with boundary, and its boundary is the set {a, b}. In order to get the correct statement of the fundamental theorem of calculus, the point b should be oriented positively, while the point a should be oriented negatively.
On a line
The one-dimensional case deals with a line which may be traversed in one of two directions. There are two orientations to a line just as there are two orientations to a circle. In the case of a line segment (a connected subset of a line), the two possible orientations result in directed line segments. An orientable surface sometimes has the selected orientation indicated by the orientation of a line perpendicular to the surface.
Alternate viewpoints
Multilinear algebra
For any n-dimensional real vector space V we can form the kth-exterior power of V, denoted ΛkV. This is a real vector space of dimension ${\tbinom {n}{k}}$. The vector space ΛnV (called the top exterior power) therefore has dimension 1. That is, ΛnV is just a real line. There is no a priori choice of which direction on this line is positive. An orientation is just such a choice. Any nonzero linear form ω on ΛnV determines an orientation of V by declaring that x is in the positive direction when ω(x) > 0. To connect with the basis point of view we say that the positively-oriented bases are those on which ω evaluates to a positive number (since ω is an n-form we can evaluate it on an ordered set of n vectors, giving an element of R). The form ω is called an orientation form. If {ei} is a privileged basis for V and {ei∗} is the dual basis, then the orientation form giving the standard orientation is e1∗ ∧ e2∗ ∧ … ∧ en∗.
The connection of this with the determinant point of view is: the determinant of an endomorphism $T:V\to V$ can be interpreted as the induced action on the top exterior power.
Lie group theory
Let B be the set of all ordered bases for V. Then the general linear group GL(V) acts freely and transitively on B. (In fancy language, B is a GL(V)-torsor). This means that as a manifold, B is (noncanonically) homeomorphic to GL(V). Note that the group GL(V) is not connected, but rather has two connected components according to whether the determinant of the transformation is positive or negative (except for GL0, which is the trivial group and thus has a single connected component; this corresponds to the canonical orientation on a zero-dimensional vector space). The identity component of GL(V) is denoted GL+(V) and consists of those transformations with positive determinant. The action of GL+(V) on B is not transitive: there are two orbits which correspond to the connected components of B. These orbits are precisely the equivalence classes referred to above. Since B does not have a distinguished element (i.e. a privileged basis) there is no natural choice of which component is positive. Contrast this with GL(V) which does have a privileged component: the component of the identity. A specific choice of homeomorphism between B and GL(V) is equivalent to a choice of a privileged basis and therefore determines an orientation.
More formally: $\pi _{0}(\operatorname {GL} (V))=(\operatorname {GL} (V)/\operatorname {GL} ^{+}(V)=\{\pm 1\}$, and the Stiefel manifold of n-frames in $V$ is a $\operatorname {GL} (V)$-torsor, so $V_{n}(V)/\operatorname {GL} ^{+}(V)$ is a torsor over $\{\pm 1\}$, i.e., its 2 points, and a choice of one of them is an orientation.
Geometric algebra
The various objects of geometric algebra are charged with three attributes or features: attitude, orientation, and magnitude.[4] For example, a vector has an attitude given by a straight line parallel to it, an orientation given by its sense (often indicated by an arrowhead) and a magnitude given by its length. Similarly, a bivector in three dimensions has an attitude given by the family of planes associated with it (possibly specified by the normal line common to these planes [5]), an orientation (sometimes denoted by a curved arrow in the plane) indicating a choice of sense of traversal of its boundary (its circulation), and a magnitude given by the area of the parallelogram defined by its two vectors.[6]
Orientation on manifolds
Main article: Orientability
Each point p on an n-dimensional differentiable manifold has a tangent space TpM which is an n-dimensional real vector space. Each of these vector spaces can be assigned an orientation. Some orientations "vary smoothly" from point to point. Due to certain topological restrictions, this is not always possible. A manifold that admits a smooth choice of orientations for its tangent spaces is said to be orientable.
See also
• Sign convention
• Rotation formalisms in three dimensions
• Chirality (mathematics)
• Right-hand rule
• Even and odd permutations
• Cartesian coordinate system
• Pseudovector
• Orientation of a vector bundle
References
1. W., Weisstein, Eric. "Vector Space Orientation". mathworld.wolfram.com. Retrieved 2017-12-08.{{cite web}}: CS1 maint: multiple names: authors list (link)
2. W., Weisstein, Eric. "Orientation-Preserving". mathworld.wolfram.com. Retrieved 2017-12-08.{{cite web}}: CS1 maint: multiple names: authors list (link)
3. Leo Dorst; Daniel Fontijne; Stephen Mann (2009). Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry (2nd ed.). Morgan Kaufmann. p. 32. ISBN 978-0-12-374942-0. The algebraic bivector is not specific on shape; geometrically it is an amount of oriented area in a specific plane, that's all.
4. B Jancewicz (1996). "Tables 28.1 & 28.2 in section 28.3: Forms and pseudoforms". In William Eric Baylis (ed.). Clifford (geometric) algebras with applications to physics, mathematics, and engineering. Springer. p. 397. ISBN 0-8176-3868-7.
5. William Anthony Granville (1904). "§178 Normal line to a surface". Elements of the differential and integral calculus. Ginn & Company. p. 275.
6. David Hestenes (1999). New foundations for classical mechanics: Fundamental Theories of Physics (2nd ed.). Springer. p. 21. ISBN 0-7923-5302-1.
External links
• "Orientation", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
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Standard part function
In nonstandard analysis, the standard part function is a function from the limited (finite) hyperreal numbers to the real numbers. Briefly, the standard part function "rounds off" a finite hyperreal to the nearest real. It associates to every such hyperreal $x$, the unique real $x_{0}$ infinitely close to it, i.e. $x-x_{0}$ is infinitesimal. As such, it is a mathematical implementation of the historical concept of adequality introduced by Pierre de Fermat,[1] as well as Leibniz's Transcendental law of homogeneity.
The standard part function was first defined by Abraham Robinson who used the notation ${}^{\circ }x$ for the standard part of a hyperreal $x$ (see Robinson 1974). This concept plays a key role in defining the concepts of the calculus, such as continuity, the derivative, and the integral, in nonstandard analysis. The latter theory is a rigorous formalization of calculations with infinitesimals. The standard part of x is sometimes referred to as its shadow.[2]
Definition
Nonstandard analysis deals primarily with the pair $\mathbb {R} \subseteq {}^{*}\mathbb {R} $, where the hyperreals ${}^{*}\mathbb {R} $ are an ordered field extension of the reals $\mathbb {R} $, and contain infinitesimals, in addition to the reals. In the hyperreal line every real number has a collection of numbers (called a monad, or halo) of hyperreals infinitely close to it. The standard part function associates to a finite hyperreal x, the unique standard real number x0 that is infinitely close to it. The relationship is expressed symbolically by writing
$\operatorname {st} (x)=x_{0}.$
The standard part of any infinitesimal is 0. Thus if N is an infinite hypernatural, then 1/N is infinitesimal, and st(1/N) = 0.
If a hyperreal $u$ is represented by a Cauchy sequence $\langle u_{n}:n\in \mathbb {N} \rangle $ in the ultrapower construction, then
$\operatorname {st} (u)=\lim _{n\to \infty }u_{n}.$
More generally, each finite $u\in {}^{*}\mathbb {R} $ defines a Dedekind cut on the subset $\mathbb {R} \subseteq {}^{*}\mathbb {R} $ (via the total order on ${}^{\ast }\mathbb {R} $) and the corresponding real number is the standard part of u.
Not internal
The standard part function "st" is not defined by an internal set. There are several ways of explaining this. Perhaps the simplest is that its domain L, which is the collection of limited (i.e. finite) hyperreals, is not an internal set. Namely, since L is bounded (by any infinite hypernatural, for instance), L would have to have a least upper bound if L were internal, but L doesn't have a least upper bound. Alternatively, the range of "st" is $\mathbb {R} \subseteq {}^{*}\mathbb {R} $, which is not internal; in fact every internal set in ${}^{*}\mathbb {R} $ that is a subset of $\mathbb {R} $ is necessarily finite.[3]
Applications
All the traditional notions of calculus can be expressed in terms of the standard part function, as follows.
Derivative
The standard part function is used to define the derivative of a function f. If f is a real function, and h is infinitesimal, and if f′(x) exists, then
$f'(x)=\operatorname {st} \left({\frac {f(x+h)-f(x)}{h}}\right).$
Alternatively, if $y=f(x)$, one takes an infinitesimal increment $\Delta x$, and computes the corresponding $\Delta y=f(x+\Delta x)-f(x)$. One forms the ratio $ {\frac {\Delta y}{\Delta x}}$. The derivative is then defined as the standard part of the ratio:
${\frac {dy}{dx}}=\operatorname {st} \left({\frac {\Delta y}{\Delta x}}\right).$
Integral
Given a function $f$ on $[a,b]$, one defines the integral $ \int _{a}^{b}f(x)\,dx$ as the standard part of an infinite Riemann sum $S(f,a,b,\Delta x)$ when the value of $\Delta x$ is taken to be infinitesimal, exploiting a hyperfinite partition of the interval [a,b].
Limit
Given a sequence $(u_{n})$, its limit is defined by $ \lim _{n\to \infty }u_{n}=\operatorname {st} (u_{H})$ where $H\in {}^{*}\mathbb {N} \setminus \mathbb {N} $ is an infinite index. Here the limit is said to exist if the standard part is the same regardless of the infinite index chosen.
Continuity
A real function $f$ is continuous at a real point $x$ if and only if the composition $\operatorname {st} \circ f$ is constant on the halo of $x$. See microcontinuity for more details.
See also
• Adequality
• Nonstandard calculus
References
1. Katz, Karin Usadi; Katz, Mikhail G. (March 2012). "A Burgessian Critique of Nominalistic Tendencies in Contemporary Mathematics and its Historiography". Foundations of Science. 17 (1): 51–89. arXiv:1104.0375. doi:10.1007/s10699-011-9223-1The authors refer to the Fermat-Robinson standard part.{{cite journal}}: CS1 maint: postscript (link)
2. Bascelli, Tiziana; Bottazzi, Emanuele; Herzberg, Frederik; Kanovei, Vladimir; Katz, Karin U.; Katz, Mikhail G.; Nowik, Tahl; Sherry, David; Shnider, Steven (1 September 2014). "Fermat, Leibniz, Euler, and the Gang: The True History of the Concepts of Limit and Shadow" (PDF). Notices of the American Mathematical Society. 61 (08): 848. doi:10.1090/noti1149.
3. Goldblatt, Robert (1998). Lectures on the Hyperreals: An Introduction to Nonstandard Analysis. New York: Springer. ISBN 978-0-387-98464-3.
Further reading
• H. Jerome Keisler. Elementary Calculus: An Infinitesimal Approach. First edition 1976; 2nd edition 1986. (This book is now out of print. The publisher has reverted the copyright to the author, who has made available the 2nd edition in .pdf format available for downloading at http://www.math.wisc.edu/~keisler/calc.html.)
• Abraham Robinson. Non-standard analysis. Reprint of the second (1974) edition. With a foreword by Wilhelmus A. J. Luxemburg. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ, 1996. xx+293 pp. ISBN 0-691-04490-2
Infinitesimals
History
• Adequality
• Leibniz's notation
• Integral symbol
• Criticism of nonstandard analysis
• The Analyst
• The Method of Mechanical Theorems
• Cavalieri's principle
Related branches
• Nonstandard analysis
• Nonstandard calculus
• Internal set theory
• Synthetic differential geometry
• Smooth infinitesimal analysis
• Constructive nonstandard analysis
• Infinitesimal strain theory (physics)
Formalizations
• Differentials
• Hyperreal numbers
• Dual numbers
• Surreal numbers
Individual concepts
• Standard part function
• Transfer principle
• Hyperinteger
• Increment theorem
• Monad
• Internal set
• Levi-Civita field
• Hyperfinite set
• Law of continuity
• Overspill
• Microcontinuity
• Transcendental law of homogeneity
Mathematicians
• Gottfried Wilhelm Leibniz
• Abraham Robinson
• Pierre de Fermat
• Augustin-Louis Cauchy
• Leonhard Euler
Textbooks
• Analyse des Infiniment Petits
• Elementary Calculus
• Cours d'Analyse
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Amitsur–Levitzki theorem
In algebra, the Amitsur–Levitzki theorem states that the algebra of n × n matrices over a commutative ring satisfies a certain identity of degree 2n. It was proved by Amitsur and Levitsky (1950). In particular matrix rings are polynomial identity rings such that the smallest identity they satisfy has degree exactly 2n.
Statement
The standard polynomial of degree n is
$S_{n}(x_{1},\dots ,x_{n})=\sum _{\sigma \in S_{n}}{\text{sgn}}(\sigma )x_{\sigma (1)}\cdots x_{\sigma (n)}$
in non-commuting variables x1, ..., xn, where the sum is taken over all n! elements of the symmetric group Sn.
The Amitsur–Levitzki theorem states that for n × n matrices A1, ..., A2n whose entries are taken from a commutative ring then
$S_{2n}(A_{1},\dots ,A_{2n})=0.$
Proofs
Amitsur and Levitzki (1950) gave the first proof.
Kostant (1958) deduced the Amitsur–Levitzki theorem from the Koszul–Samelson theorem about primitive cohomology of Lie algebras.
