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Wiedersehen pair
In mathematics—specifically, in Riemannian geometry—a Wiedersehen pair is a pair of distinct points x and y on a (usually, but not necessarily, two-dimensional) compact Riemannian manifold (M, g) such that every geodesic through x also passes through y, and the same with x and y interchanged.
For example, on an ordinary sphere where the geodesics are great circles, the Wiedersehen pairs are exactly the pairs of antipodal points.
If every point of an oriented manifold (M, g) belongs to a Wiedersehen pair, then (M, g) is said to be a Wiedersehen manifold. The concept was introduced by the Austro-Hungarian mathematician Wilhelm Blaschke and comes from the German term meaning "seeing again". As it turns out, in each dimension n the only Wiedersehen manifold (up to isometry) is the standard Euclidean n-sphere. Initially known as the Blaschke conjecture, this result was established by combined works of Berger, Kazdan, Weinstein (for even n), and Yang (odd n).
See also
• Cut locus (Riemannian manifold)
References
• Blaschke, Wilhelm (1921). Vorlesung über Differentialgeometrie I. Berlin: Springer-Verlag.
• C. T. Yang (1980). "Odd-dimensional wiedersehen manifolds are spheres". J. Differential Geom. 15 (1): 91–96. doi:10.4310/jdg/1214435386. ISSN 0022-040X.
• Chavel, Isaac (2006). Riemannian geometry: a modern introduction. New York: Cambridge University Press. pp. 328–329. ISBN 0-521-61954-8.
External links
• Weisstein, Eric W. "Wiedersehen pair". MathWorld.
• Weisstein, Eric W. "Wiedersehen surface". MathWorld.
| Wikipedia |
Wieferich prime
In number theory, a Wieferich prime is a prime number p such that p2 divides 2p − 1 − 1,[4] therefore connecting these primes with Fermat's little theorem, which states that every odd prime p divides 2p − 1 − 1. Wieferich primes were first described by Arthur Wieferich in 1909 in works pertaining to Fermat's Last Theorem, at which time both of Fermat's theorems were already well known to mathematicians.[5][6]
Wieferich prime
Named afterArthur Wieferich
Publication year1909
Author of publicationWieferich, A.
No. of known terms2
Conjectured no. of termsInfinite
Subsequence of
• Crandall numbers[1]
• Wieferich numbers[2]
• Lucas–Wieferich primes[3]
• near-Wieferich primes
First terms1093, 3511
Largest known term3511
OEIS indexA001220
Since then, connections between Wieferich primes and various other topics in mathematics have been discovered, including other types of numbers and primes, such as Mersenne and Fermat numbers, specific types of pseudoprimes and some types of numbers generalized from the original definition of a Wieferich prime. Over time, those connections discovered have extended to cover more properties of certain prime numbers as well as more general subjects such as number fields and the abc conjecture.
As of April 2023, the only known Wieferich primes are 1093 and 3511 (sequence A001220 in the OEIS).
Equivalent definitions
The stronger version of Fermat's little theorem, which a Wieferich prime satisfies, is usually expressed as a congruence relation 2p -1 ≡ 1 (mod p2). From the definition of the congruence relation on integers, it follows that this property is equivalent to the definition given at the beginning. Thus if a prime p satisfies this congruence, this prime divides the Fermat quotient ${\tfrac {2^{p-1}-1}{p}}$. The following are two illustrative examples using the primes 11 and 1093:
For p = 11, we get ${\tfrac {2^{10}-1}{11}}$ which is 93 and leaves a remainder of 5 after division by 11, hence 11 is not a Wieferich prime. For p = 1093, we get ${\tfrac {2^{1092}-1}{1093}}$ or 485439490310...852893958515 (302 intermediate digits omitted for clarity), which leaves a remainder of 0 after division by 1093 and thus 1093 is a Wieferich prime.
Wieferich primes can be defined by other equivalent congruences. If p is a Wieferich prime, one can multiply both sides of the congruence 2p−1 ≡ 1 (mod p2) by 2 to get 2p ≡ 2 (mod p2). Raising both sides of the congruence to the power p shows that a Wieferich prime also satisfies 2p2 ≡2p ≡ 2 (mod p2), and hence 2pk ≡ 2 (mod p2) for all k ≥ 1. The converse is also true: 2pk ≡ 2 (mod p2) for some k ≥ 1 implies that the multiplicative order of 2 modulo p2 divides gcd(pk − 1, φ(p2)) = p − 1, that is, 2p−1 ≡ 1 (mod p2) and thus p is a Wieferich prime. This also implies that Wieferich primes can be defined as primes p such that the multiplicative orders of 2 modulo p and modulo p2 coincide: ordp2 2 = ordp 2, (By the way, ord10932 = 364, and ord35112 = 1755).
H. S. Vandiver proved that 2p−1 ≡ 1 (mod p3) if and only if $1+{\tfrac {1}{3}}+\dots +{\tfrac {1}{p-2}}\equiv 0{\pmod {p^{2}}}$.[7]: 187
History and search status
In 1902, Meyer proved a theorem about solutions of the congruence ap − 1 ≡ 1 (mod pr).[8]: 930 [9] Later in that decade Arthur Wieferich showed specifically that if the first case of Fermat's last theorem has solutions for an odd prime exponent, then that prime must satisfy that congruence for a = 2 and r = 2.[10] In other words, if there exist solutions to xp + yp + zp = 0 in integers x, y, z and p an odd prime with p ∤ xyz, then p satisfies 2p − 1 ≡ 1 (mod p2). In 1913, Bachmann examined the residues of ${\tfrac {2^{p-1}-1}{p}}\,{\bmod {\,}}p$. He asked the question when this residue vanishes and tried to find expressions for answering this question.[11]
The prime 1093 was found to be a Wieferich prime by W. Meissner in 1913 and confirmed to be the only such prime below 2000. He calculated the smallest residue of ${\tfrac {2^{t}-1}{p}}\,{\bmod {\,}}p$ for all primes p < 2000 and found this residue to be zero for t = 364 and p = 1093, thereby providing a counterexample to a conjecture by Grave about the impossibility of the Wieferich congruence.[12] E. Haentzschel later ordered verification of the correctness of Meissner's congruence via only elementary calculations.[13]: 664 Inspired by an earlier work of Euler, he simplified Meissner's proof by showing that 10932 | (2182 + 1) and remarked that (2182 + 1) is a factor of (2364 − 1).[14] It was also shown that it is possible to prove that 1093 is a Wieferich prime without using complex numbers contrary to the method used by Meissner,[15] although Meissner himself hinted at that he was aware of a proof without complex values.[12]: 665
The prime 3511 was first found to be a Wieferich prime by N. G. W. H. Beeger in 1922[16] and another proof of it being a Wieferich prime was published in 1965 by Guy.[17] In 1960, Kravitz[18] doubled a previous record set by Fröberg[19] and in 1961 Riesel extended the search to 500000 with the aid of BESK.[20] Around 1980, Lehmer was able to reach the search limit of 6×109.[21] This limit was extended to over 2.5×1015 in 2006,[22] finally reaching 3×1015. It is now known that if any other Wieferich primes exist, they must be greater than 6.7×1015.[23]
In 2007–2016, a search for Wieferich primes was performed by the distributed computing project Wieferich@Home.[24] In 2011–2017, another search was performed by the PrimeGrid project, although later the work done in this project was claimed wasted.[25] While these projects reached search bounds above 1×1017, neither of them reported any sustainable results.
In 2020, PrimeGrid started another project that searches for Wieferich and Wall–Sun–Sun primes simultaneously. The new project uses checksums to enable independent double-checking of each subinterval, thus minimizing the risk of missing an instance because of faulty hardware.[26] The project ended in December 2022, definitely proving that a third Wieferich prime must exceed 264 (about 18×1018).[27]
It has been conjectured (as for Wilson primes) that infinitely many Wieferich primes exist, and that the number of Wieferich primes below x is approximately log(log(x)), which is a heuristic result that follows from the plausible assumption that for a prime p, the (p − 1)-th degree roots of unity modulo p2 are uniformly distributed in the multiplicative group of integers modulo p2.[28]
Properties
Connection with Fermat's Last Theorem
The following theorem connecting Wieferich primes and Fermat's Last Theorem was proven by Wieferich in 1909:[10]
Let p be prime, and let x, y, z be integers such that xp + yp + zp = 0. Furthermore, assume that p does not divide the product xyz. Then p is a Wieferich prime.
The above case (where p does not divide any of x, y or z) is commonly known as the first case of Fermat's Last Theorem (FLTI)[29][30] and FLTI is said to fail for a prime p, if solutions to the Fermat equation exist for that p, otherwise FLTI holds for p.[31] In 1910, Mirimanoff expanded[32] the theorem by showing that, if the preconditions of the theorem hold true for some prime p, then p2 must also divide 3p − 1 − 1. Granville and Monagan further proved that p2 must actually divide mp − 1 − 1 for every prime m ≤ 89.[33] Suzuki extended the proof to all primes m ≤ 113.[34]
Let Hp be a set of pairs of integers with 1 as their greatest common divisor, p being prime to x, y and x + y, (x + y)p−1 ≡ 1 (mod p2), (x + ξy) being the pth power of an ideal of K with ξ defined as cos 2π/p + i sin 2π/p. K = Q(ξ) is the field extension obtained by adjoining all polynomials in the algebraic number ξ to the field of rational numbers (such an extension is known as a number field or in this particular case, where ξ is a root of unity, a cyclotomic number field).[33]: 332 From uniqueness of factorization of ideals in Q(ξ) it follows that if the first case of Fermat's last theorem has solutions x, y, z then p divides x+y+z and (x, y), (y, z) and (z, x) are elements of Hp.[33]: 333 Granville and Monagan showed that (1, 1) ∈ Hp if and only if p is a Wieferich prime.[33]: 333
Connection with the abc conjecture and non-Wieferich primes
A non-Wieferich prime is a prime p satisfying 2p − 1 ≢ 1 (mod p2). J. H. Silverman showed in 1988 that if the abc conjecture holds, then there exist infinitely many non-Wieferich primes.[35] More precisely he showed that the abc conjecture implies the existence of a constant only depending on α such that the number of non-Wieferich primes to base α with p less than or equal to a variable X is greater than log(X) as X goes to infinity.[36]: 227 Numerical evidence suggests that very few of the prime numbers in a given interval are Wieferich primes. The set of Wieferich primes and the set of non-Wieferich primes, sometimes denoted by W2 and W2c respectively,[37] are complementary sets, so if one of them is shown to be finite, the other one would necessarily have to be infinite, because both are proper subsets of the set of prime numbers. It was later shown that the existence of infinitely many non-Wieferich primes already follows from a weaker version of the abc conjecture, called the ABC-(k, ε) conjecture.[38] Additionally, the existence of infinitely many non-Wieferich primes would also follow if there exist infinitely many square-free Mersenne numbers[39] as well as if there exists a real number ξ such that the set {n ∈ N : λ(2n − 1) < 2 − ξ} is of density one, where the index of composition λ(n) of an integer n is defined as ${\tfrac {\log n}{\log \gamma (n)}}$ and $\gamma (n)=\prod _{p\mid n}p$, meaning $\gamma (n)$ gives the product of all prime factors of n.[37]: 4
Connection with Mersenne and Fermat primes
It is known that the nth Mersenne number Mn = 2n − 1 is prime only if n is prime. Fermat's little theorem implies that if p > 2 is prime, then Mp−1 (= 2p − 1 − 1) is always divisible by p. Since Mersenne numbers of prime indices Mp and Mq are co-prime,
A prime divisor p of Mq, where q is prime, is a Wieferich prime if and only if p2 divides Mq.[40]
Thus, a Mersenne prime cannot also be a Wieferich prime. A notable open problem is to determine whether or not all Mersenne numbers of prime index are square-free. If q is prime and the Mersenne number Mq is not square-free, that is, there exists a prime p for which p2 divides Mq, then p is a Wieferich prime. Therefore, if there are only finitely many Wieferich primes, then there will be at most finitely many Mersenne numbers with prime index that are not square-free. Rotkiewicz showed a related result: if there are infinitely many square-free Mersenne numbers, then there are infinitely many non-Wieferich primes.[41]
Similarly, if p is prime and p2 divides some Fermat number Fn = 22n + 1, then p must be a Wieferich prime.[42]
In fact, there exists a natural number n and a prime p that p2 divides $\Phi _{n}(2)$ (where $\Phi _{n}(x)$ is the n-th cyclotomic polynomial) if and only if p is a Wieferich prime. For example, 10932 divides $\Phi _{364}(2)$, 35112 divides $\Phi _{1755}(2)$. Mersenne and Fermat numbers are just special situations of $\Phi _{n}(2)$. Thus, if 1093 and 3511 are only two Wieferich primes, then all $\Phi _{n}(2)$ are square-free except $\Phi _{364}(2)$ and $\Phi _{1755}(2)$ (In fact, when there exists a prime p which p2 divides some $\Phi _{n}(2)$, then it is a Wieferich prime); and clearly, if $\Phi _{n}(2)$ is a prime, then it cannot be Wieferich prime. (Any odd prime p divides only one $\Phi _{n}(2)$ and n divides p − 1, and if and only if the period length of 1/p in binary is n, then p divides $\Phi _{n}(2)$. Besides, if and only if p is a Wieferich prime, then the period length of 1/p and 1/p2 are the same (in binary). Otherwise, this is p times than that.)
For the primes 1093 and 3511, it was shown that neither of them is a divisor of any Mersenne number with prime index nor a divisor of any Fermat number, because 364 and 1755 are neither prime nor powers of 2.[43]
Connection with other equations
Scott and Styer showed that the equation px – 2y = d has at most one solution in positive integers (x, y), unless when p4 | 2ordp 2 – 1 if p ≢ 65 (mod 192) or unconditionally when p2 | 2ordp 2 – 1, where ordp 2 denotes the multiplicative order of 2 modulo p.[44]: 215, 217–218 They also showed that a solution to the equation ±ax1 ± 2y1 = ±ax2 ± 2y2 = c must be from a specific set of equations but that this does not hold, if a is a Wieferich prime greater than 1.25 x 1015.[45]: 258
Binary periodicity of p − 1
Johnson observed[46] that the two known Wieferich primes are one greater than numbers with periodic binary expansions (1092 = 0100010001002=44416; 3510 = 1101101101102=66668). The Wieferich@Home project searched for Wieferich primes by testing numbers that are one greater than a number with a periodic binary expansion, but up to a "bit pseudo-length" of 3500 of the tested binary numbers generated by combination of bit strings with a bit length of up to 24 it has not found a new Wieferich prime.[47]
Abundancy of p − 1
It has been noted (sequence A239875 in the OEIS) that the known Wieferich primes are one greater than mutually friendly numbers (the shared abundancy index being 112/39).
Connection with pseudoprimes
It was observed that the two known Wieferich primes are the square factors of all non-square free base-2 Fermat pseudoprimes up to 25×109.[48] Later computations showed that the only repeated factors of the pseudoprimes up to 1012 are 1093 and 3511.[49] In addition, the following connection exists:
Let n be a base 2 pseudoprime and p be a prime divisor of n. If ${\tfrac {2^{n-1}-1}{n}}\not \equiv 0{\pmod {p}}$, then also ${\tfrac {2^{p-1}-1}{p}}\not \equiv 0{\pmod {p}}$.[31]: 378 Furthermore, if p is a Wieferich prime, then p2 is a Catalan pseudoprime.
Connection with directed graphs
For all primes p up to 100000, L(pn+1) = L(pn) only in two cases: L(10932) = L(1093) = 364 and L(35112) = L(3511) = 1755, where L(m) is the number of vertices in the cycle of 1 in the doubling diagram modulo m. Here the doubling diagram represents the directed graph with the non-negative integers less than m as vertices and with directed edges going from each vertex x to vertex 2x reduced modulo m.[50]: 74 It was shown, that for all odd prime numbers either L(pn+1) = p · L(pn) or L(pn+1) = L(pn).[50]: 75
Properties related to number fields
It was shown that $\chi _{D_{0}}{\big (}p{\big )}=1$ and $\lambda \,\!_{p}{\big (}\mathbb {Q} {\big (}{\sqrt {D_{0}}}{\big )}{\big )}=1$ if and only if 2p − 1 ≢ 1 (mod p2) where p is an odd prime and $D_{0}<0$ is the fundamental discriminant of the imaginary quadratic field $\mathbb {Q} {\big (}{\sqrt {1-p^{2}}}{\big )}$. Furthermore, the following was shown: Let p be a Wieferich prime. If p ≡ 3 (mod 4), let $D_{0}<0$ be the fundamental discriminant of the imaginary quadratic field $\mathbb {Q} {\big (}{\sqrt {1-p}}{\big )}$ and if p ≡ 1 (mod 4), let $D_{0}<0$ be the fundamental discriminant of the imaginary quadratic field $\mathbb {Q} {\big (}{\sqrt {4-p}}{\big )}$. Then $\chi _{D_{0}}{\big (}p{\big )}=1$ and $\lambda \,\!_{p}{\big (}\mathbb {Q} {\big (}{\sqrt {D_{0}}}{\big )}{\big )}=1$ (χ and λ in this context denote Iwasawa invariants).[51]: 27
Furthermore, the following result was obtained: Let q be an odd prime number, k and p are primes such that p = 2k + 1, k ≡ 3 (mod 4), p ≡ −1 (mod q), p ≢ −1 (mod q3) and the order of q modulo k is ${\tfrac {k-1}{2}}$. Assume that q divides h+, the class number of the real cyclotomic field $\mathbb {Q} {\big (}\zeta \,\!_{p}+\zeta \,\!_{p}^{-1}{\big )}$, the cyclotomic field obtained by adjoining the sum of a p-th root of unity and its reciprocal to the field of rational numbers. Then q is a Wieferich prime.[52]: 55 This also holds if the conditions p ≡ −1 (mod q) and p ≢ −1 (mod q3) are replaced by p ≡ −3 (mod q) and p ≢ −3 (mod q3) as well as when the condition p ≡ −1 (mod q) is replaced by p ≡ −5 (mod q) (in which case q is a Wall–Sun–Sun prime) and the incongruence condition replaced by p ≢ −5 (mod q3).[53]: 376
Generalizations
Near-Wieferich primes
A prime p satisfying the congruence 2(p−1)/2 ≡ ±1 + Ap (mod p2) with small |A| is commonly called a near-Wieferich prime (sequence A195988 in the OEIS).[28][54] Near-Wieferich primes with A = 0 represent Wieferich primes. Recent searches, in addition to their primary search for Wieferich primes, also tried to find near-Wieferich primes.[23][55] The following table lists all near-Wieferich primes with |A| ≤ 10 in the interval [1×109, 3×1015].[56] This search bound was reached in 2006 in a search effort by P. Carlisle, R. Crandall and M. Rodenkirch.[22][57] Bigger entries are by PrimeGrid.
p1 or −1A
3520624567+1−6
46262476201+1+5
47004625957−1+1
58481216789−1+5
76843523891−1+1
1180032105761+1−6
12456646902457+1+2
134257821895921+1+10
339258218134349−1+2
2276306935816523−1−3
82687771042557349-1-10
3156824277937156367+1+7
The sign +1 or -1 above can be easily predicted by Euler's criterion (and the second supplement to the law of quadratic reciprocity).
Dorais and Klyve[23] used a different definition of a near-Wieferich prime, defining it as a prime p with small value of $\left|{\tfrac {\omega (p)}{p}}\right|$ where $\omega (p)={\tfrac {2^{p-1}-1}{p}}\,{\bmod {\,}}p$ is the Fermat quotient of 2 with respect to p modulo p (the modulo operation here gives the residue with the smallest absolute value). The following table lists all primes p ≤ 6.7 × 1015 with $\left|{\tfrac {\omega (p)}{p}}\right|\leq 10^{-14}$.
p$\omega (p)$$\left|{\tfrac {\omega (p)}{p}}\right|\times 10^{14}$
109300
351100
2276306935816523+60.264
3167939147662997−170.537
3723113065138349−360.967
5131427559624857−360.702
5294488110626977−310.586
6517506365514181+580.890
The two notions of nearness are related as follows. If $2^{(p-1)/2}\equiv \pm 1+Ap{\pmod {p^{2}}}$, then by squaring, clearly $2^{p-1}\equiv 1\pm 2Ap{\pmod {p^{2}}}$. So if A had been chosen with $|A|$ small, then clearly $|\!\pm 2A|=2|A|$ is also (quite) small, and an even number. However, when $\omega (p)$ is odd above, the related A from before the last squaring was not "small". For example, with $p=3167939147662997$, we have $2^{(p-1)/2}\equiv -1-1583969573831490p{\pmod {p^{2}}}$ which reads extremely non-near, but after squaring this is $2^{p-1}\equiv 1-17p{\pmod {p^{2}}}$ which is a near-Wieferich by the second definition.
Base-a Wieferich primes
Main article: Fermat quotient
A Wieferich prime base a is a prime p that satisfies
ap − 1 ≡ 1 (mod p2),[8] with a less than p but greater than 1.
Such a prime cannot divide a, since then it would also divide 1.
It's a conjecture that for every natural number a, there are infinitely many Wieferich primes in base a.
Bolyai showed that if p and q are primes, a is a positive integer not divisible by p and q such that ap−1 ≡ 1 (mod q), aq−1 ≡ 1 (mod p), then apq−1 ≡ 1 (mod pq). Setting p = q leads to ap2−1 ≡ 1 (mod p2).[58]: 284 It was shown that ap2−1 ≡ 1 (mod p2) if and only if ap−1 ≡ 1 (mod p2).[58]: 285–286
Known solutions of ap−1 ≡ 1 (mod p2) for small values of a are:[59] (checked up to 5 × 1013)
a primes p such that ap − 1 = 1 (mod p2) OEIS sequence
12, 3, 5, 7, 11, 13, 17, 19, 23, 29, ... (All primes) A000040
21093, 3511, ... A001220
311, 1006003, ... A014127
41093, 3511, ...
52, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801, ... A123692
666161, 534851, 3152573, ... A212583
75, 491531, ... A123693
83, 1093, 3511, ...
92, 11, 1006003, ...
103, 487, 56598313, ... A045616
1171, ...
122693, 123653, ... A111027
132, 863, 1747591, ... A128667
1429, 353, 7596952219, ... A234810
1529131, 119327070011, ... A242741
161093, 3511, ...
172, 3, 46021, 48947, 478225523351, ... A128668
185, 7, 37, 331, 33923, 1284043, ... A244260
193, 7, 13, 43, 137, 63061489, ... A090968
20281, 46457, 9377747, 122959073, ... A242982
212, ...
2213, 673, 1595813, 492366587, 9809862296159, ... A298951
2313, 2481757, 13703077, 15546404183, 2549536629329, ... A128669
245, 25633, ...
252, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801, ...
263, 5, 71, 486999673, 6695256707, ... A306255
2711, 1006003, ...
283, 19, 23, ...
292, ...
307, 160541, 94727075783, ... A306256
317, 79, 6451, 2806861, ... A331424
325, 1093, 3511, ...
332, 233, 47441, 9639595369, ...
3446145917691, ...
353, 1613, 3571, ...
3666161, 534851, 3152573, ...
372, 3, 77867, 76407520781, ... A331426
3817, 127, ...
398039, ...
4011, 17, 307, 66431, 7036306088681, ...
412, 29, 1025273, 138200401, ... A331427
4223, 719867822369, ...
435, 103, 13368932516573, ...
443, 229, 5851, ...
452, 1283, 131759, 157635607, ...
463, 829, ...
47...
487, 257, ...
492, 5, 491531, ...
507, ...
For more information, see[60][61][62] and.[63] (Note that the solutions to a = bk is the union of the prime divisors of k which does not divide b and the solutions to a = b)
The smallest solutions of np−1 ≡ 1 (mod p2) are
2, 1093, 11, 1093, 2, 66161, 5, 3, 2, 3, 71, 2693, 2, 29, 29131, 1093, 2, 5, 3, 281, 2, 13, 13, 5, 2, 3, 11, 3, 2, 7, 7, 5, 2, 46145917691, 3, 66161, 2, 17, 8039, 11, 2, 23, 5, 3, 2, 3, ... (The next term > 4.9×1013) (sequence A039951 in the OEIS)
There are no known solutions of np−1 ≡ 1 (mod p2) for n = 47, 72, 186, 187, 200, 203, 222, 231, 304, 311, 335, 355, 435, 454, 546, 554, 610, 639, 662, 760, 772, 798, 808, 812, 858, 860, 871, 983, 986, 1002, 1023, 1130, 1136, 1138, ....
It is a conjecture that there are infinitely many solutions of ap−1 ≡ 1 (mod p2) for every natural number a.
The bases b < p2 which p is a Wieferich prime are (for b > p2, the solutions are just shifted by k·p2 for k > 0), and there are p − 1 solutions < p2 of p and the set of the solutions congruent to p are {1, 2, 3, ..., p − 1}) (sequence A143548 in the OEIS)
p values of b < p2
2 1
3 1, 8
5 1, 7, 18, 24
7 1, 18, 19, 30, 31, 48
11 1, 3, 9, 27, 40, 81, 94, 112, 118, 120
13 1, 19, 22, 23, 70, 80, 89, 99, 146, 147, 150, 168
17 1, 38, 40, 65, 75, 110, 131, 134, 155, 158, 179, 214, 224, 249, 251, 288
19 1, 28, 54, 62, 68, 69, 99, 116, 127, 234, 245, 262, 292, 293, 299, 307, 333, 360
23 1, 28, 42, 63, 118, 130, 170, 177, 195, 255, 263, 266, 274, 334, 352, 359, 399, 411, 466, 487, 501, 528
29 1, 14, 41, 60, 63, 137, 190, 196, 221, 236, 267, 270, 374, 416, 425, 467, 571, 574, 605, 620, 645, 651, 704, 778, 781, 800, 827, 840
The least base b > 1 which prime(n) is a Wieferich prime are
5, 8, 7, 18, 3, 19, 38, 28, 28, 14, 115, 18, 51, 19, 53, 338, 53, 264, 143, 11, 306, 31, 99, 184, 53, 181, 43, 164, 96, 68, 38, 58, 19, 328, 313, 78, 226, 65, 253, 259, 532, 78, 176, 276, 143, 174, 165, 69, 330, 44, 33, 332, 94, 263, 48, 79, 171, 747, 731, 20, ... (sequence A039678 in the OEIS)
We can also consider the formula $(a+1)^{p-1}-a^{p-1}\equiv 0{\pmod {p^{2}}}$, (because of the generalized Fermat little theorem, $(a+1)^{p-1}-a^{p-1}\equiv 0{\pmod {p^{2}}}$ is true for all prime p and all natural number a such that both a and a + 1 are not divisible by p). It's a conjecture that for every natural number a, there are infinitely many primes such that $(a+1)^{p-1}-a^{p-1}\equiv 0{\pmod {p^{2}}}$.
Known solutions for small a are: (checked up to 4 × 1011) [64]
$a$ primes $p$ such that $(a+1)^{p-1}-a^{p-1}\equiv 0{\pmod {p^{2}}}$
1 1093, 3511, ...
2 23, 3842760169, 41975417117, ...
3 5, 250829, ...
4 3, 67, ...
5 3457, 893122907, ...
6 72673, 1108905403, 2375385997, ...
7 13, 819381943, ...
8 67, 139, 499, 26325777341, ...
9 67, 887, 9257, 83449, 111539, 31832131, ...
10 ...
11 107, 4637, 239357, ...
12 5, 11, 51563, 363901, 224189011, ...
13 3, ...
14 11, 5749, 17733170113, 140328785783, ...
15 292381, ...
16 4157, ...
17 751, 46070159, ...
18 7, 142671309349, ...
19 17, 269, ...
20 29, 162703, ...
21 5, 2711, 104651, 112922981, 331325567, 13315963127, ...
22 3, 7, 13, 94447, 1198427, 23536243, ...
23 43, 179, 1637, 69073, ...
24 7, 353, 402153391, ...
25 43, 5399, 21107, 35879, ...
26 7, 131, 653, 5237, 97003, ...
27 2437, 1704732131, ...
28 5, 617, 677, 2273, 16243697, ...
29 73, 101, 6217, ...
30 7, 11, 23, 3301, 48589, 549667, ...
31 3, 41, 416797, ...
32 95989, 2276682269, ...
33 139, 1341678275933, ...
34 83, 139, ...
35 ...
36 107, 137, 613, 2423, 74304856177, ...
37 5, ...
38 167, 2039, ...
39 659, 9413, ...
40 3, 23, 21029249, ...
41 31, 71, 1934399021, 474528373843, ...
42 4639, 1672609, ...
43 31, 4962186419, ...
44 36677, 17786501, ...
45 241, 26120375473, ...
46 5, 13877, ...
47 13, 311, 797, 906165497, ...
48 ...
49 3, 13, 2141, 281833, 1703287, 4805298913, ...
50 2953, 22409, 99241, 5427425917, ...
Wieferich pairs
Main article: Wieferich pair
A Wieferich pair is a pair of primes p and q that satisfy
pq − 1 ≡ 1 (mod q2) and qp − 1 ≡ 1 (mod p2)
so that a Wieferich prime p ≡ 1 (mod 4) will form such a pair (p, 2): the only known instance in this case is p = 1093. There are only 7 known Wieferich pairs.[65]
(2, 1093), (3, 1006003), (5, 1645333507), (5, 188748146801), (83, 4871), (911, 318917), and (2903, 18787) (sequence OEIS: A282293 in OEIS)
Wieferich sequence
Start with a(1) any natural number (>1), a(n) = the smallest prime p such that (a(n − 1))p − 1 = 1 (mod p2) but p2 does not divide a(n − 1) − 1 or a(n − 1) + 1. (If p2 divides a(n − 1) − 1 or a(n − 1) + 1, then the solution is a trivial solution) It is a conjecture that every natural number k = a(1) > 1 makes this sequence become periodic, for example, let a(1) = 2:
2, 1093, 5, 20771, 18043, 5, 20771, 18043, 5, ..., it gets a cycle: {5, 20771, 18043}.
(sequence A359952 in the OEIS)
Let a(1) = 83:
83, 4871, 83, 4871, 83, 4871, 83, ..., it gets a cycle: {83, 4871}.
Let a(1) = 59 (a longer sequence):
59, 2777, 133287067, 13, 863, 7, 5, 20771, 18043, 5, ..., it also gets 5.
However, there are many values of a(1) with unknown status, for example, let a(1) = 3:
3, 11, 71, 47, ? (There are no known Wieferich primes in base 47).
Let a(1) = 14:
14, 29, ? (There are no known Wieferich prime in base 29 except 2, but 22 = 4 divides 29 − 1 = 28)
Let a(1) = 39 (a longer sequence):
39, 8039, 617, 101, 1050139, 29, ? (It also gets 29)
It is unknown that values for a(1) > 1 exist such that the resulting sequence does not eventually become periodic.
When a(n − 1)=k, a(n) will be (start with k = 2): 1093, 11, 1093, 20771, 66161, 5, 1093, 11, 487, 71, 2693, 863, 29, 29131, 1093, 46021, 5, 7, 281, ?, 13, 13, 25633, 20771, 71, 11, 19, ?, 7, 7, 5, 233, 46145917691, 1613, 66161, 77867, 17, 8039, 11, 29, 23, 5, 229, 1283, 829, ?, 257, 491531, ?, ... (For k = 21, 29, 47, 50, even the next value is unknown)
Wieferich numbers
A Wieferich number is an odd natural number n satisfying the congruence 2φ(n) ≡ 1 (mod n2), where φ denotes the Euler's totient function (according to Euler's theorem, 2φ(n) ≡ 1 (mod n) for every odd natural number n). If Wieferich number n is prime, then it is a Wieferich prime. The first few Wieferich numbers are:
1, 1093, 3279, 3511, 7651, 10533, 14209, 17555, 22953, 31599, 42627, 45643, 52665, 68859, 94797, 99463, ... (sequence A077816 in the OEIS)
It can be shown that if there are only finitely many Wieferich primes, then there are only finitely many Wieferich numbers. In particular, if the only Wieferich primes are 1093 and 3511, then there exist exactly 104 Wieferich numbers, which matches the number of Wieferich numbers currently known.[2]
More generally, a natural number n is a Wieferich number to base a, if aφ(n) ≡ 1 (mod n2).[66]: 31
Another definition specifies a Wieferich number as odd natural number n such that n and ${\tfrac {2^{m}-1}{n}}$ are not coprime, where m is the multiplicative order of 2 modulo n. The first of these numbers are:[67]
21, 39, 55, 57, 105, 111, 147, 155, 165, 171, 183, 195, 201, 203, 205, 219, 231, 237, 253, 273, 285, 291, 301, 305, 309, 327, 333, 355, 357, 385, 399, ... (sequence A182297 in the OEIS)
As above, if Wieferich number q is prime, then it is a Wieferich prime.
Weak Wieferich prime
A weak Wieferich prime to base a is a prime p satisfies the condition
ap ≡ a (mod p2)
Every Wieferich prime to base a is also a weak Wieferich prime to base a. If the base a is squarefree, then a prime p is a weak Wieferich prime to base a if and only if p is a Wieferich prime to base a.
Smallest weak Wieferich prime to base n are (start with n = 0)
2, 2, 1093, 11, 2, 2, 66161, 5, 2, 2, 3, 71, 2, 2, 29, 29131, 2, 2, 3, 3, 2, 2, 13, 13, 2, 2, 3, 3, 2, 2, 7, 7, 2, 2, 46145917691, 3, 2, 2, 17, 8039, 2, 2, 23, 5, 2, 2, 3, ...
Wieferich prime with order n
For integer n ≥2, a Wieferich prime to base a with order n is a prime p satisfies the condition
ap−1 ≡ 1 (mod pn)
Clearly, a Wieferich prime to base a with order n is also a Wieferich prime to base a with order m for all 2 ≤ m ≤ n, and Wieferich prime to base a with order 2 is equivalent to Wieferich prime to base a, so we can only consider the n ≥ 3 case. However, there are no known Wieferich prime to base 2 with order 3. The first base with known Wieferich prime with order 3 is 9, where 2 is a Wieferich prime to base 9 with order 3. Besides, both 5 and 113 are Wieferich prime to base 68 with order 3.
Lucas–Wieferich primes
Let P and Q be integers. The Lucas sequence of the first kind associated with the pair (P, Q) is defined by
${\begin{aligned}U_{0}(P,Q)&=0,\\U_{1}(P,Q)&=1,\\U_{n}(P,Q)&=P\cdot U_{n-1}(P,Q)-Q\cdot U_{n-2}(P,Q)\end{aligned}}$
for all $n\geq 2$. A Lucas–Wieferich prime associated with (P, Q) is a prime p such that Up−ε(P, Q) ≡ 0 (mod p2), where ε equals the Legendre symbol $\left({\tfrac {P^{2}-4Q}{p}}\right)$. All Wieferich primes are Lucas–Wieferich primes associated with the pair (3, 2).[3]: 2088
Fibonacci–Wieferich primes
Let Q = −1. For every natural number P, the Lucas–Wieferich primes associated with (P, −1) are called P-Fibonacci–Wieferich primes or P-Wall–Sun–Sun primes. If P = 1, they are called Fibonacci–Wieferich primes. If P = 2, they are called Pell–Wieferich primes.
For example, 241 is a Lucas–Wieferich prime associated with (3, −1), so it is a 3-Fibonacci–Wieferich prime or 3-Wall–Sun–Sun prime. In fact, 3 is a P-Fibonacci–Wieferich prime if and only if P congruent to 0, 4, or 5 (mod 9), which is analogous to the statement for traditional Wieferich primes that 3 is a base-n Wieferich prime if and only if n congruent to 1 or 8 (mod 9).
Wieferich places
Let K be a global field, i.e. a number field or a function field in one variable over a finite field and let E be an elliptic curve. If v is a non-archimedean place of norm qv of K and a ∈ K, with v(a) = 0 then v(aqv − 1 − 1) ≥ 1. v is called a Wieferich place for base a, if v(aqv − 1 − 1) > 1, an elliptic Wieferich place for base P ∈ E, if NvP ∈ E2 and a strong elliptic Wieferich place for base P ∈ E if nvP ∈ E2, where nv is the order of P modulo v and Nv gives the number of rational points (over the residue field of v) of the reduction of E at v.[68]: 206
See also
• Wall–Sun–Sun prime – another type of prime number which in the broadest sense also resulted from the study of FLT
• Wolstenholme prime – another type of prime number which in the broadest sense also resulted from the study of FLT
• Wilson prime
• Table of congruences – lists other congruences satisfied by prime numbers
• PrimeGrid – primes search project
• BOINC
• Distributed computing
References
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29. Coppersmith, D. (1990), "Fermat's Last Theorem (Case I) and the Wieferich Criterion" (PDF), Mathematics of Computation, 54 (190): 895–902, Bibcode:1990MaCom..54..895C, doi:10.1090/s0025-5718-1990-1010598-2, JSTOR 2008518.
30. Cikánek, P. (1994), "A Special Extension of Wieferich's Criterion" (PDF), Mathematics of Computation, 62 (206): 923–930, Bibcode:1994MaCom..62..923C, doi:10.2307/2153550, JSTOR 3562296.
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32. Mirimanoff, D. (1910), "Sur le dernier théorème de Fermat", Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences (in French), 150: 204–206.
33. Granville, A.; Monagan, M. B. (1988), "The First Case of Fermat's Last Theorem is true for all prime exponents up to 714,591,416,091,389", Transactions of the American Mathematical Society, 306 (1): 329–359, doi:10.1090/S0002-9947-1988-0927694-5.
34. Suzuki, Jiro (1994), "On the generalized Wieferich criteria", Proceedings of the Japan Academy, Series A, 70 (7): 230–234, doi:10.3792/pjaa.70.230
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Further reading
• Haussner, R. (1926), "Über die Kongruenzen 2p−1 − 1 ≡ 0 (mod p2) für die Primzahlen p=1093 und 3511", Archiv for Mathematik og Naturvidenskab (in German), 39 (5): 7, JFM 52.0141.06, DNB 363953469
• Haussner, R. (1927), "Über numerische Lösungen der Kongruenz up−1 − 1 ≡ 0 (mod p2)", Journal für die Reine und Angewandte Mathematik (in German), 1927 (156): 223–226, doi:10.1515/crll.1927.156.223, S2CID 117969297
• Ribenboim, P. (1979), Thirteen lectures on Fermat's Last Theorem, Springer-Verlag, pp. 139, 151, ISBN 978-0-387-90432-0
• Guy, Richard K. (2004), Unsolved Problems in Number Theory (3rd ed.), Springer Verlag, p. 14, ISBN 978-0-387-20860-2
• Crandall, R. E.; Pomerance, C. (2005), Prime numbers: a computational perspective (PDF), Springer Science+Business Media, pp. 31–32, ISBN 978-0-387-25282-7
• Ribenboim, P. (1996), The new book of prime number records, New York: Springer-Verlag, pp. 333–346, ISBN 978-0-387-94457-9
External links
• Weisstein, Eric W. "Wieferich prime". MathWorld.
• Fermat-/Euler-quotients (ap−1 − 1)/pk with arbitrary k
• A note on the two known Wieferich primes
• PrimeGrid's Wieferich Prime Search project page
Prime number classes
By formula
• Fermat (22n + 1)
• Mersenne (2p − 1)
• Double Mersenne (22p−1 − 1)
• Wagstaff (2p + 1)/3
• Proth (k·2n + 1)
• Factorial (n! ± 1)
• Primorial (pn# ± 1)
• Euclid (pn# + 1)
• Pythagorean (4n + 1)
• Pierpont (2m·3n + 1)
• Quartan (x4 + y4)
• Solinas (2m ± 2n ± 1)
• Cullen (n·2n + 1)
• Woodall (n·2n − 1)
• Cuban (x3 − y3)/(x − y)
• Leyland (xy + yx)
• Thabit (3·2n − 1)
• Williams ((b−1)·bn − 1)
• Mills (⌊A3n⌋)
By integer sequence
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By property
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Base-dependent
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• Self
• Smarandache–Wellin
• Strobogrammatic
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• Tetradic
Patterns
• Twin (p, p + 2)
• Bi-twin chain (n ± 1, 2n ± 1, 4n ± 1, …)
• Triplet (p, p + 2 or p + 4, p + 6)
• Quadruplet (p, p + 2, p + 6, p + 8)
• k-tuple
• Cousin (p, p + 4)
• Sexy (p, p + 6)
• Chen
• Sophie Germain/Safe (p, 2p + 1)
• Cunningham (p, 2p ± 1, 4p ± 3, 8p ± 7, ...)
• Arithmetic progression (p + a·n, n = 0, 1, 2, 3, ...)