Swan (1963) and Swan (1969) gave a simple combinatorial proof as follows. By linearity it is enough to prove the theorem when each matrix has only one nonzero entry, which is 1. In this case each matrix can be encoded as a directed edge of a graph with n vertices. So all matrices together give a graph on n vertices with 2n directed edges. The identity holds provided that for any two vertices A and B of the graph, the number of odd Eulerian paths from A to B is the same as the number of even ones. (Here a path is called odd or even depending on whether its edges taken in order give an odd or even permutation of the 2n edges.) Swan showed that this was the case provided the number of edges in the graph is at least 2n, thus proving the Amitsur–Levitzki theorem.
Razmyslov (1974) gave a proof related to the Cayley–Hamilton theorem.
Rosset (1976) gave a short proof using the exterior algebra of a vector space of dimension 2n.
Procesi (2015) gave another proof, showing that the Amitsur–Levitzki theorem is the Cayley–Hamilton identity for the generic Grassman matrix.
References
• Amitsur, A. S.; Levitzki, Jakob (1950), "Minimal identities for algebras" (PDF), Proceedings of the American Mathematical Society, 1 (4): 449–463, doi:10.1090/S0002-9939-1950-0036751-9, ISSN 0002-9939, JSTOR 2032312, MR 0036751
• Amitsur, A. S.; Levitzki, Jakob (1951), "Remarks on Minimal identities for algebras" (PDF), Proceedings of the American Mathematical Society, 2 (2): 320–327, doi:10.2307/2032509, ISSN 0002-9939, JSTOR 2032509
• Formanek, E. (2001) [1994], "Amitsur–Levitzki theorem", Encyclopedia of Mathematics, EMS Press
• Formanek, Edward (1991), The polynomial identities and invariants of n×n matrices, Regional Conference Series in Mathematics, vol. 78, Providence, RI: American Mathematical Society, ISBN 0-8218-0730-7, Zbl 0714.16001
• Kostant, Bertram (1958), "A theorem of Frobenius, a theorem of Amitsur–Levitski and cohomology theory", J. Math. Mech., 7 (2): 237–264, doi:10.1512/iumj.1958.7.07019, MR 0092755
• Razmyslov, Ju. P. (1974), "Identities with trace in full matrix algebras over a field of characteristic zero", Mathematics of the USSR-Izvestiya, 8 (4): 727, doi:10.1070/IM1974v008n04ABEH002126, ISSN 0373-2436, MR 0506414
• Rosset, Shmuel (1976), "A new proof of the Amitsur–Levitski identity", Israel Journal of Mathematics, 23 (2): 187–188, doi:10.1007/BF02756797, ISSN 0021-2172, MR 0401804, S2CID 121625182
• Swan, Richard G. (1963), "An application of graph theory to algebra" (PDF), Proceedings of the American Mathematical Society, 14 (3): 367–373, doi:10.2307/2033801, ISSN 0002-9939, JSTOR 2033801, MR 0149468
• Swan, Richard G. (1969), "Correction to "An application of graph theory to algebra"" (PDF), Proceedings of the American Mathematical Society, 21 (2): 379–380, doi:10.2307/2037008, ISSN 0002-9939, JSTOR 2037008, MR 0255439
• Procesi, Claudio (2015), "On the theorem of Amitsur—Levitzki", Israel Journal of Mathematics, 207: 151–154, arXiv:1308.2421, Bibcode:2013arXiv1308.2421P, doi:10.1007/s11856-014-1118-8
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Wikipedia
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Standard translation
In modal logic, standard translation is a logic translation that transforms formulas of modal logic into formulas of first-order logic which capture the meaning of the modal formulas. Standard translation is defined inductively on the structure of the formula. In short, atomic formulas are mapped onto unary predicates and the objects in the first-order language are the accessible worlds. The logical connectives from propositional logic remain untouched and the modal operators are transformed into first-order formulas according to their semantics.
Definition
Standard translation is defined as follows:
• $ST_{x}(p)\equiv P(x)$, where $p$ is an atomic formula; P(x) is true when $p$ holds in world $x$.
• $ST_{x}(\top )\equiv \top $
• $ST_{x}(\bot )\equiv \bot $
• $ST_{x}(\neg \varphi )\equiv \neg ST_{x}(\varphi )$
• $ST_{x}(\varphi \wedge \psi )\equiv ST_{x}(\varphi )\wedge ST_{x}(\psi )$
• $ST_{x}(\varphi \vee \psi )\equiv ST_{x}(\varphi )\vee ST_{x}(\psi )$
• $ST_{x}(\varphi \rightarrow \psi )\equiv ST_{x}(\varphi )\rightarrow ST_{x}(\psi )$
• $ST_{x}(\Diamond _{m}\varphi )\equiv \exists y(R_{m}(x,y)\wedge ST_{y}(\varphi ))$
• $ST_{x}(\Box _{m}\varphi )\equiv \forall y(R_{m}(x,y)\rightarrow ST_{y}(\varphi ))$
In the above, $x$ is the world from which the formula is evaluated. Initially, a free variable $x$ is used and whenever a modal operator needs to be translated, a fresh variable is introduced to indicate that the remainder of the formula needs to be evaluated from that world. Here, the subscript $m$ refers to the accessibility relation that should be used: normally, $\Box $ and $\Diamond $ refer to a relation $R$ of the Kripke model but more than one accessibility relation can exist (a multimodal logic) in which case subscripts are used. For example, $\Box _{a}$ and $\Diamond _{a}$ refer to an accessibility relation $R_{a}$ and $\Box _{b}$ and $\Diamond _{b}$ to $R_{b}$ in the model. Alternatively, it can also be placed inside the modal symbol.
Example
As an example, when standard translation is applied to $\Box \Box p$, we expand the outer box to get
$\forall y(R(x,y)\rightarrow ST_{y}(\Box p))$
meaning that we have now moved from $x$ to an accessible world $y$ and we now evaluate the remainder of the formula, $\Box p$, in each of those accessible worlds.
The whole standard translation of this example becomes
$\forall y(R(x,y)\rightarrow (\forall z(R(y,z)\rightarrow P(z))))$
which precisely captures the semantics of two boxes in modal logic. The formula $\Box \Box p$ holds in $x$ when for all accessible worlds $y$ from $x$ and for all accessible worlds $z$ from $y$, the predicate $P$ is true for $z$. Note that the formula is also true when no such accessible worlds exist. In case $x$ has no accessible worlds then $R(x,y)$ is false but the whole formula is vacuously true: an implication is also true when the antecedent is false.
Standard translation and modal depth
The modal depth of a formula also becomes apparent in the translation to first-order logic. When the modal depth of a formula is k, then the first-order logic formula contains a 'chain' of k transitions from the starting world $x$. The worlds are 'chained' in the sense that these worlds are visited by going from accessible to accessible world. Informally, the number of transitions in the 'longest chain' of transitions in the first-order formula is the modal depth of the formula.
The modal depth of the formula used in the example above is two. The first-order formula indicates that the transitions from $x$ to $y$ and from $y$ to $z$ are needed to verify the validity of the formula. This is also the modal depth of the formula as each modal operator increases the modal depth by one.
References
• Patrick Blackburn and Johan van Benthem (1988), Modal Logic: A Semantic Perspective.
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Wikipedia
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Standardized rate
Standardized rates are a statistical measure of any rates in a population. These are adjusted rates that take into account the vital differences between populations that may affect their birthrates or death rates.
Examples
The most common are birth, death and unemployment rates. For example, in a community made up of primarily young couples, the birthrate might appear to be high when compared to that of other populations. However, by calculating the standardized birthrates that is by comparing the same age group in other populations), a more realistic picture of childbearing capacity will be developed.
Formula
The formula for standardized rates is as follows:
Σ(crude rate for age group × standard population for age group) / Σstandard population
See also
• Mortality ratio
References
• Medical Biostatistics, Third Edition (MedicalBiostatistics.synthasite.com), A. Indrayan (indrayan.weebly.com), Chapman & Hall/ CRC Press, 2012
• Introduction to Sociology, Bruce J. Cohen and Terri L. Orbuch
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Wikipedia
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Short-time Fourier transform
The short-time Fourier transform (STFT), is a Fourier-related transform used to determine the sinusoidal frequency and phase content of local sections of a signal as it changes over time.[1] In practice, the procedure for computing STFTs is to divide a longer time signal into shorter segments of equal length and then compute the Fourier transform separately on each shorter segment. This reveals the Fourier spectrum on each shorter segment. One then usually plots the changing spectra as a function of time, known as a spectrogram or waterfall plot, such as commonly used in software defined radio (SDR) based spectrum displays. Full bandwidth displays covering the whole range of an SDR commonly use fast Fourier transforms (FFTs) with 2^24 points on desktop computers.
Forward STFT
Continuous-time STFT
Simply, in the continuous-time case, the function to be transformed is multiplied by a window function which is nonzero for only a short period of time. The Fourier transform (a one-dimensional function) of the resulting signal is taken, then the window is slid along the time axis until the end resulting in a two-dimensional representation of the signal. Mathematically, this is written as:
$\mathbf {STFT} \{x(t)\}(\tau ,\omega )\equiv X(\tau ,\omega )=\int _{-\infty }^{\infty }x(t)w(t-\tau )e^{-i\omega t}\,dt$
where $w(\tau )$ is the window function, commonly a Hann window or Gaussian window centered around zero, and $x(t)$ is the signal to be transformed (note the difference between the window function $w$ and the frequency $\omega $). $X(\tau ,\omega )$ is essentially the Fourier transform of $x(t)w(t-\tau )$, a complex function representing the phase and magnitude of the signal over time and frequency. Often phase unwrapping is employed along either or both the time axis, $\tau $, and frequency axis, $\omega $, to suppress any jump discontinuity of the phase result of the STFT. The time index $\tau $ is normally considered to be "slow" time and usually not expressed in as high resolution as time $t$. Given that the STFT is essentially a Fourier transform times a window function, the STFT is also called windowed Fourier transform or time-dependent Fourier transform.
Discrete-time STFT
In the discrete time case, the data to be transformed could be broken up into chunks or frames (which usually overlap each other, to reduce artifacts at the boundary). Each chunk is Fourier transformed, and the complex result is added to a matrix, which records magnitude and phase for each point in time and frequency. This can be expressed as:
$\mathbf {STFT} \{x[n]\}(m,\omega )\equiv X(m,\omega )=\sum _{n=-\infty }^{\infty }x[n]w[n-m]e^{-i\omega n}$
likewise, with signal x[n] and window w[n]. In this case, m is discrete and ω is continuous, but in most typical applications the STFT is performed on a computer using the fast Fourier transform, so both variables are discrete and quantized.
The magnitude squared of the STFT yields the spectrogram representation of the power spectral density of the function:
$\operatorname {spectrogram} \{x(t)\}(\tau ,\omega )\equiv |X(\tau ,\omega )|^{2}$
See also the modified discrete cosine transform (MDCT), which is also a Fourier-related transform that uses overlapping windows.
Sliding DFT
If only a small number of ω are desired, or if the STFT is desired to be evaluated for every shift m of the window, then the STFT may be more efficiently evaluated using a sliding DFT algorithm.[2]
Inverse STFT
The STFT is invertible, that is, the original signal can be recovered from the transform by the inverse STFT. The most widely accepted way of inverting the STFT is by using the overlap-add (OLA) method, which also allows for modifications to the STFT complex spectrum. This makes for a versatile signal processing method,[3] referred to as the overlap and add with modifications method.
Continuous-time STFT
Given the width and definition of the window function w(t), we initially require the area of the window function to be scaled so that
$\int _{-\infty }^{\infty }w(\tau )\,d\tau =1.$
It easily follows that
$\int _{-\infty }^{\infty }w(t-\tau )\,d\tau =1\quad \forall \ t$
and
$x(t)=x(t)\int _{-\infty }^{\infty }w(t-\tau )\,d\tau =\int _{-\infty }^{\infty }x(t)w(t-\tau )\,d\tau .$
The continuous Fourier transform is
$X(\omega )=\int _{-\infty }^{\infty }x(t)e^{-i\omega t}\,dt.$
Substituting x(t) from above:
$X(\omega )=\int _{-\infty }^{\infty }\left[\int _{-\infty }^{\infty }x(t)w(t-\tau )\,d\tau \right]\,e^{-i\omega t}\,dt$
$=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }x(t)w(t-\tau )\,e^{-i\omega t}\,d\tau \,dt.$
Swapping order of integration:
$X(\omega )=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }x(t)w(t-\tau )\,e^{-i\omega t}\,dt\,d\tau $
$=\int _{-\infty }^{\infty }\left[\int _{-\infty }^{\infty }x(t)w(t-\tau )\,e^{-i\omega t}\,dt\right]\,d\tau $
$=\int _{-\infty }^{\infty }X(\tau ,\omega )\,d\tau .$
So the Fourier transform can be seen as a sort of phase coherent sum of all of the STFTs of x(t). Since the inverse Fourier transform is
$x(t)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }X(\omega )e^{+i\omega t}\,d\omega ,$
then x(t) can be recovered from X(τ,ω) as
$x(t)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }X(\tau ,\omega )e^{+i\omega t}\,d\tau \,d\omega .$
or
$x(t)=\int _{-\infty }^{\infty }\left[{\frac {1}{2\pi }}\int _{-\infty }^{\infty }X(\tau ,\omega )e^{+i\omega t}\,d\omega \right]\,d\tau .$
It can be seen, comparing to above that windowed "grain" or "wavelet" of x(t) is
$x(t)w(t-\tau )={\frac {1}{2\pi }}\int _{-\infty }^{\infty }X(\tau ,\omega )e^{+i\omega t}\,d\omega .$
the inverse Fourier transform of X(τ,ω) for τ fixed.
An alternative definition that is valid only in the vicinity of τ, the inverse transform is:
$x(t)={\frac {1}{w(t-\tau )}}{\frac {1}{2\pi }}\int _{-\infty }^{\infty }X(\tau ,\omega )e^{+i\omega t}\,d\omega .$
In general, the window function $w(t)$ has the following properties:
(a) even symmetry: $w(t)=w(-t)$;
(b) non-increasing (for positive time): $w(t)\geq w(s)$ if $|t|\leq |s|$;
(c) compact support: $w(t)$ is equal to zero when |t| is large.
Resolution issues
One of the pitfalls of the STFT is that it has a fixed resolution. The width of the windowing function relates to how the signal is represented—it determines whether there is good frequency resolution (frequency components close together can be separated) or good time resolution (the time at which frequencies change). A wide window gives better frequency resolution but poor time resolution. A narrower window gives good time resolution but poor frequency resolution. These are called narrowband and wideband transforms, respectively.
This is one of the reasons for the creation of the wavelet transform and multiresolution analysis, which can give good time resolution for high-frequency events and good frequency resolution for low-frequency events, the combination best suited for many real signals.
This property is related to the Heisenberg uncertainty principle, but not directly – see Gabor limit for discussion. The product of the standard deviation in time and frequency is limited. The boundary of the uncertainty principle (best simultaneous resolution of both) is reached with a Gaussian window function (or mask function), as the Gaussian minimizes the Fourier uncertainty principle. This is called the Gabor transform (and with modifications for multiresolution becomes the Morlet wavelet transform).
One can consider the STFT for varying window size as a two-dimensional domain (time and frequency), as illustrated in the example below, which can be calculated by varying the window size. However, this is no longer a strictly time-frequency representation – the kernel is not constant over the entire signal.
Examples
When the original function is:
$X(t,f)=\int _{-\infty }^{\infty }w(t-\tau )x(\tau )e^{-j2\pi f\tau }d\tau $
We can have a simple example:
w(t) = 1 for |t| smaller than or equal B
w(t) = 0 otherwise
B = window
Now the original function of the Short-time Fourier transform can be changed as
$X(t,f)=\int _{t-B}^{t+B}x(\tau )e^{-j2\pi f\tau }d\tau $
Another example:
Using the following sample signal $x(t)$ that is composed of a set of four sinusoidal waveforms joined together in sequence. Each waveform is only composed of one of four frequencies (10, 25, 50, 100 Hz). The definition of $x(t)$ is:
$x(t)={\begin{cases}\cos(2\pi 10t)&0\,\mathrm {s} \leq t<5\,\mathrm {s} \\\cos(2\pi 25t)&5\,\mathrm {s} \leq t<10\,\mathrm {s} \\\cos(2\pi 50t)&10\,\mathrm {s} \leq t<15\,\mathrm {s} \\\cos(2\pi 100t)&15\,\mathrm {s} \leq t<20\,\mathrm {s} \\\end{cases}}$
Then it is sampled at 400 Hz. The following spectrograms were produced:
The 25 ms window allows us to identify a precise time at which the signals change but the precise frequencies are difficult to identify. At the other end of the scale, the 1000 ms window allows the frequencies to be precisely seen but the time between frequency changes is blurred.
Other examples:
$w(t)=exp(\sigma -t^{2})$
Normally we call $exp(\sigma -t^{2})$ a Gaussian function or Gabor function. When we use it, the short-time Fourier transform is called the "Gabor transform".
Explanation
It can also be explained with reference to the sampling and Nyquist frequency.
Take a window of N samples from an arbitrary real-valued signal at sampling rate fs . Taking the Fourier transform produces N complex coefficients. Of these coefficients only half are useful (the last N/2 being the complex conjugate of the first N/2 in reverse order, as this is a real valued signal).
These N/2 coefficients represent the frequencies 0 to fs/2 (Nyquist) and two consecutive coefficients are spaced apart by fs/N Hz.
To increase the frequency resolution of the window the frequency spacing of the coefficients needs to be reduced. There are only two variables, but decreasing fs (and keeping N constant) will cause the window size to increase — since there are now fewer samples per unit time. The other alternative is to increase N, but this again causes the window size to increase. So any attempt to increase the frequency resolution causes a larger window size and therefore a reduction in time resolution—and vice versa.
Rayleigh frequency
As the Nyquist frequency is a limitation in the maximum frequency that can be meaningfully analysed, so is the Rayleigh frequency a limitation on the minimum frequency.
The Rayleigh frequency is the minimum frequency that can be resolved by a finite duration time window.[4][5]
Given a time window that is Τ seconds long, the minimum frequency that can be resolved is 1/Τ Hz.
The Rayleigh frequency is an important consideration in applications of the short-time Fourier transform (STFT), as well as any other method of harmonic analysis on a signal of finite record-length.[6][7]
Application
STFTs as well as standard Fourier transforms and other tools are frequently used to analyze music. The spectrogram can, for example, show frequency on the horizontal axis, with the lowest frequencies at left, and the highest at the right. The height of each bar (augmented by color) represents the amplitude of the frequencies within that band. The depth dimension represents time, where each new bar was a separate distinct transform. Audio engineers use this kind of visual to gain information about an audio sample, for example, to locate the frequencies of specific noises (especially when used with greater frequency resolution) or to find frequencies which may be more or less resonant in the space where the signal was recorded. This information can be used for equalization or tuning other audio effects.
Implementation
Original function
$X(t,f)=\int _{-\infty }^{\infty }w(t-\tau )x(\tau )e^{-j2\pi f\tau }d\tau $
Converting into the discrete form:
$t=n\Delta _{t},f=m\Delta _{f},\tau =p\Delta _{t}$
$X(n\Delta _{t},m\Delta _{f})=\sum _{-\infty }^{\infty }w((n-p)\Delta _{t})x(p\Delta _{t})e^{-j2\pi pm\Delta _{t}\Delta _{f}}\Delta _{t}$
Suppose that
$w(t)\cong 0{\text{ for }}|t|>B,{\frac {B}{\Delta _{t}}}=Q$
Then we can write the original function into
$X(n\Delta _{t},m\Delta _{f})=\sum _{p=n-Q}^{n+Q}w((n-p)\Delta _{t})x(p\Delta _{t})e^{-j2\pi pm\Delta _{t}\Delta _{f}}\Delta _{t}$
Constraints
a. Nyquist criterion (avoiding the aliasing effect):
$\Delta _{t}<{\frac {1}{2\Omega }}$, where $\Omega $ is the bandwidth of $x(\tau )w(t-\tau )$
Constraint
a. $\Delta _{t}\Delta _{f}={\tfrac {1}{N}}$, where $N$ is an integer
b. $N\geq 2Q+1$
c. Nyquist criterion (avoiding the aliasing effect):
$\Delta _{t}<{\frac {1}{2\Omega }}$, $\Omega $ is the bandwidth of $x(\tau )w(t-\tau )$
$X(n\Delta _{t},m\Delta _{f})=\sum _{p=n-Q}^{n+Q}w((n-p)\Delta _{t})x(p\Delta _{t})e^{-{\frac {2\pi jpm}{N}}}\Delta _{t}$
${\text{if we have }}q=p-(n-Q),{\text{ then }}p=(n-Q)+q$
$X(n\Delta _{t},m\Delta _{f})=\Delta _{t}e^{\frac {2\pi j(Q-n)m}{N}}\sum _{q=0}^{N-1}x_{1}(q)e^{-{\frac {2\pi jqm}{N}}}$
${\text{where }}x_{1}(q)={\begin{cases}w((Q-q)\Delta _{t})x((n-Q+q)\Delta _{t})&0\leq q\leq 2Q\\0&2Q<q<N\end{cases}}$
Constraint
a. $\Delta _{t}\Delta _{f}={\tfrac {1}{N}}$, where $N$ is an integer
b. $N\geq 2Q+1$
c. Nyquist criterion (avoiding the aliasing effect):
$\Delta _{t}<{\frac {1}{2\Omega }}$, $\Omega $ is the bandwidth of $x(\tau )w(t-\tau )$
d. Only for implementing the rectangular-STFT
Rectangular window imposes the constraint
$w((n-p)\Delta _{t})=1$
Substitution gives:
${\begin{aligned}X(n\Delta _{t},m\Delta _{f})&=\sum _{p=n-Q}^{n+Q}w((n-p)\Delta _{t})&x(p\Delta _{t})e^{-{\frac {j2\pi pm}{N}}}\Delta _{t}\\&=\sum _{p=n-Q}^{n+Q}&x(p\Delta _{t})e^{-{\frac {j2\pi pm}{N}}}\Delta _{t}\\\end{aligned}}$
Change of variable n-1 for n:
$X((n-1)\Delta _{t},m\Delta _{f})=\sum _{p=n-1-Q}^{n-1+Q}x(p\Delta _{t})e^{-{\frac {j2\pi pm}{N}}}\Delta _{t}$
Calculate $X(\min {n}\Delta _{t},m\Delta _{f})$ by the N-point FFT:
$X(n_{0}\Delta _{t},m\Delta _{f})=\Delta _{t}e^{\frac {j2\pi (Q-n_{0})m}{N}}\sum _{q=0}^{N-1}x_{1}(q)e^{-j{\frac {2\pi qm}{N}}},\qquad n_{0}=\min {(n)}$
where
$x_{1}(q)={\begin{cases}x((n-Q+q)\Delta _{t})&q\leq 2Q\\0&q>2Q\end{cases}}$
Applying the recursive formula to calculate $X(n\Delta _{t},m\Delta _{f})$
$X(n\Delta _{t},m\Delta _{f})=X((n-1)\Delta _{t},m\Delta _{f})-x((n-Q-1)\Delta _{t})e^{-{\frac {j2\pi (n-Q-1)m}{N}}}\Delta _{t}+x((n+Q)\Delta _{t})e^{-{\frac {j2\pi (n+Q)m}{N}}}\Delta _{t}$
Constraint
$\exp {(-j2\pi pm\Delta _{t}\Delta _{f})}=\exp {(-j\pi p^{2}\Delta _{t}\Delta _{f})}\cdot \exp {(j\pi (p-m)^{2}\Delta _{t}\Delta _{f})}\cdot \exp {(-j\pi m^{2}\Delta _{t}\Delta _{f})}$
so
$X(n\Delta _{t},m\Delta _{f})=\Delta _{t}\sum _{p=n-Q}^{n+Q}w((n-p)\Delta _{t})x(p\Delta _{t})e^{-j2\pi pm\Delta _{t}\Delta _{f}}$
$X(n\Delta _{t},m\Delta _{f})=\Delta _{t}e^{-j2\pi m^{2}\Delta _{t}\Delta _{f}}\sum _{p=n-Q}^{n+Q}w((n-p)\Delta _{t})x(p\Delta _{t})e^{-j\pi p^{2}\Delta _{t}\Delta _{f}}e^{j\pi (p-m)^{2}\Delta _{t}\Delta _{f}}$
Implementation comparison
Method Complexity
Direct implementation $O(TFQ)$
FFT-based $O(TN\log _{2}N)$
Recursive $O(TF)$
Chirp Z transform $O(TN\log _{2}N)$
See also
• Least-squares spectral analysis
• Spectral density estimation
• Time-frequency analysis
• Time-frequency representation
• Reassignment method
Other time-frequency transforms:
• Cone-shape distribution function
• Constant-Q transform
• Fractional Fourier transform
• Gabor transform
• Newland transform
• S transform
• Wavelet transform
• Chirplet transform
References
1. Sejdić E.; Djurović I.; Jiang J. (2009). "Time-frequency feature representation using energy concentration: An overview of recent advances". Digital Signal Processing. 19 (1): 153–183. doi:10.1016/j.dsp.2007.12.004.
2. E. Jacobsen and R. Lyons, The sliding DFT, Signal Processing Magazine vol. 20, issue 2, pp. 74–80 (March 2003).
3. Jont B. Allen (June 1977). "Short Time Spectral Analysis, Synthesis, and Modification by Discrete Fourier Transform". IEEE Transactions on Acoustics, Speech, and Signal Processing. ASSP-25 (3): 235–238. doi:10.1109/TASSP.1977.1162950.
4. https://physics.ucsd.edu/neurophysics/publications/Cold%20Spring%20Harb%20Protoc-2014-Kleinfeld-pdb.top081075.pdf
5. "What does "padding not sufficient for requested frequency resolution" mean? – FieldTrip toolbox".
6. Zeitler M, Fries P, Gielen S (2008). "Biased competition through variations in amplitude of gamma-oscillations". J Comput Neurosci. 25 (1): 89–107. doi:10.1007/s10827-007-0066-2. PMC 2441488. PMID 18293071.
7. Wingerden, Marijn van; Vinck, Martin; Lankelma, Jan; Pennartz, Cyriel M. A. (2010-05-19). "Theta-Band Phase Locking of Orbitofrontal Neurons during Reward Expectancy". Journal of Neuroscience. 30 (20): 7078–7087. doi:10.1523/JNEUROSCI.3860-09.2010. ISSN 0270-6474. PMC 6632657. PMID 20484650.