• Balanced (consecutive p − n, p, p + n)
By size
• Mega (1,000,000+ digits)
• Largest known
• list
Complex numbers
• Eisenstein prime
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Composite numbers
• Pseudoprime
• Catalan
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• Euler
• Euler–Jacobi
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• Somer–Lucas
• Strong
• Carmichael number
• Almost prime
• Semiprime
• Sphenic number
• Interprime
• Pernicious
Related topics
• Probable prime
• Industrial-grade prime
• Illegal prime
• Formula for primes
• Prime gap
First 60 primes
• 2
• 3
• 5
• 7
• 11
• 13
• 17
• 19
• 23
• 29
• 31
• 37
• 41
• 43
• 47
• 53
• 59
• 61
• 67
• 71
• 73
• 79
• 83
• 89
• 97
• 101
• 103
• 107
• 109
• 113
• 127
• 131
• 137
• 139
• 149
• 151
• 157
• 163
• 167
• 173
• 179
• 181
• 191
• 193
• 197
• 199
• 211
• 223
• 227
• 229
• 233
• 239
• 241
• 251
• 257
• 263
• 269
• 271
• 277
• 281
List of prime numbers
| Wikipedia |
Wieferich pair
In mathematics, a Wieferich pair is a pair of prime numbers p and q that satisfy
pq − 1 ≡ 1 (mod q2) and qp − 1 ≡ 1 (mod p2)
Wieferich pairs are named after German mathematician Arthur Wieferich. Wieferich pairs play an important role in Preda Mihăilescu's 2002 proof[1] of Mihăilescu's theorem (formerly known as Catalan's conjecture).[2]
Known Wieferich pairs
There are only 7 Wieferich pairs known:[3][4]
(2, 1093), (3, 1006003), (5, 1645333507), (5, 188748146801), (83, 4871), (911, 318917), and (2903, 18787). (sequence OEIS: A124121 and OEIS: A124122 in OEIS)
Wieferich triple
A Wieferich triple is a triple of prime numbers p, q and r that satisfy
pq − 1 ≡ 1 (mod q2), qr − 1 ≡ 1 (mod r2), and rp − 1 ≡ 1 (mod p2).
There are 17 known Wieferich triples:
(2, 1093, 5), (2, 3511, 73), (3, 11, 71), (3, 1006003, 3188089), (5, 20771, 18043), (5, 20771, 950507), (5, 53471161, 193), (5, 6692367337, 1601), (5, 6692367337, 1699), (5, 188748146801, 8807), (13, 863, 23), (17, 478225523351, 2311), (41, 138200401, 2953), (83, 13691, 821), (199, 1843757, 2251), (431, 2393, 54787), and (1657, 2281, 1667). (sequences OEIS: A253683, OEIS: A253684 and OEIS: A253685 in OEIS)
Barker sequence
Barker sequence or Wieferich n-tuple is a generalization of Wieferich pair and Wieferich triple. It is primes (p1, p2, p3, ..., pn) such that
p1p2 − 1 ≡ 1 (mod p22), p2p3 − 1 ≡ 1 (mod p32), p3p4 − 1 ≡ 1 (mod p42), ..., pn−1pn − 1 ≡ 1 (mod pn2), pnp1 − 1 ≡ 1 (mod p12).[5]
For example, (3, 11, 71, 331, 359) is a Barker sequence, or a Wieferich 5-tuple; (5, 188748146801, 453029, 53, 97, 76704103313, 4794006457, 12197, 3049, 41) is a Barker sequence, or a Wieferich 10-tuple.
For the smallest Wieferich n-tuple, see OEIS: A271100, for the ordered set of all Wieferich tuples, see OEIS: A317721.
Wieferich sequence
Wieferich sequence is a special type of Barker sequence. Every integer k>1 has its own Wieferich sequence. To make a Wieferich sequence of an integer k>1, start with a(1)=k, a(n) = the smallest prime p such that a(n-1)p-1 = 1 (mod p) but a(n-1) ≠ 1 or -1 (mod p). It is a conjecture that every integer k>1 has a periodic Wieferich sequence. For example, the Wieferich sequence of 2:
2, 1093, 5, 20771, 18043, 5, 20771, 18043, 5, ..., it gets a cycle: {5, 20771, 18043}. (a Wieferich triple)
The Wieferich sequence of 83:
83, 4871, 83, 4871, 83, 4871, 83, ..., it gets a cycle: {83, 4871}. (a Wieferich pair)
The Wieferich sequence of 59: (this sequence needs more terms to be periodic)
59, 2777, 133287067, 13, 863, 7, 5, 20771, 18043, 5, ... it also gets 5.
However, there are many values of a(1) with unknown status. For example, the Wieferich sequence of 3:
3, 11, 71, 47, ? (There are no known Wieferich primes in base 47).
The Wieferich sequence of 14:
14, 29, ? (There are no known Wieferich primes in base 29 except 2, but 22 = 4 divides 29 - 1 = 28)
The Wieferich sequence of 39:
39, 8039, 617, 101, 1050139, 29, ? (It also gets 29)
It is unknown that values for k exist such that the Wieferich sequence of k does not become periodic. Eventually, it is unknown that values for k exist such that the Wieferich sequence of k is finite.
When a(n - 1)=k, a(n) will be (start with k = 2): 1093, 11, 1093, 20771, 66161, 5, 1093, 11, 487, 71, 2693, 863, 29, 29131, 1093, 46021, 5, 7, 281, ?, 13, 13, 25633, 20771, 71, 11, 19, ?, 7, 7, 5, 233, 46145917691, 1613, 66161, 77867, 17, 8039, 11, 29, 23, 5, 229, 1283, 829, ?, 257, 491531, ?, ... (For k = 21, 29, 47, 50, even the next value is unknown)
See also
• Wieferich prime
• Fermat quotient
References
1. Preda Mihăilescu (2004). "Primary Cyclotomic Units and a Proof of Catalan's Conjecture". J. Reine Angew. Math. 2004 (572): 167–195. doi:10.1515/crll.2004.048. MR 2076124.
2. Jeanine Daems A Cyclotomic Proof of Catalan's Conjecture.
3. Weisstein, Eric W. "Double Wieferich Prime Pair". MathWorld.
4. OEIS: A124121, For example, currently there are two known double Wieferich prime pairs (p, q) with q = 5: (1645333507, 5) and (188748146801, 5).
5. List of all known Barker sequence
Further reading
• Bilu, Yuri F. (2004). "Catalan's conjecture (after Mihăilescu)". Astérisque. 294: vii, 1–26. Zbl 1094.11014.
• Ernvall, Reijo; Metsänkylä, Tauno (1997). "On the p-divisibility of Fermat quotients". Math. Comp. 66 (219): 1353–1365. Bibcode:1997MaCom..66.1353E. doi:10.1090/S0025-5718-97-00843-0. MR 1408373. Zbl 0903.11002.
• Steiner, Ray (1998). "Class number bounds and Catalan's equation". Math. Comp. 67 (223): 1317–1322. Bibcode:1998MaCom..67.1317S. doi:10.1090/S0025-5718-98-00966-1. MR 1468945. Zbl 0897.11009.
| Wikipedia |
Wiener's Tauberian theorem
In mathematical analysis, Wiener's tauberian theorem is any of several related results proved by Norbert Wiener in 1932.[1] They provide a necessary and sufficient condition under which any function in $L^{1}$ or $L^{2}$ can be approximated by linear combinations of translations of a given function.[2]
Informally, if the Fourier transform of a function $f$ vanishes on a certain set $Z$, the Fourier transform of any linear combination of translations of $f$ also vanishes on $Z$. Therefore, the linear combinations of translations of $f$ cannot approximate a function whose Fourier transform does not vanish on $Z$.
Wiener's theorems make this precise, stating that linear combinations of translations of $f$ are dense if and only if the zero set of the Fourier transform of $f$ is empty (in the case of $L^{1}$) or of Lebesgue measure zero (in the case of $L^{2}$).
Gelfand reformulated Wiener's theorem in terms of commutative C*-algebras, when it states that the spectrum of the $L^{1}$ group ring $L^{1}(\mathbb {R} )$ of the group $\mathbb {R} $ of real numbers is the dual group of $\mathbb {R} $. A similar result is true when $\mathbb {R} $ is replaced by any locally compact abelian group.
The condition in $L^{1}$
Let $f\in L^{1}(\mathbb {R} $ be an integrable function. The span of translations $f_{a}(x)=f(x+a)$ is dense in $L^{1}(\mathbb {R} )$ if and only if the Fourier transform of $f$ has no real zeros.
Tauberian reformulation
The following statement is equivalent to the previous result, and explains why Wiener's result is a Tauberian theorem:
Suppose the Fourier transform of $f\in L^{1}$ has no real zeros, and suppose the convolution $f*h$ tends to zero at infinity for some $h\in L^{\infty }$. Then the convolution $g*h$ tends to zero at infinity for any $g\in L^{1}$.
More generally, if
$\lim _{x\to \infty }(f*h)(x)=A\int f(x)\,dx$
for some $f\in L^{1}$ the Fourier transform of which has no real zeros, then also
$\lim _{x\to \infty }(g*h)(x)=A\int g(x)\,dx$
for any $g\in L^{1}$.
Discrete version
Wiener's theorem has a counterpart in $l^{1}(\mathbb {Z} )$: the span of the translations of $f\in l^{1}(\mathbb {Z} )$ is dense if and only if the Fourier transform
$\varphi (\theta )=\sum _{n\in \mathbb {Z} }f(n)e^{-in\theta }\,$
has no real zeros. The following statements are equivalent version of this result:
• Suppose the Fourier transform of $f\in l^{1}(\mathbb {Z} )$ has no real zeros, and for some bounded sequence $h$ the convolution $f*h$
tends to zero at infinity. Then $g*h$ also tends to zero at infinity for any $g\in l^{1}(\mathbb {Z} )$.
• Let $\varphi $ be a function on the unit circle with absolutely convergent Fourier series. Then $1/\varphi $ has absolutely convergent Fourier series
if and only if $\varphi $ has no zeros.
Gelfand (1941a, 1941b) showed that this is equivalent to the following property of the Wiener algebra $A(\mathbb {T} )$, which he proved using the theory of Banach algebras, thereby giving a new proof of Wiener's result:
• The maximal ideals of $A(\mathbb {T} )$ are all of the form
$M_{x}=\left\{f\in A(\mathbb {T} )\mid f(x)=0\right\},\quad x\in \mathbb {T} .$
The condition in L2
Let $f\in L^{2}(\mathbb {R} )$ be a square-integrable function. The span of translations $f_{a}(x)=f(x+a)$ is dense in $L^{2}(\mathbb {R} )$ if and only if the real zeros of the Fourier transform of $f$ form a set of zero Lebesgue measure.
The parallel statement in $l^{2}(\mathbb {Z} )$ is as follows: the span of translations of a sequence $f\in l^{2}(\mathbb {Z} )$ is dense if and only if the zero set of the Fourier transform
$\varphi (\theta )=\sum _{n\in \mathbb {Z} }f(n)e^{-in\theta }$
has zero Lebesgue measure.
Notes
1. See Wiener (1932).
2. see Rudin (1991).
References
• Gelfand, I. (1941a), "Normierte Ringe", Rec. Math. (Mat. Sbornik), Nouvelle Série, 9 (51): 3–24, MR 0004726
• Gelfand, I. (1941b), "Über absolut konvergente trigonometrische Reihen und Integrale", Rec. Math. (Mat. Sbornik), Nouvelle Série, 9 (51): 51–66, MR 0004727
• Rudin, W. (1991), Functional analysis, International Series in Pure and Applied Mathematics, New York: McGraw-Hill, Inc., ISBN 0-07-054236-8, MR 1157815
• Wiener, N. (1932), "Tauberian Theorems", Annals of Mathematics, 33 (1): 1–100, doi:10.2307/1968102, JSTOR 1968102
External links
• Shtern, A.I. (2001) [1994], "Wiener Tauberian theorem", Encyclopedia of Mathematics, EMS Press
| Wikipedia |
Wiener–Hopf method
The Wiener–Hopf method is a mathematical technique widely used in applied mathematics. It was initially developed by Norbert Wiener and Eberhard Hopf as a method to solve systems of integral equations, but has found wider use in solving two-dimensional partial differential equations with mixed boundary conditions on the same boundary. In general, the method works by exploiting the complex-analytical properties of transformed functions. Typically, the standard Fourier transform is used, but examples exist using other transforms, such as the Mellin transform.
In general, the governing equations and boundary conditions are transformed and these transforms are used to define a pair of complex functions (typically denoted with '+' and '−' subscripts) which are respectively analytic in the upper and lower halves of the complex plane, and have growth no faster than polynomials in these regions. These two functions will also coincide on some region of the complex plane, typically, a thin strip containing the real line. Analytic continuation guarantees that these two functions define a single function analytic in the entire complex plane, and Liouville's theorem implies that this function is an unknown polynomial, which is often zero or constant. Analysis of the conditions at the edges and corners of the boundary allows one to determine the degree of this polynomial.
Wiener–Hopf decomposition
The key step in many Wiener–Hopf problems is to decompose an arbitrary function $\Phi $ into two functions $\Phi _{\pm }$ with the desired properties outlined above. In general, this can be done by writing
$\Phi _{+}(\alpha )={\frac {1}{2\pi i}}\int _{C_{1}}\Phi (z){\frac {dz}{z-\alpha }}$
and
$\Phi _{-}(\alpha )=-{\frac {1}{2\pi i}}\int _{C_{2}}\Phi (z){\frac {dz}{z-\alpha }},$
where the contours $C_{1}$ and $C_{2}$ are parallel to the real line, but pass above and below the point $z=\alpha $, respectively.
Similarly, arbitrary scalar functions may be decomposed into a product of +/− functions, i.e. $K(\alpha )=K_{+}(\alpha )K_{-}(\alpha )$, by first taking the logarithm, and then performing a sum decomposition. Product decompositions of matrix functions (which occur in coupled multi-modal systems such as elastic waves) are considerably more problematic since the logarithm is not well defined, and any decomposition might be expected to be non-commutative. A small subclass of commutative decompositions were obtained by Khrapkov, and various approximate methods have also been developed.
Example
Consider the linear partial differential equation
${\boldsymbol {L}}_{xy}f(x,y)=0,$
where ${\boldsymbol {L}}_{xy}$ is a linear operator which contains derivatives with respect to x and y, subject to the mixed conditions on y = 0, for some prescribed function g(x),
$f=g(x){\text{ for }}x\leq 0,\quad f_{y}=0{\text{ when }}x>0$
and decay at infinity i.e. f → 0 as ${\boldsymbol {x}}\rightarrow \infty $.
Taking a Fourier transform with respect to x results in the following ordinary differential equation
${\boldsymbol {L}}_{y}{\widehat {f}}(k,y)-P(k,y){\widehat {f}}(k,y)=0,$
where ${\boldsymbol {L}}_{y}$ is a linear operator containing y derivatives only, P(k,y) is a known function of y and k and
${\widehat {f}}(k,y)=\int _{-\infty }^{\infty }f(x,y)e^{-ikx}\,{\textrm {d}}x.$
If a particular solution of this ordinary differential equation which satisfies the necessary decay at infinity is denoted F(k,y), a general solution can be written as
${\widehat {f}}(k,y)=C(k)F(k,y),$
where C(k) is an unknown function to be determined by the boundary conditions on y=0.
The key idea is to split ${\widehat {f}}$ into two separate functions, ${\widehat {f}}_{+}$ and ${\widehat {f}}_{-}$ which are analytic in the lower- and upper-halves of the complex plane, respectively,
${\widehat {f}}_{+}(k,y)=\int _{0}^{\infty }f(x,y)e^{-ikx}\,{\textrm {d}}x,$
${\widehat {f}}_{-}(k,y)=\int _{-\infty }^{0}f(x,y)e^{-ikx}\,{\textrm {d}}x.$
The boundary conditions then give
${\widehat {g\,}}(k)+{\widehat {f}}_{+}(k,0)={\widehat {f}}_{-}(k,0)+{\widehat {f}}_{+}(k,0)={\widehat {f}}(k,0)=C(k)F(k,0)$
and, on taking derivatives with respect to $y$,
${\widehat {f}}'_{-}(k,0)={\widehat {f}}'_{-}(k,0)+{\widehat {f}}'_{+}(k,0)={\widehat {f}}'(k,0)=C(k)F'(k,0).$
Eliminating $C(k)$ yields
${\widehat {g\,}}(k)+{\widehat {f}}_{+}(k,0)={\widehat {f}}'_{-}(k,0)/K(k),$
where
$K(k)={\frac {F'(k,0)}{F(k,0)}}.$
Now $K(k)$ can be decomposed into the product of functions $K^{-}$ and $K^{+}$ which are analytic in the upper and lower half-planes respectively.
To be precise, $K(k)=K^{+}(k)K^{-}(k),$ where
$\log K^{-}={\frac {1}{2\pi i}}\int _{-\infty }^{\infty }{\frac {\log(K(z))}{z-k}}\,{\textrm {d}}z,\quad \operatorname {Im} k>0,$
$\log K^{+}=-{\frac {1}{2\pi i}}\int _{-\infty }^{\infty }{\frac {\log(K(z))}{z-k}}\,{\textrm {d}}z,\quad \operatorname {Im} k<0.$
(Note that this sometimes involves scaling $K$ so that it tends to $1$ as $k\rightarrow \infty $.) We also decompose $K^{+}{\widehat {g\,}}$ into the sum of two functions $G^{+}$ and $G^{-}$ which are analytic in the lower and upper half-planes respectively, i.e.,
$K^{+}(k){\widehat {g\,}}(k)=G^{+}(k)+G^{-}(k).$
This can be done in the same way that we factorised $K(k).$ Consequently,
$G^{+}(k)+K_{+}(k){\widehat {f}}_{+}(k,0)={\widehat {f}}'_{-}(k,0)/K_{-}(k)-G^{-}(k).$
Now, as the left-hand side of the above equation is analytic in the lower half-plane, whilst the right-hand side is analytic in the upper half-plane, analytic continuation guarantees existence of an entire function which coincides with the left- or right-hand sides in their respective half-planes. Furthermore, since it can be shown that the functions on either side of the above equation decay at large k, an application of Liouville's theorem shows that this entire function is identically zero, therefore
${\widehat {f}}_{+}(k,0)=-{\frac {G^{+}(k)}{K^{+}(k)}},$
and so
$C(k)={\frac {K^{+}(k){\widehat {g\,}}(k)-G^{+}(k)}{K^{+}(k)F(k,0)}}.$
See also
• Wiener filter
• Riemann–Hilbert problem
References
• "Category:Wiener-Hopf - WikiWaves". wikiwaves.org. Retrieved 2020-05-19.
• "Wiener-Hopf method", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Fornberg, Bengt. Complex variables and analytic functions : an illustrated introduction. Piret, Cécile. Philadelphia. ISBN 978-1-61197-597-0. OCLC 1124781689.
| Wikipedia |
Wiener–Lévy theorem
Wiener–Lévy theorem is a theorem in Fourier analysis, which states that a function of an absolutely convergent Fourier series has an absolutely convergent Fourier series under some conditions. The theorem was named after Norbert Wiener and Paul Lévy.
Norbert Wiener first proved Wiener's 1/f theorem,[1] see Wiener's theorem. It states that if f has absolutely convergent Fourier series and is never zero, then its inverse 1/f also has an absolutely convergent Fourier series.
Wiener–Levy theorem
Paul Levy generalized Wiener's result,[2] showing that
Let $F(\theta )=\sum \limits _{k=-\infty }^{\infty }c_{k}e^{ik\theta },\quad \theta \in [0,2\pi ]$ be an absolutely convergent Fourier series with
$\|F\|=\sum \limits _{k=-\infty }^{\infty }|c_{k}|<\infty .$
The values of $F(\theta )$ lie on a curve $C$, and $H(t)$ is an analytic (not necessarily single-valued) function of a complex variable which is regular at every point of $C$. Then $H[F(\theta )]$ has an absolutely convergent Fourier series.
The proof can be found in the Zygmund's classic book Trigonometric Series.[3]
Example
Let $H(\theta )=\ln(\theta )$ and $F(\theta )=\sum \limits _{k=0}^{\infty }p_{k}e^{ik\theta },(\sum \limits _{k=0}^{\infty }p_{k}=1$) is characteristic function of discrete probability distribution. So $F(\theta )$ is an absolutely convergent Fourier series. If $F(\theta )$ has no zeros, then we have
$H[F(\theta )]=\ln \left(\sum \limits _{k=0}^{\infty }p_{k}e^{ik\theta }\right)=\sum _{k=0}^{\infty }c_{k}e^{ik\theta },$
where $\|H\|=\sum \limits _{k=0}^{\infty }|c_{k}|<\infty .$
The statistical application of this example can be found in discrete pseudo compound Poisson distribution[4] and zero-inflated model.
If a discrete r.v. $X$ with $\Pr(X=i)=P_{i}$, $i\in \mathbb {N} $, has the probability generating function of the form
$P(z)=\sum \limits _{i=0}^{\infty }P_{i}z^{i}=\exp \left\{\sum \limits _{i=1}^{\infty }\alpha _{i}\lambda (z^{i}-1)\right\},z=e^{ik\theta }$
where $\sum \limits _{i=1}^{\infty }\alpha _{i}=1$, $\sum \limits _{i=1}^{\infty }\left|\alpha _{i}\right|<\infty $, $\alpha _{i}\in \mathbb {R} $, and $\lambda >0$. Then $X$ is said to have the discrete pseudo compound Poisson distribution, abbreviated DPCP.
We denote it as $X\sim DPCP({\alpha _{1}}\lambda ,{\alpha _{2}}\lambda ,\cdots )$.
See also
• Wiener's theorem (disambiguation)
References
1. Wiener, N. (1932). "Tauberian Theorems". Annals of Mathematics. 33 (1): 1–100. doi:10.2307/1968102. JSTOR 1968102.
2. Lévy, P. (1935). "Sur la convergence absolue des séries de Fourier". Compositio Mathematica. 1: 1–14.
3. Zygmund, A. (2002). Trigonometric Series. Cambridge: Cambridge University Press. p. 245.
4. Huiming, Zhang; Li, Bo; G. Jay Kerns (2017). "A characterization of signed discrete infinitely divisible distributions". Studia Scientiarum Mathematicarum Hungarica. 54: 446–470. arXiv:1701.03892. doi:10.1556/012.2017.54.4.1377.
| Wikipedia |
Wiener amalgam space
In mathematics, amalgam spaces categorize functions with regard to their local and global behavior. While the concept of function spaces treating local and global behavior separately was already known earlier, Wiener amalgams, as the term is used today, were introduced by Hans Georg Feichtinger in 1980. The concept is named after Norbert Wiener.
Let $X$ be a normed space with norm $\|\cdot \|_{X}$. Then the Wiener amalgam space[1] with local component $X$ and global component $L_{m}^{p}$, a weighted $L^{p}$ space with non-negative weight $m$, is defined by
$W(X,L^{p})=\left\{f\ :\ \left(\int _{\mathbb {R} ^{d}}\|f(\cdot ){\bar {g}}(\cdot -x)\|_{X}^{p}m(x)^{p}\,dx\right)^{1/p}<\infty \right\},$ :\ \left(\int _{\mathbb {R} ^{d}}\|f(\cdot ){\bar {g}}(\cdot -x)\|_{X}^{p}m(x)^{p}\,dx\right)^{1/p}<\infty \right\},}
where $g$ is a continuously differentiable, compactly supported function, such that $\sum _{x\in \mathbb {Z^{d}} }g(z-x)=1$, for all $z\in \mathbb {R} ^{d}$. Again, the space defined is independent of $g$. As the definition suggests, Wiener amalgams are useful to describe functions showing characteristic local and global behavior.[2]
References
1. Wiener amalgam spaces for the Fundamental Identity of Gabor Analysis by Hans Georg Feichtinger and Franz Luef
2. Foundations of Time-Frequency Analysis by Karlheinz Gröchenig
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Wiener connector
In network theory, the Wiener connector is a means of maximizing efficiency in connecting specified "query vertices" in a network. Given a connected, undirected graph and a set of query vertices in a graph, the minimum Wiener connector is an induced subgraph that connects the query vertices and minimizes the sum of shortest path distances among all pairs of vertices in the subgraph. In combinatorial optimization, the minimum Wiener connector problem is the problem of finding the minimum Wiener connector. It can be thought of as a version of the classic Steiner tree problem (one of Karp's 21 NP-complete problems), where instead of minimizing the size of the tree, the objective is to minimize the distances in the subgraph.[1][2]
The minimum Wiener connector was first presented by Ruchansky et al. in 2015.[3]
The minimum Wiener connector has applications in many domains where there is a graph structure and an interest in learning about connections between sets of individuals. For example, given a set of patients infected with a viral disease, which other patients should be checked to find the culprit? Or given a set of proteins of interest, which other proteins participate in pathways with them?
The Wiener connector was named in honor of chemist Harry Wiener who first introduced the Wiener Index.
Problem definition
The Wiener index is the sum of shortest path distances in a (sub)graph. Using $d(u,v)$ to denote the shortest path between $u$ and $v$, the Wiener index of a (sub)graph $S$, denoted $W(S)$, is defined as
$W(S)=\sum _{(u,v)\in S}d(u,v)$.
The minimum Wiener connector problem is defined as follows. Given an undirected and unweighted graph with vertex set $V$ and edge set $E$ and a set of query vertices $Q\subseteq V$, find a connector $H\subseteq V$ of minimum Wiener index. More formally, the problem is to compute
$\operatorname {*} {arg\,min}_{H}W(H\cup Q)$,
that is, find a connector $H$ that minimizes the sum of shortest paths in $H$.
Relationship to Steiner tree
The minimum Wiener connector problem is related to the Steiner tree problem. In the former, the objective function in the minimization is the Wiener index of the connector, whereas in the latter, the objective function is the sum of the weights of the edges in the connector. The optimum solutions to these problems may differ, given the same graph and set of query vertices. In fact, a solution for the Steiner tree problem may be arbitrarily bad for the minimum Wiener connector problem; the graph on the right provides an example.
Computational complexity
Hardness
The problem is NP-hard, and does not admit a polynomial-time approximation scheme unless P = NP.[3] This can be proven using the inapproximability of vertex cover in bounded degree graphs.[4] Although there is no polynomial-time approximation scheme, there is a polynomial-time constant-factor approximation—an algorithm that finds a connector whose Wiener index is within a constant multiplicative factor of the Wiener index of the optimum connector. In terms of complexity classes, the minimum Wiener connector problem is in APX but is not in PTAS unless P = NP.
Exact algorithms
An exhaustive search over all possible subsets of vertices to find the one that induces the connector of minimum Wiener index yields an algorithm that finds the optimum solution in $2^{O(n)}$ time (that is, exponential time) on graphs with n vertices. In the special case that there are exactly two query vertices, the optimum solution is the shortest path joining the two vertices, so the problem can be solved in polynomial time by computing the shortest path. In fact, for any fixed constant number of query vertices, an optimum solution can be found in polynomial time.
Approximation algorithms
There is a constant-factor approximation algorithm for the minimum Wiener connector problem that runs in time $O(q(m\log n+n\log ^{2}n))$ on a graph with n vertices, m edges, and q query vertices, roughly the same time it takes to compute shortest-path distances from the query vertices to every other vertex in the graph.[3] The central approach of this algorithm is to reduce the problem to the vertex-weighted Steiner tree problem, which admits a constant-factor approximation in particular instances related to the minimum Wiener connector problem.
Behavior
The minimum Wiener connector behaves like betweenness centrality.
When the query vertices belong to the same community, the non-query vertices that form the minimum Wiener connector tend to belong to the same community and have high centrality within the community. Such vertices are likely to be influential vertices playing leadership roles in the community. In a social network, these influential vertices might be good users for spreading information or to target in a viral marketing campaign.[5]
When the query vertices belong to different communities, the non-query vertices that form the minimum Wiener connector contain vertices adjacent to edges that bridge the different communities. These vertices span a structural hole in the graph and are important.[6]
Applications
The minimum Wiener connector is useful in applications in which one wishes to learn about the relationship between a set of vertices in a graph. For example,
• in biology, it provides insight into how a set of proteins in a protein–protein interaction network are related,
• in social networks (like Twitter), it demonstrates the communities to which a set of users belong and how these communities are related,
• in computer networks, it may be useful in identifying an efficient way to route a multicast message to a set of destinations.
References
1. Hwang, Frank; Richards, Dana; Winter, Dana; Winter, Pawel (1992). "The Steiner Tree Problem". Annals of Discrete Mathematics.
2. DIMACS Steiner Tree Challenge
3. Ruchansky, Natali; Bonchi, Francesco; Garcia-Soriano, David; Gullo, Francesco; Kourtellis, Nicolas (2015). "The Minimum Wiener Connector". SIGMOD. arXiv:1504.00513. Bibcode:2015arXiv150400513R. doi:10.1145/2723372.2749449. S2CID 2856346.
4. Dinur, Irit; Safra, Samuel (2005). "On the hardness of approximating minimum vertex cover". Annals of Mathematics. 162: 439–485. doi:10.4007/annals.2005.162.439.
5. Hinz, Oliver; Skiera, Bernd; Barrot, Christian; Becker, Jan U. (2011). "Seeding Strategies for Viral Marketing: An Empirical Comparison". Journal of Marketing. 75 (6): 55–71. doi:10.1509/jm.10.0088. S2CID 53972310.
6. Lou, Tiancheng; Tang, Jie (2013). "Mining Structural Hole Spanners Through Information Diffusion in Social Networks". Proceedings of the 22nd International Conference on World Wide Web. Rio de Janeiro, Brazil: International World Wide Web Conferences Steering Committee. pp. 825–836. ISBN 9781450320351.
| Wikipedia |
Wiener deconvolution
In mathematics, Wiener deconvolution is an application of the Wiener filter to the noise problems inherent in deconvolution. It works in the frequency domain, attempting to minimize the impact of deconvolved noise at frequencies which have a poor signal-to-noise ratio.
The Wiener deconvolution method has widespread use in image deconvolution applications, as the frequency spectrum of most visual images is fairly well behaved and may be estimated easily.
Wiener deconvolution is named after Norbert Wiener.
Definition
Given a system:
$\ y(t)=(h*x)(t)+n(t)$
where $*$ denotes convolution and:
• $\ x(t)$ is some original signal (unknown) at time $\ t$.
• $\ h(t)$ is the known impulse response of a linear time-invariant system
• $\ n(t)$ is some unknown additive noise, independent of $\ x(t)$
• $\ y(t)$ is our observed signal
Our goal is to find some $\ g(t)$ so that we can estimate $\ x(t)$ as follows:
$\ {\hat {x}}(t)=(g*y)(t)$
where $\ {\hat {x}}(t)$ is an estimate of $\ x(t)$ that minimizes the mean square error
$\ \epsilon (t)=\mathbb {E} \left|x(t)-{\hat {x}}(t)\right|^{2}$,
with $\ \mathbb {E} $ denoting the expectation. The Wiener deconvolution filter provides such a $\ g(t)$. The filter is most easily described in the frequency domain:
$\ G(f)={\frac {H^{*}(f)S(f)}{|H(f)|^{2}S(f)+N(f)}}$
where:
• $\ G(f)$ and $\ H(f)$ are the Fourier transforms of $\ g(t)$ and $\ h(t)$,
• $\ S(f)=\mathbb {E} |X(f)|^{2}$ is the mean power spectral density of the original signal $\ x(t)$,
• $\ N(f)=\mathbb {E} |V(f)|^{2}$ is the mean power spectral density of the noise $\ n(t)$,
• $X(f)$, $Y(f)$, and $V(f)$ are the Fourier transforms of $x(t)$, and $y(t)$, and $n(t)$, respectively,
• the superscript ${}^{*}$ denotes complex conjugation.
The filtering operation may either be carried out in the time-domain, as above, or in the frequency domain:
$\ {\hat {X}}(f)=G(f)Y(f)$
and then performing an inverse Fourier transform on $\ {\hat {X}}(f)$ to obtain $\ {\hat {x}}(t)$.
Note that in the case of images, the arguments $\ t$ and $\ f$ above become two-dimensional; however the result is the same.
Interpretation
The operation of the Wiener filter becomes apparent when the filter equation above is rewritten:
${\begin{aligned}G(f)&={\frac {1}{H(f)}}\left[{\frac {1}{1+1/(|H(f)|^{2}\mathrm {SNR} (f))}}\right]\end{aligned}}$
Here, $\ 1/H(f)$ is the inverse of the original system, $\ \mathrm {SNR} (f)=S(f)/N(f)$ is the signal-to-noise ratio, and $\ |H(f)|^{2}\mathrm {SNR} (f)$ is the ratio of the pure filtered signal to noise spectral density. When there is zero noise (i.e. infinite signal-to-noise), the term inside the square brackets equals 1, which means that the Wiener filter is simply the inverse of the system, as we might expect. However, as the noise at certain frequencies increases, the signal-to-noise ratio drops, so the term inside the square brackets also drops. This means that the Wiener filter attenuates frequencies according to their filtered signal-to-noise ratio.
The Wiener filter equation above requires us to know the spectral content of a typical image, and also that of the noise. Often, we do not have access to these exact quantities, but we may be in a situation where good estimates can be made. For instance, in the case of photographic images, the signal (the original image) typically has strong low frequencies and weak high frequencies, while in many cases the noise content will be relatively flat with frequency.
Derivation
As mentioned above, we want to produce an estimate of the original signal that minimizes the mean square error, which may be expressed:
$\ \epsilon (f)=\mathbb {E} \left|X(f)-{\hat {X}}(f)\right|^{2}$ .
The equivalence to the previous definition of $\epsilon $, can be derived using Plancherel theorem or Parseval's theorem for the Fourier transform.
If we substitute in the expression for $\ {\hat {X}}(f)$, the above can be rearranged to
${\begin{aligned}\epsilon (f)&=\mathbb {E} \left|X(f)-G(f)Y(f)\right|^{2}\\&=\mathbb {E} \left|X(f)-G(f)\left[H(f)X(f)+V(f)\right]\right|^{2}\\&=\mathbb {E} {\big |}\left[1-G(f)H(f)\right]X(f)-G(f)V(f){\big |}^{2}\end{aligned}}$
If we expand the quadratic, we get the following:
${\begin{aligned}\epsilon (f)&={\Big [}1-G(f)H(f){\Big ]}{\Big [}1-G(f)H(f){\Big ]}^{*}\,\mathbb {E} |X(f)|^{2}\\&{}-{\Big [}1-G(f)H(f){\Big ]}G^{*}(f)\,\mathbb {E} {\Big \{}X(f)V^{*}(f){\Big \}}\\&{}-G(f){\Big [}1-G(f)H(f){\Big ]}^{*}\,\mathbb {E} {\Big \{}V(f)X^{*}(f){\Big \}}\\&{}+G(f)G^{*}(f)\,\mathbb {E} |V(f)|^{2}\end{aligned}}$
However, we are assuming that the noise is independent of the signal, therefore:
$\ \mathbb {E} {\Big \{}X(f)V^{*}(f){\Big \}}=\mathbb {E} {\Big \{}V(f)X^{*}(f){\Big \}}=0$
Substituting the power spectral densities $\ S(f)$ and $\ N(f)$, we have:
$\epsilon (f)={\Big [}1-G(f)H(f){\Big ]}{\Big [}1-G(f)H(f){\Big ]}^{*}S(f)+G(f)G^{*}(f)N(f)$
To find the minimum error value, we calculate the Wirtinger derivative with respect to $\ G(f)$ and set it equal to zero.
$\ {\frac {d\epsilon (f)}{dG(f)}}=0\Rightarrow G^{*}(f)N(f)-H(f){\Big [}1-G(f)H(f){\Big ]}^{*}S(f)=0$
This final equality can be rearranged to give the Wiener filter.
See also
• Information field theory
• Deconvolution
• Wiener filter
• Point spread function
• Blind deconvolution
• Fourier transform
Wikimedia Commons has media related to An example of Wiener deconvolution on motion blured image (and source codes in MATLAB/GNU Octave)..
References
• Rafael Gonzalez, Richard Woods, and Steven Eddins. Digital Image Processing Using Matlab. Prentice Hall, 2003.
External links
• Comparison of different deconvolution methods.
| Wikipedia |
Wiener equation
A simple mathematical representation of Brownian motion, the Wiener equation, named after Norbert Wiener,[1] assumes the current velocity of a fluid particle fluctuates randomly:
$\mathbf {v} ={\frac {d\mathbf {x} }{dt}}=g(t),$
where v is velocity, x is position, d/dt is the time derivative, and g(t) may for instance be white noise.
Since velocity changes instantly in this formalism, the Wiener equation is not suitable for short time scales. In those cases, the Langevin equation, which looks at particle acceleration, must be used.
References
1. Pesi R. Masani (6 December 2012). Norbert Wiener 1894–1964. Birkhäuser. pp. 134–. ISBN 978-3-0348-9252-0.
| Wikipedia |
Wiener sausage
In the mathematical field of probability, the Wiener sausage is a neighborhood of the trace of a Brownian motion up to a time t, given by taking all points within a fixed distance of Brownian motion. It can be visualized as a sausage of fixed radius whose centerline is Brownian motion. The Wiener sausage was named after Norbert Wiener by M. D. Donsker and S. R. Srinivasa Varadhan (1975) because of its relation to the Wiener process; the name is also a pun on Vienna sausage, as "Wiener" is German for "Viennese".
The Wiener sausage is one of the simplest non-Markovian functionals of Brownian motion. Its applications include stochastic phenomena including heat conduction. It was first described by Frank Spitzer (1964), and it was used by Mark Kac and Joaquin Mazdak Luttinger (1973, 1974) to explain results of a Bose–Einstein condensate, with proofs published by M. D. Donsker and S. R. Srinivasa Varadhan (1975).
Definitions
The Wiener sausage Wδ(t) of radius δ and length t is the set-valued random variable on Brownian paths b (in some Euclidean space) defined by
$W_{\delta }(t)({b})$ is the set of points within a distance δ of some point b(x) of the path b with 0≤x≤t.
Volume of the Wiener sausage
There has been a lot of work on the behavior of the volume (Lebesgue measure) |Wδ(t)| of the Wiener sausage as it becomes thin (δ→0); by rescaling, this is essentially equivalent to studying the volume as the sausage becomes long (t→∞).
Spitzer (1964) showed that in 3 dimensions the expected value of the volume of the sausage is
$E(|W_{\delta }(t)|)=2\pi \delta t+4\delta ^{2}{\sqrt {2\pi t}}+4\pi \delta ^{3}/3.$
In dimension d at least 3 the volume of the Wiener sausage is asymptotic to
$\delta ^{d-2}\pi ^{d/2}2t/\Gamma ((d-2)/2)$
as t tends to infinity. In dimensions 1 and 2 this formula gets replaced by ${\sqrt {8t/\pi }}$ and $2{\pi }t/\log(t)$ respectively. Whitman (1964), a student of Spitzer, proved similar results for generalizations of Wiener sausages with cross sections given by more general compact sets than balls.
References
• Donsker, M. D.; Varadhan, S. R. S. (1975), "Asymptotics for the Wiener sausage", Communications on Pure and Applied Mathematics, 28 (4): 525–565, doi:10.1002/cpa.3160280406
• Hollander, F. den (2001) [1994], "Wiener sausage", Encyclopedia of Mathematics, EMS Press
• Kac, M.; Luttinger, J. M. (1973), "Bose-Einstein condensation in the presence of impurities", J. Math. Phys., 14 (11): 1626–1628, Bibcode:1973JMP....14.1626K, doi:10.1063/1.1666234, MR 0342114
• Kac, M.; Luttinger, J. M. (1974), "Bose-Einstein condensation in the presence of impurities. II", J. Math. Phys., 15 (2): 183–186, Bibcode:1974JMP....15..183K, doi:10.1063/1.1666617, MR 0342115
• Simon, Barry (2005), Functional integration and quantum physics, Providence, RI: AMS Chelsea Publishing, ISBN 0-8218-3582-3, MR 2105995 Especially chapter 22.
• Spitzer, F. (1964), "Electrostatic capacity, heat flow and Brownian motion", Probability Theory and Related Fields, 3 (2): 110–121, doi:10.1007/BF00535970, S2CID 198179345
• Spitzer, Frank (1976), Principles of random walks, Graduate Texts in Mathematics, vol. 34, New York-Heidelberg: Springer-Verlag, p. 40, MR 0171290 (Reprint of 1964 edition)
• Sznitman, Alain-Sol (1998), Brownian motion, obstacles and random media, Springer Monographs in Mathematics, Berlin: Springer-Verlag, doi:10.1007/978-3-662-11281-6, ISBN 3-540-64554-3, MR 1717054 An advanced monograph covering the Wiener sausage.
• Whitman, Walter William (1964), Some Strong Laws for Random Walks and Brownian Motion, PhD Thesis, Cornell U.
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| Wikipedia |
Wiener series
In mathematics, the Wiener series, or Wiener G-functional expansion, originates from the 1958 book of Norbert Wiener. It is an orthogonal expansion for nonlinear functionals closely related to the Volterra series and having the same relation to it as an orthogonal Hermite polynomial expansion has to a power series. For this reason it is also known as the Wiener–Hermite expansion. The analogue of the coefficients are referred to as Wiener kernels. The terms of the series are orthogonal (uncorrelated) with respect to a statistical input of white noise. This property allows the terms to be identified in applications by the Lee–Schetzen method.