External links
• DiscreteTFDs – software for computing the short-time Fourier transform and other time-frequency distributions
• Singular Spectral Analysis – MultiTaper Method Toolkit – a free software program to analyze short, noisy time series
• kSpectra Toolkit for Mac OS X from SpectraWorks
• Time stretched short time Fourier transform for time frequency analysis of ultra wideband signals
• A BSD-licensed Matlab class to perform STFT and inverse STFT
• LTFAT – A free (GPL) Matlab / Octave toolbox to work with short-time Fourier transforms and time-frequency analysis
• Sonogram visible speech – A free (GPL)Freeware for short-time Fourier transforms and time-frequency analysis
• National Taiwan University, Time-Frequency Analysis and Wavelet Transform 2021, Professor of Jian-Jiun Ding, Department of Electrical Engineering
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Wikipedia
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Stanislas Ouaro
Stanislas Ouaro (born 19 January 1975[1]) is a Burkinabé politician and mathematician.
Stanislas Ouaro
Minister of National Education and Literacy
In office
31 January 2018 – 19 January 2019
Preceded byJean-Martin Coulibaly
Succeeded byPosition abolished
Minister of National Education, Literacy and Promotion of National Languages
Incumbent
Assumed office
24 January 2019
Preceded byPosition established
Personal details
Born (1975-01-19) 19 January 1975
Bobo-Dioulasso, Burkina Faso
Alma materUniversity of Ouagadougou
Biography
Stanislas Ouaro was born on 19 January 1975. He graduated with a doctor's degree from University of Ouagadougou in 2001 with his thesis titled Etude de problèmes elliptiques-paraboliques nonlinéaires en une dimension d'espace.[1] Before he joined government, he was the president of University of Ouaga II since 2012.[2]
On 31 January 2018, he was appointed the Minister of National Education and Literacy, replacing Jean-Martin Coulibaly.[3] On 19 January 2019, he resigned together with other members of Thieba cabinet.[4] On 24 January, he was appointed the Minister of National Education, Literacy and Promotion of National Languages.[5]
Health
During the 2020 coronavirus outbreak, on 21 March, Ouaro contracted the coronavirus.[6]
References
1. "Stanislas OUARO". Mathematicians of the African Diaspora. Retrieved 24 March 2020.
2. "Stanislas OUARO, Ministre de l'Éducation Nationale, de l'Alphabétisation et de la Promotion des langues nationales". Gouvernement du Burkina Faso (in French). Archived from the original on 24 March 2020. Retrieved 24 March 2020.
3. "Ouaro, ministre de l'Education nationale et de l'alphabétisation. Le président du l'Université de Ouaga II jusque là prend la place de Jean Martin Coulibaly" (in French). omegabf.info. 31 January 2018.
4. "Burkina Faso: Prime Minister and cabinet resign from office". 19 January 2019.
5. "Burkina Faso : La composition du premier gouvernement de Christophe Dabiré dévoilée". lefaso.net (in French). 25 January 2019.
6. "Burkina Faso Mines Minister Tests Positive for Coronavirus". Bloomberg.com. 2020-03-21. Retrieved 2020-03-21.
External links
• Stanislas Ouaro on Facebook
Authority control
International
• VIAF
National
• Poland
Academics
• MathSciNet
• Mathematics Genealogy Project
• ORCID
• zbMATH
Other
• IdRef
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Wikipedia
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Stanislav Smirnov
Stanislav Konstantinovich Smirnov (Russian: Станисла́в Константи́нович Cмирно́в; born 3 September 1970) is a Russian mathematician currently working at the University of Geneva. He was awarded the Fields Medal in 2010. His research involves complex analysis, dynamical systems and probability theory.[1][2]
Stanislav Smirnov
Born (1970-09-03) 3 September 1970
Leningrad, Soviet Union
NationalityRussian
Alma materSaint Petersburg State University
California Institute of Technology
AwardsClay Research Award (2001)
Salem Prize (2001)
Rollo Davidson Prize (2002)
EMS Prize (2004)
Fields Medal (2010)
Scientific career
FieldsMathematics
InstitutionsUniversity of Geneva
Royal Institute of Technology
Saint Petersburg State University
Yale University
Max Planck Institute for Mathematics
IAS Princeton
Skolkovo Institute of Science and Technology
ThesisSpectral Analysis of Julia Sets (1996)
Doctoral advisorNikolai Makarov
Career
Smirnov's Ph.D. was conducted at Caltech under advisor Nikolai Makarov. In 1998 he was employed as part of the faculty at the Royal Institute of Technology in Stockholm, after which he took up his second position as a professor in the Analysis, Mathematical Physics and Probability group at the University of Geneva in 2003.[3][4]
Research
Smirnov has worked on percolation theory, where he proved Cardy's formula for critical site percolation on the triangular lattice, and deduced conformal invariance.[5] The conjecture was proved in the special case of site percolation on the triangular lattice.[6] Smirnov's theorem has led to a fairly complete theory for percolation on the triangular lattice, and to its relationship to the Schramm–Loewner evolution introduced by Oded Schramm. He also established conformality for the two-dimensional critical Ising model.[7]
Awards
Smirnov was awarded the Saint Petersburg Mathematical Society Prize (1997), the Clay Research Award (2001), the Salem Prize (joint with Oded Schramm, 2001), the Göran Gustafsson Prize (2001), the Rollo Davidson Prize (2002), and the Prize of the European Mathematical Society (2004).[3] In 2010 Smirnov was awarded the Fields medal for his work on the mathematical foundations of statistical physics, particularly finite lattice models.[8] His citation read "for the proof of conformal invariance of percolation and the planar Ising model in statistical physics".[9]
Publications
• Probability and Statistical Physics in St. Petersburg, American Math Society, (2015)
References
1. "Stanislav Smirnov's publications on Google Scholar".
2. "Stanislav Smirnov's home page".
3. "La Médaille Fields pour un professeur de l'UNIGE". University of Geneva press releases (in French). University of Geneva. 19 August 2010. Retrieved 19 August 2010.
4. "Stanislav Smirnov's page at the University of Geneva" (in French). University of Geneva. 3 October 2007. Retrieved 19 August 2010.
5. "Clay Mathematics Institute". Archived from the original on 5 October 2008.
6. Smirnov, Stanislav (2001). "Critical percolation in the plane: conformal invariance, Cardy's formula, scaling limits". Comptes Rendus de l'Académie des Sciences. 333 (3): 239–244. arXiv:0909.4499. Bibcode:2001CRASM.333..239S. doi:10.1016/S0764-4442(01)01991-7.
7. "Papers of Stanislav Smirnov".
8. Cipra, Barry A. (19 August 2010). "Fields Medals, Other Top Math Prizes, Awarded". Science Now. AAAS. Archived from the original on 22 August 2010. Retrieved 19 August 2010.
9. Rehmeyer, Julie (19 August 2010). "Stanislav Smirnov profile" (PDF). International Congress of Mathematicians. Retrieved 19 August 2010.
External links
Media related to Stanislav Smirnov at Wikimedia Commons
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Stanislav Vydra
Stanislav Vydra (13 November 1741 in Hradec Králové – 2 December 1804 in Prague) was a Bohemian Jesuit priest, writer, mathematician.
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Life
Vydra entered the Jesuit novitiate of Hradec Králové in 1757. After two years in Brno, he studied philosophy and mathematics from 1762 to 1764 at Charles University. His teachers included Joseph Stepling and Jan Tesánek.
In 1765, he went as a teacher to Jičín and became Stepling's assistant a year later. He ministered as parish priest in Vilémov from 1771 to 1772. In 1772, Vydra was appointed professor of mathematics in Charles University in Prague. Here he taught until 1773. From 1789 to 1799, he was appointed to the mathematics faculty and served as dean of the Faculty of Arts. He became the rector of the university in 1800. He went blind in 1803 and died one year later. He is buried in Prague at the Olšany Cemetery in Prague.
Teachings
Stanislav Vydra taught elementary mathematics, a compulsory subject for the students at the philosophical faculty since 1752. He published “Elementa calcvli differentialis et integralis” in 1783, which became a well-known calculus textbook in Prague.[1] After his death, his pupil and successor Josef Ladislav Jandera published his book Pocátkowé Arytmetyky, which was the first text book of elementary mathematics in Bohemia.
Selected works
• Historia Matheseos in Bohemia et Moravia cultae, 1778
• Elementa Calcvli Differentialis, et Integralis, 1783
• Počátkowé Arytmetyky, 1806
References
1. "Šimerka's Czech Book of Calculus" (PDF). Retrieved October 14, 2022.
• Between elementary mathematics and national wiedergeburt – 274 sides, Broschur
• George Schuppener, Karel Macek: Stanislav Vydra (1741–1804), Leipzig University publishing house (2004)
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Stanisław Saks
Stanisław Saks (30 December 1897 – 23 November 1942) was a Polish mathematician and university tutor, a member of the Lwów School of Mathematics, known primarily for his membership in the Scottish Café circle, an extensive monograph on the theory of integrals, his works on measure theory and the Vitali–Hahn–Saks theorem.
Stanisław Saks
Born(1897-12-30)30 December 1897
Kalisz, Congress Poland
Died23 November 1942(1942-11-23) (aged 44)
Warsaw, German-occupied Poland
Cause of deathExecution
NationalityPolish
Alma materWarsaw University
Known forVitali–Hahn–Saks theorem
Denjoy–Luzin–Saks theorem
Denjoy–Young–Saks theorem
Scientific career
FieldsMathematics
Life and work
Stanisław Saks was born on 30 December 1897 in Kalisz, Congress Poland, to an assimilated Polish-Jewish family. In 1915 he graduated from a local gymnasium and joined the newly recreated Warsaw University. In 1922 he received a doctorate of his alma mater with a prestigious distinction maxima cum laude. Soon afterwards he also passed his habilitation and received the Rockefeller fellowship, which allowed him to travel to the United States. Around that time he started publishing articles in various mathematical journals, mostly the Fundamenta Mathematicae, but also in the Transactions of the American Mathematical Society. He participated in the Silesian Uprisings and was awarded the Cross of the Valorous and the Medal of Independence for his bravery. Following the end of the uprising he returned to Warsaw and resumed his academic career.
For most of it he studied the theories of functions and functionals in particular. In 1930 he published his most notable book, the Zarys teorii całki (Sketch on the Theory of the Integral), which later got expanded and translated into several languages, including English (Theory of the Integral), French (Théorie de l'Intégrale)[1] and Russian (Teoriya Integrala). Despite his successes, Saks was never awarded the title of professor and remained an ordinary tutor, initially at his alma mater and the Warsaw University of Technology, and later at the Lwów University and Wilno University. He was also an active socialist and a journalist at the Robotnik weekly (1919–1926) and later a collaborator of the Association of Socialist Youth.
Saks wrote a mathematics book with Antoni Zygmund, Analytic Functions, in 1933. It was translated into English in 1952 by E. J. Scott.[2] In the preface to the English edition, Zygmund writes:[3]
Stanislaw Saks was a man of moral as well as physical courage, of rare intelligence and wit. To his colleagues and pupils he was an inspiration not only as a mathematician but as a human being. In the period between the two world wars he exerted great influence upon a whole generation of Polish mathematicians in Warsaw and Lwów. In November 1942, at the age of 45, Saks died in a Warsaw prison, victim of a policy of extermination.
After the outbreak of World War II and the occupation of Poland by Germany, Saks joined the Polish underground. Arrested in November 1942, he was executed on 23 November 1942 by the German Gestapo in Warsaw.[4]
Publications
• Saks, Stanisław (1937). Theory of the Integral. Monografie Matematyczne. Vol. 7 (2nd ed.). Warszawa-Lwów: G.E. Stechert & Co. pp. VI+347. JFM 63.0183.05. Zbl 0017.30004.. English translation by Laurence Chisholm Young, with two additional notes by Stefan Banach.[5]
• Saks, Stanisław; Zygmund, Antoni (1965). Analytic functions. Monografie Matematyczne. Vol. 28 (Second ed.). Warsaw: Państwowe Wydawnietwo Naukowe. MR 0180658.
See also
• Lwów School of Mathematics
Notes
1. Tamarkin, J. D. (1934). "Review: Théorie de l'Intégrale by S. Saks" (PDF). Bull. Amer. Math. Soc. 40 (1): 16–18. doi:10.1090/s0002-9904-1934-05770-7.
2. Heins, Maurice (1954). "Review: Analytic Functions by S. Saks and A. Zygmund" (PDF). Bull. Amer. Math. Soc. 60 (5): 495–497. doi:10.1090/s0002-9904-1954-09846-4.
3. Saks & Zygmund 1965.
4. Czyż, Janusz (1994). Paradoxes of measures and dimensions originating in Felix Hausdorff's ideas. World Scientific. p. 34. ISBN 978-981-02-0189-0.
5. Tamarkin, J. D. (1938). "Review: Theory of the Integral by S. Saks" (PDF). Bull. Amer. Math. Soc. 44 (9, Part 1): 615–616. doi:10.1090/s0002-9904-1938-06811-5.