The Wiener series is important in nonlinear system identification. In this context, the series approximates the functional relation of the output to the entire history of system input at any time. The Wiener series has been applied mostly to the identification of biological systems, especially in neuroscience.
The name Wiener series is almost exclusively used in system theory. In the mathematical literature it occurs as the Itô expansion (1951) which has a different form but is entirely equivalent to it.
The Wiener series should not be confused with the Wiener filter, which is another algorithm developed by Norbert Wiener used in signal processing.
Wiener G-functional expressions
Given a system with an input/output pair $(x(t),y(t))$ where the input is white noise with zero mean value and power A, we can write the output of the system as sum of a series of Wiener G-functionals $y(n)=\sum _{p}(G_{p}x)(n)$
In the following the expressions of the G-functionals up to the fifth order will be given:
$(G_{0}x)(n)=k_{0}=E\left\{y(n)\right\};$
$(G_{1}x)(n)=\sum _{\tau _{1}=0}^{N_{1}-1}k_{1}(\tau _{1})x(n-\tau _{1});$
$(G_{2}x)(n)=\sum _{\tau _{1},\tau _{2}=0}^{N_{2}-1}k_{2}(\tau _{1},\tau _{2})x(n-\tau _{1})x(n-\tau _{2})-A\sum _{\tau _{1}=0}^{N_{2}-1}k_{2}(\tau _{1},\tau _{1});$
$(G_{3}x)(n)=\sum _{\tau _{1},\ldots ,\tau _{3}=0}^{N_{3}-1}k_{3}(\tau _{1},\tau _{2},\tau _{3})x(n-\tau _{1})x(n-\tau _{2})x(n-\tau _{3})-3A\sum _{\tau _{1}=0}^{N_{3}-1}\sum _{\tau _{2}=0}^{N_{3}-1}k_{3}(\tau _{1},\tau _{2},\tau _{2})x(n-\tau _{1});$
${\begin{aligned}(G_{4}x)(n)={}&\sum _{\tau _{1},\ldots ,\tau _{4}=0}^{N_{4}-1}k_{4}(\tau _{1},\tau _{2},\tau _{3},\tau _{4})x(n-\tau _{1})x(n-\tau _{2})x(n-\tau _{3})x(n-\tau _{4})+{}\\[6pt]&{}-6A\sum _{\tau _{1},\tau _{2}=0}^{N_{4}-1}\sum _{\tau _{3}=0}^{N_{4}-1}k_{4}(\tau _{1},\tau _{2},\tau _{3},\tau _{3})x(n-\tau _{1})x(n-\tau _{2})+3A^{2}\sum _{\tau _{1},\tau _{2}=0}^{N_{4}-1}k_{4}(\tau _{1},\tau _{1},\tau _{2},\tau _{2});\end{aligned}}$
${\begin{aligned}(G_{5}x)(n)={}&\sum _{\tau _{1},\ldots ,\tau _{5}=0}^{N_{5}-1}k_{5}(\tau _{1},\tau _{2},\tau _{3},\tau _{4},\tau _{5})x(n-\tau _{1})x(n-\tau _{2})x(n-\tau _{3})x(n-\tau _{4})x(n-\tau _{5})+{}\\[6pt]&{}-10A\sum _{\tau _{1},\ldots ,\tau _{3}=0}^{N_{5}-1}\sum _{\tau _{4}=0}^{N_{5}-1}k_{5}(\tau _{1},\tau _{2},\tau _{3},\tau _{4},\tau _{4})x(n-\tau _{1})x(n-\tau _{2})x(n-\tau _{3})\\[6pt]&{}+15A^{2}\sum _{\tau _{1}=0}^{N_{5}-1}\sum _{\tau _{2},\tau _{3}=0}^{N_{5}-1}k_{5}(\tau _{1},\tau _{2},\tau _{2},\tau _{3},\tau _{3})x(n-\tau _{1}).\end{aligned}}$
See also
• Volterra series
• System identification
• Spike-triggered average
References
• Wiener, Norbert (1958). Nonlinear Problems in Random Theory. Wiley and MIT Press.
• Lee and Schetzen; Schetzen‡, M. (1965). "Measurement of the Wiener kernels of a non-linear system by cross-correlation". International Journal of Control. First. 2 (3): 237–254. doi:10.1080/00207176508905543.
• Itô K "A multiple Wiener integral" J. Math. Soc. Jpn. 3 1951 157–169
• Marmarelis, P.Z.; Naka, K. (1972). "White-noise analysis of a neuron chain: an application of the Wiener theory". Science. 175 (4027): 1276–1278. doi:10.1126/science.175.4027.1276. PMID 5061252.
• Schetzen, Martin (1980). The Volterra and Wiener Theories of Nonlinear Systems. John Wiley and Sons. ISBN 978-0-471-04455-0.
• Marmarelis, P.Z. (1991). "Wiener Analysis of Nonlinear Feedback". Sensory Systems Annals of Biomedical Engineering. 19 (4): 345–382. doi:10.1007/BF02584316.
• Franz, M; Schölkopf, B. (2006). "A unifying view of Wiener and Volterra theory and polynomial kernel regression". Neural Computation. 18 (12): 3097–3118. doi:10.1162/neco.2006.18.12.3097.
• L.A. Zadeh On the representation of nonlinear operators. IRE Westcon Conv. Record pt.2 1957 105-113.
| Wikipedia |
Wiener–Araya graph
The Wiener–Araya graph is, in graph theory, a graph on 42 vertices with 67 edges. It is hypohamiltonian, which means that it does not itself have a Hamiltonian cycle but every graph formed by removing a single vertex from it is Hamiltonian. It is also planar.
Wiener-Araya graph
Vertices42
Edges67
Radius5
Diameter7
Girth4
Automorphisms2
Chromatic number3
Chromatic index4
PropertiesHypohamiltonian
Planar
Table of graphs and parameters
History
Hypohamiltonian graphs were first studied by Sousselier in Problèmes plaisants et délectables (1963).[1] In 1967, Lindgren built an infinite sequence of hypohamiltonian graphs.[2] He first cited Gaudin, Herz and Rossi,[3] then Busacker and Saaty[4] as pioneers on this topic.
From the start, the smallest hypohamiltonian graph is known: the Petersen graph. However, the hunt for the smallest planar hypohamiltonian graph continues. This question was first raised by Václav Chvátal in 1973.[5] The first candidate answer was provided in 1976 by Carsten Thomassen, who exhibited a 105-vertices construction, the 105-Thomassen graph.[6] In 1979, Hatzel improved this result with a planar hypohamiltonian graph on 57 vertices : the Hatzel graph.[7] This bound was lowered in 2007 by the 48-Zamfirescu graph.[8]
In 2009, a graph built by Gábor Wiener and Makoto Araya became (with its 42 vertices) the smallest planar hypohamiltonian graph known.[9][10] In their paper, Wiener and Araya conjectured their graph to be optimal arguing that its order (42) appears to be the answer to The Ultimate Question of Life, the Universe, and Everything from The Hitchhiker's Guide to the Galaxy, a Douglas Adams novel. However, subsequently, smaller planar hypohamiltonian graphs have been discovered.[11]
References
1. Sousselier, R. (1963), Problème no. 29: Le cercle des irascibles, vol. 7, Rev. Franç. Rech. Opérationnelle, pp. 405–406
2. Lindgren, W. F. (1967), "An infinite class of hypohamiltonian graphs", American Mathematical Monthly, 74 (9): 1087–1089, doi:10.2307/2313617, JSTOR 2313617, MR 0224501
3. Gaudin, T.; Herz, P.; Rossi (1964), "Solution du problème No. 29", Rev. Franç. Rech. Opérationnelle (in French), 8: 214–218
4. Busacker, R. G.; Saaty, T. L. (1965), Finite Graphs and Networks
5. Chvátal, V. (1973), "Flip-flops in hypo-Hamiltonian graphs", Canadian Mathematical Bulletin, 16: 33–41, doi:10.4153/cmb-1973-008-9, MR 0371722
6. Thomassen, Carsten (1976), "Planar and infinite hypohamiltonian and hypotraceable graphs", Discrete Mathematics, 14 (4): 377–389, doi:10.1016/0012-365x(76)90071-6, MR 0422086
7. Hatzel, Wolfgang (1979), "Ein planarer hypohamiltonscher Graph mit 57 Knoten", Mathematische Annalen (in German), 243 (3): 213–216, doi:10.1007/BF01424541, MR 0548801
8. Zamfirescu, Carol T.; Zamfirescu, Tudor I. (2007), "A planar hypohamiltonian graph with 48 vertices", Journal of Graph Theory, 55 (4): 338–342, doi:10.1002/jgt.20241, MR 2336805
9. Wiener, Gábor; Araya, Makoto (April 20, 2009), The ultimate question, arXiv:0904.3012, Bibcode:2009arXiv0904.3012W.
10. Wiener, Gábor; Araya, Makoto (2011), "On planar hypohamiltonian graphs", Journal of Graph Theory, 67 (1): 55–68, doi:10.1002/jgt.20513, MR 2809563, S2CID 5340663.
11. Jooyandeh, Mohammadreza; McKay, Brendan D.; Östergård, Patric R. J.; Pettersson, Ville H.; Zamfirescu, Carol T. (2017), "Planar hypohamiltonian graphs on 40 vertices", Journal of Graph Theory, 84 (2): 121–133, arXiv:1302.2698, doi:10.1002/jgt.22015, MR 3601121, S2CID 5535167
External links
• Weisstein, Eric W., "Wiener-Araya Graph", MathWorld
| Wikipedia |
Wiener–Wintner theorem
In mathematics, the Wiener–Wintner theorem, named after Norbert Wiener and Aurel Wintner, is a strengthening of the ergodic theorem, proved by Wiener and Wintner (1941).
Statement
Suppose that τ is a measure-preserving transformation of a measure space S with finite measure. If f is a real-valued integrable function on S then the Wiener–Wintner theorem states that there is a measure 0 set E such that the average
$\lim _{\ell \rightarrow \infty }{\frac {1}{2\ell +1}}\sum _{j=-\ell }^{\ell }e^{ij\lambda }f(\tau ^{j}P)$
exists for all real λ and for all P not in E.
The special case for λ = 0 is essentially the Birkhoff ergodic theorem, from which the existence of a suitable measure 0 set E for any fixed λ, or any countable set of values λ, immediately follows. The point of the Wiener–Wintner theorem is that one can choose the measure 0 exceptional set E to be independent of λ.
This theorem was even much more generalized by the Return Times Theorem.
References
• Assani, I. (2001) [1994], "Wiener–Wintner theorem", Encyclopedia of Mathematics, EMS Press
• Wiener, Norbert; Wintner, Aurel (1941), "Harmonic analysis and ergodic theory", American Journal of Mathematics, 63: 415–426, doi:10.2307/2371534, ISSN 0002-9327, JSTOR 2371534, MR 0004098
| Wikipedia |
Karl Weierstrass
Karl Theodor Wilhelm Weierstrass (German: Weierstraß [ˈvaɪɐʃtʁaːs];[1] 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics and trained as a school teacher, eventually teaching mathematics, physics, botany and gymnastics.[2] He later received an honorary doctorate and became professor of mathematics in Berlin.
Karl Weierstrass
Karl Weierstraß
Born(1815-10-31)31 October 1815
Ennigerloh, Kingdom of Prussia
Died19 February 1897(1897-02-19) (aged 81)
Berlin, Kingdom of Prussia
NationalityGerman
Alma mater
• University of Bonn
• Münster Academy
Known for
• Weierstrass function
• Weierstrass product inequality
• (ε, δ)-definition of limit
• Weierstrass–Erdmann condition
• Weierstrass theorems
• Bolzano–Weierstrass theorem
Awards
• PhD (Hon): University of Königsberg (1854)
• Copley Medal (1895)
Scientific career
FieldsMathematics
InstitutionsGewerbeinstitut, Friedrich Wilhelm University
Academic advisorsChristoph Gudermann
Doctoral students
• Nikolai Bugaev
• Georg Cantor
• Georg Frobenius
• Lazarus Fuchs
• Wilhelm Killing
• Johannes Knoblauch
• Leo Königsberger
• Ernst Kötter
• Sofia Kovalevskaya
• Mathias Lerch
• Hans von Mangoldt
• Eugen Netto
• Adolf Piltz
• Carl Runge
• Arthur Schoenflies
• Friedrich Schottky
• Hermann Schwarz
• Ludwig Stickelberger
Among many other contributions, Weierstrass formalized the definition of the continuity of a function and complex analysis, proved the intermediate value theorem and the Bolzano–Weierstrass theorem, and used the latter to study the properties of continuous functions on closed bounded intervals.
Biography
Weierstrass was born into a Roman Catholic family in Ostenfelde, a village near Ennigerloh, in the Province of Westphalia.[3]
Weierstrass was the son of Wilhelm Weierstrass, a government official, and Theodora Vonderforst both of whom were Catholic Rhinelanders. His interest in mathematics began while he was a gymnasium student at the Theodorianum in Paderborn. He was sent to the University of Bonn upon graduation to prepare for a government position. Because his studies were to be in the fields of law, economics, and finance, he was immediately in conflict with his hopes to study mathematics. He resolved the conflict by paying little heed to his planned course of study but continuing private study in mathematics. The outcome was that he left the university without a degree. He then studied mathematics at the Münster Academy (which was even then famous for mathematics) and his father was able to obtain a place for him in a teacher training school in Münster. Later he was certified as a teacher in that city. During this period of study, Weierstrass attended the lectures of Christoph Gudermann and became interested in elliptic functions.
In 1843 he taught in Deutsch Krone in West Prussia and since 1848 he taught at the Lyceum Hosianum in Braunsberg.[4] Besides mathematics he also taught physics, botany, and gymnastics.[3]
Weierstrass may have had an illegitimate child named Franz with the widow of his friend Carl Wilhelm Borchardt.[5]
After 1850 Weierstrass suffered from a long period of illness, but was able to publish mathematical articles that brought him fame and distinction. The University of Königsberg conferred an honorary doctor's degree on him on 31 March 1854. In 1856 he took a chair at the Gewerbeinstitut in Berlin (an institute to educate technical workers which would later merge with the Bauakademie to form the Technical University of Berlin). In 1864 he became professor at the Friedrich-Wilhelms-Universität Berlin, which later became the Humboldt Universität zu Berlin.
In 1870, at the age of fifty-five, Weierstrass met Sofia Kovalevsky whom he tutored privately after failing to secure her admission to the university. They had a fruitful intellectual, and kindly personal relationship that "far transcended the usual teacher-student relationship". He mentored her for four years, and regarded her as his best student, helping to secure a doctorate for her from Heidelberg University without the need for an oral thesis defense. He was immobile for the last three years of his life, and died in Berlin from pneumonia.[6]
Mathematical contributions
Soundness of calculus
Weierstrass was interested in the soundness of calculus, and at the time there were somewhat ambiguous definitions of the foundations of calculus so that important theorems could not be proven with sufficient rigour. Although Bolzano had developed a reasonably rigorous definition of a limit as early as 1817 (and possibly even earlier) his work remained unknown to most of the mathematical community until years later, and many mathematicians had only vague definitions of limits and continuity of functions.
The basic idea behind Delta-epsilon proofs is, arguably, first found in the works of Cauchy in the 1820s.[7][8] Cauchy did not clearly distinguish between continuity and uniform continuity on an interval. Notably, in his 1821 Cours d'analyse, Cauchy argued that the (pointwise) limit of (pointwise) continuous functions was itself (pointwise) continuous, a statement that is false in general. The correct statement is rather that the uniform limit of continuous functions is continuous (also, the uniform limit of uniformly continuous functions is uniformly continuous). This required the concept of uniform convergence, which was first observed by Weierstrass's advisor, Christoph Gudermann, in an 1838 paper, where Gudermann noted the phenomenon but did not define it or elaborate on it. Weierstrass saw the importance of the concept, and both formalized it and applied it widely throughout the foundations of calculus.
The formal definition of continuity of a function, as formulated by Weierstrass, is as follows:
$\displaystyle f(x)$ is continuous at $\displaystyle x=x_{0}$ if $\displaystyle \forall \ \varepsilon >0\ \exists \ \delta >0$ such that for every $x$ in the domain of $f$, $\displaystyle \ |x-x_{0}|<\delta \Rightarrow |f(x)-f(x_{0})|<\varepsilon .$ In simple English, $\displaystyle f(x)$ is continuous at a point $\displaystyle x=x_{0}$ if for each $x$ close enough to $x_{0}$, the function value $f(x)$ is very close to $f(x_{0})$, where the "close enough" restriction typically depends on the desired closeness of $f(x_{0})$ to $f(x).$ Using this definition, he proved the Intermediate Value Theorem. He also proved the Bolzano–Weierstrass theorem and used it to study the properties of continuous functions on closed and bounded intervals.
Calculus of variations
Weierstrass also made advances in the field of calculus of variations. Using the apparatus of analysis that he helped to develop, Weierstrass was able to give a complete reformulation of the theory that paved the way for the modern study of the calculus of variations. Among several axioms, Weierstrass established a necessary condition for the existence of strong extrema of variational problems. He also helped devise the Weierstrass–Erdmann condition, which gives sufficient conditions for an extremal to have a corner along a given extremum and allows one to find a minimizing curve for a given integral.
Other analytical theorems
See also: List of things named after Karl Weierstrass
• Stone–Weierstrass theorem
• Casorati–Weierstrass theorem
• Weierstrass elliptic function
• Weierstrass function
• Weierstrass M-test
• Weierstrass preparation theorem
• Lindemann–Weierstrass theorem
• Weierstrass factorization theorem
• Weierstrass–Enneper parameterization
Students
• Edmund Husserl
Honours and awards
The lunar crater Weierstrass and the asteroid 14100 Weierstrass are named after him. Also, there is the Weierstrass Institute for Applied Analysis and Stochastics in Berlin.
Selected works
• Zur Theorie der Abelschen Funktionen (1854)
• Theorie der Abelschen Funktionen (1856)
• Abhandlungen-1, Math. Werke. Bd. 1. Berlin, 1894
• Abhandlungen-2, Math. Werke. Bd. 2. Berlin, 1895
• Abhandlungen-3, Math. Werke. Bd. 3. Berlin, 1903
• Vorl. ueber die Theorie der Abelschen Transcendenten, Math. Werke. Bd. 4. Berlin, 1902
• Vorl. ueber Variationsrechnung, Math. Werke. Bd. 7. Leipzig, 1927
See also
• List of things named after Karl Weierstrass
References
1. Duden. Das Aussprachewörterbuch. 7. Auflage. Bibliographisches Institut, Berlin 2015, ISBN 978-3-411-04067-4
2. Weierstrass, Karl Theodor Wilhelm. (2018). In Helicon (Ed.), The Hutchinson unabridged encyclopedia with atlas and weather guide. [Online]. Abington: Helicon. Available from: http://libezproxy.open.ac.uk/login?url=https://search.credoreference.com/content/entry/heliconhe/weierstrass_karl_theodor_wilhelm/0?institutionId=292 [Accessed 8 July 2018].
3. O'Connor, J. J.; Robertson, E. F. (October 1998). "Karl Theodor Wilhelm Weierstrass". School of Mathematics and Statistics, University of St Andrews, Scotland. Retrieved 7 September 2014.
4. Elstrodt, Jürgen (2016), König, Wolfgang; Sprekels, Jürgen (eds.), "Die prägenden Jahre im Leben von Karl Weierstraß", Karl Weierstraß (1815–1897) (in German), Wiesbaden: Springer Fachmedien Wiesbaden, pp. 11–51, doi:10.1007/978-3-658-10619-5_2, ISBN 978-3-658-10618-8, retrieved 2023-08-12
5. Biermann, Kurt-R.; Schubring, Gert (1996). "Einige Nachträge zur Biographie von Karl Weierstraß. (German) [Some postscripts to the biography of Karl Weierstrass]". History of mathematics. San Diego, CA: Academic Press. pp. 65–91.
6. Dictionary of scientific biography. Gillispie, Charles Coulston,, American Council of Learned Societies. New York. 1970. p. 223. ISBN 978-0-684-12926-6. OCLC 89822.{{cite book}}: CS1 maint: location missing publisher (link) CS1 maint: others (link)
7. Grabiner, Judith V. (March 1983), "Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus" (PDF), The American Mathematical Monthly, 90 (3): 185–194, doi:10.2307/2975545, JSTOR 2975545, archived (PDF) from the original on 2014-11-29
8. Cauchy, A.-L. (1823), "Septième Leçon – Valeurs de quelques expressions qui se présentent sous les formes indéterminées ${\frac {\infty }{\infty }},\infty ^{0},\ldots $ Relation qui existe entre le rapport aux différences finies et la fonction dérivée", Résumé des leçons données à l'école royale polytechnique sur le calcul infinitésimal, Paris, archived from the original on 2009-05-04, retrieved 2009-05-01, p. 44. {{citation}}: External link in |postscript= (help)CS1 maint: postscript (link)
External links
Wikimedia Commons has media related to Karl Weierstrass.
Wikiquote has quotations related to Karl Weierstrass.
• O'Connor, John J.; Robertson, Edmund F., "Karl Weierstrass", MacTutor History of Mathematics Archive, University of St Andrews
• Digitalized versions of Weierstrass's original publications are freely available online from the library of the Berlin Brandenburgische Akademie der Wissenschaften.
• Works by Karl Weierstrass at Project Gutenberg
• Works by or about Karl Weierstrass at Internet Archive
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| Wikipedia |
Borchers algebra
In mathematics, a Borchers algebra or Borchers–Uhlmann algebra or BU-algebra is the tensor algebra of a vector space, often a space of smooth test functions. They were studied by H. J. Borchers (1962), who showed that the Wightman distributions of a quantum field could be interpreted as a state, called a Wightman functional, on a Borchers algebra. A Borchers algebra with a state can often be used to construct an O*-algebra.
The Borchers algebra of a quantum field theory has an ideal called the locality ideal, generated by elements of the form ab−ba for a and b having spacelike-separated support. The Wightman functional of a quantum field theory vanishes on the locality ideal, which is equivalent to the locality axiom for quantum field theory.
References
• Borchers, H.-J. (1962), "On structure of the algebra of field operators", Nuovo Cimento, 24 (2): 214–236, doi:10.1007/BF02745645, MR 0142320, S2CID 122439590
External links
• Yngvason, Jakob (2009), The Borchers-Uhlmann Algebra and its Descendants (PDF)
| Wikipedia |
Eugene Wigner
Eugene Paul "E. P." Wigner (Hungarian: Wigner Jenő Pál, pronounced [ˈviɡnɛr ˈjɛnøː ˈpaːl]; November 17, 1902 – January 1, 1995) was a Hungarian-American theoretical physicist who also contributed to mathematical physics. He received the Nobel Prize in Physics in 1963 "for his contributions to the theory of the atomic nucleus and the elementary particles, particularly through the discovery and application of fundamental symmetry principles".[1]
Eugene Wigner
Wigner in 1963
Born
Wigner Jenő Pál
(1902-11-17)November 17, 1902
Budapest, Kingdom of Hungary, Austria-Hungary
DiedJanuary 1, 1995(1995-01-01) (aged 92)
Princeton, New Jersey, U.S.
Citizenship
• Hungary (by birth)
• United States (naturalized 1937)
Alma materBudapest University of Technology and Economics
Technical University of Berlin
Known for
• Bargmann–Wigner equations
• Law of conservation of parity
• Wigner D-matrix
• Wigner–Eckart theorem
• Wigner's friend
• Wigner semicircle distribution
• Wigner's classification
• Wigner distribution function
• Wigner quasiprobability distribution
• Wigner crystal
• Wigner effect
• Wigner energy
• Relativistic Breit–Wigner distribution
• Modified Wigner distribution function
• Wigner–d'Espagnat inequality
• Gabor–Wigner transform
• Wigner's theorem
• Jordan–Wigner transformation
• Newton–Wigner localization
• Wigner–Inonu contraction
• Wigner–Seitz cell
• Wigner–Seitz radius
• Thomas–Wigner rotation
• Wigner–Weyl transform
• Wigner–Wilkins spectrum
• 6-j symbol
• 9-j symbol
Spouses
Amelia Frank
(m. 1936; died 1937)
Mary Annette Wheeler
(m. 1941; died 1977)
Eileen Clare-Patton Hamilton
(m. 1979)
Children3
Awards
• Medal for Merit (1946)
• Franklin Medal (1950)
• Enrico Fermi Award (1958)
• Atoms for Peace Award (1959)
• Max Planck Medal (1961)
• Nobel Prize in Physics (1963)
• National Medal of Science (1969)
• Albert Einstein Award (1972)
• Wigner Medal (1978)
Scientific career
Fields
• Theoretical physics
• Atomic physics
• Nuclear physics
• Solid-state physics
Institutions
• University of Göttingen
• University of Wisconsin–Madison
• Princeton University
• Manhattan Project
ThesisBildung und Zerfall von Molekülen (1925)
Doctoral advisorMichael Polanyi
Other academic advisors
• László Rátz
• Richard Becker
Doctoral students
• John Bardeen
• Victor Frederick Weisskopf
• Marcos Moshinsky
• Abner Shimony
• Edwin Thompson Jaynes
• Frederick Seitz
• Conyers Herring
• Frederick Tappert
• J O Hirschfelder
Signature
A graduate of the Technical University of Berlin, Wigner worked as an assistant to Karl Weissenberg and Richard Becker at the Kaiser Wilhelm Institute in Berlin, and David Hilbert at the University of Göttingen. Wigner and Hermann Weyl were responsible for introducing group theory into physics, particularly the theory of symmetry in physics. Along the way he performed ground-breaking work in pure mathematics, in which he authored a number of mathematical theorems. In particular, Wigner's theorem is a cornerstone in the mathematical formulation of quantum mechanics. He is also known for his research into the structure of the atomic nucleus. In 1930, Princeton University recruited Wigner, along with John von Neumann, and he moved to the United States, where he obtained citizenship in 1937.
Wigner participated in a meeting with Leo Szilard and Albert Einstein that resulted in the Einstein–Szilard letter, which prompted President Franklin D. Roosevelt to initiate the Manhattan Project to develop atomic bombs. Wigner was afraid that the German nuclear weapon project would develop an atomic bomb first. During the Manhattan Project, he led a team whose task was to design nuclear reactors to convert uranium into weapons grade plutonium. At the time, reactors existed only on paper, and no reactor had yet gone critical. Wigner was disappointed that DuPont was given responsibility for the detailed design of the reactors, not just their construction. He became director of research and development at the Clinton Laboratory (now the Oak Ridge National Laboratory) in early 1946, but became frustrated with bureaucratic interference by the Atomic Energy Commission, and returned to Princeton.
In the postwar period, he served on a number of government bodies, including the National Bureau of Standards from 1947 to 1951, the mathematics panel of the National Research Council from 1951 to 1954, the physics panel of the National Science Foundation, and the influential General Advisory Committee of the Atomic Energy Commission from 1952 to 1957 and again from 1959 to 1964. In later life, he became more philosophical, and published The Unreasonable Effectiveness of Mathematics in the Natural Sciences, his best-known work outside technical mathematics and physics.
Early life and education
Wigner Jenő Pál was born in Budapest, Austria-Hungary on November 17, 1902, to middle class Jewish parents, Elisabeth Elsa Einhorn and Antal Anton Wigner, a leather tanner. He had an older sister, Berta, known as Biri, and a younger sister Margit, known as Manci,[2] who later married British theoretical physicist Paul Dirac.[3] He was home schooled by a professional teacher until the age of 9, when he started school at the third grade. During this period, Wigner developed an interest in mathematical problems.[4] At the age of 11, Wigner contracted what his doctors believed to be tuberculosis. His parents sent him to live for six weeks in a sanatorium in the Austrian mountains, before the doctors concluded that the diagnosis was mistaken.[5]
Wigner's family was Jewish, but not religiously observant, and his Bar Mitzvah was a secular one. From 1915 through 1919, he studied at the secondary grammar school called Fasori Evangélikus Gimnázium, the school his father had attended. Religious education was compulsory, and he attended classes in Judaism taught by a rabbi.[6] A fellow student was János von Neumann, who was a year behind Wigner. They both benefited from the instruction of the noted mathematics teacher László Rátz.[7] In 1919, to escape the Béla Kun communist regime, the Wigner family briefly fled to Austria, returning to Hungary after Kun's downfall.[8] Partly as a reaction to the prominence of Jews in the Kun regime, the family converted to Lutheranism.[9] Wigner explained later in his life that his family decision to convert to Lutheranism "was not at heart a religious decision but an anti-communist one".[9]
After graduating from the secondary school in 1920, Wigner enrolled at the Budapest University of Technical Sciences, known as the Műegyetem. He was not happy with the courses on offer,[10] and in 1921 enrolled at the Technische Hochschule Berlin (now Technical University of Berlin), where he studied chemical engineering.[11] He also attended the Wednesday afternoon colloquia of the German Physical Society. These colloquia featured leading researchers including Max Planck, Max von Laue, Rudolf Ladenburg, Werner Heisenberg, Walther Nernst, Wolfgang Pauli, and Albert Einstein.[12] Wigner also met the physicist Leó Szilárd, who at once became Wigner's closest friend.[13] A third experience in Berlin was formative. Wigner worked at the Kaiser Wilhelm Institute for Physical Chemistry and Electrochemistry (now the Fritz Haber Institute), and there he met Michael Polanyi, who became, after László Rátz, Wigner's greatest teacher. Polanyi supervised Wigner's DSc thesis, Bildung und Zerfall von Molekülen ("Formation and Decay of Molecules").[14]
Middle years
Wigner returned to Budapest, where he went to work at his father's tannery, but in 1926, he accepted an offer from Karl Weissenberg at the Kaiser Wilhelm Institute in Berlin. Weissenberg wanted someone to assist him with his work on x-ray crystallography, and Polanyi had recommended Wigner. After six months as Weissenberg's assistant, Wigner went to work for Richard Becker for two semesters. Wigner explored quantum mechanics, studying the work of Erwin Schrödinger. He also delved into the group theory of Ferdinand Frobenius and Eduard Ritter von Weber.[15]
Wigner received a request from Arnold Sommerfeld to work at the University of Göttingen as an assistant to the great mathematician David Hilbert. This proved a disappointment, as the aged Hilbert's abilities were failing, and his interests had shifted to logic. Wigner nonetheless studied independently.[16] He laid the foundation for the theory of symmetries in quantum mechanics and in 1927 introduced what is now known as the Wigner D-matrix.[17] Wigner and Hermann Weyl were responsible for introducing group theory into quantum mechanics. The latter had written a standard text, Group Theory and Quantum Mechanics (1928), but it was not easy to understand, especially for younger physicists. Wigner's Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra (1931) made group theory accessible to a wider audience.[18]
In these works, Wigner laid the foundation for the theory of symmetries in quantum mechanics.[19] Wigner's theorem proved by Wigner in 1931, is a cornerstone of the mathematical formulation of quantum mechanics. The theorem specifies how physical symmetries such as rotations, translations, and CPT symmetry are represented on the Hilbert space of states. According to the theorem, any symmetry transformation is represented by a linear and unitary or antilinear and antiunitary transformation of Hilbert space. The representation of a symmetry group on a Hilbert space is either an ordinary representation or a projective representation.[20][21]
In the late 1930s, Wigner extended his research into atomic nuclei. By 1929, his papers were drawing notice in the world of physics. In 1930, Princeton University recruited Wigner for a one-year lectureship, at 7 times the salary that he had been drawing in Europe. Princeton recruited von Neumann at the same time. Jenő Pál Wigner and János von Neumann had collaborated on three papers together in 1928 and two in 1929. They anglicized their first names to "Eugene" and "John", respectively.[22] When their year was up, Princeton offered a five-year contract as visiting professors for half the year. The Technische Hochschule responded with a teaching assignment for the other half of the year. This was very timely, since the Nazis soon rose to power in Germany.[23] At Princeton in 1934, Wigner introduced his sister Margit "Manci" Wigner to the physicist Paul Dirac, with whom she remarried.[24]
Princeton did not rehire Wigner when his contract ran out in 1936.[25] Through Gregory Breit, Wigner found new employment at the University of Wisconsin. There, he met his first wife, Amelia Frank, who was a physics student there. However, she died unexpectedly in 1937, leaving Wigner distraught. He therefore accepted a 1938 offer from Princeton to return there.[26] Wigner became a naturalized citizen of the United States on January 8, 1937, and he brought his parents to the United States.[27]
Manhattan Project
Although he was a professed political amateur, on August 2, 1939, he participated in a meeting with Leó Szilárd and Albert Einstein that resulted in the Einstein–Szilárd letter, which prompted President Franklin D. Roosevelt to initiate the Manhattan Project to develop atomic bombs.[28] Wigner was afraid that the German nuclear weapon project would develop an atomic bomb first, and even refused to have his fingerprints taken because they could be used to track him down if Germany won.[29] "Thoughts of being murdered," he later recalled, "focus your mind wonderfully."[29]
On June 4, 1941, Wigner married his second wife, Mary Annette Wheeler, a professor of physics at Vassar College, who had completed her Ph.D. at Yale University in 1932. After the war she taught physics on the faculty of Rutgers University's Douglass College in New Jersey until her retirement in 1964. They remained married until her death in November 1977.[30][31] They had two children, David Wigner and Martha Wigner Upton.[32]
During the Manhattan Project, Wigner led a team that included J. Ernest Wilkins Jr., Alvin M. Weinberg, Katharine Way, Gale Young and Edward Creutz. The group's task was to design the production nuclear reactors that would convert uranium into weapons grade plutonium. At the time, reactors existed only on paper, and no reactor had yet gone critical. In July 1942, Wigner chose a conservative 100 MW design, with a graphite neutron moderator and water cooling.[33] Wigner was present at a converted rackets court under the stands at the University of Chicago's abandoned Stagg Field on December 2, 1942, when the world's first atomic reactor, Chicago Pile One (CP-1) achieved a controlled nuclear chain reaction.[34]
Wigner was disappointed that DuPont was given responsibility for the detailed design of the reactors, not just their construction. He threatened to resign in February 1943, but was talked out of it by the head of the Metallurgical Laboratory, Arthur Compton, who sent him on vacation instead. As it turned out, a design decision by DuPont to give the reactor additional load tubes for more uranium saved the project when neutron poisoning became a problem.[35] Without the additional tubes, the reactor could have been run at 35% power until the boron impurities in the graphite were burned up and enough plutonium produced to run the reactor at full power; but this would have set the project back a year.[36] During the 1950s, he would even work for DuPont on the Savannah River Site.[35] Wigner did not regret working on the Manhattan Project, and sometimes wished the atomic bomb had been ready a year earlier.[37]
An important discovery Wigner made during the project was the Wigner effect. This is a swelling of the graphite moderator caused by the displacement of atoms by neutron radiation.[38] The Wigner effect was a serious problem for the reactors at the Hanford Site in the immediate post-war period, and resulted in production cutbacks and a reactor being shut down entirely.[39] It was eventually discovered that it could be overcome by controlled heating and annealing.[40]
Through Manhattan project funding, Wigner and Leonard Eisenbud also developed an important general approach to nuclear reactions, the Wigner–Eisenbud R-matrix theory, which was published in 1947.[41]
Later years
Wigner was elected to the American Philosophical Society in 1944 and the United States National Academy of Sciences in 1945.[42][43] He accepted a position as the director of research and development at the Clinton Laboratory (now the Oak Ridge National Laboratory) in Oak Ridge, Tennessee in early 1946. Because he did not want to be involved in administrative duties, he became co-director of the laboratory, with James Lum handling the administrative chores as executive director.[44] When the newly created Atomic Energy Commission (AEC) took charge of the laboratory's operations at the start of 1947, Wigner feared that many of the technical decisions would be made in Washington.[45] He also saw the Army's continuation of wartime security policies at the laboratory as a "meddlesome oversight", interfering with research.[46] One such incident occurred in March 1947, when the AEC discovered that Wigner's scientists were conducting experiments with a critical mass of uranium-235 when the director of the Manhattan Project, Major General Leslie R. Groves, Jr., had forbidden such experiments in August 1946 after the death of Louis Slotin at the Los Alamos Laboratory. Wigner argued that Groves's order had been superseded, but was forced to terminate the experiments, which were completely different from the one that killed Slotin.[47]
Feeling unsuited to a managerial role in such an environment, he left Oak Ridge in 1947 and returned to Princeton University,[48] although he maintained a consulting role with the facility for many years.[45] In the postwar period, he served on a number of government bodies, including the National Bureau of Standards from 1947 to 1951, the mathematics panel of the National Research Council from 1951 to 1954, the physics panel of the National Science Foundation, and the influential General Advisory Committee of the Atomic Energy Commission from 1952 to 1957 and again from 1959 to 1964.[49] He also contributed to civil defense.[50]
Wigner was elected to the American Academy of Arts and Sciences in 1950.[51]
Near the end of his life, Wigner's thoughts turned more philosophical. In 1960, he published a now classic article on the philosophy of mathematics and of physics, which has become his best-known work outside technical mathematics and physics, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences".[52] He argued that biology and cognition could be the origin of physical concepts, as we humans perceive them, and that the happy coincidence that mathematics and physics were so well matched, seemed to be "unreasonable" and hard to explain.[52] His original paper has provoked and inspired many responses across a wide range of disciplines. These included Richard Hamming in Computer Science,[53] Arthur Lesk in Molecular Biology,[54] Peter Norvig in data mining,[55] Max Tegmark in Physics,[56] Ivor Grattan-Guinness in Mathematics,[57] and Vela Velupillai in Economics.[58]
Turning to philosophical questions about the theory of quantum mechanics, Wigner developed a thought experiment (later called Wigner's Friend paradox) to illustrate his belief that consciousness is foundational to the quantum mechanical measurement process. He thereby followed an ontological approach that sets human's consciousness at the center: "All that quantum mechanics purports to provide are probability connections between subsequent impressions (also called 'apperceptions') of the consciousness".[59]
Measurements are understood as the interactions which create the impressions in our consciousness (and as a result modify the wave function of the "measured" physical system), an idea which has been called the "consciousness causes collapse" interpretation.
Interestingly, Hugh Everett III (a student of Wigner) discussed Wigner's thought experiment in the introductory part of his 1957 dissertation as an "amusing, but extremely hypothetical drama".[60] In an early draft of Everett's work, one also finds a drawing of the Wigner's Friend situation,[61] which must be seen as the first evidence on paper of the thought experiment that was later assigned to be Wigner's. This suggests that Everett must at least have discussed the problem together with Wigner.
In November 1963, Wigner called for the allocation of 10% of the national defense budget to be spent on nuclear blast shelters and survival resources, arguing that such an expenditure would be less costly than disarmament. Wigner considered a recent Woods Hole study's conclusion that a nuclear strike would kill 20% of Americans to be a very modest projection and that the country could recover from such an attack more quickly than Germany had recovered from the devastation of World War II.[62]
Wigner was awarded the Nobel Prize in Physics in 1963 "for his contributions to the theory of the atomic nucleus and the elementary particles, particularly through the discovery and application of fundamental symmetry principles".[1] The prize was shared that year, with the other half of the award divided between Maria Goeppert-Mayer and J. Hans D. Jensen.[1] Wigner professed that he had never considered the possibility that this might occur, and added: "I never expected to get my name in the newspapers without doing something wicked."[63] He also won the Franklin Medal in 1950,[64] the Enrico Fermi award in 1958,[65] the Atoms for Peace Award in 1959,[66] the Max Planck Medal in 1961,[67] the National Medal of Science in 1969,[68] the Albert Einstein Award in 1972,[69] the Golden Plate Award of the American Academy of Achievement in 1974,[70] and the eponymous Wigner Medal in 1978.[71] In 1968 he gave the Josiah Willard Gibbs lecture.[72][73]
After his retirement from Princeton in 1971, Wigner prepared the first edition of Symmetries and Reflections, a collection of philosophical essays, and became more involved in international and political meetings; around this time he became a leader[74] and vocal defender[75] of the Unification Church's annual International Conference on the Unity of the Sciences.
Mary died in November 1977. In 1979, Wigner married his third wife, Eileen Clare-Patton (Pat) Hamilton, the widow of physicist Donald Ross Hamilton, the dean of the graduate school at Princeton University, who had died in 1972.[76] In 1992, at the age of 90, he published his memoirs, The Recollections of Eugene P. Wigner with Andrew Szanton. In it, Wigner said: "The full meaning of life, the collective meaning of all human desires, is fundamentally a mystery beyond our grasp. As a young man, I chafed at this state of affairs. But by now I have made peace with it. I even feel a certain honor to be associated with such a mystery."[77] In his collection of essays 'Philosophical Reflections and Syntheses' (1995), he commented: "It was not possible to formulate the laws of quantum mechanics in a fully consistent way without reference to consciousness."[78]
Wigner was credited as a member of the advisory board for the Western Goals Foundation, a private domestic intelligence agency created in the US in 1979 to "fill the critical gap caused by the crippling of the FBI, the disabling of the House Un-American Activities Committee and the destruction of crucial government files".[79]
Wigner died of pneumonia at the University Medical Center in Princeton, New Jersey on 1 January 1995.[80] He was survived by his wife Eileen (died 2010) and children Erika, David and Martha, and his sisters Bertha and Margit.[69]
Publications
• 1958 (with Alvin M. Weinberg). Physical Theory of Neutron Chain Reactors University of Chicago Press. ISBN 0-226-88517-8
• 1959. Group Theory and its Application to the Quantum Mechanics of Atomic Spectra. New York: Academic Press. Translation by J. J. Griffin of 1931, Gruppentheorie und ihre Anwendungen auf die Quantenmechanik der Atomspektren, Vieweg Verlag, Braunschweig.
• 1970 Symmetries and Reflections: Scientific Essays. Indiana University Press, Bloomington ISBN 0-262-73021-9
• 1992 (as told to Andrew Szanton). The Recollections of Eugene P. Wigner. Plenum. ISBN 0-306-44326-0
• 1995 (with Jagdish Mehra and Arthur Wightman, eds.). Philosophical Reflections and Syntheses. Springer, Berlin ISBN 3-540-63372-3
Selected contributions
Theoretical physics
• Bargmann–Wigner equations
• Jordan–Wigner transformation
• Newton–Wigner localization
• Polynomial Wigner–Ville distribution
• Relativistic Breit–Wigner distribution
• Thomas–Wigner rotation
• Wigner–Eckart theorem
• Wigner–Inonu contraction
• Wigner–Seitz cell
• Wigner–Seitz radius
• Wigner–Weyl transform
• Wigner–Wilkins spectrum
• Wigner's classification
• Wigner quasi-probability distribution
• Wigner's friend
• Wigner's theorem
• Wigner crystal
• Wigner D-matrix
• Wigner effect
• Wigner energy
• Wigner lattice
• Wigner's disease
• Thomas–Wigner rotation
• Von Neumann–Wigner interpretation
• Wigner–Witmer correlation rules
Mathematics
• Gabor–Wigner transform
• Modified Wigner distribution function
• Wigner distribution function
• Wigner semicircle distribution
• Wigner rotation
• Wigner quasi-probability distribution
• Wigner semicircle distribution
• 6-j symbol
• 9-j symbol
• Wigner 3-j symbols
• Wigner–İnönü group contraction
• Wigner surmise
See also
• List of things named after Eugene Wigner
• The Martians (scientists)
• List of Jewish Nobel laureates
Notes
1. "The Nobel Prize in Physics 1963". Nobel Foundation. Retrieved May 19, 2015.
2. Szanton 1992, pp. 9–12.
3. Szanton 1992, pp. 164–166.
4. Szanton 1992, pp. 14–15.
5. Szanton 1992, pp. 22–24.
6. Szanton 1992, pp. 33–34, 47.
7. Szanton 1992, pp. 49–53.
8. Szanton 1992, pp. 40–43.
9. Szanton 1992, p. 38.
10. Szanton 1992, p. 59.
11. Szanton 1992, pp. 64–65.
12. Szanton 1992, pp. 68–75.
13. Szanton 1992, pp. 93–94.
14. Szanton 1992, pp. 76–84.
15. Szanton 1992, pp. 101–106.
16. Szanton 1992, pp. 109–112.
17. Wigner, E. (1927). "Einige Folgerungen aus der Schrödingerschen Theorie für die Termstrukturen". Zeitschrift für Physik (in German). 43 (9–10): 624–652. Bibcode:1927ZPhy...43..624W. doi:10.1007/BF01397327. S2CID 124334051.
18. Szanton 1992, pp. 116–119.
19. Wightman, A.S. (1995). "Eugene Paul Wigner 1902–1995" (PDF). Notices of the American Mathematical Society. 42 (7): 769–771.
20. Wigner 1931, pp. 251–254.
21. Wigner 1959, pp. 233–236.
22. Szanton 1992, pp. 127–132.
23. Szanton 1992, pp. 136, 153–155.
24. Szanton 1992, pp. 163–166.
25. Szanton 1992, pp. 171–172.
26. Szanton 1992, pp. 173–178.
27. Szanton 1992, pp. 184–185.
28. Szanton 1992, pp. 197–202.
29. Szanton 1992, p. 215.
30. Szanton 1992, pp. 205–207.
31. "Obituary: Mary Wigner". Physics Today. 31 (7): 58. July 1978. Bibcode:1978PhT....31g..58.. doi:10.1063/1.2995119. Archived from the original on 2013-09-27.
32. "Wigner Biography". St Andrews University. Retrieved August 10, 2013.
33. Szanton 1992, pp. 217–218.
34. "Chicago Pile 1 Pioneers". Los Alamos National Laboratory. Archived from the original on February 4, 2012. Retrieved August 10, 2013.
35. Szanton 1992, pp. 233–235.
36. Wigner & Weinberg 1992, p. 8.
37. Szanton 1992, p. 249.
38. Wigner, E. P. (1946). "Theoretical Physics in the Metallurgical Laboratory of Chicago". Journal of Applied Physics. 17 (11): 857–863. Bibcode:1946JAP....17..857W. doi:10.1063/1.1707653.
39. Rhodes 1995, p. 277.
40. Wilson, Richard (November 8, 2002). "A young Scientist's Meetings with Wigner in America". Budapest: Wigner Symposium, Hungarian Academy of Sciences. Retrieved May 16, 2015.
41. Leal, L. C. "Brief Review of R-Matrix Theory" (PDF). Retrieved August 12, 2013.
• The original paper is Wigner, E. P.; Eisenbud, L. (1 July 1947). "Higher Angular Momenta and Long Range Interaction in Resonance Reactions". Physical Review. 72 (1): 29–41. Bibcode:1947PhRv...72...29W. doi:10.1103/PhysRev.72.29.
42. "APS Member History". search.amphilsoc.org. Retrieved 2023-04-03.
43. "Eugene P. Wigner". www.nasonline.org. Retrieved 2023-04-03.
44. Johnson & Schaffer 1994, p. 31.
45. Seitz, Frederick; Vogt, Erich; Weinberg, Alvin M. "Eugene Paul Wigner". Biographical Memoirs. National Academies Press. Retrieved 20 August 2013.
46. "ORNL History. Chapter 2: High-Flux Years. Section: Research and Regulations". ORNL Review. Oak Ridge National Laboratory's Communications and Community Outreach. Archived from the original on 16 March 2013. Retrieved 20 August 2013. Oak Ridge at that time was so terribly bureaucratized that I am sorry to say I could not stand it.
47. Hewlett & Duncan 1969, pp. 38–39.
48. Johnson & Schaffer 1994, p. 49.
49. Szanton 1992, p. 270.
50. Szanton 1992, pp. 288–290.
51. "Eugene Paul Wigner". American Academy of Arts & Sciences. 9 February 2023. Retrieved 2023-04-03.
52. Wigner, E. P. (1960). "The unreasonable effectiveness of mathematics in the natural sciences. Richard Courant lecture in mathematical sciences delivered at New York University, May 11, 1959". Communications on Pure and Applied Mathematics. 13 (1): 1–14. Bibcode:1960CPAM...13....1W. doi:10.1002/cpa.3160130102. S2CID 6112252. Archived from the original on February 28, 2011. Retrieved December 24, 2008.
53. Hamming, R. W. (1980). "The Unreasonable Effectiveness of Mathematics". The American Mathematical Monthly. 87 (2): 81–90. doi:10.2307/2321982. hdl:10945/55827. JSTOR 2321982. Archived from the original on 2007-02-03. Retrieved 2015-08-28.
54. Lesk, A. M. (2000). "The unreasonable effectiveness of mathematics in molecular biology". The Mathematical Intelligencer. 22 (2): 28–37. doi:10.1007/BF03025372. S2CID 120102813.
55. Halevy, A.; Norvig, P.; Pereira, F. (2009). "The Unreasonable Effectiveness of Data" (PDF). IEEE Intelligent Systems. 24 (2): 8–12. doi:10.1109/MIS.2009.36. S2CID 14300215.
56. Tegmark, Max (2008). "The Mathematical Universe". Foundations of Physics. 38 (2): 101–150. arXiv:0704.0646. Bibcode:2008FoPh...38..101T. doi:10.1007/s10701-007-9186-9. S2CID 9890455.
57. Grattan-Guinness, I. (2008). "Solving Wigner's mystery: The reasonable (though perhaps limited) effectiveness of mathematics in the natural sciences". The Mathematical Intelligencer. 30 (3): 7–17. doi:10.1007/BF02985373. S2CID 123174309.
58. Velupillai, K. V. (2005). "The unreasonable ineffectiveness of mathematics in economics" (PDF). Cambridge Journal of Economics. 29 (6): 849–872. CiteSeerX 10.1.1.194.6586. doi:10.1093/cje/bei084. Archived from the original (PDF) on 2005-03-11. Retrieved 2017-10-24.
59. Wigner, E. P. (1995), "Remarks on the Mind-Body Question", Philosophical Reflections and Syntheses, Berlin, Heidelberg: Springer Berlin Heidelberg, pp. 247–260, doi:10.1007/978-3-642-78374-6_20, ISBN 978-3-540-63372-3, retrieved 2021-12-01
60. Everett, Hugh (1957-07-01). ""Relative State" Formulation of Quantum Mechanics". Reviews of Modern Physics. 29 (3): 454–462. Bibcode:1957RvMP...29..454E. doi:10.1103/RevModPhys.29.454. ISSN 0034-6861.
61. Barrett, Jeffrey A.; Byrne, Peter, eds. (2012-05-20). The Everett Interpretation of Quantum Mechanics. doi:10.1515/9781400842742. ISBN 9781400842742.
62. Lyons, R. (1963, November 22). Asks Better Civil Defense for Atomic Victory. New York Daily News, p. 6.
63. Szanton 1992, p. 147.
64. "Eugene P. Wigner". The Franklin Institute. 2014-01-15. Retrieved May 19, 2015.
65. "Eugene P. Wigner, 1958". United States Department of Energy Office of Science. Retrieved May 19, 2015.
66. "Guide to Atoms for Peace Awards Records MC.0010". Massachusetts Institute of Technology. Archived from the original on August 5, 2015. Retrieved May 19, 2015.
67. "Preisträger Max Planck nach Jahren" (in German). Deutschen Physikalischen Gesellschaft. Archived from the original on September 23, 2015. Retrieved May 19, 2015.
68. "The President's National Medal of Science: Recipient Details". United States National Science Foundation. Retrieved May 19, 2015.
69. "Eugene P. Wigner". Princeton University.
70. "Golden Plate Awardees of the American Academy of Achievement". www.achievement.org. American Academy of Achievement.
71. "The Wigner Medal". University of Texas. Retrieved May 19, 2015.
72. "Josiah Willard Gibbs Lectures". American Mathematical Society. Retrieved May 15, 2015.
73. Wigner, Eugene P (1968). "Problems of symmetry in old and new physics". Bulletin of the American Mathematical Society. 75 (5): 793–815. doi:10.1090/S0002-9904-1968-12047-6. MR 1566474.
74. Seitz, Frederick; Vogt, Erich; Weinberg, Alvin. "Eugene Paul Wigner 1902-1995: A biographical memoir" (PDF). National Academy of Sciences. National Academies Press. Retrieved 9 May 2023.
75. Johnson, Thomas (9 November 1975). "'Unification' Science Parley Is Defended". The New York Times. Retrieved 9 May 2023.
76. Szanton 1992, p. 305.
77. Szanton 1992, p. 318.
78. Wigner, Mehra & Wightman 1995, p. 14.
79. Staff writer (Jan. 2, 1989). "Western Goals Foundation." Interhemispheric Resource Center/International Relations Center. Archived from the original.
80. Broad, William J. (January 4, 1995). "Eugene Wigner, 92, Quantum Theorist Who Helped Usher In Atomic Age, Dies". The New York Times. Retrieved May 19, 2015.
References
• Hewlett, Richard G.; Duncan, Francis (1969). Atomic Shield, 1947–1952 (PDF). A History of the United States Atomic Energy Commission. University Park, Pennsylvania: Pennsylvania State University Press. ISBN 978-0-520-07187-2. OCLC 3717478. Retrieved 7 March 2015.
• Johnson, Leland; Schaffer, Daniel (1994). Oak Ridge National Laboratory: the first fifty years. Knoxville: University of Tennessee Press. ISBN 978-0-87049-853-4.
• Rhodes, Richard (1995). Dark Sun: The Making of the Hydrogen Bomb. New York: Simon & Schuster. ISBN 978-0-684-80400-2.
• N. Mukunda (1995) "Eugene Paul Wigner – A tribute", Current Science 69(4): 375–85 MR1347799
• Szanton, Andrew (1992). The Recollections of Eugene P. Wigner. Plenum. ISBN 978-0-306-44326-8.
• Wigner, E. P. (1931). Gruppentheorie und ihre Anwendung auf die Quanten mechanik der Atomspektren (in German). Braunschweig, Germany: Friedrich Vieweg und Sohn. ASIN B000K1MPEI.
• Wigner, E. P. (1959). Group Theory and its Application to the Quantum Mechanics of Atomic Spectra. translation from German by J. J. Griffin. New York: Academic Press. ISBN 978-0-12-750550-3.
• Wigner, E. P.; Weinberg, Alvin M. (1992). The collected works of Eugene Paul Wigner, Volume 5, Part A, Nuclear energy. Berlin: Springer. ISBN 978-0-387-55343-6.
• Wigner, Eugene Paul; Mehra, Jagdish; Wightman, A. S. (1995). Volume 7, Part B, Philosophical Reflections and Syntheses. Berlin: Springer. ISBN 978-3-540-63372-3.
External links
Wikimedia Commons has media related to Eugene Wigner.
Wikiquote has quotations related to Eugene Wigner.
• 1964 Audio Interview with Eugene Wigner by Stephane Groueff Voices of the Manhattan Project
• O'Connor, John J.; Robertson, Edmund F., "Eugene Wigner", MacTutor History of Mathematics Archive, University of St Andrews
• Eugene Wigner at the Mathematics Genealogy Project
• EPW contributions to the theory of the atomic nucleus and the elementary particles, particularly through the discovery and application of fundamental symmetry principles at the Wayback Machine (archived July 9, 2011)
• 1984 interview with Wigner, in: The Princeton University Mathematics Community in the 1930s. at the Wayback Machine (archived October 5, 2012)
• Oral history interview transcript with Eugene Wigner 21 November 1963, American Institute of Physics, Niels Bohr Library & Archives at the Wayback Machine (archived October 1, 2013)
• Archived March 26, 2015, at the Wayback Machine
• Wigner Jenö Iskolás Évei by Radnai Gyula, ELTE, Fizikai Szemle 2007/2 – 62.o. (Hungarian). Description of the childhood and especially of the school-years in Budapest, with some interesting photos too.
• Interview with Eugene P. Wigner on John von Neumann at the Charles Babbage Institute, University of Minnesota, Minneapolis – Wigner talks about his association with John von Neumann during their school years in Hungary, their graduate studies in Berlin, and their appointments to Princeton in 1930. Wigner discusses von Neumann's contributions to the theory of quantum mechanics, Wigner's own work in this area, and von Neumann's interest in the application of theory to the atomic bomb project.
• Works by or about Eugene Wigner at Internet Archive
• Eugene Wigner on Nobelprize.org including the Nobel Lecture, December 12, 1963 Events, Laws of Nature, and Invariance Principles
Laureates of the Nobel Prize in Physics
1901–1925
• 1901: Röntgen
• 1902: Lorentz / Zeeman
• 1903: Becquerel / P. Curie / M. Curie
• 1904: Rayleigh
• 1905: Lenard
• 1906: J. J. Thomson
• 1907: Michelson
• 1908: Lippmann
• 1909: Marconi / Braun
• 1910: Van der Waals
• 1911: Wien
• 1912: Dalén
• 1913: Kamerlingh Onnes
• 1914: Laue
• 1915: W. L. Bragg / W. H. Bragg
• 1916
• 1917: Barkla
• 1918: Planck
• 1919: Stark
• 1920: Guillaume
• 1921: Einstein
• 1922: N. Bohr
• 1923: Millikan
• 1924: M. Siegbahn
• 1925: Franck / Hertz
1926–1950
• 1926: Perrin
• 1927: Compton / C. Wilson
• 1928: O. Richardson
• 1929: De Broglie
• 1930: Raman
• 1931
• 1932: Heisenberg
• 1933: Schrödinger / Dirac
• 1934
• 1935: Chadwick
• 1936: Hess / C. D. Anderson
• 1937: Davisson / G. P. Thomson
• 1938: Fermi
• 1939: Lawrence
• 1940
• 1941
• 1942
• 1943: Stern
• 1944: Rabi
• 1945: Pauli
• 1946: Bridgman
• 1947: Appleton
• 1948: Blackett
• 1949: Yukawa
• 1950: Powell
1951–1975
• 1951: Cockcroft / Walton
• 1952: Bloch / Purcell
• 1953: Zernike
• 1954: Born / Bothe
• 1955: Lamb / Kusch
• 1956: Shockley / Bardeen / Brattain
• 1957: C. N. Yang / T. D. Lee
• 1958: Cherenkov / Frank / Tamm
• 1959: Segrè / Chamberlain
• 1960: Glaser
• 1961: Hofstadter / Mössbauer
• 1962: Landau
• 1963: Wigner / Goeppert Mayer / Jensen
• 1964: Townes / Basov / Prokhorov
• 1965: Tomonaga / Schwinger / Feynman
• 1966: Kastler
• 1967: Bethe
• 1968: Alvarez
• 1969: Gell-Mann
• 1970: Alfvén / Néel
• 1971: Gabor
• 1972: Bardeen / Cooper / Schrieffer
• 1973: Esaki / Giaever / Josephson
• 1974: Ryle / Hewish
• 1975: A. Bohr / Mottelson / Rainwater
1976–2000
• 1976: Richter / Ting
• 1977: P. W. Anderson / Mott / Van Vleck
• 1978: Kapitsa / Penzias / R. Wilson
• 1979: Glashow / Salam / Weinberg
• 1980: Cronin / Fitch
• 1981: Bloembergen / Schawlow / K. Siegbahn
• 1982: K. Wilson
• 1983: Chandrasekhar / Fowler
• 1984: Rubbia / Van der Meer
• 1985: von Klitzing
• 1986: Ruska / Binnig / Rohrer
• 1987: Bednorz / Müller
• 1988: Lederman / Schwartz / Steinberger
• 1989: Ramsey / Dehmelt / Paul
• 1990: Friedman / Kendall / R. Taylor
• 1991: de Gennes
• 1992: Charpak
• 1993: Hulse / J. Taylor
• 1994: Brockhouse / Shull
• 1995: Perl / Reines
• 1996: D. Lee / Osheroff / R. Richardson
• 1997: Chu / Cohen-Tannoudji / Phillips
• 1998: Laughlin / Störmer / Tsui
• 1999: 't Hooft / Veltman
• 2000: Alferov / Kroemer / Kilby
2001–
present
• 2001: Cornell / Ketterle / Wieman
• 2002: Davis / Koshiba / Giacconi
• 2003: Abrikosov / Ginzburg / Leggett
• 2004: Gross / Politzer / Wilczek
• 2005: Glauber / Hall / Hänsch
• 2006: Mather / Smoot
• 2007: Fert / Grünberg
• 2008: Nambu / Kobayashi / Maskawa
• 2009: Kao / Boyle / Smith
• 2010: Geim / Novoselov
• 2011: Perlmutter / Schmidt / Riess
• 2012: Wineland / Haroche
• 2013: Englert / Higgs
• 2014: Akasaki / Amano / Nakamura
• 2015: Kajita / McDonald
• 2016: Thouless / Haldane / Kosterlitz
• 2017: Weiss / Barish / Thorne
• 2018: Ashkin / Mourou / Strickland
• 2019: Peebles / Mayor / Queloz
• 2020: Penrose / Genzel / Ghez
• 2021: Parisi / Hasselmann / Manabe
• 2022: Aspect / Clauser / Zeilinger
1963 Nobel Prize laureates
Chemistry
• Karl Ziegler (Germany)
• Giulio Natta (Italy)
Literature (1963)
• Giorgos Seferis (Greece)
Peace
• International Committee of the Red Cross (Switzerland)
• International Federation of Red Cross and Red Crescent Societies (Switzerland)
Physics
• Eugene Wigner (United States)
• Maria Goeppert Mayer (United States)
• J. Hans D. Jensen (Germany)
Physiology or Medicine
• John Eccles (Great Britain)
• Alan Lloyd Hodgkin (Great Britain)
• Andrew Huxley (Australia)
Nobel Prize recipients
1958
1959
1960
1961
1962
1963
1964
1965
1966
1967
1968
Hungarian or Hungarian-American Nobel Laureates
Chemistry
• Zsigmondy Richárd (1925)
• Hevesy György (1943)
• Polányi János (1986)
• Oláh György (1994)
• Avram Hershko (2004)
Literature
• Kertész Imre (2002)
Physics
• Lénárd Fülöp (1905)
• Wigner Jenő (1963)
• Gábor Dénes (1971)
Physiology or Medicine
• Bárány Róbert (1914)
• Szent-Györgyi Albert (1937)
• Békésy György (1961)
Economic Sciences
• Harsányi János (1994)
United States National Medal of Science laureates
Behavioral and social science
1960s
1964
Neal Elgar Miller
1980s
1986
Herbert A. Simon
1987
Anne Anastasi
George J. Stigler
1988
Milton Friedman
1990s
1990
Leonid Hurwicz
Patrick Suppes
1991
George A. Miller
1992
Eleanor J. Gibson
1994
Robert K. Merton
1995
Roger N. Shepard
1996
Paul Samuelson
1997
William K. Estes
1998
William Julius Wilson
1999
Robert M. Solow
2000s
2000
Gary Becker
2003
R. Duncan Luce
2004
Kenneth Arrow
2005
Gordon H. Bower
2008
Michael I. Posner
2009
Mortimer Mishkin
2010s
2011
Anne Treisman
2014
Robert Axelrod
2015
Albert Bandura
Biological sciences
1960s
1963
C. B. van Niel
1964
Theodosius Dobzhansky
Marshall W. Nirenberg
1965
Francis P. Rous
George G. Simpson
Donald D. Van Slyke
1966
Edward F. Knipling
Fritz Albert Lipmann
William C. Rose
Sewall Wright
1967
Kenneth S. Cole
Harry F. Harlow
Michael Heidelberger
Alfred H. Sturtevant
1968
Horace Barker
Bernard B. Brodie
Detlev W. Bronk
Jay Lush
Burrhus Frederic Skinner
1969
Robert Huebner
Ernst Mayr
1970s
1970
Barbara McClintock
Albert B. Sabin
1973
Daniel I. Arnon
Earl W. Sutherland Jr.
1974
Britton Chance
Erwin Chargaff
James V. Neel
James Augustine Shannon
1975
Hallowell Davis
Paul Gyorgy
Sterling B. Hendricks
Orville Alvin Vogel
1976
Roger Guillemin
Keith Roberts Porter
Efraim Racker
E. O. Wilson
1979
Robert H. Burris
Elizabeth C. Crosby
Arthur Kornberg
Severo Ochoa
Earl Reece Stadtman
George Ledyard Stebbins
Paul Alfred Weiss
1980s
1981
Philip Handler
1982
Seymour Benzer
Glenn W. Burton
Mildred Cohn
1983
Howard L. Bachrach
Paul Berg
Wendell L. Roelofs
Berta Scharrer
1986
Stanley Cohen
Donald A. Henderson
Vernon B. Mountcastle
George Emil Palade
Joan A. Steitz
1987
Michael E. DeBakey
Theodor O. Diener
Harry Eagle
Har Gobind Khorana
Rita Levi-Montalcini
1988
Michael S. Brown
Stanley Norman Cohen
Joseph L. Goldstein
Maurice R. Hilleman
Eric R. Kandel
Rosalyn Sussman Yalow
1989
Katherine Esau
Viktor Hamburger
Philip Leder
Joshua Lederberg
Roger W. Sperry
Harland G. Wood
1990s
1990
Baruj Benacerraf
Herbert W. Boyer
Daniel E. Koshland Jr.
Edward B. Lewis
David G. Nathan
E. Donnall Thomas
1991
Mary Ellen Avery
G. Evelyn Hutchinson
Elvin A. Kabat
Robert W. Kates
Salvador Luria
Paul A. Marks
Folke K. Skoog
Paul C. Zamecnik
1992
Maxine Singer
Howard Martin Temin
1993
Daniel Nathans
Salome G. Waelsch
1994
Thomas Eisner
Elizabeth F. Neufeld
1995
Alexander Rich
1996
Ruth Patrick
1997
James Watson
Robert A. Weinberg
1998
Bruce Ames
Janet Rowley
1999
David Baltimore
Jared Diamond
Lynn Margulis
2000s
2000
Nancy C. Andreasen
Peter H. Raven
Carl Woese
2001
Francisco J. Ayala
George F. Bass
Mario R. Capecchi
Ann Graybiel
Gene E. Likens
Victor A. McKusick
Harold Varmus
2002
James E. Darnell
Evelyn M. Witkin
2003
J. Michael Bishop
Solomon H. Snyder
Charles Yanofsky
2004
Norman E. Borlaug
Phillip A. Sharp
Thomas E. Starzl
2005
Anthony Fauci
Torsten N. Wiesel
2006
Rita R. Colwell
Nina Fedoroff
Lubert Stryer
2007
Robert J. Lefkowitz
Bert W. O'Malley
2008
Francis S. Collins
Elaine Fuchs
J. Craig Venter
2009
Susan L. Lindquist
Stanley B. Prusiner
2010s
2010
Ralph L. Brinster
Rudolf Jaenisch
2011
Lucy Shapiro
Leroy Hood
Sallie Chisholm
2012
May Berenbaum
Bruce Alberts
2013
Rakesh K. Jain
2014
Stanley Falkow
Mary-Claire King
Simon Levin
Chemistry
1960s
1964
Roger Adams
1980s
1982
F. Albert Cotton
Gilbert Stork
1983
Roald Hoffmann
George C. Pimentel
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| Wikipedia |
Wigner's theorem
Wigner's theorem, proved by Eugene Wigner in 1931,[2] is a cornerstone of the mathematical formulation of quantum mechanics. The theorem specifies how physical symmetries such as rotations, translations, and CPT are represented on the Hilbert space of states.
The physical states in a quantum theory are represented by unit vectors in Hilbert space up to a phase factor, i.e. by the complex line or ray the vector spans. In addition, by the Born rule the absolute value of the unit vectors inner product, or equivalently the cosine squared of the angle between the lines the vectors span, corresponds to the transition probability. Ray space, in mathematics known as projective Hilbert space, is the space of all unit vectors in Hilbert space up to the equivalence relation of differing by a phase factor. By Wigner's theorem, any transformation of ray space that preserves the absolute value of the inner products can be represented by a unitary or antiunitary transformation of Hilbert space, which is unique up to a phase factor. As a consequence, the representation of a symmetry group on ray space can be lifted to a projective representation or sometimes even an ordinary representation on Hilbert space.
Rays and ray space
It is a postulate of quantum mechanics that vectors in Hilbert space that are scalar nonzero multiples of each other represent the same pure state. A ray belonging to the vector $\Psi \in H\setminus \{0\}$ is the complex line through the origin [3][4]
${\underline {\Psi }}=\left\{\lambda \Psi :\lambda \in \mathbb {C} \right\}$ :\lambda \in \mathbb {C} \right\}} .
Two nonzero vectors $\Psi _{1},\Psi _{2}$ define the same ray, if and only if they differ by some nonzero complex number: $\Psi _{1}=\lambda \Psi _{2}$, $\lambda \in \mathbb {C} ^{*}=\mathbb {C} \setminus \{0\}$. Alternatively, we can consider a ray ${\underline {\Psi }}$ as a set of vectors with norm 1 that span the same line, a unit ray, by intersecting the line ${\underline {\Psi }}$ with the unit sphere [5]
$SH=\{\Phi \in H\mid \|\Phi \|^{2}=1\}$.
Two unit vectors $\Psi _{1},\Psi _{2}$ then define the same unit ray ${\underline {\Psi _{1}}}={\underline {\Psi _{2}}}$ if they differ by a phase factor: $\Psi _{1}=e^{i\alpha }\Psi _{2}$. This is the more usual picture in physics. The set of rays is in one to one correspondence with the set of unit rays and we can identify them. There is also a one-to-one correspondence between physical pure states $\rho $ and (unit) rays ${\underline {\Phi }}$ given by
$\rho =P_{\Phi }={\frac {|\Phi \rangle \langle \Phi |}{\langle \Phi |\Phi \rangle }}$
where $P_{\Phi }$ is the orthogonal projection on the line ${\underline {\Phi }}$. In either interpretation, if $\Phi \in {\underline {\Psi }}$ or $P_{\Phi }=P_{\Psi }$ then $\Phi $ is a representative of ${\underline {\Psi }}$.[nb 1]
The space of all rays is called ray space. It can be defined in several ways. One may define an equivalence relation $\approx $ on $H\setminus \{0\}$ by
$\Psi \approx \Phi \Leftrightarrow \Psi =\lambda \Phi ,\quad \lambda \in \mathbb {C} \setminus \{0\},$ and define ray space as
$\mathbb {P} H=H\setminus \{0\}/{\approx }.$
Alternatively define a relation ≅ as an equivalence relation on the sphere $SH$. The unit ray space $\mathbb {P} H$, is an incarnation of ray space defined (making no notational distinction with ray space) as the set of equivalence classes
$\mathbb {P} H=SH/{\cong }.$
A third equivalent definition of ray space is as pure state ray space i.e. as density matrices that are orthogonal projections of rank 1
$\mathbb {P} H=\{P\in B(H)\mid P^{2}=P=P^{\dagger },\mathbb {tr} (P)=1\}.$
Each of these definitions make it clear that ray space is nothing but another name for projective Hilbert space.[6] If $\dim(H)=N$ is finite, $\mathbb {P} H$ has real dimension $2N-2$. In fact, $\mathbb {P} H$ is a compact complex manifold of dimension $N-1$ which (by choosing a basis) is isomorphic to the projective space $\mathbb {C} \mathbb {P} ^{N-1}=\mathbb {P} (\mathbb {C} ^{N})$. For example, the Bloch sphere
$\mathbb {P} (\lambda _{1}|+\rangle +\lambda _{2}|-\rangle ,\ (\lambda _{1},\lambda _{2})\in \mathbb {C} ^{2}\setminus \{0\})$
is isomorphic to the Riemann sphere $\mathbb {C} \mathbb {P} ^{1}$.
Ray space (i.e. projective space) takes a little getting used to, but is a very well studied object that predates quantum mechanics going back to the study of perspective by renaissance artists. It is not a vector space with well-defined linear combinations of rays. However, for every two vectors $\Psi _{1},\Psi _{2}$ and ratio of complex numbers $(\lambda _{1}:\lambda _{2})$ (i.e. element of $\mathbb {C} \mathbb {P} ^{1}$) there is a well defined ray ${\underline {\lambda _{1}\Psi _{1}+\lambda _{2}\Psi _{2}}}$. Moreover, for distinct rays ${\underline {\Psi }}_{1},{\underline {\Psi }}_{2}$ (i.e. linearly independent lines) there is a projective line of rays of the form ${\underline {\lambda _{1}\Psi _{1}+\lambda _{2}\Psi _{2}}}$ in $\mathbb {P} H$: all 1 dimensional complex lines in the 2 complex dimensional plane spanned by $\Psi _{1}$ and $\Psi _{2}$ in $H$).
The Hilbert space structure on $H$ defines additional structure on ray space. Define the ray correlation (or ray product)
${\underline {\Psi }}\cdot {\underline {\Phi }}={\frac {\left|\left\langle \Psi ,\Phi \right\rangle \right|}{\|\Phi \|\|\Psi \|}}={\sqrt {\mathrm {tr} (P_{\Psi }P_{\Phi })}},$
where $\langle \,,\,\rangle $ is the Hilbert space inner product, and $\Psi ,\Phi $ are representatives of ${\underline {\Phi }}$ and ${\underline {\Psi }}$. Note that the righthand side is independent of the choice of representatives. The physical significance of this definition is that according to the Born rule, another postulate of quantum mechanics, the transition probabilities between normalised states $\Psi $ and $\Phi $ in Hilbert space is given by
$P(\Psi \rightarrow \Phi )=|\langle \Psi ,\Phi \rangle |^{2}=\left({\underline {\Psi }}\cdot {\underline {\Phi }}\right)^{2}$
i.e. we can define Born's rule on ray space by.
$P({\underline {\Psi }}\to {\underline {\Phi }}):=\left({\underline {\Psi }}\cdot {\underline {\Phi }}\right)^{2}.$
Geometrically, we can define an angle $\theta $ with $0\leq \theta \leq \pi /2$ between the lines ${\underline {\Phi }}$ and ${\underline {\Psi }}$ by $\cos(\theta )=({\underline {\Psi }}\cdot {\underline {\Phi }})$. The angle then turns out to satisfy the triangle inequality and defines a metric structure on ray space which comes from a Riemannian metric, the Fubini-Study metric.
Symmetry transformations
Loosely speaking, a symmetry transformation is a change in which "nothing happens"[7] or a "change in our point of view"[8] that does not change the outcomes of possible experiments. For example, translating a system in a homogeneous environment should have no qualitative effect on the outcomes of experiments made on the system. Likewise for rotating a system in an isotropic environment. This becomes even clearer when one considers the mathematically equivalent passive transformations, i.e. simply changes of coordinates and let the system be. Usually, the domain and range Hilbert spaces are the same. An exception would be (in a non-relativistic theory) the Hilbert space of electron states that is subjected to a charge conjugation transformation. In this case the electron states are mapped to the Hilbert space of positron states and vice versa. However this means that the symmetry acts on the direct sum of the Hilbert spaces.
A transformation of a physical system is a transformation of states, hence mathematically a transformation, not of the Hilbert space, but of its ray space. Hence, in quantum mechanics, a transformation of a physical system gives rise to a bijective ray transformation $T$
${\begin{aligned}T:\mathbb {P} H&\to \mathbb {P} H\\{\underline {\Psi }}&\mapsto T{\underline {\Psi }}.\\\end{aligned}}$
Since the composition of two physical transformations and the reversal of a physical transformation are also physical transformations, the set of all ray transformations so obtained is a group acting on $\mathbb {P} H$. Not all bijections of $\mathbb {P} H$ are permissible as symmetry transformations, however. Physical transformations must preserve Born's rule.
For a physical transformation, the transition probabilities in the transformed and untransformed systems should be preserved:
$P({\underline {\Psi }}\rightarrow {\underline {\Phi }})=\left({\underline {\Psi }}\cdot {\underline {\Phi }}\right)^{2}=\left(T{\underline {\Psi }}\cdot T{\underline {\Phi }}\right)^{2}=P\left(T\Psi \rightarrow T\Phi \right)$
A bijective ray transformation $\mathbb {P} H\to \mathbb {P} H$ is called a symmetry transformation iff[9]
$T{\underline {\Psi }}\cdot T{\underline {\Phi }}={\underline {\Psi }}\cdot {\underline {\Phi }},\quad \forall {\underline {\Psi }},{\underline {\Phi }}\in \mathbb {P} H.$
A geometric interpretation, is that a symmetry transformation is an isometry of ray space.
Some facts about symmetry transformations that can be verified using the definition:
• The product of two symmetry transformations, i.e. two symmetry transformations applied in succession, is a symmetry transformation.
• Any symmetry transformation has an inverse.
• The identity transformation is a symmetry transformation.
• Multiplication of symmetry transformations is associative.
The set of symmetry transformations thus forms a group, the symmetry group of the system. Some important frequently occurring subgroups in the symmetry group of a system are realizations of
• The symmetric group with its subgroups. This is important on the exchange of particle labels.
• The Poincaré group. It encodes the fundamental symmetries of spacetime [NB: a symmetry is defined above as a map on the ray space describing a given system, the notion of symmetry of spacetime has not been defined and is not clear].
• Internal symmetry groups like SU(2) and SU(3). They describe so called internal symmetries, like isospin and color charge peculiar to quantum mechanical systems.
These groups are also referred to as symmetry groups of the system.
Statement of Wigner's theorem
Preliminaries
Some preliminary definitions are needed to state the theorem. A transformation $U:H\to K$ of Hilbert spaces is unitary if it is bijective and
$\langle U\Psi ,U\Phi \rangle =\langle \Psi ,\Phi \rangle .$
Since
$\langle U(\lambda _{1}\Psi _{1}+\lambda _{2}\Psi _{2}),\Phi '\rangle =\langle \lambda _{1}\Psi _{1}+\lambda _{2}\Psi _{2},U^{-1}\Phi '\rangle =\lambda _{1}\langle \Psi _{1},U^{-1}\Phi '\rangle +\lambda _{2}\langle \Psi _{2},U^{-1}\Phi '\rangle =$
$=\lambda _{1}\langle U\Psi _{1},\Phi '\rangle +\lambda _{2}\langle U\Psi _{2},\Phi '\rangle =\langle \lambda _{1}U\Psi _{1}+\lambda _{2}U\Psi _{2},\Phi '\rangle $
for all $\Phi '\in K$, a unitary transformation is automatically linear and $U^{\dagger }=U^{-1}$.
Likewise, a transformation $A:H\to K$ is antiunitary if it is bijective and
$\langle A\Psi ,A\Phi \rangle =\langle \Psi ,\Phi \rangle ^{*}=\langle \Phi ,\Psi \rangle .$
As above, an antiunitary transformation is necessarily antilinear.[nb 2] Both variants are real linear and additive.
Given a unitary transformation $U:H\to K$ of Hilbert spaces, define
${\begin{aligned}T_{U}:\mathbb {P} H&\to \mathbb {P} K\\{\underline {\Psi }}&\mapsto {\underline {U\Psi }}\\\end{aligned}}$
This is a symmetry transformation since
$T{\underline {\Psi }}\cdot T{\underline {\Phi }}={\frac {\left|\langle U\Psi ,U\Phi \rangle \right|}{\|U\Psi \|\|U\Phi \|}}={\frac {\left|\langle \Psi ,\Phi \rangle \right|}{\|\Psi \|\|\Phi \|}}={\underline {\Psi }}\cdot {\underline {\Phi }}.$
In the same way an antiunitary transformation of Hilbert space induces a symmetry transformation. One says that a transformation $U:H\to K$ of Hilbert spaces is compatible with the transformation $T:\mathbb {P} H\to \mathbb {P} K$ of ray spaces if $T=T_{U}$ or equivalently
$U\Psi \in T{\underline {\Psi }}$
for all $\Psi \in H\setminus \{0\}$.[10]
Transformations of Hilbert space induced by either a unitary linear transformation or an antiunitary antilinear operator are obviously compatible with the transformations or ray space they induce as described.
Statement
Wigner's theorem states a converse of the above:[11]
Wigner's theorem (1931) — If $H$ and $K$ are Hilbert spaces and if
$T:\mathbb {P} H\to \mathbb {P} K$
is a symmetry transformation, then there exists a unitary or antiunitary transformation $V:H\to K$ which is compatible with $T$. If $\dim(H)\geq 2$ , $V$ is either unitary or antiunitary. If $\dim(H)=1$ (and $\mathbb {P} H$ and $\mathbb {P} K$ consist of a single point), all unitary transformations $U:H\to K$ and all antiunitary transformations $A:H\to K$ are compatible with $T$. If $V_{1}$ and $V_{2}$ are both compatible with $T$ then $V_{1}=e^{i\alpha }V_{2}$ for some $\alpha \in \mathbb {R} $
Proofs can be found in Wigner (1931, 1959), Bargmann (1964) and Weinberg (2002). Antiunitary and antilinear transformations are less prominent in physics. They are all related to a reversal of the direction of the flow of time.[12]
Remark 1: The significance of the uniqueness part of the theorem is that it specifies the degree of uniqueness of the representation on $H$. For example, one might be tempted to believe that
$V\Psi =Ue^{i\alpha (\Psi )}\Psi ,\alpha (\Psi )\in \mathbb {R} ,\Psi \in H\quad ({\text{wrong unless }}\alpha (\Psi ){\text{ is const.}})$
would be admissible, with $\alpha (\Psi )\neq \alpha (\Phi )$ for $\langle \Psi ,\Phi \rangle =0$ but this is not the case according to the theorem.[nb 3][13] In fact such a $V$ would not be additive.