References
• O'Connor, John J.; Robertson, Edmund F., "Stanisław Saks", MacTutor History of Mathematics Archive, University of St Andrews
• Zygmund, Antoni (1987), "Stanislaw Saks, 1897–1942", The Mathematical Intelligencer, Springer New York, 9: 36–41, doi:10.1007/BF03023571, ISSN 0343-6993, S2CID 119349092
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Stanisław Knapowski
Stanisław Knapowski (May 19, 1931 – September 28, 1967) was a Polish mathematician who worked on prime numbers and number theory. Knapowski published 53 papers despite dying at only 36 years old.[1]
Stanisław Knapowski
Knapowski expanded on the distribution of primes
Born(1931-05-19)May 19, 1931
Poznań, Poland
DiedSeptember 28, 1967(1967-09-28) (aged 36)
Florida, USA
CitizenshipPolish
EducationPoznań University, Wrocław University and Adam Mickiewicz University
Known forPrime numbers and number theory
AwardsMazurkiewicz Prize, Rockefeller Scholarship
Scientific career
FieldsMathematician
Life and education
Stanisław Knapowski was the son of Zofia Krysiewicz and Roch Knapowski. His father, Roch Knapowski was a lawyer in Poznań but later taught at Poznań University. The family moved to the Kielce province in south-eastern Poland after the German invasion of 1939 but returned to Poznań after the war.[1]
Stanisław completed his high school education in 1949 excelling at math and continued on at Poznań University to study mathematics. Later in 1952 he continued his studies at University of Wrocław and earned his master's degree in 1954.
Knapowski was appointment an assistant at Adam Mickiewicz University in Poznań under Władysław Orlicz and worked towards his doctorate. He studied under the direction of Pál Turán starting in Lublin in 1956. He published many of his papers with Turán and Turán wrote a short biography of his life and work in 1971 after his death.[2] Knapowski began to work in this area and finished his doctorate in 1957 “Zastosowanie metod Turaná w analitycznej teorii liczb” ("Certain applications of Turan's methods in the analytical theory of numbers").
Knapowski had the opportunity to work abroad. He spent a year in Cambridge and worked with Louis J. Mordell and listened to classes by J.W.S. Cassels and Albert Ingham. He then moved on to Belgium, France and The Netherlands.
Knapowski returned to Poznań to finish another thesis to complete a post-doctoral qualification needed to lecture at a German university.[1][2] "On new "explicit formulas" in prime number theory" in 1960.[3] In 1962 the Polish Mathematical Society awarded him their Mazurkiewicz Prize and he moved to Tulane University in New Orleans, United States. After a very short return to Poland, he left again and taught in Marburg in Germany, Gainesville, Florida and Miami, Florida.[2][4]
Death
Knapowski was a good classical pianist. He was also an avid driver, but he died in a traffic accident where he lost control of his car, leaving the Miami airport.[2]
Work
Knapowski expanded on the work of others in several fields of number theory, prime number theorem, modular arithmetic and non-Euclidean geometry.
Number of times the Δ(n) prime sign changes
Main articles: Prime number theorem and Prime-counting function
Mathematicians work on primality tests to develop easier ways to find prime numbers when finding them by trial division is not practical. This has many applications in cybersecurity. There is no formula to calculate prime numbers. However, the distribution of primes can be statistically modelled. The prime number theorem, which was proven at the end of the 19th century, says that the probability of a randomly chosen number being prime is inversely proportional to its number of digits (logarithm). At the start of the 19th century, Adrien-Marie Legendre and Carl Friedrich Gauss suggested that as $x$ goes very large, the number of primes up to $x$ is asymptotic to $x/\log x$, where $\log x$ is the natural logarithm of $x$.
$\operatorname {li} (n)=\int _{0}^{n}{\frac {dt}{\log t}}$
where the integral is evaluated at $t=n$, also fits the distribution.
The prime-counting function $\pi (n)$ is defined as the number of primes not greater than $n$.[5]
And $\Delta (n)=\pi (n)-\operatorname {li} (n)$
Bernhard Riemann stated that $\Delta (n)$ was always negative but J.E. Littlewood later disproved this. In 1914 J.E. Littlewood proved that there are arbitrarily large values of x for which
$\pi (x)>\operatorname {Li} (x)+{\frac {1}{3}}{\frac {\sqrt {x}}{\log x}}\log \log \log x,$
and that there are also arbitrarily large values of x for which
$\pi (x)<\operatorname {Li} (x)-{\frac {1}{3}}{\frac {\sqrt {x}}{\log x}}\log \log \log x.$
Thus the difference π(x) − Li(x) changes sign infinitely many times.
Stanley Skewes then added an upper bound on the smallest natural number $x$:
$\pi (x)>\operatorname {li} (x),$
Knapowski followed this up and published a paper on the number of times $\Delta (n)$ changes sign in the interval $\Delta (n)$.[6]
Modular arithmetic
Main article: Modular arithmetic
Knapowski worked in other areas of number theory. One area was on the distribution of prime numbers in different residue classes modulo $k$.
Modular arithmetic modifies usual arithmetic by only using the numbers $\{0,1,2,\dots ,n-1\}$, for a natural number $n$ called the modulus. Any other natural number can be mapped into this system by replacing it by its remainder after division by $n$.[7]
The distribution of the primes looks random, without a pattern. Take a list of consecutive prime numbers and divide them by another prime (like 7) and keep only the remainder (this is called reducing them modulo 7). The result is a sequence of integers from 1 to 6. Knapowski worked to determine the parameters of this modular distribution[8]
Other areas of research
• Non-Euclidean geometry[9]
• On prime numbers in arithmetical progression[10]
• Möbius function[11]
• On a theorem of Hecke[12]
• On Linnik's theorem concerning exceptional L-zeros” (1961)
• Comparative prime number theory[8]
• On Siegel's Theorem[13]
References
1. "Stanislaw Knapowski biography". www-history.mcs.st-andrews.ac.uk. Archived from the original on 2018-07-08. Retrieved 2019-01-04.
2. Turán, Paul (1971). "Stanisław Knapowski (19 V 193 – 28 IX 1967)". Colloquium Mathematicum. 23 (2): 309–321. doi:10.4064/cm-23-2-309-321. ISSN 0010-1354.
3. Knapowski, Stanisław (1960). "On new "explicit formulas" in prime number theory II" (PDF). Acta Arithmetica. 6: 23–35. doi:10.4064/aa-6-1-23-35. Archived (PDF) from the original on 2018-07-23. Retrieved 2019-01-09.
4. Browkin, J. (2017). "Stanisław Knapowski". Wiadomości Matematyczne. 14 (1). doi:10.14708/wm.v14i1.1966. ISSN 2543-991X.
5. Crandall & Pomerance 2005 harvnb error: no target: CITEREFCrandallPomerance2005 (help), p. 6 Archived 2019-03-23 at the Wayback Machine.
6. Knapowski, Stanisław (1962). "On sign-changes of the difference π(x)-li(x)". Acta Arithmetica. 7 (2): 107–119. doi:10.4064/aa-7-2-107-119. ISSN 0065-1036.
7. Kraft & Washington (2014) harvtxt error: no target: CITEREFKraftWashington2014 (help), Proposition 5.3 Archived 2019-03-23 at the Wayback Machine, p. 96.
8. Knapowski, Stanisław (1955). "On the greatest prime factors of certain products". Annales Polonici Mathematici. 2 (1): 56–63. doi:10.4064/ap-2-1-56-63. ISSN 0066-2216.
9. Coxeter, H. S. M.; Kulczycki, S.; Knapowski, S. (1962). "Non-Euclidean Geometry". The American Mathematical Monthly. 69 (9): 937. doi:10.2307/2311278. ISSN 0002-9890. JSTOR 2311278.
10. Knapowski, Stanisław (1958). "On prime numbers in an arithmetical progression". Acta Arithmetica. 4 (1): 57–70. doi:10.4064/aa-4-1-57-70. ISSN 0065-1036.
11. Knapowski, Stanisław (1958). "On the Möbius function". Acta Arithmetica. 4 (3): 209–216. doi:10.4064/aa-4-3-209-216. ISSN 0065-1036.
12. Knapowski, S. (1969). "On a theorem of Hecke". Journal of Number Theory. 1 (2): 235–251. Bibcode:1969JNT.....1..235K. doi:10.1016/0022-314X(69)90043-2. ISSN 0022-314X.
13. Knapowski, Stanisław (1968). "On Siegel's Theorem". Acta Arithmetica. 14 (4): 417–424. doi:10.4064/aa-14-4-417-424. ISSN 0065-1036.
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Stanisław Krystyn Zaremba
Stanisław Krystyn Zaremba (August 15, 1903, in Cracow – January 14, 1990, in Aberystwyth, Wales) was a Polish climber, mountaineer and mathematician. He was the son of Stanisław Zaremba (1863–1942), also a mathematician.
Zaremba is known for his contributions to low-discrepancy sequences, low-discrepancy sets of points, and their application to Quasi-Monte Carlo methods for numerical integration. He is one of the namesakes of the Hlawka–Zaremba formula in the theory of low-discrepancy sequences.
Education and career
Zaremba studied mathematics at the Jagiellonian University (1921-1924, 1926-1929) and in Paris (1924–26). After his studies he was an assistant at Vilnius University and since 1936 he was an associate professor at Jagiellonian University. He then worked as a lecturer in Stalinabad (nowadays Dushanbe in Tajikistan), England, Wales, United States and Canada, eventually settling in Wales, where he died in 1990. He visited Poland well into his old age, lecturing at the Jagiellonian University in 1981.
Mountaineering
Zaremba was an active climber in the years 1925-1927, especially during the winter. Its author is many first to enter the Tatra Mountains. The new roads he won first in the summer, among others. Little Kozi Wierch, Wieliczka Peak, Frog Turtle Muster, Mięguszowiecki Peak, Durny Peak. In winter he made his way to Hruby Wierch, Zadni Kościelec, Jaworowy Peak. Apart from the Tatras, he also climbed the Alps, the Pyrenees and the mountains of Lebanon, Iran, Wales and Tajikistan.
He was the editor of the magazine Taternik and an honorary member of the Polish Alpine Association.
One of the Wrocław's dwarfs (Alpinki group) is named Zarembek in his honor.
See also
• Zaremba's conjecture
Selected publications
• Zaremba, S. C. (1966), "Good lattice points, discrepancy, and numerical integration", Annali di Matematica Pura ed Applicata, 73: 293–317, doi:10.1007/BF02415091, MR 0218018, S2CID 123200672
• Zaremba, S. C. (1968), "The mathematical basis of Monte Carlo and quasi-Monte Carlo methods", SIAM Review, 10 (3): 303–314, Bibcode:1968SIAMR..10..303Z, doi:10.1137/1010056, MR 0233489
• Halton, J. H.; Zaremba, S. C. (1969), "The extreme and $L^{2}$ discrepancies of some plane sets", Monatsh. Math., 73 (4): 316–328, doi:10.1007/BF01298982, MR 0252329, S2CID 123387557
• Erdős, Paul; Zaremba, S. K. (1973), "The arithmetic function $\textstyle \sum _{d|n}\log d/d$", Demonstratio Mathematica, 6: 575–579 (1974), MR 0352025
References
Zofia, Witold Henryk Paryscy: Great encyclopaedia of Tatra, article "Zaremba, Stanislaw Krystyn"
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Stanisław Mazur
Stanisław Mieczysław Mazur (1 January 1905, Lwów – 5 November 1981, Warsaw) was a Polish mathematician and a member of the Polish Academy of Sciences.
Stanisław Mazur
Stanisław Mazur in 1966
Born(1905-01-01)1 January 1905
Lwów, Galicia and Lodomeria, Austria-Hungary (now Lviv, Ukraine)
Died5 November 1981(1981-11-05) (aged 76)
Warsaw, Poland
Scientific career
FieldsMathematics
Doctoral advisorStefan Banach
Mazur made important contributions to geometrical methods in linear and nonlinear functional analysis and to the study of Banach algebras. He was also interested in summability theory, infinite games and computable functions.
Lwów and Warsaw
Mazur was a student of Stefan Banach at University of Lwów. His doctorate, under Banach's supervision, was awarded in 1935.[1] Mazur, with Juliusz Schauder, was an Invited Speaker of the ICM in 1936 in Oslo.[2]
Mazur was a close collaborator with Banach at Lwów and was a member of the Lwów School of Mathematics, where he participated in the mathematical activities at the Scottish Café. On 6 November 1936, he posed the "basis problem" of determining whether every Banach space has a Schauder basis, with Mazur promising a "live goose" as a reward: 37 years later and in a ceremony that was broadcast throughout Poland, Mazur awarded a live goose to Per Enflo for constructing a counter-example.
From 1948 Mazur worked at the University of Warsaw.
See also
• Approximation problem
• Approximation property
• Banach–Mazur theorem
• Banach–Mazur game
• Compact operator
• Gelfand–Mazur theorem
• Mazur–Ulam theorem
• Schauder basis
References
1. Stanisław Mazur at the Mathematics Genealogy Project
2. Mazur, S.; Schauder, J. (1937). "Über ein Prinzip in der Variationsrechnung". Comptes rendus du Congrès international des mathématiciens: Oslo, 1936. Vol. 2. p. 65.
External links
• O'Connor, John J.; Robertson, Edmund F., "Stanisław Mazur", MacTutor History of Mathematics Archive, University of St Andrews
• Stanisław Mazur at the Mathematics Genealogy Project
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Stanisław Świerczkowski
Stanisław (Stash) Świerczkowski (16 July 1932 – 30 September 2015) was a Polish mathematician famous for his solutions to two iconic problems posed by Hugo Steinhaus: the three-gap theorem and the Non-Tetratorus Theorem.
Stanisław Świerczkowski
Born(1932-07-16)16 July 1932
Toruń, Poland
Died30 September 2015(2015-09-30) (aged 83)
Australia
NationalityPolish
Alma materUniversity of Wroclaw,
Known forThree-gap theorem, Non-Tetratorus theorem
Children2
AwardsFoundation of Alfred Jurzykowski (1996)[1]
Scientific career
FieldsMathematics
InstitutionsGlasgow University, University of Sussex, University of Washington, Australian National University, Sultan Qaboos University, Queen's University, University of Colorado at Boulder
Doctoral advisorHugo Steinhaus
Early life and education
Stanisław (Stash) Świerczkowski was born in Toruń, Poland. His parents were divorced during his infancy. When war broke out his father was captured in Soviet-controlled Poland and murdered in the 1940 Katyń Massacre. He belonged to the Polish nobility; Świerczkowski's mother belonged to the upper middle class and would have probably suffered deportation and murder by the Nazis. However she had German connections and was able to gain relatively privileged class 2 Volksliste citizenship. At the end of the war Świerczkowski's mother was forced into hiding near Toruń until she was confident that she could win exoneration from the Soviet-controlled government for her Volksliste status and be rehabilitated as a Polish citizen. Meanwhile, Świerczkowski lived in a rented room in Toruń and attended school there.
Świerczkowski won a university place to study astronomy at the University of Wrocław but switched to mathematics to avoid the drudgery of astronomical calculations. He discovered a natural ability through his friendship with Jan Mycielski and was able to remain at Wrocław to complete his masters under Jan Mikusiński. He graduated with a PhD in 1960, his dissertation including the now-famous Three-Distance Theorem, which he proved in 1956 in answer to a question of Hugo Steinhaus.
Noted mathematical results
The three-gap theorem[2] says: take arbitrarily finitely many integer multiples of an irrational number between zero and one and plot them as points around a circle of unit circumference; then at most three different distances will occur between consecutive points. This answered a question of Hugo Steinhaus. The theorem belongs to the field of Diophantine approximation since the smallest of the three distances observed may be used to give a rational approximation to the chosen irrational number. It has been extended and generalised in many ways.[3]
The Non-Tetratorus Theorem, published by Świerczkowski in 1958,[4] states that it is impossible to construct a closed chain (torus) of regular tetrahedra, placed face to face. Again this answered a question of Hugo Steinhaus. The result is attractive and counter-intuitive, since the tetrahedron is unique among the Platonic solids in having this property. Recent work [5] by Michael Elgersma and Stan Wagon has sparked new interest in this result by showing that one can create chains of tetrahedra that are arbitrarily close to being closed.
In 1964, in a joint work with Jan Mycielski, he established one of the early results on the axiom of determinacy (AD), namely that AD implies that all sets of real numbers are Lebesgue measurable.[6]
Świerczkowski's last mathematical work[7] was on proving Gödel's incompleteness theorems using hereditarily finite sets instead of encoding of finite sequences of natural numbers. It is these proofs that were the basis for the production, in 2015, of mechanised proofs of Gödel's two famous theorems.[8]
Career
Świerczkowski had a very migratory career. He was allowed abroad from Poland to study at Dundee University where his work with Alexander Murray MacBeath would later attract the attention of André Weil. He then took up a research fellowship at Glasgow University before being obliged to return to Poland. When the Polish Academy of Sciences granted him a passport to attend a conference in Stuttgart he used this as an opportunity to leave Poland for good in 1961, first resuming his fellowship in Glasgow before taking a job in the recently created University of Sussex. In 1963 he visited André Weil at the Institute for Advanced Study and thereafter, between 1964 and 1973, held posts at the University of Washington, the Australian National University and Queen's University in Canada. In 1973 he left mathematics, moved to the Netherlands and built a yacht in which he sailed around the world for ten years. The period 1986 to 1997 was again spent teaching mathematics, at Sultan Qaboos University. His last post was at the University of Colorado at Boulder (1998–2001). Thereafter he retired to Tasmania.
References
1. Jerzy Krzywicki (2000). "Nagrody Fundacji Jurzykowskiego w matematyce" (PDF). Roczniki Polskiego Towarzystwa Matematycznego Seria II: Wiadomo Sci Matematyczne XXXVI. 73: 115–138.
2. Three-Distance Theorem at Theorem of the Day
3. Alessandri, P., and Berthé, V. (1998). "Three distance theorems and combinatorics on words". L'Enseignement Mathématique. 44: 103–132.{{cite journal}}: CS1 maint: multiple names: authors list (link)
4. Świerczkowski, S. (1958). "On a free group of rotations of the Euclidean space". Indagationes Mathematicae. 61: 376–378. doi:10.1016/s1385-7258(58)50051-1.
5. Elgersma, M. & Wagon, S. (2017). "An asymptotically closed loop of tetrahedra". The Mathematical Intelligencer. 39 (3): 40–45. doi:10.1007/s00283-016-9696-4. S2CID 253818257.
6. Mycielski J. and Świerczkowski, S. (1964). "On the Lebesgue measurability and the axiom of determinateness". Fund. Math. 54: 67–71. doi:10.4064/fm-54-1-67-71.
7. Świerczkowski, S. (2003). "Finite sets and Gödel's incompleteness theorems". Dissertationes Mathematicae. 422: 1–58. doi:10.4064/dm422-0-1.
8. Lawrence C. Paulson (2015). "A Mechanised Proof of Gödel's Incompleteness Theorems Using Nominal Isabelle". Journal of Automated Reasoning. 55 (1): 1–37. arXiv:2104.13792. CiteSeerX 10.1.1.697.5227. doi:10.1007/s10817-015-9322-8. S2CID 254604706.
External links
Wikimedia Commons has media related to Stanisław Świerczkowski.
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• Stanisław Świerczkowski at the Mathematics Genealogy Project
• O'Connor, John J.; Robertson, Edmund F., "Stanisław Świerczkowski", MacTutor History of Mathematics Archive, University of St Andrews
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Stanisława Nikodym
Stanisława Nikodym (née Liliental; 2 July 1897 — 25 March 1988) was a Polish mathematician and artist. She is known for her results in continuum theory, especially on Jordanian continuums.
Stanisława Nikodym
Stanisława Liliental as a student at Warsaw University, 1916
Born
Stanisława Dorota Liliental
(1897-07-02)July 2, 1897
Warsaw
DiedMarch 25, 1988(1988-03-25) (aged 90)
Warsaw
NationalityPolish
CitizenshipUnited States
Alma materWarsaw University
SpouseOtto M. Nikodym
Scientific career
Fieldscontinuum theory
InstitutionsWarsaw Polytechnic, Kenyon College
ThesisO rozcinaniu płaszczyźny przez zbiory spójne i kontinua (1925)
Doctoral advisorStefan Mazurkiewicz
Life
Stanisława Dorota Liliental was born in Warsaw to Regina Lilientalowa, an ethnographer, and Nathan Liliental. Stanisława had a younger brother, Antoni (born 1908). She attended Helena Skłodowska-Szalay's primary school, and went to Warsaw's private school for women for 7 years. She joined the Warsaw University in 1916, reading mathematics under Stefan Mazurkiewicz, Zygmunt Janiszewski, and Wacław Sierpiński [1]
She married Otto M. Nikodym, a mathematician, in 1924, and joined him at Krakow. Supervised by Mazurkiewicz, she was awarded a doctoral degree from the Jagiellonian University in 1925. She was the first woman in Poland to obtain a PhD in mathematics.[2]
Receiving government funding to study in Paris, she and Otto attended the Sorbonne for two years from 1926. In 1930, they returned to Warsaw. She took up a job at the Warsaw Polytechnic, working with Franciszek Leja till 1936, when he left for Krakow.[3]
Her brother, Antoni, a chemist and officer in the Polish army, was murdered by Russians during the Katyn Massacre in 1940.[1] Under the German Nazi occupation of Poland, unnecessary occupations, including higher education, were suppressed. The Nikodyms conducted clandestine classes in mathematics despite the danger of punishment.[4] In 1944's Warsaw Uprising, she and her husband lost their possessions, including several unpublished mathematical works. They moved to Belgium for a congress of mathematicians in 1946, and Otto gave lectures in various European cities, before they emigrated to the United States, settling in Gambier, Ohio.[5]
After her husband's death in 1974, she donated their papers and her paintings to the Briscoe Center for American History at the University of Texas, Austin.[6] Stanisława Nikodym died in Warsaw in 1988.[5]
Career
Mathematics
While on leave from university in 1918–1919, Stanisława taught mathematics to soldiers in the Polish army.[2]
Her doctoral thesis was titled On disconnecting the plane by connected sets and continua.[1]
She published three books and several articles before the Second World War broke out.[3]
Among her findings were necessary and sufficient conditions for a subcontinuum of a Jordanian continuum to be Jordanian. She also established that if the intersection and union of two closed sets are Jordanian continua, then so are the sets themselves.[3]
In the 1940s, she taught mathematics at Kenyon College in Gambier, Ohio, where her husband was also a member of the faculty.[7]
Art
As a student, Liliental participated in open-air painting in Sandomierz. She painted the cityscape several times in watercolour over many years in the inter-war period from 1922. The resulting works were presented at an exhibition in 1933, after which she donated them to the District Museum in Sandomierz, where they remain to this day.[8]
Selected works
• Nikodym, Stanisława (1925). "Sur les coupures du plan faites par les ensembles connexes et les continus". Fundamenta Mathematicae. 7: 15–23. doi:10.4064/fm-7-1-15-23.
• Nikodym, Otto; Nikodym, Stanisława (1936). Wstęp do rachunku rózniczkowego (in Polish). Warsaw: Nasza Księgarnia.
• Nikodym, Stanisława (1948). Wzory i krótkie repetytorium matematyki. Poznan: Księgarnia W. Wilka.
• Nikodym, Otto; Nikodym, Stanisława (1954). "Sur l'extension des corps algébriques abstraits par un procédé généralisé de Cantor". Rendiconti dell'Accademia Nazionale dei Lincei (Classe di Scienze Fisiche, Matematiche e Naturali). 8. 17.
• Nikodym, Otto; Nikodym, Stanisława (1957). "Some theorems on divisibility of infinite cardinals". Archiv der Mathematik. 8 (2): 96–103. doi:10.1007/BF01900432. S2CID 115982796.
References
1. Ciesielska 2018, p. 115.
2. Domoradzki & Stawiska 2018, p. 24.
3. Ciesielska 2018, p. 116.
4. Szymanski 1990, p. 30.
5. Ciesielska 2018, p. 117.
6. Piotrowski 2014, p. 73.
7. Stamp 1995.
8. Piotrowski 2014, p. 72.
Bibliography
• Ciesielska, Danuta (2018). "A mathematician and a painter Stanisława Nikodym and her husband Otton Nikodym". In Kjeldsen, T.H.; Oswald, N.; Tobies, R. (eds.). Women in Mathematics: Historical and Modern Perspectives. Mathematisches Forschungsinstitut Oberwolfach. doi:10.4171/OWR/2017/2.
• Domoradzki, Stanisław; Stawiska, Małgorzata (2018). "Polish Mathematics and Mathematicians in World War I". arXiv:1804.02448. doi:10.4467/2543702XSHS.19.004.11010. S2CID 119624038. {{cite journal}}: Cite journal requires |journal= (help)
• Piotrowski, Walerian (2014). "Jeszcze w sprawie biografii Ottona i Stanisławy Nikodymów". Wiadomości Matematyczne (in Polish). 50 (1).
• Stamp, Tom (1995). "Let Us Now Praise Not-So-Famous Women". Kenyon College Alumni Bulletin (Spring).
• Szymanski, Waclaw (1990). "Who Was Otto Nikodym?". Mathematical Intelligencer. 12 (2).
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Stanko Bilinski
Stanko Bilinski (22 April 1909 in Našice – 6 April 1998 in Zagreb) was a Croatian mathematician and academician. He was a professor at the University of Zagreb and a fellow of the Croatian Academy of Sciences and Arts.[1]
In 1960, he discovered a rhombic dodecahedron of the second kind, the Bilinski dodecahedron. Like the standard rhombic dodecahedron, this convex polyhedron has 12 congruent rhombus sides, but they are differently shaped and arranged. Bilinski's discovery corrected a 75-year-old omission in Evgraf Fedorov's classification of convex polyhedra with congruent rhombic faces.[2]
References
1. Stanko Bilinski, F.C.A., Mathematician, Croatian Academy of Sciences and Arts, retrieved 2016-05-26
2. Grünbaum, Branko (2010), "The Bilinski dodecahedron and assorted parallelohedra, zonohedra, monohedra, isozonohedra, and otherhedra" (PDF), The Mathematical Intelligencer, 32 (4): 5–15, doi:10.1007/s00283-010-9138-7, hdl:1773/15593, MR 2747698, S2CID 120403108, archived from the original (PDF) on 2015-04-02.
Further reading
• "In memoriam: Stanko Bilinski (22.4.1909.–6.4.1998.)" (PDF). Glasnik Matematički (in Croatian). Croatian Mathematical Society. 33 (2): 323–333. December 1998.
• "Stanko Bilinski (1909. – 2009.)" (PDF). miš – matematika i škola (in Croatian).
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Stanley's reciprocity theorem
In combinatorial mathematics, Stanley's reciprocity theorem, named after MIT mathematician Richard P. Stanley, states that a certain functional equation is satisfied by the generating function of any rational cone (defined below) and the generating function of the cone's interior.