Remark 2: Whether $T$ must be represented by a unitary or antiunitary operator is determined by topology. If $\dim _{\mathbb {C} }(\mathbb {P} H)=\dim _{\mathbb {C} }(\mathbb {P} K)\geq 1$, the second cohomology $H^{2}(\mathbb {P} H)$ has a unique generator $c_{\mathbb {P} H}$ such that for a (equivalently for every) complex projective line $L\subset \mathbb {P} H$, one has $c_{\mathbb {P} H}\cap [L]=\deg _{L}(c_{\mathbb {P} H}|_{L})=1$. Since $T$ is a homeomorphism, $T^{*}c_{\mathbb {P} K}$ also generates $H^{2}(\mathbb {P} H)$ and so we have $T^{*}c_{\mathbb {P} K}=\pm c_{\mathbb {P} H}$. If $U:H\to K$ is unitary, then $T_{U}^{*}c_{\mathbb {P} K}=c_{\mathbb {P} H}$ while if $A:H\to K$ is anti linear then $T_{A}^{*}c_{\mathbb {P} K}=-c_{\mathbb {P} H}$.
Remark 3: Wigner's theorem is in close connection with the fundamental theorem of projective geometry[14]
Representations and projective representations
If G is a symmetry group (in this latter sense of being embedded as a subgroup of the symmetry group of the system acting on ray space), and if f, g, h ∈ G with fg = h, then
$T(f)T(g)=T(h),$
where the T are ray transformations. From the uniqueness part of Wigner's theorem, one has for the compatible representatives U,
$U(f)U(g)=\omega (f,g)U(fg)=e^{i\xi (f,g)}U(fg),$
where ω(f, g) is a phase factor.[nb 4]
The function ω is called a 2-cocycle or Schur multiplier. A map U:G → GL(V) satisfying the above relation for some vector space V is called a projective representation or a ray representation. If ω(f, g) = 1, then it is called a representation.
One should note that the terminology differs between mathematics and physics. In the linked article, term projective representation has a slightly different meaning, but the term as presented here enters as an ingredient and the mathematics per se is of course the same. If the realization of the symmetry group, g → T(g), is given in terms of action on the space of unit rays S = PH, then it is a projective representation G → PGL(H) in the mathematical sense, while its representative on Hilbert space is a projective representation G → GL(H) in the physical sense.
Applying the last relation (several times) to the product fgh and appealing to the known associativity of multiplication of operators on H, one finds
${\begin{aligned}\omega (f,g)\omega (fg,h)&=\omega (g,h)\omega (f,gh),\\\xi (f,g)+\xi (fg,h)&=\xi (g,h)+\xi (f,gh)\quad (\operatorname {mod} 2\pi ).\end{aligned}}$
They also satisfy
${\begin{aligned}\omega (g,e)&=\omega (e,g)=1,\\\xi (g,e)&=\xi (e,g)=0\quad (\operatorname {mod} 2\pi ),\\\omega \left(g,g^{-1}\right)&=\omega (g^{-1},g),\\\xi \left(g,g^{-1}\right)&=\xi (g^{-1},g).\\\end{aligned}}$
Upon redefinition of the phases,
$U(g)\mapsto {\hat {U}}(g)=\eta (g)U(g)=e^{i\zeta (g)}U(g),$
which is allowed by last theorem, one finds[15][16]
${\begin{aligned}{\hat {\omega }}(g,h)&=\omega (g,h)\eta (g)\eta (h)\eta (gh)^{-1},\\{\hat {\xi }}(g,h)&=\xi (g,h)+\zeta (g)+\zeta (h)-\zeta (gh)\quad (\operatorname {mod} 2\pi ),\end{aligned}}$
where the hatted quantities are defined by
${\hat {U}}(f){\hat {U}}(g)={\hat {\omega }}(f,g){\hat {U}}(fg)=e^{i{\hat {\xi }}(f,g)}{\hat {U}}(fg).$
Utility of phase freedom
The following rather technical theorems and many more can be found, with accessible proofs, in Bargmann (1954).
The freedom of choice of phases can be used to simplify the phase factors. For some groups the phase can be eliminated altogether.
Theorem — If G is semisimple and simply connected, then ω(g, h) = 1 is possible.[17]
In the case of the Lorentz group and its subgroup the rotation group SO(3), phases can, for projective representations, be chosen such that ω(g, h) = ± 1. For their respective universal covering groups, SL(2,C) and Spin(3), it is according to the theorem possible to have ω(g, h) = 1, i.e. they are proper representations.
The study of redefinition of phases involves group cohomology. Two functions related as the hatted and non-hatted versions of ω above are said to be cohomologous. They belong to the same second cohomology class, i.e. they are represented by the same element in H2(G), the second cohomology group of G. If an element of H2(G) contains the trivial function ω = 0, then it is said to be trivial.[16] The topic can be studied at the level of Lie algebras and Lie algebra cohomology as well.[18][19]
Assuming the projective representation g → T(g) is weakly continuous, two relevant theorems can be stated. An immediate consequence of (weak) continuity is that the identity component is represented by unitary operators.[nb 5]
Theorem: (Wigner 1939) — The phase freedom can be used such that in a some neighborhood of the identity the map g → U(g) is strongly continuous.[20]
Theorem (Bargmann) — In a sufficiently small neighborhood of e, the choice ω(g1, g2) ≡ 1 is possible for semisimple Lie groups (such as SO(n), SO(3,1) and affine linear groups, (in particular the Poincaré group). More precisely, this is exactly the case when the second cohomology group H2(g, R) of the Lie algebra g of G is trivial.[20]
See also
• Particle physics and representation theory
Remarks
1. Here the possibility of superselection rules is ignored. It may be the case that a system cannot be prepared in specific states. For instance, superposition of states with different spin is generally believed impossible. Likewise, states being superpositions of states with different charge are considered impossible. Minor complications due to those issues are treated in Bogoliubov, Logunov & Todorov (1975)
2. Bäurle & de Kerf (1999, p. 342) This is stated but not proved.
3. There is an exception to this. If a superselection rule is in effect, then the phase may depend on in which sector of $H$ the element $\Psi $ resides, see Weinberg 2002, p. 53
4. Again there is an exception. If a superselection rule is in effect, then the phase may depend on in which sector of H h resides on which the operators act, see Weinberg 2002, p. 53
5. This is made plausible as follows. In an open neighborhood in the vicinity of the identity all operators can be expressed as squares. Whether an operator is unitary or antiunitary its square is unitary. Hence they are all unitary in a sufficiently small neighborhood. Such a neighborhood generates the identity.
Notes
1. Seitz, Vogt & Weinberg 2000
2. Wigner 1931, pp. 251–254 (in German),
Wigner 1959, pp. 233–236 (English translation).
3. Weinberg 2002, p. 49
4. Bäuerle & de Kerf 1999, p. 341 harvnb error: no target: CITEREFBäuerlede_Kerf1999 (help)
5. Simon et al. 2008
6. This approach is used in Bargmann 1964, which serves as a basis reference for the proof outline to be given below.
7. de Kerf & Bäuerle 1999 harvnb error: no target: CITEREFde_KerfBäuerle1999 (help)
8. Weinberg 2002, p. 50
9. de Kerf & Van Groesen 1999, p. 342 harvnb error: no target: CITEREFde_KerfVan_Groesen1999 (help)
10. Bargmann 1964
11. de Kerf & Van Groesen 1999, p. 343 harvnb error: no target: CITEREFde_KerfVan_Groesen1999 (help)
12. Weinberg 2002, p. 51
13. de Kerf & Van Groesen 1999, p. 344 harvnb error: no target: CITEREFde_KerfVan_Groesen1999 (help) This is stated but not proved.
14. Faure 2002
15. de Kerf & Van Groesen 1999, p. 346 harvnb error: no target: CITEREFde_KerfVan_Groesen1999 (help) There is an error in this formula in the book.
16. Weinberg 2002, p. 82
17. Weinberg 2002, Appendix B, Chapter 2
18. Bäurle & de Kerf 1999, pp. 347–349
19. Weinberg 2002, Section 2.7.
20. Straumann 2014
References
• Bargmann, V. (1954). "On unitary ray representations of continuous groups". Ann. of Math. 59 (1): 1–46. doi:10.2307/1969831. JSTOR 1969831.
• Bargmann, V. (1964). "Note on Wigner's Theorem on Symmetry Operations". Journal of Mathematical Physics. 5 (7): 862–868. Bibcode:1964JMP.....5..862B. doi:10.1063/1.1704188.
• Bogoliubov, N. N.; Logunov, A.A.; Todorov, I. T. (1975). Introduction to axiomatic quantum field theory. Mathematical Physics Monograph Series. Vol. 18. Translated to English by Stephan A. Fulling and Ludmila G. Popova. New York: Benjamin. ASIN B000IM4HLS.
• Bäurle, C. G. A.; de Kerf, E.A. (1999). E.A. Van Groesen; E. M. De Jager (eds.). Lie algebras Part 1:Finite and infinite dimensional Lie algebras and their applications in physics. Studies in mathematical physics. Vol. 1 (2nd ed.). Amsterdam: North-Holland. ISBN 0-444-88776-8.
• Faure, Claude-Alain (2002). "An Elementary Proof of the Fundamental Theorem of Projective Geometry". Geometriae Dedicata. 90: 145–151. doi:10.1023/A:1014933313332. S2CID 115770315.
• Seitz, F.; Vogt, E.; Weinberg, A. M. (2000). "Eugene Paul Wigner. 17 November 1902 -- 1 January 1995". Biogr. Mem. Fellows R. Soc. 46: 577–592. doi:10.1098/rsbm.1999.0102.
• Simon, R.; Mukunda, N.; Chaturvedi, S.; Srinivasan, V.; Hamhalter, J. (2008). "Two elementary proofs of the Wigner theorem on symmetry in quantum mechanics". Phys. Lett. A. 372 (46): 6847–6852. arXiv:0808.0779. Bibcode:2008PhLA..372.6847S. doi:10.1016/j.physleta.2008.09.052. S2CID 53858196.
• Straumann, N. (2014). "Unitary Representations of the inhomogeneous Lorentz Group and their Significance in Quantum Physics". In A. Ashtekar; V. Petkov (eds.). Springer Handbook of Spacetime. Springer Handbooks. pp. 265–278. arXiv:0809.4942. Bibcode:2014shst.book..265S. CiteSeerX 10.1.1.312.401. doi:10.1007/978-3-642-41992-8_14. ISBN 978-3-642-41991-1. S2CID 18493194.
• Weinberg, S. (2002), The Quantum Theory of Fields, vol. I, Cambridge University Press, ISBN 978-0-521-55001-7
• Wigner, E. P. (1931). Gruppentheorie und ihre Anwendung auf die Quanten mechanik der Atomspektren (in German). Braunschweig, Germany: Friedrich Vieweg und Sohn. pp. 251–254. ASIN B000K1MPEI.
• Wigner, E. P. (1959). Group Theory and its Application to the Quantum Mechanics of Atomic Spectra. translation from German by J. J. Griffin. New York: Academic Press. pp. 233–236. ISBN 978-0-1275-0550-3.
Further reading
• Mouchet, Amaury (2013). "An alternative proof of Wigner theorem on quantum transformations based on elementary complex analysis". Physics Letters A. 377 (39): 2709–2711. arXiv:1304.1376. Bibcode:2013PhLA..377.2709M. doi:10.1016/j.physleta.2013.08.017. S2CID 42994708.
• Molnar, Lajos (1999). "An Algebraic Approach to Wigner's Unitary-Antiunitary Theorem" (PDF). J. Austral. Math. Soc. Ser. A. 65 (3): 354–369. arXiv:math/9808033. Bibcode:1998math......8033M. doi:10.1017/s144678870003593x. S2CID 119593689.
| Wikipedia |
Group contraction
In theoretical physics, Eugene Wigner and Erdal İnönü have discussed[1] the possibility to obtain from a given Lie group a different (non-isomorphic) Lie group by a group contraction with respect to a continuous subgroup of it. That amounts to a limiting operation on a parameter of the Lie algebra, altering the structure constants of this Lie algebra in a nontrivial singular manner, under suitable circumstances.[2][3]
For example, the Lie algebra of the 3D rotation group SO(3), [X1, X2] = X3, etc., may be rewritten by a change of variables Y1 = εX1, Y2 = εX2, Y3 = X3, as
[Y1, Y2] = ε2 Y3, [Y2, Y3] = Y1, [Y3, Y1] = Y2.
The contraction limit ε → 0 trivializes the first commutator and thus yields the non-isomorphic algebra of the plane Euclidean group, E2 ~ ISO(2). (This is isomorphic to the cylindrical group, describing motions of a point on the surface of a cylinder. It is the little group, or stabilizer subgroup, of null four-vectors in Minkowski space.) Specifically, the translation generators Y1, Y2, now generate the Abelian normal subgroup of E2 (cf. Group extension), the parabolic Lorentz transformations.
Similar limits, of considerable application in physics (cf. correspondence principles), contract
• the de Sitter group SO(4, 1) ~ Sp(2, 2) to the Poincaré group ISO(3, 1), as the de Sitter radius diverges: R → ∞; or
• the super-anti-de Sitter algebra to the super-Poincaré algebra as the AdS radius diverges R → ∞; or
• the Poincaré group to the Galilei group, as the speed of light diverges: c → ∞;[4] or
• the Moyal bracket Lie algebra (equivalent to quantum commutators) to the Poisson bracket Lie algebra, in the classical limit as the Planck constant vanishes: ħ → 0.
Notes
1. Inönü & Wigner 1953
2. Segal 1951, p. 221
3. Saletan 1961, p. 1
4. Gilmore 2006
References
• Dooley, A. H.; Rice, J. W. (1985). "On contractions of semisimple Lie groups" (PDF). Transactions of the American Mathematical Society. 289 (1): 185–202. doi:10.2307/1999695. ISSN 0002-9947. JSTOR 1999695. MR 0779059.
• Gilmore, Robert (2006). Lie Groups, Lie Algebras, and Some of Their Applications. Dover Books on Mathematics. Dover Publications. ISBN 0486445291. MR 1275599.
• Inönü, E.; Wigner, E. P. (1953). "On the Contraction of Groups and Their Representations". Proc. Natl. Acad. Sci. 39 (6): 510–24. Bibcode:1953PNAS...39..510I. doi:10.1073/pnas.39.6.510. PMC 1063815. PMID 16589298.
• Saletan, E. J. (1961). "Contraction of Lie Groups". Journal of Mathematical Physics. 2 (1): 1–21. Bibcode:1961JMP.....2....1S. doi:10.1063/1.1724208.
• Segal, I. E. (1951). "A class of operator algebras which are determined by groups". Duke Mathematical Journal. 18: 221. doi:10.1215/S0012-7094-51-01817-0.
| Wikipedia |
Wigner D-matrix
The Wigner D-matrix is a unitary matrix in an irreducible representation of the groups SU(2) and SO(3). It was introduced in 1927 by Eugene Wigner, and plays a fundamental role in the quantum mechanical theory of angular momentum. The complex conjugate of the D-matrix is an eigenfunction of the Hamiltonian of spherical and symmetric rigid rotors. The letter D stands for Darstellung, which means "representation" in German.
Definition of the Wigner D-matrix
Let Jx, Jy, Jz be generators of the Lie algebra of SU(2) and SO(3). In quantum mechanics, these three operators are the components of a vector operator known as angular momentum. Examples are the angular momentum of an electron in an atom, electronic spin, and the angular momentum of a rigid rotor.
In all cases, the three operators satisfy the following commutation relations,
$[J_{x},J_{y}]=iJ_{z},\quad [J_{z},J_{x}]=iJ_{y},\quad [J_{y},J_{z}]=iJ_{x},$
where i is the purely imaginary number and Planck's constant ħ has been set equal to one. The Casimir operator
$J^{2}=J_{x}^{2}+J_{y}^{2}+J_{z}^{2}$
commutes with all generators of the Lie algebra. Hence, it may be diagonalized together with Jz.
This defines the spherical basis used here. That is, there is a complete set of kets (i.e. orthonormal basis of joint eigenvectors labelled by quantum numbers that define the eigenvalues) with
$J^{2}|jm\rangle =j(j+1)|jm\rangle ,\quad J_{z}|jm\rangle =m|jm\rangle ,$
where j = 0, 1/2, 1, 3/2, 2, ... for SU(2), and j = 0, 1, 2, ... for SO(3). In both cases, m = −j, −j + 1, ..., j.
A 3-dimensional rotation operator can be written as
${\mathcal {R}}(\alpha ,\beta ,\gamma )=e^{-i\alpha J_{z}}e^{-i\beta J_{y}}e^{-i\gamma J_{z}},$
where α, β, γ are Euler angles (characterized by the keywords: z-y-z convention, right-handed frame, right-hand screw rule, active interpretation).
The Wigner D-matrix is a unitary square matrix of dimension 2j + 1 in this spherical basis with elements
$D_{m'm}^{j}(\alpha ,\beta ,\gamma )\equiv \langle jm'|{\mathcal {R}}(\alpha ,\beta ,\gamma )|jm\rangle =e^{-im'\alpha }d_{m'm}^{j}(\beta )e^{-im\gamma },$
where
$d_{m'm}^{j}(\beta )=\langle jm'|e^{-i\beta J_{y}}|jm\rangle =D_{m'm}^{j}(0,\beta ,0)$
is an element of the orthogonal Wigner's (small) d-matrix.
That is, in this basis,
$D_{m'm}^{j}(\alpha ,0,0)=e^{-im'\alpha }\delta _{m'm}$
is diagonal, like the γ matrix factor, but unlike the above β factor.
Wigner (small) d-matrix
Wigner gave the following expression:[1]
$d_{m'm}^{j}(\beta )=[(j+m')!(j-m')!(j+m)!(j-m)!]^{\frac {1}{2}}\sum _{s=s_{\mathrm {min} }}^{s_{\mathrm {max} }}\left[{\frac {(-1)^{m'-m+s}\left(\cos {\frac {\beta }{2}}\right)^{2j+m-m'-2s}\left(\sin {\frac {\beta }{2}}\right)^{m'-m+2s}}{(j+m-s)!s!(m'-m+s)!(j-m'-s)!}}\right].$
The sum over s is over such values that the factorials are nonnegative, i.e. $s_{\mathrm {min} }=\mathrm {max} (0,m-m')$, $s_{\mathrm {max} }=\mathrm {min} (j+m,j-m')$.
Note: The d-matrix elements defined here are real. In the often-used z-x-z convention of Euler angles, the factor $(-1)^{m'-m+s}$ in this formula is replaced by $(-1)^{s}i^{m-m'},$ causing half of the functions to be purely imaginary. The realness of the d-matrix elements is one of the reasons that the z-y-z convention, used in this article, is usually preferred in quantum mechanical applications.
The d-matrix elements are related to Jacobi polynomials $P_{k}^{(a,b)}(\cos \beta )$ with nonnegative $a$ and $b.$[2] Let
$k=\min(j+m,j-m,j+m',j-m').$
If
$k={\begin{cases}j+m:&a=m'-m;\quad \lambda =m'-m\\j-m:&a=m-m';\quad \lambda =0\\j+m':&a=m-m';\quad \lambda =0\\j-m':&a=m'-m;\quad \lambda =m'-m\\\end{cases}}$
Then, with $b=2j-2k-a,$ the relation is
$d_{m'm}^{j}(\beta )=(-1)^{\lambda }{\binom {2j-k}{k+a}}^{\frac {1}{2}}{\binom {k+b}{b}}^{-{\frac {1}{2}}}\left(\sin {\frac {\beta }{2}}\right)^{a}\left(\cos {\frac {\beta }{2}}\right)^{b}P_{k}^{(a,b)}(\cos \beta ),$
where $a,b\geq 0.$
Properties of the Wigner D-matrix
The complex conjugate of the D-matrix satisfies a number of differential properties that can be formulated concisely by introducing the following operators with $(x,y,z)=(1,2,3),$
${\begin{aligned}{\hat {\mathcal {J}}}_{1}&=i\left(\cos \alpha \cot \beta {\frac {\partial }{\partial \alpha }}+\sin \alpha {\partial \over \partial \beta }-{\cos \alpha \over \sin \beta }{\partial \over \partial \gamma }\right)\\{\hat {\mathcal {J}}}_{2}&=i\left(\sin \alpha \cot \beta {\partial \over \partial \alpha }-\cos \alpha {\partial \over \partial \beta }-{\sin \alpha \over \sin \beta }{\partial \over \partial \gamma }\right)\\{\hat {\mathcal {J}}}_{3}&=-i{\partial \over \partial \alpha }\end{aligned}}$
which have quantum mechanical meaning: they are space-fixed rigid rotor angular momentum operators.
Further,
${\begin{aligned}{\hat {\mathcal {P}}}_{1}&=i\left({\cos \gamma \over \sin \beta }{\partial \over \partial \alpha }-\sin \gamma {\partial \over \partial \beta }-\cot \beta \cos \gamma {\partial \over \partial \gamma }\right)\\{\hat {\mathcal {P}}}_{2}&=i\left(-{\sin \gamma \over \sin \beta }{\partial \over \partial \alpha }-\cos \gamma {\partial \over \partial \beta }+\cot \beta \sin \gamma {\partial \over \partial \gamma }\right)\\{\hat {\mathcal {P}}}_{3}&=-i{\partial \over \partial \gamma },\\\end{aligned}}$
which have quantum mechanical meaning: they are body-fixed rigid rotor angular momentum operators.
The operators satisfy the commutation relations
$\left[{\mathcal {J}}_{1},{\mathcal {J}}_{2}\right]=i{\mathcal {J}}_{3},\qquad {\hbox{and}}\qquad \left[{\mathcal {P}}_{1},{\mathcal {P}}_{2}\right]=-i{\mathcal {P}}_{3},$
and the corresponding relations with the indices permuted cyclically. The ${\mathcal {P}}_{i}$ satisfy anomalous commutation relations (have a minus sign on the right hand side).
The two sets mutually commute,
$\left[{\mathcal {P}}_{i},{\mathcal {J}}_{j}\right]=0,\quad i,j=1,2,3,$
and the total operators squared are equal,
${\mathcal {J}}^{2}\equiv {\mathcal {J}}_{1}^{2}+{\mathcal {J}}_{2}^{2}+{\mathcal {J}}_{3}^{2}={\mathcal {P}}^{2}\equiv {\mathcal {P}}_{1}^{2}+{\mathcal {P}}_{2}^{2}+{\mathcal {P}}_{3}^{2}.$
Their explicit form is,
${\mathcal {J}}^{2}={\mathcal {P}}^{2}=-{\frac {1}{\sin ^{2}\beta }}\left({\frac {\partial ^{2}}{\partial \alpha ^{2}}}+{\frac {\partial ^{2}}{\partial \gamma ^{2}}}-2\cos \beta {\frac {\partial ^{2}}{\partial \alpha \partial \gamma }}\right)-{\frac {\partial ^{2}}{\partial \beta ^{2}}}-\cot \beta {\frac {\partial }{\partial \beta }}.$
The operators ${\mathcal {J}}_{i}$ act on the first (row) index of the D-matrix,
${\begin{aligned}{\mathcal {J}}_{3}D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*}&=m'D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*}\\({\mathcal {J}}_{1}\pm i{\mathcal {J}}_{2})D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*}&={\sqrt {j(j+1)-m'(m'\pm 1)}}D_{m'\pm 1,m}^{j}(\alpha ,\beta ,\gamma )^{*}\end{aligned}}$
The operators ${\mathcal {P}}_{i}$ act on the second (column) index of the D-matrix,
${\mathcal {P}}_{3}D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*}=mD_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*},$
and, because of the anomalous commutation relation the raising/lowering operators are defined with reversed signs,
$({\mathcal {P}}_{1}\mp i{\mathcal {P}}_{2})D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*}={\sqrt {j(j+1)-m(m\pm 1)}}D_{m',m\pm 1}^{j}(\alpha ,\beta ,\gamma )^{*}.$
Finally,
${\mathcal {J}}^{2}D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*}={\mathcal {P}}^{2}D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*}=j(j+1)D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*}.$
In other words, the rows and columns of the (complex conjugate) Wigner D-matrix span irreducible representations of the isomorphic Lie algebras generated by $\{{\mathcal {J}}_{i}\}$ and $\{-{\mathcal {P}}_{i}\}$.
An important property of the Wigner D-matrix follows from the commutation of ${\mathcal {R}}(\alpha ,\beta ,\gamma )$ with the time reversal operator T,
$\langle jm'|{\mathcal {R}}(\alpha ,\beta ,\gamma )|jm\rangle =\langle jm'|T^{\dagger }{\mathcal {R}}(\alpha ,\beta ,\gamma )T|jm\rangle =(-1)^{m'-m}\langle j,-m'|{\mathcal {R}}(\alpha ,\beta ,\gamma )|j,-m\rangle ^{*},$
or
$D_{m'm}^{j}(\alpha ,\beta ,\gamma )=(-1)^{m'-m}D_{-m',-m}^{j}(\alpha ,\beta ,\gamma )^{*}.$
Here, we used that $T$ is anti-unitary (hence the complex conjugation after moving $T^{\dagger }$ from ket to bra), $T|jm\rangle =(-1)^{j-m}|j,-m\rangle $ and $(-1)^{2j-m'-m}=(-1)^{m'-m}$.
A further symmetry implies
$(-1)^{m'-m}D_{mm'}^{j}(\alpha ,\beta ,\gamma )=D_{m'm}^{j}(\gamma ,\beta ,\alpha )~.$
Orthogonality relations
The Wigner D-matrix elements $D_{mk}^{j}(\alpha ,\beta ,\gamma )$ form a set of orthogonal functions of the Euler angles $\alpha ,\beta ,$ and $\gamma $:
$\int _{0}^{2\pi }d\alpha \int _{0}^{\pi }d\beta \sin \beta \int _{0}^{2\pi }d\gamma \,\,D_{m'k'}^{j'}(\alpha ,\beta ,\gamma )^{\ast }D_{mk}^{j}(\alpha ,\beta ,\gamma )={\frac {8\pi ^{2}}{2j+1}}\delta _{m'm}\delta _{k'k}\delta _{j'j}.$
This is a special case of the Schur orthogonality relations.
Crucially, by the Peter–Weyl theorem, they further form a complete set.
The fact that $D_{mk}^{j}(\alpha ,\beta ,\gamma )$ are matrix elements of a unitary transformation from one spherical basis $|lm\rangle $ to another ${\mathcal {R}}(\alpha ,\beta ,\gamma )|lm\rangle $ is represented by the relations:[3]
$\sum _{k}D_{m'k}^{j}(\alpha ,\beta ,\gamma )^{*}D_{mk}^{j}(\alpha ,\beta ,\gamma )=\delta _{m,m'},$
$\sum _{k}D_{km'}^{j}(\alpha ,\beta ,\gamma )^{*}D_{km}^{j}(\alpha ,\beta ,\gamma )=\delta _{m,m'}.$
The group characters for SU(2) only depend on the rotation angle β, being class functions, so, then, independent of the axes of rotation,
$\chi ^{j}(\beta )\equiv \sum _{m}D_{mm}^{j}(\beta )=\sum _{m}d_{mm}^{j}(\beta )={\frac {\sin \left({\frac {(2j+1)\beta }{2}}\right)}{\sin \left({\frac {\beta }{2}}\right)}},$
and consequently satisfy simpler orthogonality relations, through the Haar measure of the group,[4]
${\frac {1}{\pi }}\int _{0}^{2\pi }d\beta \sin ^{2}\left({\frac {\beta }{2}}\right)\chi ^{j}(\beta )\chi ^{j'}(\beta )=\delta _{j'j}.$
The completeness relation (worked out in the same reference, (3.95)) is
$\sum _{j}\chi ^{j}(\beta )\chi ^{j}(\beta ')=\delta (\beta -\beta '),$
whence, for $\beta '=0,$
$\sum _{j}\chi ^{j}(\beta )(2j+1)=\delta (\beta ).$
Kronecker product of Wigner D-matrices, Clebsch-Gordan series
The set of Kronecker product matrices
$\mathbf {D} ^{j}(\alpha ,\beta ,\gamma )\otimes \mathbf {D} ^{j'}(\alpha ,\beta ,\gamma )$
forms a reducible matrix representation of the groups SO(3) and SU(2). Reduction into irreducible components is by the following equation:[3]
$D_{mk}^{j}(\alpha ,\beta ,\gamma )D_{m'k'}^{j'}(\alpha ,\beta ,\gamma )=\sum _{J=|j-j'|}^{j+j'}\langle jmj'm'|J\left(m+m'\right)\rangle \langle jkj'k'|J\left(k+k'\right)\rangle D_{\left(m+m'\right)\left(k+k'\right)}^{J}(\alpha ,\beta ,\gamma )$
The symbol $\langle j_{1}m_{1}j_{2}m_{2}|j_{3}m_{3}\rangle $ is a Clebsch–Gordan coefficient.
Relation to spherical harmonics and Legendre polynomials
For integer values of $l$, the D-matrix elements with second index equal to zero are proportional to spherical harmonics and associated Legendre polynomials, normalized to unity and with Condon and Shortley phase convention:
$D_{m0}^{\ell }(\alpha ,\beta ,\gamma )={\sqrt {\frac {4\pi }{2\ell +1}}}Y_{\ell }^{m*}(\beta ,\alpha )={\sqrt {\frac {(\ell -m)!}{(\ell +m)!}}}\,P_{\ell }^{m}(\cos {\beta })\,e^{-im\alpha }.$
This implies the following relationship for the d-matrix:
$d_{m0}^{\ell }(\beta )={\sqrt {\frac {(\ell -m)!}{(\ell +m)!}}}\,P_{\ell }^{m}(\cos {\beta }).$
A rotation of spherical harmonics $\langle \theta ,\phi |\ell m'\rangle $ then is effectively a composition of two rotations,
$\sum _{m'=-\ell }^{\ell }Y_{\ell }^{m'}(\theta ,\phi )~D_{m'~m}^{\ell }(\alpha ,\beta ,\gamma ).$
When both indices are set to zero, the Wigner D-matrix elements are given by ordinary Legendre polynomials:
$D_{0,0}^{\ell }(\alpha ,\beta ,\gamma )=d_{0,0}^{\ell }(\beta )=P_{\ell }(\cos \beta ).$
In the present convention of Euler angles, $\alpha $ is a longitudinal angle and $\beta $ is a colatitudinal angle (spherical polar angles in the physical definition of such angles). This is one of the reasons that the z-y-z convention is used frequently in molecular physics. From the time-reversal property of the Wigner D-matrix follows immediately
$\left(Y_{\ell }^{m}\right)^{*}=(-1)^{m}Y_{\ell }^{-m}.$
There exists a more general relationship to the spin-weighted spherical harmonics:
$D_{ms}^{\ell }(\alpha ,\beta ,-\gamma )=(-1)^{s}{\sqrt {\frac {4\pi }{2{\ell }+1}}}{}_{s}Y_{\ell }^{m}(\beta ,\alpha )e^{is\gamma }.$[5]
Connection with transition probability under rotations
The absolute square of an element of the D-matrix,
$F_{mm'}(\beta )=|D_{mm'}^{j}(\alpha ,\beta ,\gamma )|^{2},$
gives the probability that a system with spin $j$ prepared in a state with spin projection $m$ along some direction will be measured to have a spin projection $m'$ along a second direction at an angle $\beta $ to the first direction. The set of quantities $F_{mm'}$ itself forms a real symmetric matrix, that depends only on the Euler angle $\beta $, as indicated.
Remarkably, the eigenvalue problem for the $F$ matrix can be solved completely:[6][7]
$\sum _{m'=-j}^{j}F_{mm'}(\beta )f_{\ell }^{j}(m')=P_{\ell }(\cos \beta )f_{\ell }^{j}(m)\qquad (\ell =0,1,\ldots ,2j).$
Here, the eigenvector, $f_{\ell }^{j}(m)$, is a scaled and shifted discrete Chebyshev polynomial, and the corresponding eigenvalue, $P_{\ell }(\cos \beta )$, is the Legendre polynomial.
Relation to Bessel functions
In the limit when $\ell \gg m,m^{\prime }$ we have
$D_{mm'}^{\ell }(\alpha ,\beta ,\gamma )\approx e^{-im\alpha -im'\gamma }J_{m-m'}(\ell \beta )$
where $J_{m-m'}(\ell \beta )$ is the Bessel function and $\ell \beta $ is finite.
List of d-matrix elements
Using sign convention of Wigner, et al. the d-matrix elements $d_{m'm}^{j}(\theta )$ for j = 1/2, 1, 3/2, and 2 are given below.
for j = 1/2
${\begin{aligned}d_{{\frac {1}{2}},{\frac {1}{2}}}^{\frac {1}{2}}&=\cos {\frac {\theta }{2}}\\[6pt]d_{{\frac {1}{2}},-{\frac {1}{2}}}^{\frac {1}{2}}&=-\sin {\frac {\theta }{2}}\end{aligned}}$
for j = 1
${\begin{aligned}d_{1,1}^{1}&={\frac {1}{2}}(1+\cos \theta )\\[6pt]d_{1,0}^{1}&=-{\frac {1}{\sqrt {2}}}\sin \theta \\[6pt]d_{1,-1}^{1}&={\frac {1}{2}}(1-\cos \theta )\\[6pt]d_{0,0}^{1}&=\cos \theta \end{aligned}}$
for j = 3/2
${\begin{aligned}d_{{\frac {3}{2}},{\frac {3}{2}}}^{\frac {3}{2}}&={\frac {1}{2}}(1+\cos \theta )\cos {\frac {\theta }{2}}\\[6pt]d_{{\frac {3}{2}},{\frac {1}{2}}}^{\frac {3}{2}}&=-{\frac {\sqrt {3}}{2}}(1+\cos \theta )\sin {\frac {\theta }{2}}\\[6pt]d_{{\frac {3}{2}},-{\frac {1}{2}}}^{\frac {3}{2}}&={\frac {\sqrt {3}}{2}}(1-\cos \theta )\cos {\frac {\theta }{2}}\\[6pt]d_{{\frac {3}{2}},-{\frac {3}{2}}}^{\frac {3}{2}}&=-{\frac {1}{2}}(1-\cos \theta )\sin {\frac {\theta }{2}}\\[6pt]d_{{\frac {1}{2}},{\frac {1}{2}}}^{\frac {3}{2}}&={\frac {1}{2}}(3\cos \theta -1)\cos {\frac {\theta }{2}}\\[6pt]d_{{\frac {1}{2}},-{\frac {1}{2}}}^{\frac {3}{2}}&=-{\frac {1}{2}}(3\cos \theta +1)\sin {\frac {\theta }{2}}\end{aligned}}$
for j = 2[8]
${\begin{aligned}d_{2,2}^{2}&={\frac {1}{4}}\left(1+\cos \theta \right)^{2}\\[6pt]d_{2,1}^{2}&=-{\frac {1}{2}}\sin \theta \left(1+\cos \theta \right)\\[6pt]d_{2,0}^{2}&={\sqrt {\frac {3}{8}}}\sin ^{2}\theta \\[6pt]d_{2,-1}^{2}&=-{\frac {1}{2}}\sin \theta \left(1-\cos \theta \right)\\[6pt]d_{2,-2}^{2}&={\frac {1}{4}}\left(1-\cos \theta \right)^{2}\\[6pt]d_{1,1}^{2}&={\frac {1}{2}}\left(2\cos ^{2}\theta +\cos \theta -1\right)\\[6pt]d_{1,0}^{2}&=-{\sqrt {\frac {3}{8}}}\sin 2\theta \\[6pt]d_{1,-1}^{2}&={\frac {1}{2}}\left(-2\cos ^{2}\theta +\cos \theta +1\right)\\[6pt]d_{0,0}^{2}&={\frac {1}{2}}\left(3\cos ^{2}\theta -1\right)\end{aligned}}$
Wigner d-matrix elements with swapped lower indices are found with the relation:
$d_{m',m}^{j}=(-1)^{m-m'}d_{m,m'}^{j}=d_{-m,-m'}^{j}.$
Symmetries and special cases
${\begin{aligned}d_{m',m}^{j}(\pi )&=(-1)^{j-m}\delta _{m',-m}\\[6pt]d_{m',m}^{j}(\pi -\beta )&=(-1)^{j+m'}d_{m',-m}^{j}(\beta )\\[6pt]d_{m',m}^{j}(\pi +\beta )&=(-1)^{j-m}d_{m',-m}^{j}(\beta )\\[6pt]d_{m',m}^{j}(2\pi +\beta )&=(-1)^{2j}d_{m',m}^{j}(\beta )\\[6pt]d_{m',m}^{j}(-\beta )&=d_{m,m'}^{j}(\beta )=(-1)^{m'-m}d_{m',m}^{j}(\beta )\end{aligned}}$
See also
• Clebsch–Gordan coefficients
• Tensor operator
• Symmetries in quantum mechanics
References
1. Wigner, E. P. (1951) [1931]. Gruppentheorie und ihre Anwendungen auf die Quantenmechanik der Atomspektren. Braunschweig: Vieweg Verlag. OCLC 602430512. Translated into English by Group Theory and its Application to the Quantum Mechanics of Atomic Spectra. Translated by Griffin, J.J. Elsevier. 2013 [1959]. ISBN 978-1-4832-7576-5.
2. Biedenharn, L. C.; Louck, J. D. (1981). Angular Momentum in Quantum Physics. Reading: Addison-Wesley. ISBN 0-201-13507-8.
3. Rose, Morris Edgar (1995) [1957]. Elementary theory of angular momentum. Dover. ISBN 0-486-68480-6. OCLC 31374243.
4. Schwinger, J. (January 26, 1952). On Angular Momentum (Technical report). Harvard University, Nuclear Development Associates. doi:10.2172/4389568. NYO-3071, TRN: US200506%%295.
5. Shiraishi, M. (2013). "Appendix A: Spin-Weighted Spherical Harmonic Function" (PDF). Probing the Early Universe with the CMB Scalar, Vector and Tensor Bispectrum (PhD). Nagoya University. pp. 153–4. ISBN 978-4-431-54180-6.
6. Meckler, A. (1958). "Majorana formula". Physical Review. 111 (6): 1447. doi:10.1103/PhysRev.111.1447.
7. Mermin, N.D.; Schwarz, G.M. (1982). "Joint distributions and local realism in the higher-spin Einstein-Podolsky-Rosen experiment". Foundations of Physics. 12 (2): 101. doi:10.1007/BF00736844. S2CID 121648820.
8. Edén, M. (2003). "Computer simulations in solid-state NMR. I. Spin dynamics theory". Concepts in Magnetic Resonance Part A. 17A (1): 117–154. doi:10.1002/cmr.a.10061.
External links
• Amsler, C.; et al. (Particle Data Group) (2008). "PDG Table of Clebsch-Gordan Coefficients, Spherical Harmonics, and d-Functions" (PDF). Physics Letters B667.
| Wikipedia |
Wigner distribution function
The Wigner distribution function (WDF) is used in signal processing as a transform in time-frequency analysis.
The WDF was first proposed in physics to account for quantum corrections to classical statistical mechanics in 1932 by Eugene Wigner, and it is of importance in quantum mechanics in phase space (see, by way of comparison: Wigner quasi-probability distribution, also called the Wigner function or the Wigner–Ville distribution).
Given the shared algebraic structure between position-momentum and time-frequency conjugate pairs, it also usefully serves in signal processing, as a transform in time-frequency analysis, the subject of this article. Compared to a short-time Fourier transform, such as the Gabor transform, the Wigner distribution function provides the highest possible temporal vs frequency resolution which is mathematically possible within the limitations of the uncertainty principle. The downside is the introduction of large cross terms between every pair of signal components and between positive and negative frequencies, which makes the original formulation of the function a poor fit for most analysis applications. Subsequent modifications have been proposed which preserve the sharpness of the Wigner distribution function but largely suppress cross terms.
Mathematical definition
There are several different definitions for the Wigner distribution function. The definition given here is specific to time-frequency analysis. Given the time series $x[t]$, its non-stationary auto-covariance function is given by
$C_{x}(t_{1},t_{2})=\left\langle \left(x[t_{1}]-\mu [t_{1}]\right)\left(x[t_{2}]-\mu [t_{2}]\right)^{*}\right\rangle ,$
where $\langle \cdots \rangle $ denotes the average over all possible realizations of the process and $\mu (t)$ is the mean, which may or may not be a function of time. The Wigner function $W_{x}(t,f)$ is then given by first expressing the autocorrelation function in terms of the average time $t=(t_{1}+t_{2})/2$ and time lag $\tau =t_{1}-t_{2}$, and then Fourier transforming the lag.
$W_{x}(t,f)=\int _{-\infty }^{\infty }C_{x}\left(t+{\frac {\tau }{2}},t-{\frac {\tau }{2}}\right)\,e^{-2\pi i\tau f}\,d\tau .$
So for a single (mean-zero) time series, the Wigner function is simply given by
$W_{x}(t,f)=\int _{-\infty }^{\infty }x\left(t+{\frac {\tau }{2}}\right)\,x^{*}\left(t-{\frac {\tau }{2}}\right)\,e^{-2\pi i\tau f}\,d\tau .$
The motivation for the Wigner function is that it reduces to the spectral density function at all times $t$ for stationary processes, yet it is fully equivalent to the non-stationary autocorrelation function. Therefore, the Wigner function tells us (roughly) how the spectral density changes in time.