Definitions
A rational cone is the set of all d-tuples
(a1, ..., ad)
of nonnegative integers satisfying a system of inequalities
$M\left[{\begin{matrix}a_{1}\\\vdots \\a_{d}\end{matrix}}\right]\geq \left[{\begin{matrix}0\\\vdots \\0\end{matrix}}\right]$
where M is a matrix of integers. A d-tuple satisfying the corresponding strict inequalities, i.e., with ">" rather than "≥", is in the interior of the cone.
The generating function of such a cone is
$F(x_{1},\dots ,x_{d})=\sum _{(a_{1},\dots ,a_{d})\in {\rm {cone}}}x_{1}^{a_{1}}\cdots x_{d}^{a_{d}}.$
The generating function Fint(x1, ..., xd) of the interior of the cone is defined in the same way, but one sums over d-tuples in the interior rather than in the whole cone.
It can be shown that these are rational functions.
Formulation
Stanley's reciprocity theorem states that for a rational cone as above, we have
$F(1/x_{1},\dots ,1/x_{d})=(-1)^{d}F_{\rm {int}}(x_{1},\dots ,x_{d}).$
Matthias Beck and Mike Develin have shown how to prove this by using the calculus of residues. Develin has said that this amounts to proving the result "without doing any work".
Stanley's reciprocity theorem generalizes Ehrhart-Macdonald reciprocity for Ehrhart polynomials of rational convex polytopes.
See also
• Ehrhart polynomial
References
• Stanley, Richard P. (1974). "Combinatorial reciprocity theorems" (PDF). Advances in Mathematics. 14 (2): 194–253. doi:10.1016/0001-8708(74)90030-9.
• Beck, M.; Develin, M. (2004). "On Stanley's reciprocity theorem for rational cones". arXiv:math.CO/0409562.
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Stanley Skewes
Stanley Skewes (/skjuːz/; 1899–1988) was a South African mathematician, best known for his discovery of the Skewes's number in 1933. He was one of John Edensor Littlewood's students at Cambridge University.[1][2] Skewes's numbers contributed to the refinement of the theory of prime numbers.
Stanley Skewes
Born1899
Germiston, South African Republic
Died1988 (Aged 89)
Cape Town, South Africa
Alma materUniversity of Cape Town
University of Cambridge
Known forSkewes's numbers
SpouseEna Allen
Scientific career
Academic advisorsJohn Edensor Littlewood
Academic career
Skewes obtained a degree in civil engineering from the University of Cape Town before emigrating to England. He studied mathematics at Cambridge University and obtained a PhD in mathematics in 1938.[3]
He discovered the first Skewes's number in 1933[1][2]. This is also referred to as the Riemann true Skewes's number[4] owing to its relationship to the Riemann hypothesis as related to prime number theory. He discovered the second Skewes's number in 1955.[2][5] This number was applicable if the Riemann hypothesis is false. Since his original discovery the numbers have been further refined.
Publications
• Skewes, S. (1933). "On the difference π(x) − Li(x) (I)". Journal of the London Mathematical Society. 8 (4): 277–283. doi:10.1112/jlms/s1-8.4.277. Archived May 19, 2007, at the Wayback Machine
• Skewes, S. (1955). "On the difference π(x) − Li(x) (II)". Proceedings of the London Mathematical Society. 5 (17): 48–70. doi:10.1112/plms/s3-5.1.48. Archived May 19, 2007, at the Wayback Machine
Personal life
Stanley Skewes was born in Germiston, South Africa in 1899. His parents were Henry (Harry) Skewes, a tin miner and assayer from Cury, Cornwall, England and Emily Moyle, who was American by birth. His parents moved from Redruth, Cornwall in 1894 to the Transvaal, South Africa.
He married Ena Allen. She was the daughter of the head chef at King's College, Cambridge, and a talented opera singer. Among his contemporaries at Cambridge was Alan Turing. They rowed together at Cambridge. Although Skewes returned to South Africa, he revisited Cambridge and Cornwall. He was also a keen rugby player in his youth. [2]
Skewes and his number are discussed by Isaac Asimov in his book Of Matters Great and Small[6][7] and in the 20th edition of the Guinness Book of Records.[7][8] A memorandum written by Skewes on his retirement was kept in a glass case in the department of mathematics at the University of Cape Town. The memorandum discuses Skewes's number and further development of it.[2]
He died in 1988 in Cape Town, South Africa.
References
Wikimedia Commons has media related to Stanley Skewes (mathematician).
1. Peter Borwein (2008). The Riemann Hypothesis: A Resource for the Aficionado and Virtuoso Alike. Springer Science & Business Media. p. 375. ISBN 978-0-387-72125-5.
2. Skues, Keith (1983). Cornish heritage. W. Shaw. ISBN 9780907961000.
3. "S. Skewes - Mathematics Genealogy Project". Mathematics Genealogy Project. North Dakota State University - Department of Mathematics. Retrieved 11 February 2019.
4. Bays, C.; Hudson, R.H. (2000). "A new bound for the smallest x with π(x) > li(x)" (PDF). Mathematics of Computation. 69 (231): 1285–1296. doi:10.1090/S0025-5718-99-01104-7.
5. Igor Ushakov (2007). Histories of Scientific Insights. Lulu.com. pp. 235–. ISBN 978-1-4303-2849-0.
6. Of Matters Great and Small. Ace Books. 1980. ISBN 978-0441610723. Retrieved 11 February 2019.
7. HP Williams. "Stanley Skewes and the Skewes Number". London School Of Economics. Archived from the original on 24 June 2021. Retrieved 11 February 2019.
8. Guinness Book of Records (20 ed.). Norris and Ross McWhirter. 1973. ISBN 9780900424137.
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Stanley decomposition
In commutative algebra, a Stanley decomposition is a way of writing a ring in terms of polynomial subrings. They were introduced by Richard Stanley (1982).
Definition
Suppose that a ring R is a quotient of a polynomial ring k[x1,...] over a field by some ideal. A Stanley decomposition of R is a representation of R as a direct sum (of vector spaces)
$R=\bigoplus _{\alpha }x_{\alpha }k(X_{\alpha })$
where each xα is a monomial and each Xα is a finite subset of the generators.
See also
• Rees decomposition
• Hironaka decomposition
References
• Stanley, Richard P. (1982), "Linear Diophantine equations and local cohomology", Invent. Math., 68 (2): 175–193, doi:10.1007/bf01394054, MR 0666158
• Sturmfels, Bernd; White, Neil (1991), "Computing combinatorial decompositions of rings", Combinatorica, 11 (3): 275–293, doi:10.1007/BF01205079, MR 1122013
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Stanley symmetric function
In mathematics and especially in algebraic combinatorics, the Stanley symmetric functions are a family of symmetric functions introduced by Richard Stanley (1984) in his study of the symmetric group of permutations.
Formally, the Stanley symmetric function Fw(x1, x2, ...) indexed by a permutation w is defined as a sum of certain fundamental quasisymmetric functions. Each summand corresponds to a reduced decomposition of w, that is, to a way of writing w as a product of a minimal possible number of adjacent transpositions. They were introduced in the course of Stanley's enumeration of the reduced decompositions of permutations, and in particular his proof that the permutation w0 = n(n − 1)...21 (written here in one-line notation) has exactly
${\frac {{\binom {n}{2}}!}{1^{n-1}\cdot 3^{n-2}\cdot 5^{n-3}\cdots (2n-3)^{1}}}$
reduced decompositions. (Here ${\binom {n}{2}}$ denotes the binomial coefficient n(n − 1)/2 and ! denotes the factorial.)
Properties
The Stanley symmetric function Fw is homogeneous with degree equal to the number of inversions of w. Unlike other nice families of symmetric functions, the Stanley symmetric functions have many linear dependencies and so do not form a basis of the ring of symmetric functions. When a Stanley symmetric function is expanded in the basis of Schur functions, the coefficients are all non-negative integers.
The Stanley symmetric functions have the property that they are the stable limit of Schubert polynomials
$F_{w}(x)=\lim _{n\to \infty }{\mathfrak {S}}_{1^{n}\times w}(x)$
where we treat both sides as formal power series, and take the limit coefficientwise.
References
• Stanley, Richard P. (1984), "On the number of reduced decompositions of elements of Coxeter groups" (PDF), European Journal of Combinatorics, 5 (4): 359–372, doi:10.1016/s0195-6698(84)80039-6, ISSN 0195-6698, MR 0782057
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Stanley–Reisner ring
In mathematics, a Stanley–Reisner ring, or face ring, is a quotient of a polynomial algebra over a field by a square-free monomial ideal. Such ideals are described more geometrically in terms of finite simplicial complexes. The Stanley–Reisner ring construction is a basic tool within algebraic combinatorics and combinatorial commutative algebra.[1] Its properties were investigated by Richard Stanley, Melvin Hochster, and Gerald Reisner in the early 1970s.
Definition and properties
Given an abstract simplicial complex Δ on the vertex set {x1,...,xn} and a field k, the corresponding Stanley–Reisner ring, or face ring, denoted k[Δ], is obtained from the polynomial ring k[x1,...,xn] by quotienting out the ideal IΔ generated by the square-free monomials corresponding to the non-faces of Δ:
$I_{\Delta }=(x_{i_{1}}\ldots x_{i_{r}}:\{i_{1},\ldots ,i_{r}\}\notin \Delta ),\quad k[\Delta ]=k[x_{1},\ldots ,x_{n}]/I_{\Delta }.$
The ideal IΔ is called the Stanley–Reisner ideal or the face ideal of Δ.[2]
Properties
• The Stanley–Reisner ring k[Δ] is multigraded by Zn, where the degree of the variable xi is the ith standard basis vector ei of Zn.
• As a vector space over k, the Stanley–Reisner ring of Δ admits a direct sum decomposition
$k[\Delta ]=\bigoplus _{\sigma \in \Delta }k[\Delta ]_{\sigma },$
whose summands k[Δ]σ have a basis of the monomials (not necessarily square-free) supported on the faces σ of Δ.
• The Krull dimension of k[Δ] is one larger than the dimension of the simplicial complex Δ.
• The multigraded, or fine, Hilbert series of k[Δ] is given by the formula
$H(k[\Delta ];x_{1},\ldots ,x_{n})=\sum _{\sigma \in \Delta }\prod _{i\in \sigma }{\frac {x_{i}}{1-x_{i}}}.$
• The ordinary, or coarse, Hilbert series of k[Δ] is obtained from its multigraded Hilbert series by setting the degree of every variable xi equal to 1:
$H(k[\Delta ];t,\ldots ,t)={\frac {1}{(1-t)^{n}}}\sum _{i=0}^{d}f_{i-1}t^{i}(1-t)^{n-i},$
where d = dim(Δ) + 1 is the Krull dimension of k[Δ] and fi is the number of i-faces of Δ. If it is written in the form
$H(k[\Delta ];t,\ldots ,t)={\frac {h_{0}+h_{1}t+\cdots +h_{d}t^{d}}{(1-t)^{d}}}$
then the coefficients (h0, ..., hd) of the numerator form the h-vector of the simplicial complex Δ.
Examples
It is common to assume that every vertex {xi} is a simplex in Δ. Thus none of the variables belongs to the Stanley–Reisner ideal IΔ.
• Δ is a simplex {x1,...,xn}. Then IΔ is the zero ideal and
$k[\Delta ]=k[x_{1},\ldots ,x_{n}]$
is the polynomial algebra in n variables over k.
• The simplicial complex Δ consists of n isolated vertices {x1}, ..., {xn}. Then
$I_{\Delta }=\{x_{i}x_{j}:1\leq i<j\leq n\}$
and the Stanley–Reisner ring is the following truncation of the polynomial ring in n variables over k:
$k[\Delta ]=k\oplus \bigoplus _{1\leq i\leq n}x_{i}k[x_{i}].$
• Generalizing the previous two examples, let Δ be the d-skeleton of the simplex {x1,...,xn}, thus it consists of all (d + 1)-element subsets of {x1,...,xn}. Then the Stanley–Reisner ring is following truncation of the polynomial ring in n variables over k:
$k[\Delta ]=k\oplus \bigoplus _{0\leq r\leq d}\bigoplus _{i_{0}<\ldots <i_{r}}x_{i_{0}}\ldots x_{i_{r}}k[x_{i_{0}},\ldots ,x_{i_{r}}].$
• Suppose that the abstract simplicial complex Δ is a simplicial join of abstract simplicial complexes Δ′ on x1,...,xm and Δ′′ on xm+1,...,xn. Then the Stanley–Reisner ring of Δ is the tensor product over k of the Stanley–Reisner rings of Δ′ and Δ′′:
$k[\Delta ]\simeq k[\Delta ']\otimes _{k}k[\Delta ''].$
Cohen–Macaulay condition and the upper bound conjecture
The face ring k[Δ] is a multigraded algebra over k all of whose components with respect to the fine grading have dimension at most 1. Consequently, its homology can be studied by combinatorial and geometric methods. An abstract simplicial complex Δ is called Cohen–Macaulay over k if its face ring is a Cohen–Macaulay ring.[3] In his 1974 thesis, Gerald Reisner gave a complete characterization of such complexes. This was soon followed up by more precise homological results about face rings due to Melvin Hochster. Then Richard Stanley found a way to prove the Upper Bound Conjecture for simplicial spheres, which was open at the time, using the face ring construction and Reisner's criterion of Cohen–Macaulayness. Stanley's idea of translating difficult conjectures in algebraic combinatorics into statements from commutative algebra and proving them by means of homological techniques was the origin of the rapidly developing field of combinatorial commutative algebra.
Reisner's criterion
A simplicial complex Δ is Cohen–Macaulay over k if and only if for all simplices σ ∈ Δ, all reduced simplicial homology groups of the link of σ in Δ with coefficients in k are zero, except the top dimensional one:[3]
${\tilde {H}}_{i}(\operatorname {link} _{\Delta }(\sigma );k)=0\quad {\text{for all}}\quad i<\dim \operatorname {link} _{\Delta }(\sigma ).$
A result due to Munkres then shows that the Cohen–Macaulayness of Δ over k is a topological property: it depends only on the homeomorphism class of the simplicial complex Δ. Namely, let |Δ| be the geometric realization of Δ. Then the vanishing of the simplicial homology groups in Reisner's criterion is equivalent to the following statement about the reduced and relative singular homology groups of |Δ|:
${\text{For all }}p\in |\Delta |{\text{ and for all }}i<\dim |\Delta |=d-1,\quad {\tilde {H}}_{i}(\operatorname {|} \Delta |;k)=H_{i}(\operatorname {|} \Delta |,\operatorname {|} \Delta |-p;k)=0.$
In particular, if the complex Δ is a simplicial sphere, that is, |Δ| is homeomorphic to a sphere, then it is Cohen–Macaulay over any field. This is a key step in Stanley's proof of the Upper Bound Conjecture. By contrast, there are examples of simplicial complexes whose Cohen–Macaulayness depends on the characteristic of the field k.
References
1. Miller & Sturmfels (2005) p.19
2. Miller & Sturmfels (2005) pp.3–5
3. Miller & Sturmfels (2005) p.101
• Melvin Hochster, Cohen-Macaulay rings, combinatorics, and simplicial complexes. Ring theory, II (Proc. Second Conf., Univ. Oklahoma, Norman, Okla., 1975), pp. 171–223. Lecture Notes in Pure and Appl. Math., Vol. 26, Dekker, New York, 1977
• Stanley, Richard (1996). Combinatorics and commutative algebra. Progress in Mathematics. Vol. 41 (Second ed.). Boston, MA: Birkhäuser Boston. ISBN 0-8176-3836-9. Zbl 0838.13008.
• Bruns, Winfried; Herzog, Jürgen (1993). Cohen–Macaulay rings. Cambridge Studies in Advanced Mathematics. Vol. 39. Cambridge University Press. ISBN 0-521-41068-1. Zbl 0788.13005.
• Miller, Ezra; Sturmfels, Bernd (2005). Combinatorial commutative algebra. Graduate Texts in Mathematics. Vol. 227. New York, NY: Springer-Verlag. ISBN 0-387-23707-0. Zbl 1090.13001.
Further reading
• Panov, Taras E. (2008). "Cohomology of face rings, and torus actions". In Young, Nicholas; Choi, Yemon (eds.). Surveys in contemporary mathematics. London Mathematical Society Lecture Note Series. Vol. 347. Cambridge University Press. pp. 165–201. ISBN 978-0-521-70564-6. Zbl 1140.13018.
External links
• "Stanley–Reisner ring", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
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Star-free language
A regular language is said to be star-free if it can be described by a regular expression constructed from the letters of the alphabet, the empty set symbol, all boolean operators – including complementation – and concatenation but no Kleene star.[1] For instance, the language of words over the alphabet $\{a,\,b\}$ that do not have consecutive a's can be defined by $(\emptyset ^{c}aa\emptyset ^{c})^{c}$, where $X^{c}$ denotes the complement of a subset $X$ of $\{a,\,b\}^{*}$. The condition is equivalent to having generalized star height zero.
An example of a regular language which is not star-free is $\{(aa)^{n}\mid n\geq 0\}$,[2] i.e. the language of strings consisting of an even number of "a".
Marcel-Paul Schützenberger characterized star-free languages as those with aperiodic syntactic monoids.[3][4] They can also be characterized logically as languages definable in FO[<], the first-order logic over the natural numbers with the less-than relation,[5] as the counter-free languages[6] and as languages definable in linear temporal logic.[7]
All star-free languages are in uniform AC0.
See also
• Star height
• Generalized star height problem
• Star height problem
References
1. Lawson (2004) p.235
2. Arto Salomaa (1981). Jewels of Formal Language Theory. Computer Science Press. p. 53. ISBN 978-0-914894-69-8.
3. Marcel-Paul Schützenberger (1965). "On finite monoids having only trivial subgroups" (PDF). Information and Computation. 8 (2): 190–194. doi:10.1016/s0019-9958(65)90108-7.
4. Lawson (2004) p.262
5. Straubing, Howard (1994). Finite automata, formal logic, and circuit complexity. Progress in Theoretical Computer Science. Basel: Birkhäuser. p. 79. ISBN 3-7643-3719-2. Zbl 0816.68086.
6. McNaughton, Robert; Papert, Seymour (1971). Counter-free Automata. Research Monograph. Vol. 65. With an appendix by William Henneman. MIT Press. ISBN 0-262-13076-9. Zbl 0232.94024.
7. Kamp, Johan Antony Willem (1968). Tense Logic and the Theory of Linear Order. University of California at Los Angeles (UCLA).
• Lawson, Mark V. (2004). Finite automata. Chapman and Hall/CRC. ISBN 1-58488-255-7. Zbl 1086.68074.
• Diekert, Volker; Gastin, Paul (2008). "First-order definable languages". In Jörg Flum; Erich Grädel; Thomas Wilke (eds.). Logic and automata: history and perspectives (PDF). Amsterdam University Press. ISBN 978-90-5356-576-6.
Automata theory: formal languages and formal grammars
Chomsky hierarchyGrammarsLanguagesAbstract machines
• Type-0
• —
• Type-1
• —
• —
• —
• —
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• Unrestricted
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Each category of languages, except those marked by a *, is a proper subset of the category directly above it. Any language in each category is generated by a grammar and by an automaton in the category in the same line.
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Wikipedia
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Star domain
In geometry, a set $S$ in the Euclidean space $\mathbb {R} ^{n}$ is called a star domain (or star-convex set, star-shaped set or radially convex set) if there exists an $s_{0}\in S$ such that for all $s\in S,$ the line segment from $s_{0}$ to $s$ lies in $S.$ This definition is immediately generalizable to any real, or complex, vector space.
Intuitively, if one thinks of $S$ as a region surrounded by a wall, $S$ is a star domain if one can find a vantage point $s_{0}$ in $S$ from which any point $s$ in $S$ is within line-of-sight. A similar, but distinct, concept is that of a radial set.
Definition
Given two points $x$ and $y$ in a vector space $X$ (such as Euclidean space $\mathbb {R} ^{n}$), the convex hull of $\{x,y\}$ is called the closed interval with endpoints $x$ and $y$ and it is denoted by
$\left[x,y\right]~:=~\left\{tx+(1-t)y:0\leq t\leq 1\right\}~=~x+(y-x)[0,1],$
where $z[0,1]:=\{zt:0\leq t\leq 1\}$ for every vector $z.$
A subset $S$ of a vector space $X$ is said to be star-shaped at $s_{0}\in S$ if for every $s\in S,$ the closed interval $\left[s_{0},s\right]\subseteq S.$ A set $S$ is star shaped and is called a star domain if there exists some point $s_{0}\in S$ such that $S$ is star-shaped at $s_{0}.$
A set that is star-shaped at the origin is sometimes called a star set.[1] Such sets are closed related to Minkowski functionals.
Examples
• Any line or plane in $\mathbb {R} ^{n}$ is a star domain.
• A line or a plane with a single point removed is not a star domain.
• If $A$ is a set in $\mathbb {R} ^{n},$ the set $B=\{ta:a\in A,t\in [0,1]\}$ obtained by connecting all points in $A$ to the origin is a star domain.
• Any non-empty convex set is a star domain. A set is convex if and only if it is a star domain with respect to any point in that set.
• A cross-shaped figure is a star domain but is not convex.
• A star-shaped polygon is a star domain whose boundary is a sequence of connected line segments.
Properties
• The closure of a star domain is a star domain, but the interior of a star domain is not necessarily a star domain.
• Every star domain is a contractible set, via a straight-line homotopy. In particular, any star domain is a simply connected set.
• Every star domain, and only a star domain, can be "shrunken into itself"; that is, for every dilation ratio $r<1,$ the star domain can be dilated by a ratio $r$ such that the dilated star domain is contained in the original star domain.[2]
• The union and intersection of two star domains is not necessarily a star domain.
• A non-empty open star domain $S$ in $\mathbb {R} ^{n}$ is diffeomorphic to $\mathbb {R} ^{n}.$
• Given $W\subseteq X,$ the set $\bigcap _{|u|=1}uW$ (where $u$ ranges over all unit length scalars) is a balanced set whenever $W$ is a star shaped at the origin (meaning that $0\in W$ and $rw\in W$ for all $0\leq r\leq 1$ and $w\in W$).
See also
• Absolutely convex set – convex and balanced setPages displaying wikidata descriptions as a fallback
• Absorbing set – Set that can be "inflated" to reach any point
• Art gallery problem – Mathematical problem
• Balanced set – Construct in functional analysis
• Bounded set (topological vector space) – Generalization of boundedness
• Convex set – In geometry, set whose intersection with every line is a single line segment
• Minkowski functional – Function made from a set
• Radial set
• Star polygon – Regular non-convex polygon
• Symmetric set – Property of group subsets (mathematics)
References
1. Schechter 1996, p. 303.
2. Drummond-Cole, Gabriel C. "What polygons can be shrinked into themselves?". Math Overflow. Retrieved 2 October 2014.
• Ian Stewart, David Tall, Complex Analysis. Cambridge University Press, 1983, ISBN 0-521-28763-4, MR0698076
• C.R. Smith, A characterization of star-shaped sets, American Mathematical Monthly, Vol. 75, No. 4 (April 1968). p. 386, MR0227724, JSTOR 2313423
• Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
• Schechter, Eric (1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. ISBN 978-0-12-622760-4. OCLC 175294365.
External links
Wikimedia Commons has media related to Star-shaped sets.
• Humphreys, Alexis. "Star convex". MathWorld.
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Wikipedia
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Star coloring
In the mathematical field of graph theory, a star coloring of a graph G is a (proper) vertex coloring in which every path on four vertices uses at least three distinct colors. Equivalently, in a star coloring, the induced subgraphs formed by the vertices of any two colors has connected components that are star graphs. Star coloring has been introduced by Grünbaum (1973). The star chromatic number $\chi _{s}(G)$ of G is the fewest colors needed to star color G.
One generalization of star coloring is the closely related concept of acyclic coloring, where it is required that every cycle uses at least three colors, so the two-color induced subgraphs are forests. If we denote the acyclic chromatic number of a graph G by $\chi _{a}(G)$, we have that $\chi _{a}(G)\leq \chi _{s}(G)$, and in fact every star coloring of G is an acyclic coloring.
The star chromatic number has been proved to be bounded on every proper minor closed class by Nešetřil & Ossona de Mendez (2003). This results was further generalized by Nešetřil & Ossona de Mendez (2006) to all low-tree-depth colorings (standard coloring and star coloring being low-tree-depth colorings with respective parameter 1 and 2).
Complexity
It was demonstrated by Albertson et al. (2004) that it is NP-complete to determine whether $\chi _{s}(G)\leq 3$, even when G is a graph that is both planar and bipartite. Coleman & Moré (1984) showed that finding an optimal star coloring is NP-hard even when G is a bipartite graph.
References
• Albertson, Michael O.; Chappell, Glenn G.; Kierstead, Hal A.; Kündgen, André; Ramamurthi, Radhika (2004), "Coloring with no 2-Colored P4's", The Electronic Journal of Combinatorics, 11 (1), doi:10.37236/1779, MR 2056078.
• Coleman, Thomas F.; Moré, Jorge (1984), "Estimation of sparse Hessian matrices and graph coloring problems" (PDF), Mathematical Programming, 28 (3): 243–270, doi:10.1007/BF02612334, hdl:1813/6374, MR 0736293.
• Fertin, Guillaume; Raspaud, André; Reed, Bruce (2004), "Star coloring of graphs", Journal of Graph Theory, 47 (3): 163–182, doi:10.1002/jgt.20029, MR 2089462.
• Grünbaum, Branko (1973), "Acyclic colorings of planar graphs", Israel Journal of Mathematics, 14 (4): 390–408, doi:10.1007/BF02764716, MR 0317982.
• Nešetřil, Jaroslav; Ossona de Mendez, Patrice (2003), "Colorings and homomorphisms of minor closed classes", Discrete & Computational Geometry: The Goodman-Pollack Festschrift, Algorithms & Combinatorics, vol. 25, Springer-Verlag, pp. 651–664, MR 2038495.
• Nešetřil, Jaroslav; Ossona de Mendez, Patrice (2006), "Tree depth, subgraph coloring and homomorphism bounds", European Journal of Combinatorics, 27 (6): 1022–1041, doi:10.1016/j.ejc.2005.01.010, MR 2226435.
External links
• Star colorings and acyclic colorings (1973), present at the Research Experiences for Graduate Students (REGS) at the University of Illinois, 2008.
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