Time-frequency analysis example
Here are some examples illustrating how the WDF is used in time-frequency analysis.
Constant input signal
When the input signal is constant, its time-frequency distribution is a horizontal line along the time axis. For example, if x(t) = 1, then
$W_{x}(t,f)=\int _{-\infty }^{\infty }e^{-i2\pi \tau \,f}\,d\tau =\delta (f).$
Sinusoidal input signal
When the input signal is a sinusoidal function, its time-frequency distribution is a horizontal line parallel to the time axis, displaced from it by the sinusoidal signal's frequency. For example, if x(t) = e i2πkt, then
${\begin{aligned}W_{x}(t,f)&=\int _{-\infty }^{\infty }e^{i2\pi k\left(t+{\frac {\tau }{2}}\right)}e^{-i2\pi k\left(t-{\frac {\tau }{2}}\right)}e^{-i2\pi \tau \,f}\,d\tau \\&=\int _{-\infty }^{\infty }e^{-i2\pi \tau \left(f-k\right)}\,d\tau \\&=\delta (f-k).\end{aligned}}$
Chirp input signal
When the input signal is a linear chirp function, the instantaneous frequency is a linear function. This means that the time frequency distribution should be a straight line. For example, if
$x(t)=e^{i2\pi kt^{2}}$ ,
then its instantaneous frequency is
${\frac {1}{2\pi }}{\frac {d(2\pi kt^{2})}{dt}}=2kt~,$
and its WDF
${\begin{aligned}W_{x}(t,f)&=\int _{-\infty }^{\infty }e^{i2\pi k\left(t+{\frac {\tau }{2}}\right)^{2}}e^{-i2\pi k\left(t-{\frac {\tau }{2}}\right)^{2}}e^{-i2\pi \tau \,f}\,d\tau \\&=\int _{-\infty }^{\infty }e^{i4\pi kt\tau }e^{-i2\pi \tau f}\,d\tau \\&=\int _{-\infty }^{\infty }e^{-i2\pi \tau (f-2kt)}\,d\tau \\&=\delta (f-2kt)~.\end{aligned}}$
Delta input signal
When the input signal is a delta function, since it is only non-zero at t=0 and contains infinite frequency components, its time-frequency distribution should be a vertical line across the origin. This means that the time frequency distribution of the delta function should also be a delta function. By WDF
${\begin{aligned}W_{x}(t,f)&=\int _{-\infty }^{\infty }\delta \left(t+{\frac {\tau }{2}}\right)\delta \left(t-{\frac {\tau }{2}}\right)e^{-i2\pi \tau \,f}\,d\tau \\&=4\int _{-\infty }^{\infty }\delta (2t+\tau )\delta (2t-\tau )e^{-i2\pi \tau f}\,d\tau \\&=4\delta (4t)e^{i4\pi tf}\\&=\delta (t)e^{i4\pi tf}\\&=\delta (t).\end{aligned}}$
The Wigner distribution function is best suited for time-frequency analysis when the input signal's phase is 2nd order or lower. For those signals, WDF can exactly generate the time frequency distribution of the input signal.
Boxcar function
$x(t)={\begin{cases}1&|t|<1/2\\0&{\text{otherwise}}\end{cases}}\qquad $ ,
the rectangular function ⇒
$W_{x}(t,f)={\begin{cases}{\frac {1}{\pi f}}\sin(2\pi f\{1-2|t|\})&|t|<1/2\\0&{\mbox{otherwise}}\end{cases}}$
Cross term property
The Wigner distribution function is not a linear transform. A cross term ("time beats") occurs when there is more than one component in the input signal, analogous in time to frequency beats.[1] In the ancestral physics Wigner quasi-probability distribution, this term has important and useful physics consequences, required for faithful expectation values. By contrast, the short-time Fourier transform does not have this feature. Negative features of the WDF are reflective of the Gabor limit of the classical signal and physically unrelated to any possible underlay of quantum structure.
The following are some examples that exhibit the cross-term feature of the Wigner distribution function.
• $x(t)={\begin{cases}\cos(2\pi t)&t\leq -2\\\cos(4\pi t)&-2<t\leq 2\\\cos(3\pi t)&t>2\end{cases}}$
• $x(t)=e^{it^{3}}$
In order to reduce the cross-term difficulty, several approaches have been proposed in the literature,[2][3] some of them leading to new transforms as the modified Wigner distribution function, the Gabor–Wigner transform, the Choi-Williams distribution function and Cohen's class distribution.
Properties of the Wigner distribution function
The Wigner distribution function has several evident properties listed in the following table.
Projection property
${\begin{aligned}|x(t)|^{2}&=\int _{-\infty }^{\infty }W_{x}(t,f)\,df\\|X(f)|^{2}&=\int _{-\infty }^{\infty }W_{x}(t,f)\,dt\end{aligned}}$
Energy property
$\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }W_{x}(t,f)\,df\,dt=\int _{-\infty }^{\infty }|x(t)|^{2}\,dt=\int _{-\infty }^{\infty }|X(f)|^{2}\,df$
Recovery property
${\begin{aligned}\int _{-\infty }^{\infty }W_{x}\left({\frac {t}{2}},f\right)e^{i2\pi ft}\,df&=x(t)x^{*}(0)\\\int _{-\infty }^{\infty }W_{x}\left(t,{\frac {f}{2}}\right)e^{i2\pi ft}\,dt&=X(f)X^{*}(0)\end{aligned}}$
Mean condition frequency and mean condition time
${\begin{aligned}X(f)&=|X(f)|e^{i2\pi \psi (f)},\quad x(t)=|x(t)|e^{i2\pi \phi (t)},\\{\text{if }}\phi '(t)&=|x(t)|^{-2}\int _{-\infty }^{\infty }fW_{x}(t,f)\,df\\{\text{ and }}-\psi '(f)&=|X(f)|^{-2}\int _{-\infty }^{\infty }tW_{x}(t,f)\,dt\end{aligned}}$
Moment properties
${\begin{aligned}\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }t^{n}W_{x}(t,f)\,dt\,df&=\int _{-\infty }^{\infty }t^{n}|x(t)|^{2}\,dt\\\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }f^{n}W_{x}(t,f)\,dt\,df&=\int _{-\infty }^{\infty }f^{n}|X(f)|^{2}\,df\end{aligned}}$
Real properties
$W_{x}^{*}(t,f)=W_{x}(t,f)$
Region properties
${\begin{aligned}{\text{If }}x(t)&=0{\text{ for }}t>t_{0}{\text{ then }}W_{x}(t,f)=0{\text{ for }}t>t_{0}\\{\text{If }}x(t)&=0{\text{ for }}t<t_{0}{\text{ then }}W_{x}(t,f)=0{\text{ for }}t<t_{0}\end{aligned}}$
Multiplication theorem
${\begin{aligned}{\text{If }}y(t)&=x(t)h(t)\\{\text{then }}W_{y}(t,f)&=\int _{-\infty }^{\infty }W_{x}(t,\rho )W_{h}(t,f-\rho )\,d\rho \end{aligned}}$
Convolution theorem
${\begin{aligned}{\text{If }}y(t)&=\int _{-\infty }^{\infty }x(t-\tau )h(\tau )\,d\tau \\{\text{then }}W_{y}(t,f)&=\int _{-\infty }^{\infty }W_{x}(\rho ,f)W_{h}(t-\rho ,f)\,d\rho \end{aligned}}$
Correlation theorem
${\begin{aligned}{\text{If }}y(t)&=\int _{-\infty }^{\infty }x(t+\tau )h^{*}(\tau )\,d\tau {\text{ then }}\\W_{y}(t,\omega )&=\int _{-\infty }^{\infty }W_{x}(\rho ,\omega )W_{h}(-t+\rho ,\omega )\,d\rho \end{aligned}}$
Time-shifting covariance
${\begin{aligned}{\text{If }}y(t)&=x(t-t_{0})\\{\text{then }}W_{y}(t,f)&=W_{x}(t-t_{0},f)\end{aligned}}$
Modulation covariance
${\begin{aligned}{\text{If }}y(t)&=e^{i2\pi f_{0}t}x(t)\\{\text{then }}W_{y}(t,f)&=W_{x}(t,f-f_{0})\end{aligned}}$
Scale covariance
${\begin{aligned}{\text{If }}y(t)&={\sqrt {a}}x(at){\text{ for some }}a>0{\text{ then }}\\{\text{then }}W_{y}(t,f)&=W_{x}(at,{\frac {f}{a}})\end{aligned}}$
Windowed Wigner Distribution Function
When a signal is not time limited, its Wigner Distribution Function is hard to implement. Thus, we add a new function(mask) to its integration part, so that we only have to implement part of the original function instead of integrating all the way from negative infinity to positive infinity. Original function: $W_{x}(t,f)=\int _{-\infty }^{\infty }x\left(t+{\frac {\tau }{2}}\right)\cdot x^{*}\left(t-{\frac {\tau }{2}}\right)e^{-j2\pi \tau f}\cdot d\tau $ Function with mask: $W_{x}(t,f)=\int _{-\infty }^{\infty }w(\tau )x\left(t+{\frac {\tau }{2}}\right)\cdot x^{*}\left(t-{\frac {\tau }{2}}\right)e^{-j2\pi \tau f}\cdot d\tau $ $w(\tau )$ is real and time-limited
Implementation
According to definition:
${\begin{aligned}W_{x}(t,f)=\int _{-\infty }^{\infty }w(\tau )x\left(t+{\frac {\tau }{2}}\right)\cdot x^{*}\left(t-{\frac {\tau }{2}}\right)e^{-j2\pi \tau f}\cdot d\tau \\W_{x}(t,f)=2\int _{-\infty }^{\infty }w(2\tau ')x\left(t+\tau '\right)\cdot x^{*}\left(t-\tau '\right)e^{-j4\pi \tau 'f}\cdot d\tau '\\W_{x}(n\Delta _{t},m\Delta _{f})=2\sum _{p=-\infty }^{\infty }w(2p\Delta _{t})x((n+p)\Delta _{t})x^{\ast }((n-p)\Delta _{t})e^{-j4\pi mp\Delta _{t}\Delta _{f}}\Delta _{t}\end{aligned}}$
Suppose that $w(t)=0$ for $|t|>B\rightarrow w(2p\Delta _{t})=0$ for $p<-Q$ and $p>Q$
${\begin{aligned}W_{x}(n\Delta _{t},m\Delta _{f})=2\sum _{p=-Q}^{Q}w(2p\Delta _{t})x((n+p)\Delta _{t})x^{\ast }((n-p)\Delta _{t})e^{-j4\pi mp\Delta _{t}\Delta _{f}}\Delta _{t}\end{aligned}}$
We take $x(t)=\delta (t-t_{1})+\delta (t-t_{2})$ as example
${\begin{aligned}W_{x}(t,f)=\int _{-\infty }^{\infty }w(\tau )x\left(t+{\frac {\tau }{2}}\right)\cdot x^{*}\left(t-{\frac {\tau }{2}}\right)e^{-j2\pi \tau f}\cdot d\tau \,,\end{aligned}}$
where $w(\tau )$ is a real function
And then we compare the difference between two conditions.
Ideal: $W_{x}(t,f)=0,{\text{ for }}t\neq t_{2},t_{1}$
When mask function $w(\tau )=1$, which means no mask function.
$y(t,\tau )=x(t+{\frac {\tau }{2}})$ $y^{*}(t,-\tau )=x^{*}(t-{\frac {\tau }{2}})$
$W_{x}(t,f)=\int _{-\infty }^{\infty }x(t+{\frac {\tau }{2}})x^{*}(t-{\frac {\tau }{2}}e^{-j2\pi \tau f}d\tau $
$=\int _{-\infty }^{\infty }[\delta (t+{\frac {\tau }{2}}-t_{1})+\delta (t+{\frac {\tau }{2}}-t_{2})][\delta (t-{\frac {\tau }{2}}-t_{1})+\delta (t-{\frac {\tau }{2}}-t_{2})]e^{-j2\pi \tau f}\cdot d\tau $
$=4\int _{-\infty }^{\infty }[\delta (2t+\tau -2t_{1})+\delta (2t+\tau -2t_{2})][\delta (2t-\tau -2t_{1})+\delta (2t-\tau -2t_{2})]e^{j2\pi \tau f}\cdot d\tau $
3 Conditions
Then we consider the condition with mask function:
We can see that $w(\tau )$ have value only between –B to B, thus conducting with $w(\tau )$ can remove cross term of the function. But if x(t) is not a Delta function nor a narrow frequency function, instead, it is a function with wide frequency or ripple. The edge of the signal may still exist between –B and B, which still cause the cross term problem.
for example:
See also
• Time-frequency representation
• Short-time Fourier transform
• Spectrogram
• Gabor transform
• Autocorrelation
• Gabor–Wigner transform
• Modified Wigner distribution function
• Optical equivalence theorem
• Polynomial Wigner–Ville distribution
• Cohen's class distribution function
• Wigner quasi-probability distribution
• Transformation between distributions in time-frequency analysis
• Bilinear time–frequency distribution
References
1. F. Hlawatsch and P. Flandrin, "The interference structure of the Wigner distribution and related time-frequency signal representations", in W. Mecklenbräuker and F. Hlawatsch, The Wigner Distribution - Theory and Applications in Signal Processing
2. B. Boashah (Ed.), Time Frequency Signal Analysis and Processing, Elsevier, 2003
3. P. Flandrin, Time-Frequency/Time-Scale Analysis, Elsevier, 1998
Further reading
• Wigner, E. (1932). "On the Quantum Correction for Thermodynamic Equilibrium" (PDF). Physical Review. 40 (5): 749–759. Bibcode:1932PhRv...40..749W. doi:10.1103/PhysRev.40.749. hdl:10338.dmlcz/141466.
• J. Ville, 1948. "Théorie et Applications de la Notion de Signal Analytique", Câbles et Transmission, 2, 61–74 .
• T. A. C. M. Classen and W. F. G. Mecklenbrauker, 1980. "The Wigner distribution-a tool for time-frequency signal analysis; Part I," Philips J. Res., vol. 35, pp. 217–250.
• L. Cohen (1989): Proceedings of the IEEE 77 pp. 941–981, Time-frequency distributions---a review
• L. Cohen, Time-Frequency Analysis, Prentice-Hall, New York, 1995. ISBN 978-0135945322
• S. Qian and D. Chen, Joint Time-Frequency Analysis: Methods and Applications, Chap. 5, Prentice Hall, N.J., 1996.
• B. Boashash, "Note on the Use of the Wigner Distribution for Time Frequency Signal Analysis", IEEE Transactions on Acoustics, Speech, and Signal Processing, Vol. 36, No. 9, pp. 1518–1521, Sept. 1988. doi:10.1109/29.90380. B. Boashash, editor,Time-Frequency Signal Analysis and Processing – A Comprehensive Reference, Elsevier Science, Oxford, 2003, ISBN 0-08-044335-4.
• F. Hlawatsch, G. F. Boudreaux-Bartels: "Linear and quadratic time-frequency signal representation," IEEE Signal Processing Magazine, pp. 21–67, Apr. 1992.
• R. L. Allen and D. W. Mills, Signal Analysis: Time, Frequency, Scale, and Structure, Wiley- Interscience, NJ, 2004.
• Jian-Jiun Ding, Time frequency analysis and wavelet transform class notes, the Department of Electrical Engineering, National Taiwan University (NTU), Taipei, Taiwan, 2015.
• Kakofengitis, D., & Steuernagel, O. (2017). "Wigner's quantum phase space current in weakly anharmonic weakly excited two-state systems" European Physical Journal Plus 14.07.2017
External links
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| Wikipedia |
Wigner semicircle distribution
The Wigner semicircle distribution, named after the physicist Eugene Wigner, is the probability distribution on [−R, R] whose probability density function f is a scaled semicircle (i.e., a semi-ellipse) centered at (0, 0):
$f(x)={2 \over \pi R^{2}}{\sqrt {R^{2}-x^{2}\,}}\,$
Wigner semicircle
Probability density function
Cumulative distribution function
Parameters $R>0\!$ radius (real)
Support $x\in [-R;+R]\!$
PDF ${\frac {2}{\pi R^{2}}}\,{\sqrt {R^{2}-x^{2}}}\!$
CDF ${\frac {1}{2}}+{\frac {x{\sqrt {R^{2}-x^{2}}}}{\pi R^{2}}}+{\frac {\arcsin \!\left({\frac {x}{R}}\right)}{\pi }}\!$
for $-R\leq x\leq R$
Mean $0\,$
Median $0\,$
Mode $0\,$
Variance ${\frac {R^{2}}{4}}\!$
Skewness $0\,$
Ex. kurtosis $-1\,$
Entropy $\ln(\pi R)-{\frac {1}{2}}\,$
MGF $2\,{\frac {I_{1}(R\,t)}{R\,t}}$
CF $2\,{\frac {J_{1}(R\,t)}{R\,t}}$
for −R ≤ x ≤ R, and f(x) = 0 if |x| > R. The parameter R is commonly referred to as the "radius" parameter of the distribution.
The Wigner distribution also coincides with a scaled beta distribution. That is, if Y is a beta-distributed random variable with parameters α = β = 3/2, then the random variable X = 2RY – R exhibits a Wigner semicircle distribution with radius R.
The distribution arises as the limiting distribution of the eigenvalues of many random symmetric matrices, that is, as the dimensions of the random matrix approach infinity. The distribution of the spacing or gaps between eigenvalues is addressed by the similarly named Wigner surmise.
General properties
The Chebyshev polynomials of the third kind are orthogonal polynomials with respect to the Wigner semicircle distribution.
For positive integers n, the 2n-th moment of this distribution is
$E(X^{2n})=\left({R \over 2}\right)^{2n}C_{n}\,$
where X is any random variable with this distribution and Cn is the nth Catalan number
$C_{n}={1 \over n+1}{2n \choose n},\,$
so that the moments are the Catalan numbers if R = 2. (Because of symmetry, all of the odd-order moments are zero.)
Making the substitution $x=R\cos(\theta )$ into the defining equation for the moment generating function it can be seen that:
$M(t)={\frac {2}{\pi }}\int _{0}^{\pi }e^{Rt\cos(\theta )}\sin ^{2}(\theta )\,d\theta $
which can be solved (see Abramowitz and Stegun §9.6.18) to yield:
$M(t)=2\,{\frac {I_{1}(Rt)}{Rt}}$
where $I_{1}(z)$ is the modified Bessel function. Similarly, the characteristic function is given by:[1][2][3]
$\varphi (t)=2\,{\frac {J_{1}(Rt)}{Rt}}$
where $J_{1}(z)$ is the Bessel function. (See Abramowitz and Stegun §9.1.20), noting that the corresponding integral involving $\sin(Rt\cos(\theta ))$ is zero.)
In the limit of $R$ approaching zero, the Wigner semicircle distribution becomes a Dirac delta function.
Relation to free probability
In free probability theory, the role of Wigner's semicircle distribution is analogous to that of the normal distribution in classical probability theory. Namely, in free probability theory, the role of cumulants is occupied by "free cumulants", whose relation to ordinary cumulants is simply that the role of the set of all partitions of a finite set in the theory of ordinary cumulants is replaced by the set of all noncrossing partitions of a finite set. Just as the cumulants of degree more than 2 of a probability distribution are all zero if and only if the distribution is normal, so also, the free cumulants of degree more than 2 of a probability distribution are all zero if and only if the distribution is Wigner's semicircle distribution.
Related distributions
Wigner (spherical) parabolic distribution
Wigner parabolic
Parameters $R>0\!$ radius (real)
Support $x\in [-R;+R]\!$
PDF ${\frac {3}{4R^{3}}}\,(R^{2}-x^{2})$
CDF ${\frac {1}{4R^{3}}}\,(2R-x)\,(R+x)^{2}$
MGF $3\,{\frac {i_{1}(R\,t)}{R\,t}}$
CF $3\,{\frac {j_{1}(R\,t)}{R\,t}}$
The parabolic probability distribution supported on the interval [−R, R] of radius R centered at (0, 0):
$f(x)={3 \over \ 4R^{3}}{(R^{2}-x^{2})}\,$
for −R ≤ x ≤ R, and f(x) = 0 if |x| > R.
Example. The joint distribution is
$\int _{0}^{\pi }\int _{0}^{+2\pi }\int _{0}^{R}f_{X,Y,Z}(x,y,z)R^{2}\,dr\sin(\theta )\,d\theta \,d\phi =1;$
$f_{X,Y,Z}(x,y,z)={\frac {3}{4\pi }}$
Hence, the marginal PDF of the spherical (parametric) distribution is:[4]
$f_{X}(x)=\int _{-{\sqrt {1-y^{2}-x^{2}}}}^{+{\sqrt {1-y^{2}-x^{2}}}}\int _{-{\sqrt {1-x^{2}}}}^{+{\sqrt {1-x^{2}}}}f_{X,Y,Z}(x,y,z)\,dy\,dz;$
$f_{X}(x)=\int _{-{\sqrt {1-x^{2}}}}^{+{\sqrt {1-x^{2}}}}2{\sqrt {1-y^{2}-x^{2}}}\,dy\,;$
$f_{X}(x)={3 \over \ 4}{(1-x^{2})}\,;$ such that R=1
The characteristic function of a spherical distribution becomes the pattern multiplication of the expected values of the distributions in X, Y and Z.
The parabolic Wigner distribution is also considered the monopole moment of the hydrogen like atomic orbitals.
Wigner n-sphere distribution
The normalized N-sphere probability density function supported on the interval [−1, 1] of radius 1 centered at (0, 0):
$f_{n}(x;n)={(1-x^{2})^{(n-1)/2}\Gamma (1+n/2) \over {\sqrt {\pi }}\Gamma ((n+1)/2)}\,(n>=-1)$,
for −1 ≤ x ≤ 1, and f(x) = 0 if |x| > 1.
Example. The joint distribution is
$\int _{-{\sqrt {1-y^{2}-x^{2}}}}^{+{\sqrt {1-y^{2}-x^{2}}}}\int _{-{\sqrt {1-x^{2}}}}^{+{\sqrt {1-x^{2}}}}\int _{0}^{1}f_{X,Y,Z}(x,y,z){{\sqrt {1-x^{2}-y^{2}-z^{2}}}^{(n)}}dxdydz=1;$
$f_{X,Y,Z}(x,y,z)={\frac {3}{4\pi }}$
Hence, the marginal PDF distribution is [4]
$f_{X}(x;n)={(1-x^{2})^{(n-1)/2)}\Gamma (1+n/2) \over \ {\sqrt {\pi }}\Gamma ((n+1)/2)}\,;$ such that R=1
The cumulative distribution function (CDF) is
$F_{X}(x)={2x\Gamma (1+n/2)_{2}F_{1}(1/2,(1-n)/2;3/2;x^{2}) \over \ {\sqrt {\pi }}\Gamma ((n+1)/2)}\,;$ such that R=1 and n >= -1
The characteristic function (CF) of the PDF is related to the beta distribution as shown below
$CF(t;n)={_{1}F_{1}(n/2,;n;jt/2)}\,\urcorner (\alpha =\beta =n/2);$
In terms of Bessel functions this is
$CF(t;n)={\Gamma (n/2+1)J_{n/2}(t)/(t/2)^{(n/2)}}\,\urcorner (n>=-1);$
Raw moments of the PDF are
$\mu '_{N}(n)=\int _{-1}^{+1}x^{N}f_{X}(x;n)dx={(1+(-1)^{N})\Gamma (1+n/2) \over \ {2{\sqrt {\pi }}}\Gamma ((2+n+N)/2)};$
Central moments are
$\mu _{0}(x)=1$
$\mu _{1}(n)=\mu _{1}'(n)$
$\mu _{2}(n)=\mu _{2}'(n)-\mu _{1}'^{2}(n)$
$\mu _{3}(n)=2\mu _{1}'^{3}(n)-3\mu _{1}'(n)\mu _{2}'(n)+\mu _{3}'(n)$
$\mu _{4}(n)=-3\mu _{1}'^{4}(n)+6\mu _{1}'^{2}(n)\mu _{2}'(n)-4\mu '_{1}(n)\mu '_{3}(n)+\mu '_{4}(n)$
The corresponding probability moments (mean, variance, skew, kurtosis and excess-kurtosis) are:
$\mu (x)=\mu _{1}'(x)=0$
$\sigma ^{2}(n)=\mu _{2}'(n)-\mu ^{2}(n)=1/(2+n)$
$\gamma _{1}(n)=\mu _{3}/\mu _{2}^{3/2}=0$
$\beta _{2}(n)=\mu _{4}/\mu _{2}^{2}=3(2+n)/(4+n)$
$\gamma _{2}(n)=\mu _{4}/\mu _{2}^{2}-3=-6/(4+n)$
Raw moments of the characteristic function are:
$\mu '_{N}(n)=\mu '_{N;E}(n)+\mu '_{N;O}(n)=\int _{-1}^{+1}cos^{N}(xt)f_{X}(x;n)dx+\int _{-1}^{+1}sin^{N}(xt)f_{X}(x;n)dx;$
For an even distribution the moments are [5]
$\mu _{1}'(t;n:E)=CF(t;n)$
$\mu _{1}'(t;n:O)=0$
$\mu _{1}'(t;n)=CF(t;n)$
$\mu _{2}'(t;n:E)=1/2(1+CF(2t;n))$
$\mu _{2}'(t;n:O)=1/2(1-CF(2t;n))$
$\mu '_{2}(t;n)=1$
$\mu _{3}'(t;n:E)=(CF(3t)+3CF(t;n))/4$
$\mu _{3}'(t;n:O)=0$
$\mu _{3}'(t;n)=(CF(3t;n)+3CF(t;n))/4$
$\mu _{4}'(t;n:E)=(3+4CF(2t;n)+CF(4t;n))/8$
$\mu _{4}'(t;n:O)=(3-4CF(2t;n)+CF(4t;n))/8$
$\mu _{4}'(t;n)=(3+CF(4t;n))/4$
Hence, the moments of the CF (provided N=1) are
$\mu (t;n)=\mu _{1}'(t)=CF(t;n)=_{0}F_{1}({2+n \over 2},-{t^{2} \over 4})$
$\sigma ^{2}(t;n)=1-|CF(t;n)|^{2}=1-|_{0}F_{1}({2+n \over 2},-t^{2}/4)|^{2}$
$\gamma _{1}(n)={\mu _{3} \over \mu _{2}^{3/2}}={_{0}F_{1}({2+n \over 2},-9{t^{2} \over 4})-_{0}F_{1}({2+n \over 2},-{t^{2} \over 4})+8|_{0}F_{1}({2+n \over 2},-{t^{2} \over 4})|^{3} \over 4(1-|_{0}F_{1}({2+n \over 2},-{t^{2} \over 4}))^{2}|^{(3/2)}}$
$\beta _{2}(n)={\mu _{4} \over \mu _{2}^{2}}={3+_{0}F_{1}({2+n \over 2},-4t^{2})-(4_{0}F_{1}({2+n \over 2},-{t^{2} \over 4})(_{0}F_{1}({2+n \over 2},-9{t^{2} \over 4}))+3_{0}F_{1}({2+n \over 2},-{t^{2} \over 4})(-1+|_{0}F_{1}({2+n \over 2},-{t^{2} \over 4}|^{2})) \over 4(-1+|_{0}F_{1}({2+n \over 2},-{t^{2} \over 4}))^{2}|^{2}}$
$\gamma _{2}(n)=\mu _{4}/\mu _{2}^{2}-3={-9+_{0}F_{1}({2+n \over 2},-4t^{2})-(4_{0}F_{1}({2+n \over 2},-t^{2}/4)(_{0}F_{1}({2+n \over 2},-9{t^{2} \over 4}))-9_{0}F_{1}({2+n \over 2},-{t^{2} \over 4})+6|_{0}F_{1}({2+n \over 2},-{t^{2} \over 4}|^{3}) \over 4(-1+|_{0}F_{1}({2+n \over 2},-{t^{2} \over 4}))^{2}|^{2}}$
Skew and Kurtosis can also be simplified in terms of Bessel functions.
The entropy is calculated as
$H_{N}(n)=\int _{-1}^{+1}f_{X}(x;n)\ln(f_{X}(x;n))dx$
The first 5 moments (n=-1 to 3), such that R=1 are
$\ -\ln(2/\pi );n=-1$
$\ -\ln(2);n=0$
$\ -1/2+\ln(\pi );n=1$
$\ 5/3-\ln(3);n=2$
$\ -7/4-\ln(1/3\pi );n=3$
N-sphere Wigner distribution with odd symmetry applied
The marginal PDF distribution with odd symmetry is [4]
$f{_{X}}(x;n)={(1-x^{2})^{(n-1)/2)}\Gamma (1+n/2) \over \ {\sqrt {\pi }}\Gamma ((n+1)/2)}\operatorname {sgn}(x)\,;$ such that R=1
Hence, the CF is expressed in terms of Struve functions
$CF(t;n)={\Gamma (n/2+1)H_{n/2}(t)/(t/2)^{(n/2)}}\,\urcorner (n>=-1);$
"The Struve function arises in the problem of the rigid-piston radiator mounted in an infinite baffle, which has radiation impedance given by" [6]
$Z={\rho c\pi a^{2}[R_{1}(2ka)-iX_{1}(2ka)],}$
$R_{1}={1-{2J_{1}(x) \over 2x},}$
$X_{1}={{2H_{1}(x) \over x},}$
Example (Normalized Received Signal Strength): quadrature terms
The normalized received signal strength is defined as
$|R|={{1 \over N}|}\sum _{k=1}^{N}\exp[ix_{n}t]|$
and using standard quadrature terms
$x={1 \over N}\sum _{k=1}^{N}\cos(x_{n}t)$
$y={1 \over N}\sum _{k=1}^{N}\sin(x_{n}t)$
Hence, for an even distribution we expand the NRSS, such that x = 1 and y = 0, obtaining
${\sqrt {x^{2}+y^{2}}}=x+{3 \over 2}y^{2}-{3 \over 2}xy^{2}+{1 \over 2}x^{2}y^{2}+O(y^{3})+O(y^{3})(x-1)+O(y^{3})(x-1)^{2}+O(x-1)^{3}$
The expanded form of the Characteristic function of the received signal strength becomes [7]
$E[x]={1 \over N}CF(t;n)$
$E[y^{2}]={1 \over 2N}(1-CF(2t;n))$
$E[x^{2}]={1 \over 2N}(1+CF(2t;n))$
$E[xy^{2}]={t^{2} \over 3N^{2}}CF(t;n)^{3}+({N-1 \over 2N^{2}})(1-tCF(2t;n))CF(t;n)$
$E[x^{2}y^{2}]={1 \over 8N^{3}}(1-CF(4t;n))+({N-1 \over 4N^{3}})(1-CF(2t;n)^{2})+({N-1 \over 3N^{3}})t^{2}CF(t;n)^{4}+({(N-1)(N-2) \over N^{3}})CF(t;n)^{2}(1-CF(2t;n))$
See also
• Wigner surmise
• The Wigner semicircle distribution is the limit of the Kesten–McKay distributions, as the parameter d tends to infinity.
• In number-theoretic literature, the Wigner distribution is sometimes called the Sato–Tate distribution. See Sato–Tate conjecture.
• Marchenko–Pastur distribution or Free Poisson distribution
References
1. Buchanan, Kristopher; Flores, Carlos; Wheeland, Sara; Jensen, Jeffrey; Grayson, David; Huff, Gregory (2017). "Transmit beamforming for radar applications using circularly tapered random arrays". 2017 IEEE Radar Conference (Radar Conf). pp. 0112–0117. doi:10.1109/RADAR.2017.7944181. ISBN 978-1-4673-8823-8. S2CID 38429370.
2. Ryan, Buchanan (29 May 2014). Theory and Applications of Aperiodic (Random) Phased Arrays (Thesis). hdl:1969.1/157918.
3. Overturf, Drew; Buchanan, Kristopher; Jensen, Jeffrey; Wheeland, Sara; Huff, Gregory (2017). "Investigation of beamforming patterns from volumetrically distributed phased arrays". MILCOM 2017 - 2017 IEEE Military Communications Conference (MILCOM). pp. 817–822. doi:10.1109/MILCOM.2017.8170756. ISBN 978-1-5386-0595-0. S2CID 11591305. https://ieeexplore.ieee.org/abstract/document/8170756/
4. Buchanan, K.; Huff, G. H. (July 2011). "A comparison of geometrically bound random arrays in euclidean space". 2011 IEEE International Symposium on Antennas and Propagation (APSURSI). pp. 2008–2011. doi:10.1109/APS.2011.5996900. ISBN 978-1-4244-9563-4. S2CID 10446533.
5. Thomas M. Cover (1963). "Antenna pattern distribution from random array" (PDF) (MEMORANDUM RM-3502--PR). Santa Monica: The RAND Corporation. Archived (PDF) from the original on September 4, 2021.
6. W., Weisstein, Eric. "Struve Function". mathworld.wolfram.com. Retrieved 2017-07-28.{{cite web}}: CS1 maint: multiple names: authors list (link)
7. "Advanced Beamforming for Distributed and Multi-Beam Networks" (PDF).
• Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972.
External links
• Eric W. Weisstein et al., Wigner's semicircle
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| Wikipedia |
Wigner surmise
In mathematical physics, the Wigner surmise is a statement about the probability distribution of the spaces between points in the spectra of nuclei of heavy atoms, which have many degrees of freedom, or quantum systems with few degrees of freedom but chaotic classical dynamics. It was proposed by Eugene Wigner in probability theory.[1] The surmise was a result of Wigner's introduction of random matrices in the field of nuclear physics. The surmise consists of two postulates:
• In a simple sequence (spin and parity are same), the probability density function for a spacing is given by,
$p_{w}(s)={\frac {\pi s}{2}}e^{-\pi s^{2}/4}.$
Here, $s={\frac {S}{D}}$ where S is a particular spacing and D is the mean distance between neighboring intervals.[2]
• In a mixed sequence (spin and parity are different), the probability density function can be obtained by randomly superimposing simple sequences.
The above result is exact for $2\times 2$ real symmetric matrices $M$, with elements that are independent standard gaussian random variables, with joint distribution proportional to
$e^{-{\frac {1}{2}}{\rm {Tr}}(M^{2})}=e^{-{\frac {1}{2}}{\rm {Tr}}\left({\begin{array}{cc}a&b\\b&c\\\end{array}}\right)^{2}}=e^{-{\frac {1}{2}}a^{2}-{\frac {1}{2}}c^{2}-b^{2}}.$
In practice, it is a good approximation for the actual distribution for real symmetric matrices of any dimension. The corresponding result for complex hermitian matrices (which is also exact in the $2\times 2$ case and a good approximation in general) with distribution proportional to $e^{-{\frac {1}{2}}{\rm {Tr}}(MM^{\dagger })}$, is given by
$p_{w}(s)={\frac {32s^{2}}{\pi ^{2}}}e^{-4s^{2}/\pi }.$
History
During the conference on Neutron Physics by Time-of-Flight, held at Gatlinburg, Tennessee, November 1 and 2, 1956, Wigner delivered a presentation on the theoretical arrangement of neighboring neutron resonances (with matching spin and parity) in heavy nuclei. In the presentation he gave the following guess:[3][4]
Perhaps I am now too courageous when I try to guess the distribution of the distances between successive levels (of energies of heavy nuclei). Theoretically, the situation is quite simple if one attacks the problem in a simpleminded fashion. The question is simply what are the distances of the characteristic values of a symmetric matrix with random coefficients.
— Eugene Wigner, Results and theory of resonance absorption
[5]
See also
• Wigner semicircle distribution
References
1. Mehta, Madan Lal (6 October 2004). Random Matrices By Madan Lal Mehta. p. 13. ISBN 9780080474113.
2. Benenti, Giuliano; Casati, Giulio; Strini, Giuliano (2004). Principles of Quantum Computation and Information. p. 406. ISBN 9789812563453.
3. Conference on Neutron Physics by Time-of-Flight (1957) [1956]. Conference on Neutron Physics by Time-of-Flight, held at Gatlinburg, Tennessee, November 1 and 2, 1956; Oak Ridge National Laboratory Report ORNL-2309. Oak Ridge National Laboratory. p. 67.
4. Porter, Charles E. (1965). Statistical Theories of Spectra: Fluctuations. Elsevier Science & Technology Books. p. 208. ISBN 978-0-12-562356-8.
5. Barrett, Owen; Firk, Frank W. K.; Miller, Steven J.; Turnage-Butterbaugh, Caroline (2016), "From Quantum Systems to L-Functions: Pair Correlation Statistics and Beyond", Open Problems in Mathematics, Cham: Springer International Publishing, pp. 123–171, ISBN 978-3-319-32160-8, retrieved 2023-05-13
| Wikipedia |
Wigner quasiprobability distribution
The Wigner quasiprobability distribution (also called the Wigner function or the Wigner–Ville distribution, after Eugene Wigner and Jean-André Ville) is a quasiprobability distribution. It was introduced by Eugene Wigner in 1932[1] to study quantum corrections to classical statistical mechanics. The goal was to link the wavefunction that appears in Schrödinger's equation to a probability distribution in phase space.
It is a generating function for all spatial autocorrelation functions of a given quantum-mechanical wavefunction ψ(x). Thus, it maps[2] on the quantum density matrix in the map between real phase-space functions and Hermitian operators introduced by Hermann Weyl in 1927,[3] in a context related to representation theory in mathematics (see Weyl quantization). In effect, it is the Wigner–Weyl transform of the density matrix, so the realization of that operator in phase space. It was later rederived by Jean Ville in 1948 as a quadratic (in signal) representation of the local time-frequency energy of a signal,[4] effectively a spectrogram.
In 1949, José Enrique Moyal, who had derived it independently, recognized it as the quantum moment-generating functional,[5] and thus as the basis of an elegant encoding of all quantum expectation values, and hence quantum mechanics, in phase space (see Phase-space formulation). It has applications in statistical mechanics, quantum chemistry, quantum optics, classical optics and signal analysis in diverse fields, such as electrical engineering, seismology, time–frequency analysis for music signals, spectrograms in biology and speech processing, and engine design.
Relation to classical mechanics
A classical particle has a definite position and momentum, and hence it is represented by a point in phase space. Given a collection (ensemble) of particles, the probability of finding a particle at a certain position in phase space is specified by a probability distribution, the Liouville density. This strict interpretation fails for a quantum particle, due to the uncertainty principle. Instead, the above quasiprobability Wigner distribution plays an analogous role, but does not satisfy all the properties of a conventional probability distribution; and, conversely, satisfies boundedness properties unavailable to classical distributions.
For instance, the Wigner distribution can and normally does take on negative values for states which have no classical model—and is a convenient indicator of quantum-mechanical interference. (See below for a characterization of pure states whose Wigner functions are non-negative.) Smoothing the Wigner distribution through a filter of size larger than ħ (e.g., convolving with a phase-space Gaussian, a Weierstrass transform, to yield the Husimi representation, below), results in a positive-semidefinite function, i.e., it may be thought to have been coarsened to a semi-classical one.[lower-alpha 1]
Regions of such negative value are provable (by convolving them with a small Gaussian) to be "small": they cannot extend to compact regions larger than a few ħ, and hence disappear in the classical limit. They are shielded by the uncertainty principle, which does not allow precise location within phase-space regions smaller than ħ, and thus renders such "negative probabilities" less paradoxical.
Definition and meaning
The Wigner distribution W(x,p) of a pure state is defined as
$W(x,p)~{\stackrel {\text{def}}{=}}~{\frac {1}{\pi \hbar }}\int _{-\infty }^{\infty }\psi ^{*}(x+y)\psi (x-y)e^{2ipy/\hbar }\,dy,$
where ψ is the wavefunction, and x and p are position and momentum, but could be any conjugate variable pair (e.g. real and imaginary parts of the electric field or frequency and time of a signal). Note that it may have support in x even in regions where ψ has no support in x ("beats").
It is symmetric in x and p:
$W(x,p)={\frac {1}{\pi \hbar }}\int _{-\infty }^{\infty }\varphi ^{*}(p+q)\varphi (p-q)e^{-2ixq/\hbar }\,dq,$
where φ is the normalized momentum-space wave function, proportional to the Fourier transform of ψ.
In 3D,
$W({\vec {r}},{\vec {p}})={\frac {1}{(2\pi )^{3}}}\int \psi ^{*}({\vec {r}}+\hbar {\vec {s}}/2)\psi ({\vec {r}}-\hbar {\vec {s}}/2)e^{i{\vec {p}}\cdot {\vec {s}}}\,d^{3}s.$
In the general case, which includes mixed states, it is the Wigner transform of the density matrix:
$W(x,p)={\frac {1}{\pi \hbar }}\int _{-\infty }^{\infty }\langle x-y|{\hat {\rho }}|x+y\rangle e^{2ipy/\hbar }\,dy,$
where ⟨x|ψ⟩ = ψ(x). This Wigner transformation (or map) is the inverse of the Weyl transform, which maps phase-space functions to Hilbert-space operators, in Weyl quantization.
Thus, the Wigner function is the cornerstone of quantum mechanics in phase space.
In 1949, José Enrique Moyal elucidated how the Wigner function provides the integration measure (analogous to a probability density function) in phase space, to yield expectation values from phase-space c-number functions g(x, p) uniquely associated to suitably ordered operators Ĝ through Weyl's transform (see Wigner–Weyl transform and property 7 below), in a manner evocative of classical probability theory.
Specifically, an operator's Ĝ expectation value is a "phase-space average" of the Wigner transform of that operator:
$\langle {\hat {G}}\rangle =\int dx\,dp\,W(x,p)g(x,p).$
Mathematical properties
1. W(x, p) is a real-valued function.
2. The x and p probability distributions are given by the marginals:
$\int _{-\infty }^{\infty }dp\,W(x,p)=\langle x|{\hat {\rho }}|x\rangle .$ If the system can be described by a pure state, one gets $\int _{-\infty }^{\infty }dp\,W(x,p)=|\psi (x)|^{2}.$
$\int _{-\infty }^{\infty }dx\,W(x,p)=\langle p|{\hat {\rho }}|p\rangle .$ If the system can be described by a pure state, one has $\int _{-\infty }^{\infty }dx\,W(x,p)=|\varphi (p)|^{2}.$
$\int _{-\infty }^{\infty }dx\int _{-\infty }^{\infty }dp\,W(x,p)=\operatorname {Tr} ({\hat {\rho }}).$
Typically the trace of the density matrix ${\hat {\rho }}$ is equal to 1.
3. W(x, p) has the following reflection symmetries:
• Time symmetry: $\psi (x)\to \psi (x)^{*}\Rightarrow W(x,p)\to W(x,-p).$
• Space symmetry: $\psi (x)\to \psi (-x)\Rightarrow W(x,p)\to W(-x,-p).$
4. W(x, p) is Galilei-covariant:
$\psi (x)\to \psi (x+y)\Rightarrow W(x,p)\to W(x+y,p).$
It is not Lorentz-covariant.
5. The equation of motion for each point in the phase space is classical in the absence of forces:
${\frac {\partial W(x,p)}{\partial t}}={\frac {-p}{m}}{\frac {\partial W(x,p)}{\partial x}}.$
In fact, it is classical even in the presence of harmonic forces.
6. State overlap is calculated as
$|\langle \psi |\theta \rangle |^{2}=2\pi \hbar \int _{-\infty }^{\infty }dx\int _{-\infty }^{\infty }dp\,W_{\psi }(x,p)W_{\theta }(x,p).$
7. Operator expectation values (averages) are calculated as phase-space averages of the respective Wigner transforms:
$g(x,p)\equiv \int _{-\infty }^{\infty }dy\,\left\langle x-{\frac {y}{2}}\right|{\hat {G}}\left|x+{\frac {y}{2}}\right\rangle e^{ipy/\hbar },$
$\langle \psi |{\hat {G}}|\psi \rangle =\operatorname {Tr} ({\hat {\rho }}{\hat {G}})=\int _{-\infty }^{\infty }dx\int _{-\infty }^{\infty }dp\,W(x,p)g(x,p).$
8. For W(x, p) to represent physical (positive) density matrices, it must satisfy
$\int _{-\infty }^{\infty }dx\,\int _{-\infty }^{\infty }dp\,W(x,p)W_{\theta }(x,p)\geq 0$
for all pure states |θ⟩.
9. By virtue of the Cauchy–Schwarz inequality, for a pure state, it is constrained to be bounded:
$-{\frac {2}{h}}\leq W(x,p)\leq {\frac {2}{h}}.$
This bound disappears in the classical limit, ħ → 0. In this limit, W(x, p) reduces to the probability density in coordinate space x, usually highly localized, multiplied by δ-functions in momentum: the classical limit is "spiky". Thus, this quantum-mechanical bound precludes a Wigner function which is a perfectly localized δ-function in phase space, as a reflection of the uncertainty principle.[6]
10. The Wigner transformation is simply the Fourier transform of the antidiagonals of the density matrix, when that matrix is expressed in a position basis.[7]
Examples
See also: Phase-space formulation § Simple harmonic oscillator
Let $|m\rangle \equiv {\frac {a^{\dagger m}}{\sqrt {m!}}}|0\rangle $ be the $m$-th Fock state of a quantum harmonic oscillator. Groenewold (1946) discovered its associated Wigner function, in dimensionless variables:
$W_{|m\rangle }(x,p)={\frac {(-1)^{m}}{\pi }}e^{-(x^{2}+p^{2})}L_{m}{\big (}2(p^{2}+x^{2}){\big )},$
where $L_{m}(x)$ denotes the $m$-th Laguerre polynomial.
This may follow from the expression for the static eigenstate wavefunctions,
$u_{m}(x)=\pi ^{-1/4}H_{m}(x)e^{-x^{2}/2},$
where $H_{m}$ is the $m$-th Hermite polynomial. From the above definition of the Wigner function, upon a change of integration variables,
$W_{|m\rangle }(x,p)={\frac {(-1)^{m}}{\pi ^{3/2}2^{m}m!}}e^{-x^{2}-p^{2}}\int _{-\infty }^{\infty }d\zeta \,e^{-\zeta ^{2}}H_{m}(\zeta -ip+x)H_{m}(\zeta -ip-x).$
The expression then follows from the integral relation between Hermite and Laguerre polynomials.[8]
Evolution equation for Wigner function
Main articles: Wigner–Weyl transform and Phase space formulation
The Wigner transformation is a general invertible transformation of an operator Ĝ on a Hilbert space to a function g(x, p) on phase space and is given by
$g(x,p)=\int _{-\infty }^{\infty }ds\,e^{ips/\hbar }\left\langle x-{\frac {s}{2}}\right|{\hat {G}}\left|x+{\frac {s}{2}}\right\rangle .$
Hermitian operators map to real functions. The inverse of this transformation, from phase space to Hilbert space, is called the Weyl transformation:
$\langle x|{\hat {G}}|y\rangle =\int _{-\infty }^{\infty }{\frac {dp}{h}}e^{ip(x-y)/\hbar }g\left({\frac {x+y}{2}},p\right)$
(not to be confused with the distinct Weyl transformation in differential geometry).
The Wigner function W(x, p) discussed here is thus seen to be the Wigner transform of the density matrix operator ρ̂. Thus the trace of an operator with the density matrix Wigner-transforms to the equivalent phase-space integral overlap of g(x, p) with the Wigner function.
The Wigner transform of the von Neumann evolution equation of the density matrix in the Schrödinger picture is Moyal's evolution equation for the Wigner function:
${\frac {\partial W(x,p,t)}{\partial t}}=-\{\{W(x,p,t),H(x,p)\}\},$
where H(x, p) is the Hamiltonian, and {{⋅, ⋅}} is the Moyal bracket. In the classical limit, ħ → 0, the Moyal bracket reduces to the Poisson bracket, while this evolution equation reduces to the Liouville equation of classical statistical mechanics.
Formally, the classical Liouville equation can be solved in terms of the phase-space particle trajectories which are solutions of the classical Hamilton equations. This technique of solving partial differential equations is known as the method of characteristics. This method transfers to quantum systems, where the characteristics' "trajectories" now determine the evolution of Wigner functions. The solution of the Moyal evolution equation for the Wigner function is represented formally as
$W(x,p,t)=W{\big (}\star {\big (}x_{-t}(x,p),p_{-t}(x,p){\big )},0{\big )},$
where $x_{t}(x,p)$ and $p_{t}(x,p)$ are the characteristic trajectories subject to the quantum Hamilton equations with initial conditions $x_{t=0}(x,p)=x$ and $p_{t=0}(x,p)=p$, and where $\star $-product composition is understood for all argument functions.
Since $\star $-composition of functions is thoroughly nonlocal (the "quantum probability fluid" diffuses, as observed by Moyal), vestiges of local trajectories in quantum systems are barely discernible in the evolution of the Wigner distribution function.[lower-alpha 2] In the integral representation of $\star $-products, successive operations by them have been adapted to a phase-space path integral, to solve the evolution equation for the Wigner function[9] (see also [10][11][12]). This non-local feature of Moyal time evolution[13] is illustrated in the gallery below, for Hamiltonians more complex than the harmonic oscillator. In the classical limit, the trajectory nature of the time evolution of Wigner functions becomes more and more distinct. At ħ = 0, the characteristics' trajectories reduce to the classical trajectories of particles in phase space.
Examples of Wigner-function time evolutions
• Pure state in a Morse potential. The green dashed lines represent level set of the Hamiltonian.
• Pure state in a quartic potential. The solid lines represent the level set of the Hamiltonian.
• Tunnelling of a wave packet through a potential barrier. The solid lines represent the level set of the Hamiltonian.
• Long-time evolution of a mixed state ρ in an anharmonic potential well. Marginals are plotted on the right (p) and top (x).
• An equilibrium mixed state ρ (evolves to itself), in the same anharmonic potential.
Harmonic-oscillator time evolution
In the special case of the quantum harmonic oscillator, however, the evolution is simple and appears identical to the classical motion: a rigid rotation in phase space with a frequency given by the oscillator frequency. This is illustrated in the gallery below. This same time evolution occurs with quantum states of light modes, which are harmonic oscillators.
Examples of Wigner-function time evolutions in a quantum harmonic oscillator
• A coherent state.[14]
• Combined ground state and 1st excited state.[14]
• A cat state; the marginals are plotted on the right (p) and underneath (x).
Classical limit
The Wigner function allows one to study the classical limit, offering a comparison of the classical and quantum dynamics in phase space.[15][16]
It has been suggested that the Wigner function approach can be viewed as a quantum analogy to the operatorial formulation of classical mechanics introduced in 1932 by Bernard Koopman and John von Neumann: the time evolution of the Wigner function approaches, in the limit ħ → 0, the time evolution of the Koopman–von Neumann wavefunction of a classical particle.[17]
Positivity of the Wigner function
As already noted, the Wigner function of quantum state typically takes some negative values. Indeed, for a pure state in one variable, if $W(x,p)\geq 0$ for all $x$ and $p$, then the wave function must have the form
$\psi (x)=e^{-ax^{2}+bx+c}$
for some complex numbers $a,b,c$ with $\operatorname {Re} (a)>0$ (Hudson's theorem[18]). Note that $a$ is allowed to be complex, so that $\psi $ is not necessarily a Gaussian wave packet in the usual sense. Thus, pure states with non-negative Wigner functions are not necessarily minimum-uncertainty states in the sense of the Heisenberg uncertainty formula; rather, they give equality in the Schrödinger uncertainty formula, which includes an anticommutator term in addition to the commutator term. (With careful definition of the respective variances, all pure-state Wigner functions lead to Heisenberg's inequality all the same.)
In higher dimensions, the characterization of pure states with non-negative Wigner functions is similar; the wave function must have the form
$\psi (x)=e^{-(x,Ax)+b\cdot x+c},$
where $A$ is a symmetric complex matrix whose real part is positive-definite, $b$ is a complex vector, and c is a complex number.[19] The Wigner function of any such state is a Gaussian distribution on phase space.
Soto and Claverie[19] give an elegant proof of this characterization, using the Segal–Bargmann transform. The reasoning is as follows. The Husimi Q function of $\psi $ may be computed as the squared magnitude of the Segal–Bargmann transform of $\psi $, multiplied by a Gaussian. Meanwhile, the Husimi Q function is the convolution of the Wigner function with a Gaussian. If the Wigner function of $\psi $ is non-negative everywhere on phase space, then the Husimi Q function will be strictly positive everywhere on phase space. Thus, the Segal–Bargmann transform $F(x+ip)$ of $\psi $ will be nowhere zero. Thus, by a standard result from complex analysis, we have
$F(x+ip)=e^{g(x+ip)}$
for some holomorphic function $g$. But in order for $F$ to belong to the Segal–Bargmann space—that is, for $F$ to be square-integrable with respect to a Gaussian measure—$g$ must have at most quadratic growth at infinity. From this, elementary complex analysis can be used to show that $g$ must actually be a quadratic polynomial. Thus, we obtain an explicit form of the Segal–Bargmann transform of any pure state whose Wigner function is non-negative. We can then invert the Segal–Bargmann transform to obtain the claimed form of the position wave function.
There does not appear to be any simple characterization of mixed states with non-negative Wigner functions.
The Wigner function in relation to other interpretations of quantum mechanics
It has been shown that the Wigner quasiprobability distribution function can be regarded as an ħ-deformation of another phase-space distribution function that describes an ensemble of de Broglie–Bohm causal trajectories.[20] Basil Hiley has shown that the quasi-probability distribution may be understood as the density matrix re-expressed in terms of a mean position and momentum of a "cell" in phase space, and the de Broglie–Bohm interpretation allows one to describe the dynamics of the centers of such "cells".[21][22]
There is a close connection between the description of quantum states in terms of the Wigner function and a method of quantum states reconstruction in terms of mutually unbiased bases.[23]
Uses of the Wigner function outside quantum mechanics
• In the modelling of optical systems such as telescopes or fibre telecommunications devices, the Wigner function is used to bridge the gap between simple ray tracing and the full wave analysis of the system. Here p/ħ is replaced with k = |k| sin θ ≈ |k|θ in the small-angle (paraxial) approximation. In this context, the Wigner function is the closest one can get to describing the system in terms of rays at position x and angle θ while still including the effects of interference.[24] If it becomes negative at any point, then simple ray tracing will not suffice to model the system. That is to say, negative values of this function are a symptom of the Gabor limit of the classical light signal and not of quantum features of light associated with ħ.
• In signal analysis, a time-varying electrical signal, mechanical vibration, or sound wave are represented by a Wigner function. Here, x is replaced with the time, and p/ħ is replaced with the angular frequency ω = 2πf, where f is the regular frequency.
• In ultrafast optics, short laser pulses are characterized with the Wigner function using the same f and t substitutions as above. Pulse defects such as chirp (the change in frequency with time) can be visualized with the Wigner function. See adjacent figure.
• In quantum optics, x and p/ħ are replaced with the X and P quadratures, the real and imaginary components of the electric field (see coherent state).
Measurements of the Wigner function
• Quantum tomography
• Frequency-resolved optical gating
Other related quasiprobability distributions
Main article: Quasiprobability distribution
The Wigner distribution was the first quasiprobability distribution to be formulated, but many more followed, formally equivalent and transformable to and from it (see Transformation between distributions in time–frequency analysis). As in the case of coordinate systems, on account of varying properties, several such have with various advantages for specific applications:
• Glauber P representation
• Husimi Q representation
Nevertheless, in some sense, the Wigner distribution holds a privileged position among all these distributions, since it is the only one whose requisite star-product drops out (integrates out by parts to effective unity) in the evaluation of expectation values, as illustrated above, and so can be visualized as a quasiprobability measure analogous to the classical ones.
Historical note
As indicated, the formula for the Wigner function was independently derived several times in different contexts. In fact, apparently, Wigner was unaware that even within the context of quantum theory, it had been introduced previously by Heisenberg and Dirac,[25][26] albeit purely formally: these two missed its significance, and that of its negative values, as they merely considered it as an approximation to the full quantum description of a system such as the atom. (Incidentally, Dirac would later become Wigner's brother-in-law, marrying his sister Manci.) Symmetrically, in most of his legendary 18-month correspondence with Moyal in the mid-1940s, Dirac was unaware that Moyal's quantum-moment generating function was effectively the Wigner function, and it was Moyal who finally brought it to his attention.[27]
See also
• Heisenberg group
• Wigner–Weyl transform
• Phase space formulation
• Moyal bracket
• Negative probability
• Optical equivalence theorem
• Modified Wigner distribution function
• Cohen's class distribution function
• Wigner distribution function
• Transformation between distributions in time–frequency analysis
• Squeezed coherent state
• Bilinear time–frequency distribution
• Continuous-variable quantum information
Footnotes
1. Specifically, since this convolution is invertible, in fact, no information has been sacrificed, and the full quantum entropy has not increased yet. However, if this resulting Husimi distribution is then used as a plain measure in a phase-space integral evaluation of expectation values without the requisite star product of the Husimi representation, then, at that stage, quantum information has been forfeited and the distribution is a semi-classical one, effectively. That is, depending on its usage in evaluating expectation values, the very same distribution may serve as a quantum or a classical distribution function.
2. Quantum characteristics should not be confused with trajectories of the Feynman path integral, or trajectories of the de Broglie–Bohm theory. This three-fold ambiguity allows better understanding of the position of Niels Bohr, who vigorously but counterproductively opposed the notion of trajectory in the atomic physics. At the 1948 Pocono Conference, e.g., he said to Richard Feynman: "... one could not talk about the trajectory of an electron in the atom, because it was something not observable". ("The Beat of a Different Drum: The Life and Science of Richard Feynman", by Jagdish Mehra (Oxford, 1994, pp. 245–248)). Arguments of this kind were widely used in the past by Ernst Mach in his criticism of an atomic theory of physics and later, in the 1960s, by Geoffrey Chew, Tullio Regge and others to motivate replacing the local quantum field theory by the S-matrix theory. Today, statistical physics entirely based on atomistic concepts is included in standard courses, the S-matrix theory went out of fashion, while the Feynman path-integral method has been recognized as the most efficient method in gauge theories.
References
1. E. P. Wigner (1932). "On the quantum correction for thermodynamic equilibrium". Physical Review. 40 (5): 749–759. Bibcode:1932PhRv...40..749W. doi:10.1103/PhysRev.40.749. hdl:10338.dmlcz/141466.
2. H. J. Groenewold (1946). "On the principles of elementary quantum mechanics". Physica. 12 (7): 405–460. Bibcode:1946Phy....12..405G. doi:10.1016/S0031-8914(46)80059-4.
3. H. Weyl (1927). "Quantenmechanik und gruppentheorie". Zeitschrift für Physik. 46 (1–2): 1. Bibcode:1927ZPhy...46....1W. doi:10.1007/BF02055756. S2CID 121036548.; H. Weyl, Gruppentheorie und Quantenmechanik (Leipzig: Hirzel) (1928); H. Weyl, The Theory of Groups and Quantum Mechanics (Dover, New York, 1931).
4. J. Ville, "Théorie et Applications de la Notion de Signal Analytique", Câbles et Transmission, 2, 61–74 (1948).
5. Moyal, J. E. (1949). "Quantum mechanics as a statistical theory". Mathematical Proceedings of the Cambridge Philosophical Society. Cambridge University Press (CUP). 45 (1): 99–124. doi:10.1017/s0305004100000487. ISSN 0305-0041. S2CID 124183640.
6. Curtright, T. L.; Zachos, C. K. (2012). "Quantum Mechanics in Phase Space". Asia Pacific Physics Newsletter. 1: 37. arXiv:1104.5269. doi:10.1142/S2251158X12000069. S2CID 119230734.; C. Zachos, D. Fairlie, and T. Curtright, Quantum Mechanics in Phase Space (World Scientific, Singapore, 2005). ISBN 978-981-238-384-6.
7. Hawkes, Peter W. (2018). Advances in Imaging and Electron Physics. Academic Press. p. 47. ISBN 9780128155424.
8. Schleich, Wolfgang P. (2001-02-09). Quantum Optics in Phase Space (1st ed.). Wiley. p. 105. doi:10.1002/3527602976. ISBN 978-3-527-29435-0.
9. B. Leaf (1968). "Weyl transform in nonrelativistic quantum dynamics". Journal of Mathematical Physics. 9 (5): 769–781. Bibcode:1968JMP.....9..769L. doi:10.1063/1.1664640.
10. P. Sharan (1979). "Star-product representation of path integrals". Physical Review D. 20 (2): 414–418. Bibcode:1979PhRvD..20..414S. doi:10.1103/PhysRevD.20.414.
11. M. S. Marinov (1991). "A new type of phase-space path integral". Physics Letters A. 153 (1): 5–11. Bibcode:1991PhLA..153....5M. doi:10.1016/0375-9601(91)90352-9.
12. B. Segev: Evolution kernels for phase space distributions. In: M. A. Olshanetsky; Arkady Vainshtein (2002). Multiple Facets of Quantization and Supersymmetry: Michael Marinov Memorial Volume. World Scientific. pp. 68–90. ISBN 978-981-238-072-2. Retrieved 26 October 2012. See especially section 5. "Path integral for the propagator" on pages 86–89. Also online.
13. M. Oliva, D. Kakofengitis, and O. Steuernagel (2018). "Anharmonic quantum mechanical systems do not feature phase space trajectories". Physica A. 502: 201–210. arXiv:1611.03303. Bibcode:2018PhyA..502..201O. doi:10.1016/j.physa.2017.10.047. S2CID 53691877.{{cite journal}}: CS1 maint: multiple names: authors list (link)
14. Curtright, T. L., Time-dependent Wigner Functions.
15. See, for example: Wojciech H. Zurek, Decoherence and the transition from quantum to classical – revisited, Los Alamos Science, 27, 2002, arXiv:quant-ph/0306072, pp. 15 ff.
16. See, for example: C. Zachos, D. Fairlie, T. Curtright, Quantum mechanics in phase space: an overview with selected papers, World Scientific, 2005. ISBN 978-981-4520-43-0.
17. Bondar, Denys I.; Cabrera, Renan; Zhdanov, Dmitry V.; Rabitz, Herschel A. (2013). "Wigner phase-space distribution as a wave function". Physical Review A. 88 (5): 052108. arXiv:1202.3628. doi:10.1103/PhysRevA.88.052108. ISSN 1050-2947. S2CID 119155284.
18. Hudson, Robin L. (1974). "When is the Wigner quasi-probability density non-negative?". Reports on Mathematical Physics. 6 (2): 249–252. Bibcode:1974RpMP....6..249H. doi:10.1016/0034-4877(74)90007-X.
19. F. Soto and P. Claverie, "When is the Wigner function of multidimensional systems nonnegative?", Journal of Mathematical Physics 24 (1983) 97–100.
20. Dias, Nuno Costa; Prata, João Nuno (2002). "Bohmian trajectories and quantum phase space distributions". Physics Letters A. 302 (5–6): 261–272. arXiv:quant-ph/0208156v1. doi:10.1016/s0375-9601(02)01175-1. ISSN 0375-9601. S2CID 39936409.
21. B. J. Hiley: Phase space descriptions of quantum phenomena, in: A. Khrennikov (ed.): Quantum Theory: Re-consideration of Foundations–2, pp. 267–286, Växjö University Press, Sweden, 2003 (PDF).
22. B. Hiley: Moyal's characteristic function, the density matrix and von Neumann's idempotent (preprint).
23. F. C. Khanna, P. A. Mello, M. Revzen, Classical and Quantum Mechanical State Reconstruction, arXiv:1112.3164v1 [quant-ph] (submitted December 14, 2011).
24. Bazarov, Ivan V. (2012-05-03). "Synchrotron radiation representation in phase space". Physical Review Special Topics - Accelerators and Beams. American Physical Society (APS). 15 (5): 050703. doi:10.1103/physrevstab.15.050703. ISSN 1098-4402. S2CID 53489256.
25. W. Heisenberg (1931). "Über die inkohärente Streuung von Röntgenstrahlen". Physikalische Zeitschrift. 32: 737–740.
26. Dirac, P. A. M. (1930). "Note on Exchange Phenomena in the Thomas Atom". Mathematical Proceedings of the Cambridge Philosophical Society. Cambridge University Press (CUP). 26 (3): 376–385. doi:10.1017/s0305004100016108. ISSN 0305-0041. S2CID 97185766.
27. Ann Moyal, (2006), "Maverick Mathematician: The Life and Science of J. E. Moyal", ANU E-press, 2006, ISBN 1-920942-59-9.
Further reading
• M. Levanda and V. Fleurov, "Wigner quasi-distribution function for charged particles in classical electromagnetic fields", Annals of Physics, 292, 199–231 (2001). arXiv:cond-mat/0105137.
External links
• wigner Wigner function implementation in QuTiP.
• Quantum Optics Gallery.
• Sonogram Visible Speech GPL-licensed freeware for the Wigner quasiprobability distribution of signal files.
| Wikipedia |
Wijsman convergence
Wijsman convergence is a variation of Hausdorff convergence suitable for work with unbounded sets. Intuitively, Wijsman convergence is to convergence in the Hausdorff metric as pointwise convergence is to uniform convergence.
History
The convergence was defined by Robert Wijsman.[1] The same definition was used earlier by Zdeněk Frolík.[2] Yet earlier, Hausdorff in his book Grundzüge der Mengenlehre defined so called closed limits; for proper metric spaces it is the same as Wijsman convergence.
Definition
Let (X, d) be a metric space and let Cl(X) denote the collection of all d-closed subsets of X. For a point x ∈ X and a set A ∈ Cl(X), set
$d(x,A)=\inf _{a\in A}d(x,a).$
A sequence (or net) of sets Ai ∈ Cl(X) is said to be Wijsman convergent to A ∈ Cl(X) if, for each x ∈ X,
$d(x,A_{i})\to d(x,A).$
Wijsman convergence induces a topology on Cl(X), known as the Wijsman topology.
Properties
• The Wijsman topology depends very strongly on the metric d. Even if two metrics are uniformly equivalent, they may generate different Wijsman topologies.
• Beer's theorem: if (X, d) is a complete, separable metric space, then Cl(X) with the Wijsman topology is a Polish space, i.e. it is separable and metrizable with a complete metric.
• Cl(X) with the Wijsman topology is always a Tychonoff space. Moreover, one has the Levi-Lechicki theorem: (X, d) is separable if and only if Cl(X) is either metrizable, first-countable or second-countable.
• If the pointwise convergence of Wijsman convergence is replaced by uniform convergence (uniformly in x), then one obtains Hausdorff convergence, where the Hausdorff metric is given by
$d_{\mathrm {H} }(A,B)=\sup _{x\in X}{\big |}d(x,A)-d(x,B){\big |}.$
The Hausdorff and Wijsman topologies on Cl(X) coincide if and only if (X, d) is a totally bounded space.
See also
• Hausdorff distance
• Kuratowski convergence
• Vietoris topology
• Hemicontinuity
References
Notes
1. Wijsman, Robert A. (1966). "Convergence of sequences of convex sets, cones and functions. II". Trans. Amer. Math. Soc. American Mathematical Society. 123 (1): 32–45. doi:10.2307/1994611. JSTOR 1994611. MR0196599
2. Z. Frolík, Concerning topological convergence of sets, Czechoskovak Math. J. 10 (1960), 168–180
Bibliography
• Beer, Gerald (1993). Topologies on closed and closed convex sets. Mathematics and its Applications 268. Dordrecht: Kluwer Academic Publishers Group. pp. xii+340. ISBN 0-7923-2531-1. MR1269778
• Beer, Gerald (1994). "Wijsman convergence: a survey". Set-Valued Anal. 2 (1–2): 77–94. doi:10.1007/BF01027094. MR1285822
External links
• Som Naimpally (2001) [1994], "Wijsman convergence", Encyclopedia of Mathematics, EMS Press
| Wikipedia |
Wiktor Eckhaus
Wiktor Eckhaus (28 June 1930 – 1 October 2000) was a Polish–Dutch mathematician, known for his work on the field of differential equations. He was Professor Emeritus of Applied Mathematics at the Utrecht University.
Wiktor Eckhaus
Born(1930-06-28)28 June 1930[1]
Stanisławów, Poland
Died1 October 2000(2000-10-01) (aged 70)
Amstelveen,[2] Netherlands
NationalityNetherlands[3]
Alma materMassachusetts Institute of Technology
Known forEckhaus instability
Eckhaus equation
Scientific career
Fieldsmathematics,
aerodynamics
InstitutionsUtrecht University,
Delft University of Technology,
National Aerospace Laboratory
Doctoral advisorLeon Trilling
Biography
Eckhaus was born into a wealthy family, and raised in Warsaw where his father was managing a fur company. During the German occupation of Poland, he, his mother and sister had to hide because of their Jewish descent. His father, after being a prisoner of war, joined the Russian Army. After the war, in 1947, the re-united family came to Amsterdam – via a refugee camp in Austria.
Wiktor passed the state exam of the Hogere Burgerschool in 1948, and started to study aeronautics at the Delft University of Technology. Following his graduation he worked with the National Aerospace Laboratory in Amsterdam, from 1953 till 1957. In the period 1957–1960 he worked at the Massachusetts Institute of Technology, where Eckhaus earned a PhD in 1959 under Leon Trilling on a dissertation entitled "Some problems of unsteady flow with discontinuities".
In 1960, he became a "maître de recherches" (senior research fellow) at the Department of Mechanics of the Sorbonne. In 1964 he was a visiting professor at the University of Amsterdam and the Mathematical Centre. Thereafter, in 1965, he became professor at the Delft University of Technology, in pure and applied mathematics and mechanics. From 1972 until his retirement in 1994, Eckhaus was professor of applied mathematics at the Utrecht University.
Initially he studied the flow around airfoils, leading to his research on the stability of solutions to (weakly nonlinear) differential equations. This resulted in what is now known as the Eckhaus instability criterion and Eckhaus instability, appearing for instance as a secondary instability in models of Rayleigh–Bénard convection. Later, Eckhaus worked on singular perturbation theory and soliton equations.
In 1983 he treated strongly singular relaxation oscillations – called "canards" (French for "ducks") – resulting in his most-read paper "Relaxation oscillations including a standard chase on French ducks".[4] Eckhaus used standard methods of analysis, on a problem qualified before, by Marc Diener, as an example of a problem only treatable through the use of non-standard analysis.[5]
He became a member of the Royal Netherlands Academy of Arts and Sciences in 1987.[2]
Publications
• Eckhaus, W. (1965), Studies in nonlinear stability theory, Springer Tracts in Natural Philosophy, vol. 6, Springer, ISBN 978-3-642-88319-4
• —— (1973), Matched asymptotic expansions and singular perturbations, Mathematics Studies, vol. 6, North Holland, ISBN 978-0-7204-2600-7
• —— (1979), Asymptotic analysis of singular perturbations, Studies in Mathematics and its Applications, vol. 9, North Holland, ISBN 978-0-444-85306-6
• ——; van Harten, A. (1981), The inverse scattering transformation and the theory of solitons – An introduction, Mathematics Studies, vol. 50, North Holland, ISBN 978-0-444-55731-5
• —— (1975), "New approach to the asymptotic theory of nonlinear oscillations and wave propagation", Journal of Mathematical Analysis and Applications, 49 (3): 575–611, doi:10.1016/0022-247X(75)90200-0
• —— (1983), "Relaxation oscillations including a standard chase on French ducks", in Verhulst, F. (ed.), Asymptotic Analysis II – Surveys and New Trends, Lecture Notes in Mathematics, vol. 985, Springer, pp. 449–494, doi:10.1007/BFb0062381, ISBN 978-3-540-12286-9
• —— (1993), "The Ginzburg–Landau manifold is an attractor", Journal of Nonlinear Science, 3 (1): 329–348, Bibcode:1993JNS.....3..329E, doi:10.1007/BF02429869, S2CID 122662589
• ——; de Jager, E.M. (1966), "Asymptotic solutions of singular perturbation problems for linear differential equations of elliptic type", Arch. Rat. Mech. Anal., 23 (1): 26–86, Bibcode:1966ArRMA..23...26E, doi:10.1007/BF00281135, S2CID 843282
• —— (1997), Witus en de jaren van angst – Een reconstructie [Witus and the years of fear – A reconstruction] (Autobiography) (in Dutch), Bas Lubberhuizen, ISBN 9789073978690
Notes
1. See Eckhaus (1997) and Doelman et al. (2001) for a discussion on his date and place of birth.
2. Wiktor Eckhaus (1929 – 2000), Koninklijke Nederlandse Akademie van Wetenschappen, retrieved 2014-09-06
3. Zitting 1964-1965-8184 – Naturalisatie van van den Berg, Lilli Elfriede en 28 anderen [Session 1964-1965-8184 – Naturalisation of van den Berg, Lilli Elfriede and 28 others] (pdf) (in Dutch), Staten Generaal, 14 August 1965, retrieved 2014-09-06
4. Eckhaus (1983)
5. Martin Wechselberger (ed.). "Canards". Scholarpedia.
References
• Doelman, A.; Duistermaat, H.; Grasman, J.; van Harten, A. (2001), "In memoriam Wiktor Eckhaus" (PDF), Nieuw Archief voor Wiskunde (in Dutch), Koninklijk Wiskundig Genootschap, 2 (1): 18–20.
Also appeared as: Duistermaat, J.J.; Doelman, A.; Grasman, J.; van Harten, A. (2003), "Levensbericht W. Eckhaus" [Eulogy W. Eckhaus] (PDF), Levensberichten en Herdenkingen (in Dutch), Amsterdam: Koninklijke Nederlandse Akademie van Wetenschappen: 19–24.
External links
• Wiktor Eckhaus at the Mathematics Genealogy Project
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Wilberd van der Kallen
Wilberd Leo Johan van der Kallen (born 15 January 1947 in Nieuwer-Amstel)[1] is a Dutch mathematician.
W. L. J. van der Kallen completed his undergraduate study of mathematics and physics at Utrecht University.[2] There he received his PhD in 1973 with thesis advisor T. A. Springer and thesis Infinitesimally central extensions of Chevalley groups.[3] In 1969 van der Kallen became a teaching assistant in Utrecht University's Mathematics Department and has spent his career there, eventually becoming a tenured professor. His research deals with algebraic K-theory and the representation theory of algebraic groups, among other topics. He has frequently been a visiting professor at Northwestern University in Evanston, Illinois and at the Tata Institute of Fundamental Research in Mumbai.[2]
He is the author or coauthor of over 60 research articles.[2] In 1977 he published an analogue of a 1977 theorem of Andrei Suslin[4][5] and a generalization of a 1969 theorem of Hideya Matsumoto.[6] In 1978 van der Kallen was an invited speaker at the International Congress of Mathematicians in Helsinki.[7] His 1980 paper Homology stability for linear groups[8] has over 200 citations. His 1977 paper Rational and generic cohomology, written with 3 other mathematicians,[9] has over 240 citations.
Books
• van der Kallen, Wilberd (1993). Lectures on Frobenius Splittings and B-modules. ISBN 978-81-85198-60-6; 98 pages{{cite book}}: CS1 maint: postscript (link)
• van der Kallen, Wilberd (15 November 2006). Infinitesimally Central Extensions of Chevalley Groups. Lecture Notes in Mathematics, vol. 356. Springer. ISBN 978-3-540-37857-0; pbk reprint of 1973 original, 154 pages{{cite book}}: CS1 maint: postscript (link)
• Cohen, Arjeh M.; Hesselink, Wim H.; van der Kallen, Wilberd L. J.; Strooker, Jan R., eds. (15 November 2006). Algebraic Groups. Utrecht 1986: Proceedings of a Symposium in Honour of T.A. Springer. Lecture Notes in Mathematics, vol. 1271. Springer. ISBN 978-3-540-47834-8; reprint of 1987 original{{cite book}}: CS1 maint: postscript (link)
References
1. "XI 5 Wilberd van der Kallen" (PDF). deel 10 : Albert (X 7) — wilberdk.home.xs4all.nl.
2. "Wilberd van der Kallen (with links to publication list)". Geometry and Quantum Theory (GQT) (a national Dutch mathematical research cluster).
3. Wilberd L. J. van der Kallen at the Mathematics Genealogy Project
4. van der Kallen, Wilberd (1977). "Another presentation for Steinberg groups" (PDF). Indagationes Mathematicae (Proceedings). 80 (4): 304–312. doi:10.1016/1385-7258(77)90026-9.
5. Suslin, A. A. (1977). "On the Structure of the Special Linear Group over Polynomial Rings". Mathematics of the USSR-Izvestiya. 11 (2): 221–238. Bibcode:1977IzMat..11..221S. doi:10.1070/IM1977v011n02ABEH001709.
6. van der Kallen, W. (1977). "The $K_{2}$ of rings with many units" (PDF). Annales scientifiques de l'École Normale Supérieure. 10 (4): 473–515. doi:10.24033/asens.1334.
7. van der Kallen, W. (1978). "Generators and relations in algebraic K-theory" (PDF). Proceedings of the International Conference of Mathematicians at Helsinki. Vol. 1. pp. 305–310.
8. van der Kallen, Wilberd (1980). "Homology stability for linear groups". Inventiones Mathematicae. 60 (3): 269–295. Bibcode:1980InMat..60..269K. doi:10.1007/BF01390018. S2CID 54671324.
9. Cline, Edward; Parshall, Brian; Scott, Leonard; van der Kallen, W. (197). "Rational and generic cohomology" (PDF). Inventiones Mathematicae. 39 (2): 143–163. doi:10.1007/BF01390106. S2CID 14358269.
External links
• "Wilberd van der Kallen". Universiteit Utrecht (webspace.science.uu.nl).
• "Slides of some talks".
• "Linear Algebraic Groups: an overview by Wilberd van der Kallen". YouTube. matsciencechannel. August 20, 2013.
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| Wikipedia |
Gibbs phenomenon
In mathematics, the Gibbs phenomenon is the oscillatory behavior of the Fourier series of a piecewise continuously differentiable periodic function around a jump discontinuity. The $ N$th partial Fourier series of the function (formed by summing the $ N$ lowest constituent sinusoids of the Fourier series of the function) produces large peaks around the jump which overshoot and undershoot the function values. As more sinusoids are used, this approximation error approaches a limit of about 9% of the jump, though the infinite Fourier series sum does eventually converge almost everywhere (pointwise convergence on continuous points) except points of discontinuity.[1]
The Gibbs phenomenon was observed by experimental physicists and was believed to be due to imperfections in the measuring apparatus,[2] but it is in fact a mathematical result. It is one cause of ringing artifacts in signal processing.
Description
The Gibbs phenomenon is a behavior of the Fourier series of a function with a jump discontinuity and is described as the following:
As more Fourier series constituents or components are taken, the Fourier series shows the first overshoot in the oscillatory behavior around the jump point approaching ~ 9% of the (full) jump and this oscillation does not disappear but gets closer to the point so that the integral of the oscillation approaches to zero (i.e., zero energy in the oscillation).
At the jump point, the Fourier series gives the average of the function's both side limits toward the point.
Square wave example
The three pictures on the right demonstrate the Gibbs phenomenon for a square wave (with peak-to-peak amplitude of $ c$ from $ -c/2$ to $ c/2$ and the periodicity $ L$) whose $ N$th partial Fourier series is
${\frac {2c}{\pi }}\left(\sin(\omega x)+{\frac {1}{3}}\sin(3\omega x)+\cdots +{\frac {1}{N-1}}\sin((N-1)\omega x)\right)$
where $ \omega =2\pi /L$. More precisely, this square wave is the function $ f(x)$ which equals ${\tfrac {c}{2}}$ between $ 2n(L/2)$ and $ (2n+1)(L/2)$ and $ -{\tfrac {c}{2}}$ between $ (2n+1)(L/2)$ and $ (2n+2)(L/2)$ for every integer $ n$; thus, this square wave has a jump discontinuity of peak-to-peak height $ c$ at every integer multiple of $ L/2$.
As more sinusoidal terms are added (i.e., increasing $ N$), the error of the partial Fourier series converges to a fixed height. But because the width of the error continues to narrow, the area of the error – and hence the energy of the error – converges to 0.[3] The square wave analysis reveals that the error exceeds the height (from zero) ${\tfrac {c}{2}}$ of the square wave by
${\frac {c}{\pi }}\int _{0}^{\pi }{\frac {\sin(t)}{t}}\ dt-{\frac {c}{2}}=c\cdot (0.089489872236\dots ).$
(OEIS: A243268)
or about 9% of the full jump $ c$. More generally, at any discontinuity of a piecewise continuously differentiable function with a jump of $ c$, the $ N$th partial Fourier series of the function will (for a very large $ N$ value) overshoot this jump by an error approaching $ c\cdot (0.089489872236\dots )$ at one end and undershoot it by the same amount at the other end; thus the "full jump" in the partial Fourier series will be about 18% larger than the full jump in the original function. At the discontinuity, the partial Fourier series will converge to the midpoint of the jump (regardless of the actual value of the original function at the discontinuity) as a consequence of Dirichlet's theorem.[4] The quantity
$\int _{0}^{\pi }{\frac {\sin t}{t}}\ dt=(1.851937051982\dots )={\frac {\pi }{2}}+\pi \cdot (0.089489872236\dots )$
(OEIS: A036792)
is sometimes known as the Wilbraham–Gibbs constant.
History
The Gibbs phenomenon was first noticed and analyzed by Henry Wilbraham in an 1848 paper.[5] The paper attracted little attention until 1914 when it was mentioned in Heinrich Burkhardt's review of mathematical analysis in Klein's encyclopedia.[6] In 1898, Albert A. Michelson developed a device that could compute and re-synthesize the Fourier series.[7] A widespread myth says that when the Fourier coefficients for a square wave were input to the machine, the graph would oscillate at the discontinuities, and that because it was a physical device subject to manufacturing flaws, Michelson was convinced that the overshoot was caused by errors in the machine. In fact the graphs produced by the machine were not good enough to exhibit the Gibbs phenomenon clearly, and Michelson may not have noticed it as he made no mention of this effect in his paper (Michelson & Stratton 1898) about his machine or his later letters to Nature.[8]
Inspired by correspondence in Nature between Michelson and A. E. H. Love about the convergence of the Fourier series of the square wave function, J. Willard Gibbs published a note in 1898 pointing out the important distinction between the limit of the graphs of the partial sums of the Fourier series of a sawtooth wave and the graph of the limit of those partial sums. In his first letter Gibbs failed to notice the Gibbs phenomenon, and the limit that he described for the graphs of the partial sums was inaccurate. In 1899 he published a correction in which he described the overshoot at the point of discontinuity (Nature, April 27, 1899, p. 606). In 1906, Maxime Bôcher gave a detailed mathematical analysis of that overshoot, coining the term "Gibbs phenomenon"[9] and bringing the term into widespread use.[8]
After the existence of Henry Wilbraham's paper became widely known, in 1925 Horatio Scott Carslaw remarked, "We may still call this property of Fourier's series (and certain other series) Gibbs's phenomenon; but we must no longer claim that the property was first discovered by Gibbs."[10]
Explanation
Informally, the Gibbs phenomenon reflects the difficulty inherent in approximating a discontinuous function by a finite series of continuous sinusoidal waves. It is important to put emphasis on the word finite, because even though every partial sum of the Fourier series overshoots around each discontinuity it is approximating, the limit of summing an infinite number of sinusoidal waves does not. The overshoot peaks moves closer and closer to the discontinuity as more terms are summed, so convergence is possible.
There is no contradiction (between the overshoot error converging to a non-zero height even though the infinite sum has no overshoot), because the overshoot peaks move toward the discontinuity. The Gibbs phenomenon thus exhibits pointwise convergence, but not uniform convergence. For a piecewise continuously differentiable (class C1) function, the Fourier series converges to the function at every point except at jump discontinuities. At jump discontinuities, the infinite sum will converge to the jump discontinuity's midpoint (i.e. the average of the values of the function on either side of the jump), as a consequence of Dirichlet's theorem.[4]
The Gibbs phenomenon is closely related to the principle that the smoothness of a function controls the decay rate of its Fourier coefficients. Fourier coefficients of smoother functions will more rapidly decay (resulting in faster convergence), whereas Fourier coefficients of discontinuous functions will slowly decay (resulting in slower convergence). For example, the discontinuous square wave has Fourier coefficients $({\tfrac {1}{1}},{\scriptstyle {\text{0}}},{\tfrac {1}{3}},{\scriptstyle {\text{0}}},{\tfrac {1}{5}},{\scriptstyle {\text{0}}},{\tfrac {1}{7}},{\scriptstyle {\text{0}}},{\tfrac {1}{9}},{\scriptstyle {\text{0}}},\dots )$ that decay only at the rate of ${\tfrac {1}{n}}$, while the continuous triangle wave has Fourier coefficients $({\tfrac {1}{1^{2}}},{\scriptstyle {\text{0}}},{\tfrac {-1}{3^{2}}},{\scriptstyle {\text{0}}},{\tfrac {1}{5^{2}}},{\scriptstyle {\text{0}}},{\tfrac {-1}{7^{2}}},{\scriptstyle {\text{0}}},{\tfrac {1}{9^{2}}},{\scriptstyle {\text{0}}},\dots )$ that decay at a much faster rate of ${\tfrac {1}{n^{2}}}$.
This only provides a partial explanation of the Gibbs phenomenon, since Fourier series with absolutely convergent Fourier coefficients would be uniformly convergent by the Weierstrass M-test and would thus be unable to exhibit the above oscillatory behavior. By the same token, it is impossible for a discontinuous function to have absolutely convergent Fourier coefficients, since the function would thus be the uniform limit of continuous functions and therefore be continuous, a contradiction. See Convergence of Fourier series § Absolute convergence.
Solutions
In practice, the difficulties associated with the Gibbs phenomenon can be ameliorated by using a smoother method of Fourier series summation, such as Fejér summation or Riesz summation, or by using sigma-approximation. Using a continuous wavelet transform, the wavelet Gibbs phenomenon never exceeds the Fourier Gibbs phenomenon.[11] Also, using the discrete wavelet transform with Haar basis functions, the Gibbs phenomenon does not occur at all in the case of continuous data at jump discontinuities,[12] and is minimal in the discrete case at large change points. In wavelet analysis, this is commonly referred to as the Longo phenomenon. In the polynomial interpolation setting, the Gibbs phenomenon can be mitigated using the S-Gibbs algorithm.[13]
Formal mathematical description of the Gibbs phenomenon
Let $ f:{\mathbb {R} }\to {\mathbb {R} }$ be a piecewise continuously differentiable function which is periodic with some period $ L>0$. Suppose that at some point $ x_{0}$, the left limit $ f(x_{0}^{-})$ and right limit $ f(x_{0}^{+})$ of the function $ f$ differ by a non-zero jump of $ c$:
$f(x_{0}^{+})-f(x_{0}^{-})=c\neq 0.$
For each positive integer $ N$ ≥ 1, let $ S_{N}f(x)$ be the $ N$th partial Fourier series ($ S_{N}$ can be treated as a mathematical operator on functions.)
$S_{N}f(x):=\sum _{-N\leq n\leq N}{\widehat {f}}(n)e^{\frac {i2\pi nx}{L}}={\frac {1}{2}}a_{0}+\sum _{n=1}^{N}\left(a_{n}\cos \left({\frac {2\pi nx}{L}}\right)+b_{n}\sin \left({\frac {2\pi nx}{L}}\right)\right),$
where the Fourier coefficients $ {\widehat {f}}(n),a_{n},b_{n}$ for integers $ n$ are given by the usual formulae
${\widehat {f}}(n):={\frac {1}{L}}\int _{0}^{L}f(x)e^{-{\frac {i2\pi nx}{L}}}\,dx$
$a_{0}:={\frac {1}{L}}\int _{0}^{L}f(x)\ dx$
$a_{n}:={\frac {2}{L}}\int _{0}^{L}f(x)\cos \left({\frac {2\pi nx}{L}}\right)\,dx$
$b_{n}:={\frac {2}{L}}\int _{0}^{L}f(x)\sin \left({\frac {2\pi nx}{L}}\right)\,dx.$
Then we have
$\lim _{N\to \infty }S_{N}f\left(x_{0}+{\frac {L}{2N}}\right)=f(x_{0}^{+})+c\cdot (0.089489872236\dots )$
and
$\lim _{N\to \infty }S_{N}f\left(x_{0}-{\frac {L}{2N}}\right)=f(x_{0}^{-})-c\cdot (0.089489872236\dots )$
but
$\lim _{N\to \infty }S_{N}f(x_{0})={\frac {f(x_{0}^{-})+f(x_{0}^{+})}{2}}.$
More generally, if $ x_{N}$ is any sequence of real numbers which converges to $ x_{0}$ as $ N\to \infty $, and if the jump of $ a$ is positive then
$\limsup _{N\to \infty }S_{N}f(x_{N})\leq f(x_{0}^{+})+c\cdot (0.089489872236\dots )$
and
$\liminf _{N\to \infty }S_{N}f(x_{N})\geq f(x_{0}^{-})-c\cdot (0.089489872236\dots ).$
If instead the jump of $ c$ is negative, one needs to interchange limit superior ($ \limsup $) with limit inferior ($ \liminf $), and also interchange the $ \leq $ and $ \geq $ signs, in the above two inequalities.
Proof of the Gibbs phenomenon in a general case
Stated again, let $ f:{\mathbb {R} }\to {\mathbb {R} }$ be a piecewise continuously differentiable function which is periodic with some period $ L>0$, and this function has multiple jump discontinuity points denoted $ x_{i}$ where $ i=0,1,2,$ and so on. At each discontinuity, the amount of the vertical full jump is $ c_{i}$.
Then, $ f$ can be expressed as the sum of a continuous function $ f_{c}$ and a multi-step function $ f_{s}$ which is the sum of step functions such as[14]
$f=f_{c}+f_{s},$
$f_{s}=f_{s_{1}}+f_{s_{2}}+f_{s_{3}}+\cdots ,$
$f_{s_{i}}(x)={\begin{cases}0&{\text{if }}x\leq x_{i},\\c_{i},&{\text{if }}x>x_{i}.\end{cases}}$
$ S_{N}f(x)$ as the $ N$th partial Fourier series of $ f=f_{c}+f_{s}=f_{c}+\left(f_{s_{1}}+f_{s_{2}}+f_{s_{3}}+\ldots \right)$ will converge well at all $ x$ points except points near discontinuities $ x_{i}$. Around each discontinuity point $ x_{i}$, $ f_{s_{i}}$ will only have the Gibbs phenomenon of its own (the maximum oscillatory convergence error of ~ 9 % of the jump $c_{i}$, as shown in the square wave analysis) because other functions are continuous ($f_{c}$) or flat zero ($f_{s_{j}}$ where $j\neq i$) around that point. This proves how the Gibbs phenomenon occurs at every discontinuity.
Signal processing explanation
From a signal processing point of view, the Gibbs phenomenon is the step response of a low-pass filter, and the oscillations are called ringing or ringing artifacts. Truncating the Fourier transform of a signal on the real line, or the Fourier series of a periodic signal (equivalently, a signal on the circle), corresponds to filtering out the higher frequencies with an ideal (brick-wall) low-pass filter. This can be represented as convolution of the original signal with the impulse response of the filter (also known as the kernel), which is the sinc function. Thus, the Gibbs phenomenon can be seen as the result of convolving a Heaviside step function (if periodicity is not required) or a square wave (if periodic) with a sinc function: the oscillations in the sinc function cause the ripples in the output.
In the case of convolving with a Heaviside step function, the resulting function is exactly the integral of the sinc function, the sine integral; for a square wave the description is not as simply stated. For the step function, the magnitude of the undershoot is thus exactly the integral of the left tail until the first negative zero: for the normalized sinc of unit sampling period, this is $ \int _{-\infty }^{-1}{\frac {\sin(\pi x)}{\pi x}}\,dx.$ The overshoot is accordingly of the same magnitude: the integral of the right tail or (equivalently) the difference between the integral from negative infinity to the first positive zero minus 1 (the non-overshooting value).
The overshoot and undershoot can be understood thus: kernels are generally normalized to have integral 1, so they result in a mapping of constant functions to constant functions – otherwise they have gain. The value of a convolution at a point is a linear combination of the input signal, with coefficients (weights) the values of the kernel.
If a kernel is non-negative, such as for a Gaussian kernel, then the value of the filtered signal will be a convex combination of the input values (the coefficients (the kernel) integrate to 1, and are non-negative), and will thus fall between the minimum and maximum of the input signal – it will not undershoot or overshoot. If, on the other hand, the kernel assumes negative values, such as the sinc function, then the value of the filtered signal will instead be an affine combination of the input values and may fall outside of the minimum and maximum of the input signal, resulting in undershoot and overshoot, as in the Gibbs phenomenon.
Taking a longer expansion – cutting at a higher frequency – corresponds in the frequency domain to widening the brick-wall, which in the time domain corresponds to narrowing the sinc function and increasing its height by the same factor, leaving the integrals between corresponding points unchanged. This is a general feature of the Fourier transform: widening in one domain corresponds to narrowing and increasing height in the other. This results in the oscillations in sinc being narrower and taller, and (in the filtered function after convolution) yields oscillations that are narrower (and thus with smaller area) but which do not have reduced magnitude: cutting off at any finite frequency results in a sinc function, however narrow, with the same tail integrals. This explains the persistence of the overshoot and undershoot.
• Oscillations can be interpreted as convolution with a sinc.
• Higher cutoff makes the sinc narrower but taller, with the same magnitude tail integrals, yielding higher frequency oscillations, but whose magnitude does not vanish.
Thus, the features of the Gibbs phenomenon are interpreted as follows:
• the undershoot is due to the impulse response having a negative tail integral, which is possible because the function takes negative values;
• the overshoot offsets this, by symmetry (the overall integral does not change under filtering);
• the persistence of the oscillations is because increasing the cutoff narrows the impulse response but does not reduce its integral – the oscillations thus move towards the discontinuity, but do not decrease in magnitude.
Square wave analysis
We examine the $ N$th partial Fourier series $ S_{N}f(x)$ of a square wave $ f(x)$ with the periodicity $ L$ and a discontinuity of a vertical "full" jump $ c$ from $ y=y_{0}$ at $ x=x_{0}$. Because the case of odd $ N$ is very similar, let us just deal with the case when $ N$ is even:
$S_{N}f(x)=\left(y_{0}+{\frac {c}{2}}\right)+{\frac {2c}{\pi }}\left(\sin(\omega (x-x_{0}))+{\frac {1}{3}}\sin(3\omega (x-x_{0}))+\cdots +{\frac {1}{N-1}}\sin((N-1)\omega (x-x_{0}))\right)$
with $ \omega ={\frac {2\pi }{L}}$. ($ N=2N'$ where $ N'$ is the number of non-zero sinusoidal Fourier series components so there are literatures using $ N'$ instead of $ N$.) Substituting $ x=x_{0}$ (a point of discontinuity), we obtain
$S_{N}f(x_{0})=\left(y_{0}+{\frac {c}{2}}\right)={\frac {f(0^{-})+f(0^{+})}{2}}={\frac {y_{0}+(y_{0}+c)}{2}}$
as claimed above. (The first term that only survives is the average of the Fourier series.)
Next, we find the first maximum of the oscillation around the discontinuity $ x=x_{0}$ by checking the first and second derivatives of $ S_{N}f(x)$. The first condition for the maximum is that the first derivative equals to zero as
${\frac {d}{dx}}S_{N}f(x)={\frac {2c}{\pi }}\left(\cos(\omega (x-x_{0}))+\cos(3\omega (x-x_{0}))+\cdots +\cos((N-1)\omega (x-x_{0}))\right)={\frac {c}{\pi }}{\frac {\sin(N\omega (x-x_{0}))}{\sin(\omega (x-x_{0}))}}=0$
where the 2nd equality is from one of Lagrange's trigonometric identities. Solving this condition gives $ x-x_{0}=k\pi /(N\omega )=kL/(2N)$ for integers $ k$ excluding multiples of $ N\omega $ to avoid the zero denominator, so $ k=1,2,\ldots ,N\omega -1,N\omega +1,\ldots $ and their negatives are allowed.
The second derivative of $ S_{N}f(x)$ at $ x-x_{0}=kL/(2N)$ is
${\frac {d^{2}}{dx^{2}}}S_{N}f(x)={\frac {c\omega }{\pi }}\left({\frac {N\cos(N\omega (x-x_{0}))\sin(\omega (x-x_{0}))-\sin(N\omega (x-x_{0}))\cos(\omega (x-x_{0}))}{\sin ^{2}(\omega (x-x_{0}))}}\right),$
$\left.{\frac {d^{2}}{dx^{2}}}S_{N}f(x)\right\vert _{x_{0}+kL/(2N)}={\begin{cases}{\frac {2c}{L}}{\frac {N}{\sin(k\pi /N)}},&{\text{if }}k{\text{ is even,}}\\[4pt]{\frac {2c}{L}}{\frac {-N}{\sin(k\pi /N)}},&{\text{if }}k{\text{ is odd.}}\end{cases}}$
Thus, the first maximum occurs at $ x=x_{0}+L/(2N)$ ($ k=1$) and $ S_{N}f(x)$ at this $ x$ value is
$S_{N}f\left(x_{0}+{\frac {L}{2N}}\right)=\left(y_{0}+{\frac {c}{2}}\right)+{\frac {2c}{\pi }}\left(\sin \left({\frac {\pi }{N}}\right)+{\frac {1}{3}}\sin \left({\frac {3\pi }{N}}\right)+\cdots +{\frac {1}{N-1}}\sin \left({\frac {(N-1)\pi }{N}}\right)\right)$
If we introduce the normalized sinc function $ \operatorname {sinc} (x)={\frac {\sin(\pi x)}{\pi x}}$ for $ x\neq 0$, we can rewrite this as
$S_{N}f\left(x_{0}+{\frac {L}{2N}}\right)=(y_{0}+{\frac {c}{2}})+c\left[{\frac {2}{N}}\operatorname {sinc} \left({\frac {1}{N}}\right)+{\frac {2}{N}}\operatorname {sinc} \left({\frac {3}{N}}\right)+\cdots +{\frac {2}{N}}\operatorname {sinc} \left({\frac {(N-1)}{N}}\right)\right].$
For a sufficiently large $ N$, the expression in the square brackets is a Riemann sum approximation to the integral $ \int _{0}^{1}\operatorname {sinc} (x)\ dx$ (more precisely, it is a midpoint rule approximation with spacing ${\tfrac {2}{N}}$). Since the sinc function is continuous, this approximation converges to the integral as $N\to \infty $. Thus, we have
${\begin{aligned}\lim _{N\to \infty }S_{N}f\left(x_{0}+{\frac {L}{2N}}\right)&=(y_{0}+{\frac {c}{2}})+c\int _{0}^{1}\operatorname {sinc} (x)\,dx\\[8pt]&=(y_{0}+{\frac {c}{2}})+{\frac {c}{\pi }}\int _{x=0}^{1}{\frac {\sin(\pi x)}{\pi x}}\,d(\pi x)\\[8pt]&=(y_{0}+{\frac {c}{2}})+{\frac {c}{\pi }}\int _{0}^{\pi }{\frac {\sin(t)}{t}}\ dt\quad =\quad (y_{0}+c)+c\cdot (0.089489872236\dots ),\end{aligned}}$
which was claimed in the previous section. A similar computation shows
$\lim _{N\to \infty }S_{N}f\left(x_{0}-{\frac {L}{2N}}\right)=-c\int _{0}^{1}\operatorname {sinc} (x)\,dx=y_{0}-c\cdot (0.089489872236\dots ).$
Consequences
The Gibbs phenomenon is undesirable because it causes artifacts, namely clipping from the overshoot and undershoot, and ringing artifacts from the oscillations. In the case of low-pass filtering, these can be reduced or eliminated by using different low-pass filters.
In MRI, the Gibbs phenomenon causes artifacts in the presence of adjacent regions of markedly differing signal intensity. This is most commonly encountered in spinal MRIs where the Gibbs phenomenon may simulate the appearance of syringomyelia.
The Gibbs phenomenon manifests as a cross pattern artifact in the discrete Fourier transform of an image,[15] where most images (e.g. micrographs or photographs) have a sharp discontinuity between boundaries at the top / bottom and left / right of an image. When periodic boundary conditions are imposed in the Fourier transform, this jump discontinuity is represented by continuum of frequencies along the axes in reciprocal space (i.e. a cross pattern of intensity in the Fourier transform).
And although this article mainly focused on the difficulty with trying to construct discontinuities without artifacts in the time domain with only a partial Fourier series, it is also important to consider that because the inverse Fourier transform is extremely similar to the Fourier transform, there equivalently is difficulty with trying to construct discontinuities in the frequency domain using only a partial Fourier series. Thus for instance because idealized brick-wall and rectangular filters have discontinuities in the frequency domain, their exact representation in the time domain necessarily requires an infinitely-long sinc filter impulse response, since a finite impulse response will result in Gibbs rippling in the frequency response near cut-off frequencies, though this rippling can be reduced by windowing finite impulse response filters (at the expense of wider transition bands).[16]
See also
• Mach bands
• Pinsky phenomenon
• Runge's phenomenon (a similar phenomenon in polynomial approximations)
• σ-approximation which adjusts a Fourier summation to eliminate the Gibbs phenomenon which would otherwise occur at discontinuities
• Sine integral
Notes
1. H. S. Carslaw (1930). "Chapter IX". Introduction to the theory of Fourier's series and integrals (Third ed.). New York: Dover Publications Inc.
2. Vretblad 2000 Section 4.7.
3. "6.7: Gibbs Phenomena". Engineering LibreTexts. 2020-05-24. Retrieved 2022-03-03.
4. M. Pinsky (2002). Introduction to Fourier Analysis and Wavelets. United states of America: Brooks/Cole. p. 27.
5. Wilbraham, Henry (1848) "On a certain periodic function", The Cambridge and Dublin Mathematical Journal, 3 : 198–201.
6. Encyklopädie der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen (PDF). Vol. II T. 1 H 1. Wiesbaden: Vieweg+Teubner Verlag. 1914. p. 1049. Retrieved 14 September 2016.
7. Hammack, Bill; Kranz, Steve; Carpenter, Bruce (2014-10-29). Albert Michelson's Harmonic Analyzer: A Visual Tour of a Nineteenth Century Machine that Performs Fourier Analysis. Articulate Noise Books. ISBN 9780983966173. Retrieved 14 September 2016.
8. Hewitt, Edwin; Hewitt, Robert E. (1979). "The Gibbs-Wilbraham phenomenon: An episode in Fourier analysis". Archive for History of Exact Sciences. 21 (2): 129–160. doi:10.1007/BF00330404. S2CID 119355426. Available on-line at: National Chiao Tung University: Open Course Ware: Hewitt & Hewitt, 1979. Archived 2016-03-04 at the Wayback Machine
9. Bôcher, Maxime (April 1906) "Introduction to the theory of Fourier's series", Annals of Mathethematics, second series, 7 (3) : 81–152. The Gibbs phenomenon is discussed on pages 123–132; Gibbs's role is mentioned on page 129.
10. Carslaw, H. S. (1 October 1925). "A historical note on Gibbs' phenomenon in Fourier's series and integrals". Bulletin of the American Mathematical Society. 31 (8): 420–424. doi:10.1090/s0002-9904-1925-04081-1. ISSN 0002-9904. Retrieved 14 September 2016.
11. Rasmussen, Henrik O. "The Wavelet Gibbs Phenomenon". In Wavelets, Fractals and Fourier Transforms, Eds M. Farge et al., Clarendon Press, Oxford, 1993.
12. Susan E., Kelly (1995). "Gibbs Phenomenon for Wavelets" (PDF). Applied and Computational Harmonic Analysis (3). Archived from the original (PDF) on 2013-09-09. Retrieved 2012-03-31.
13. De Marchi, Stefano; Marchetti, Francesco; Perracchione, Emma; Poggiali, Davide (2020). "Polynomial interpolation via mapped bases without resampling". J. Comput. Appl. Math. 364: 112347. doi:10.1016/j.cam.2019.112347. ISSN 0377-0427. S2CID 199688130.
14. Fay, Temple H.; Kloppers, P. Hendrik (2001). "The Gibbs' phenomenon". International Journal of Mathematical Education in Science and Technology. 32 (1): 73–89. doi:10.1080/00207390117151.
15. R. Hovden, Y. Jiang, H.L. Xin, L.F. Kourkoutis (2015). "Periodic Artifact Reduction in Fourier Transforms of Full Field Atomic Resolution Images". Microscopy and Microanalysis. 21 (2): 436–441. arXiv:2210.09024. doi:10.1017/S1431927614014639. PMID 25597865. S2CID 22435248.{{cite journal}}: CS1 maint: multiple names: authors list (link)
16. "Gibbs phenomenon | RecordingBlogs". www.recordingblogs.com. Retrieved 2022-03-05.
References
• Gibbs, J. Willard (1898), "Fourier's Series", Nature, 59 (1522): 200, doi:10.1038/059200b0, ISSN 0028-0836, S2CID 4004787
• Gibbs, J. Willard (1899), "Fourier's Series", Nature, 59 (1539): 606, doi:10.1038/059606a0, ISSN 0028-0836, S2CID 13420929
• Michelson, A. A.; Stratton, S. W. (1898), "A new harmonic analyser", Philosophical Magazine, 5 (45): 85–91
• Zygmund, Antoni (1959). Trigonometric Series (2nd ed.). Cambridge University Press. Volume 1, Volume 2.
• Wilbraham, Henry (1848), "On a certain periodic function", The Cambridge and Dublin Mathematical Journal, 3: 198–201
• Paul J. Nahin, Dr. Euler's Fabulous Formula, Princeton University Press, 2006. Ch. 4, Sect. 4.
• Vretblad, Anders (2000), Fourier Analysis and its Applications, Graduate Texts in Mathematics, vol. 223, New York: Springer Publishing, p. 93, ISBN 978-0-387-00836-3
External links
• Media related to Gibbs phenomenon at Wikimedia Commons
• "Gibbs phenomenon", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Weisstein, Eric W., "Gibbs Phenomenon". From MathWorld—A Wolfram Web Resource.
• Prandoni, Paolo, "Gibbs Phenomenon".
• Radaelli-Sanchez, Ricardo, and Richard Baraniuk, "Gibbs Phenomenon". The Connexions Project. (Creative Commons Attribution License)
• Horatio S Carslaw : Introduction to the theory of Fourier's series and integrals.pdf (introductiontot00unkngoog.pdf ) at archive.org
• A Python implementation of the S-Gibbs algorithm mitigating the Gibbs Phenomenon https://github.com/pog87/FakeNodes.
| Wikipedia |
Wild arc
In geometric topology, a wild arc is an embedding of the unit interval into 3-dimensional space not equivalent to the usual one in the sense that there does not exist an ambient isotopy taking the arc to a straight line segment. Antoine (1920) found the first example of a wild arc, and Fox & Artin (1948) found another example called the Fox-Artin arc whose complement is not simply connected.
See also
• Wild knot
• Horned sphere
Further reading
• Antoine, L. (1920), "Sur la possibilité d'étendre l'homéomorphie de deux figures à leurs voisinages", C. R. Acad. Sci. Paris (in French), 171: 661
• Fox, Ralph H.; Harrold, O. G. (1962), "The Wilder arcs", Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961), Prentice Hall, pp. 184–187, MR 0140096
• Fox, Ralph H.; Artin, Emil (1948), "Some wild cells and spheres in three-dimensional space", Annals of Mathematics, Second Series, 49 (4): 979–990, doi:10.2307/1969408, ISSN 0003-486X, JSTOR 1969408, MR 0027512
• Hocking, John Gilbert; Young, Gail Sellers (1988) [1961]. Topology. Dover. pp. 176–177. ISBN 0-486-65676-4.
• McPherson, James M. (1973), "Wild arcs in three-space. I. Families of Fox–Artin arcs", Pacific Journal of Mathematics, 45 (2): 585–598, doi:10.2140/pjm.1973.45.585, ISSN 0030-8730, MR 0343276
Topology
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| Wikipedia |
Wild number
Originally, wild numbers are the numbers supposed to belong to a fictional sequence of numbers imagined to exist in the mathematical world of the mathematical fiction The Wild Numbers authored by Philibert Schogt, a Dutch philosopher and mathematician.[1] Even though Schogt has given a definition of the wild number sequence in his novel, it is couched in a deliberately imprecise language that the definition turns out to be no definition at all. However, the author claims that the first few members of the sequence are 11, 67, 2, 4769, 67. Later, inspired by this wild and erratic behaviour of the fictional wild numbers, American mathematician J. C. Lagarias used the terminology to describe a precisely defined sequence of integers which shows somewhat similar wild and erratic behaviour. Lagaria's wild numbers are connected with the Collatz conjecture and the concept of the 3x + 1 semigroup.[2][3] The original fictional sequence of wild numbers has found a place in the On-Line Encyclopedia of Integer Sequences.[4]
The wild number problem
In the novel The Wild Numbers, The Wild Number Problem is formulated as follows:
• Beauregard had defined a number of deceptively simple operations, which, when applied to a whole number, at first resulted in fractions. But if the same steps were repeated often enough, the eventual outcome was once again a whole number. Or, as Beauregard cheerfully observed: “In all numbers lurks a wild number, guaranteed to emerge when you provoke them long enough” . 0 yielded the wild number 11, 1 brought forth 67, 2 itself, 3 suddenly manifested itself as 4769, 4, surprisingly, brought forth 67 again. Beauregard himself had found fifty different wild numbers. The money prize was now awarded to whoever found a new one.[5]
But it has not been specified what those "deceptively simple operations" are. Consequently, there is no way of knowing how those numbers 11, 67, etc. were obtained and no way of finding what the next wild number would be.
History of The Wild Number Problem
The novel The Wild Numbers has constructed a fictitious history for The Wild Number Problem. The important milestones in this history can be summarised as follows.
DateEvent
1823Anatole Millechamps de Beauregard poses the Wild Number Problem in its original form.
1830sThe problem is generalised: How many wild numbers are there?
Are there infinitely many wild numbers?
It was conjectured that all numbers are wild.
1907Heinrich Riedel disproves the conjecture by showing that 3 is not a wild number.
Later he also proves that there are infinitely many non-wild numbers.
Early 1960sDimitri Arkanov sparks renewed interest in the almost forgotten problem
by discovering a fundamental relationship between wild numbers and prime numbers.
The presentIsaac Swift finds a solution.
Real wild numbers
In mathematics, the multiplicative semigroup, denoted by W0, generated by the set $\left\{{\frac {3n+2}{2n+1}}:n\geq 0\right\}$ is called the Wooley semigroup in honour of the American mathematician Trevor D. Wooley. The multiplicative semigroup, denoted by W, generated by the set $\left\{{\frac {1}{2}}\right\}\cup \left\{{\frac {3n+2}{2n+1}}:n\geq 0\right\}$ is called the wild semigroup. The set of integers in W0 is itself a multiplicative semigroup. It is called the Wooley integer semigroup and members of this semigroup are called Wooley integers. Similarly, the set of integers in W is itself a multiplicative semigroup. It is called the wild integer semigroup and members of this semigroup are called wild numbers.[6]
The wild numbers in OEIS
The On-Line Encyclopedia of Integer Sequences (OEIS) has an entry with the identifying number A058883 relating to the wild numbers. According to OEIS, "apparently these are completely fictional and there is no mathematical explanation". However, the OEIS has some entries relating to pseudo-wild numbers carrying well-defined mathematical explanations.[4]
Sequences of pseudo-wild numbers
Even though the sequence of wild numbers is entirely fictional, several mathematicians have tried to find rules that would generate the sequence of the fictional wild numbers. All these attempts have resulted in failures. However, in the process, certain new sequences of integers were created having similar wild and erratic behavior. These well-defined sequences are referred to as sequences of pseudo-wild numbers. A good example of this is the one discovered by the Dutch mathematician Floor van Lamoen. This sequence is defined as follows:[7][8]
For a rational number p/q let
$f(p/q)={\frac {pq}{{\text{ sum of digits of }}p{\text{ and }}q}}$.
For a positive integer n, the n-th pseudo-wild number is the number obtained by iterating f, starting at n/1, until an integer is reached, and if no integer is reached the pseudo-wild number is 0.
For example, taking n=2, we have
${\frac {2}{1}},{\frac {2}{3}},{\frac {6}{5}},{\frac {30}{11}},66$
and so the second pseudo-wild number is 66. The first few pseudo-wild numbers are
66, 66, 462, 180, 66, 31395, 714, 72, 9, 5.
References
1. Philibert Schogt (March 23, 2000). The Wild Numbers: A Novel (First ed.). Four Walls Eight Windows. ISBN 978-1568581668.
2. Emmer, Michele (2013). Imagine Math 2: Between Culture and Mathematics. Springer Science & Business Media. pp. 37–38. ISBN 9788847028890.
3. Applegate, David; Lagarias, Jeffrey C. (2006). "The 3x + 1 semigroup". Journal of Number Theory. 117 (1): 146–159. doi:10.1016/j.jnt.2005.06.010. MR 2204740.
4. "A058883 : The "Wild Numbers", from the novel of the same title (Version 1)". OEIS. The OEIS Foundation. Retrieved 19 March 2016.
5. Philibert Schogt (March 23, 2000). The Wild Numbers: A Novel (First ed.). Four Walls Eight Windows. p. 34. ISBN 978-1568581668.
6. Jeffrey C. Lagarias (February 2006). "Wild and Wooley Numbers" (PDF). American Mathematical Monthly. 113 (2): 98–108. doi:10.2307/27641862. JSTOR 27641862. Retrieved 28 March 2016.
7. Schogt, Philibert (2012). "The Wild Number Problem: Math or fiction?". arXiv:1211.6583 [math.HO].
8. "A059175". OEIS. The OEIS Foundation. Retrieved 30 March 2016.
| Wikipedia |
Wild problem
In the mathematical areas of linear algebra and representation theory, a problem is wild if it contains the problem of classifying pairs of square matrices up to simultaneous similarity.[1][2][3] Examples of wild problems are classifying indecomposable representations of any quiver that is neither a Dynkin quiver (i.e. the underlying undirected graph of the quiver is a (finite) Dynkin diagram) nor a Euclidean quiver (i.e., the underlying undirected graph of the quiver is an affine Dynkin diagram).[4]
Necessary and sufficient conditions have been proposed to check the simultaneously block triangularization and diagonalization of a finite set of matrices under the assumption that each matrix is diagonalizable over the field of the complex numbers.[5]
See also
• Semi-invariant of a quiver
References
1. Nazarova, L. A. (1974), "Representations of partially ordered sets of infinite type", Funkcional'nyi Analiz i ego Priloženija, 8 (4): 93–94, MR 0354455
2. Gabriel, P.; Nazarova, L. A.; Roĭter, A. V.; Sergeĭchuk, V. V.; Vossieck, D. (1993), "Tame and wild subspace problems", Akademīya Nauk Ukraïni, 45 (3): 313–352, doi:10.1007/BF01061008, MR 1238674, S2CID 122603779
3. Shavarovskiĭ, B. Z. (2004), "On some "tame" and "wild" aspects of the problem of semiscalar equivalence of polynomial matrices", Matematicheskie Zametki, 76 (1): 119–132, doi:10.1023/B:MATN.0000036747.26055.cb, MR 2099848, S2CID 120324840
4. Drozd, Yuriy A.; Golovashchuk, Natalia S.; Zembyk, Vasyl V. (2017), "Representations of nodal algebras of type E", Algebra and Discrete Mathematics, 23 (1): 16–34, hdl:123456789/155928, MR 3634499
5. Mesbahi, Afshin; Haeri, Mohammad (2015), "Conditions on decomposing linear systems with more than one matrix to block triangular or diagonal form", IEEE Transactions on Automatic Control, 60 (1): 233–239, doi:10.1109/TAC.2014.2326292, MR 3299432, S2CID 27053281
| Wikipedia |
Wiles's proof of Fermat's Last Theorem
Wiles's proof of Fermat's Last Theorem is a proof by British mathematician Andrew Wiles of a special case of the modularity theorem for elliptic curves. Together with Ribet's theorem, it provides a proof for Fermat's Last Theorem. Both Fermat's Last Theorem and the modularity theorem were almost universally considered inaccessible to prove by contemporaneous mathematicians, meaning that they were believed to be impossible to prove using current knowledge.[1]: 203–205, 223, 226
Wiles first announced his proof on 23 June 1993 at a lecture in Cambridge entitled "Modular Forms, Elliptic Curves and Galois Representations".[2] However, in September 1993 the proof was found to contain an error. One year later on 19 September 1994, in what he would call "the most important moment of [his] working life", Wiles stumbled upon a revelation that allowed him to correct the proof to the satisfaction of the mathematical community. The corrected proof was published in 1995.[3]
Wiles's proof uses many techniques from algebraic geometry and number theory, and has many ramifications in these branches of mathematics. It also uses standard constructions of modern algebraic geometry, such as the category of schemes and Iwasawa theory, and other 20th-century techniques which were not available to Fermat. The proof's method of identification of a deformation ring with a Hecke algebra (now referred to as an R=T theorem) to prove modularity lifting theorems has been an influential development in algebraic number theory.
Together, the two papers which contain the proof are 129 pages long,[4][5] and consumed over seven years of Wiles's research time. John Coates described the proof as one of the highest achievements of number theory, and John Conway called it "the proof of the [20th] century."[6] Wiles's path to proving Fermat's Last Theorem, by way of proving the modularity theorem for the special case of semistable elliptic curves, established powerful modularity lifting techniques and opened up entire new approaches to numerous other problems. For proving Fermat's Last Theorem, he was knighted, and received other honours such as the 2016 Abel Prize. When announcing that Wiles had won the Abel Prize, the Norwegian Academy of Science and Letters described his achievement as a "stunning proof".[3]
Precursors to Wiles's proof
Fermat's Last Theorem and progress prior to 1980
Main article: Fermat's Last Theorem
Fermat's Last Theorem, formulated in 1637, states that no three positive integers a, b, and c can satisfy the equation
$a^{n}+b^{n}=c^{n}$
if n is an integer greater than two (n > 2).
Over time, this simple assertion became one of the most famous unproved claims in mathematics. Between its publication and Andrew Wiles's eventual solution over 350 years later, many mathematicians and amateurs attempted to prove this statement, either for all values of n > 2, or for specific cases. It spurred the development of entire new areas within number theory. Proofs were eventually found for all values of n up to around 4 million, first by hand, and later by computer. However, no general proof was found that would be valid for all possible values of n, nor even a hint how such a proof could be undertaken.
The Taniyama–Shimura–Weil conjecture
Main article: Modularity theorem
Separately from anything related to Fermat's Last Theorem, in the 1950s and 1960s Japanese mathematician Goro Shimura, drawing on ideas posed by Yutaka Taniyama, conjectured that a connection might exist between elliptic curves and modular forms. These were mathematical objects with no known connection between them. Taniyama and Shimura posed the question whether, unknown to mathematicians, the two kinds of object were actually identical mathematical objects, just seen in different ways.
They conjectured that every rational elliptic curve is also modular. This became known as the Taniyama–Shimura conjecture. In the West, this conjecture became well known through a 1967 paper by André Weil, who gave conceptual evidence for it; thus, it is sometimes called the Taniyama–Shimura–Weil conjecture.
By around 1980, much evidence had been accumulated to form conjectures about elliptic curves, and many papers had been written which examined the consequences if the conjecture were true, but the actual conjecture itself was unproven and generally considered inaccessible—meaning that mathematicians believed a proof of the conjecture was probably impossible using current knowledge.
For decades, the conjecture remained an important but unsolved problem in mathematics. Around 50 years after first being proposed, the conjecture was finally proven and renamed the modularity theorem, largely as a result of Andrew Wiles's work described below.
Frey's curve
On yet another separate branch of development, in the late 1960s, Yves Hellegouarch came up with the idea of associating hypothetical solutions (a, b, c) of Fermat's equation with a completely different mathematical object: an elliptic curve.[7] The curve consists of all points in the plane whose coordinates (x, y) satisfy the relation
$y^{2}=x(x-a^{n})(x+b^{n}).$
Such an elliptic curve would enjoy very special properties due to the appearance of high powers of integers in its equation and the fact that an + bn = cn would be an nth power as well.
In 1982–1985, Gerhard Frey called attention to the unusual properties of this same curve, now called a Frey curve. He showed that it was likely that the curve could link Fermat and Taniyama, since any counterexample to Fermat's Last Theorem would probably also imply that an elliptic curve existed that was not modular. Frey showed that there were good reasons to believe that any set of numbers (a, b, c, n) capable of disproving Fermat's Last Theorem could also probably be used to disprove the Taniyama–Shimura–Weil conjecture. Therefore, if the Taniyama–Shimura–Weil conjecture were true, no set of numbers capable of disproving Fermat could exist, so Fermat's Last Theorem would have to be true as well.
Mathematically, the conjecture says that each elliptic curve with rational coefficients can be constructed in an entirely different way, not by giving its equation but by using modular functions to parametrise coordinates x and y of the points on it. Thus, according to the conjecture, any elliptic curve over Q would have to be a modular elliptic curve, yet if a solution to Fermat's equation with non-zero a, b, c and n greater than 2 existed, the corresponding curve would not be modular, resulting in a contradiction. If the link identified by Frey could be proven, then in turn, it would mean that a disproof of Fermat's Last Theorem would disprove the Taniyama–Shimura–Weil conjecture, or by contraposition, a proof of the latter would prove the former as well.[8]
Ribet's theorem
Main article: Ribet's theorem
To complete this link, it was necessary to show that Frey's intuition was correct: that a Frey curve, if it existed, could not be modular. In 1985, Jean-Pierre Serre provided a partial proof that a Frey curve could not be modular. Serre did not provide a complete proof of his proposal; the missing part (which Serre had noticed early on[9]: 1 ) became known as the epsilon conjecture (sometimes written ε-conjecture; now known as Ribet's theorem). Serre's main interest was in an even more ambitious conjecture, Serre's conjecture on modular Galois representations, which would imply the Taniyama–Shimura–Weil conjecture. However his partial proof came close to confirming the link between Fermat and Taniyama.
In the summer of 1986, Ken Ribet succeeded in proving the epsilon conjecture, now known as Ribet's theorem. His article was published in 1990. In doing so, Ribet finally proved the link between the two theorems by confirming, as Frey had suggested, that a proof of the Taniyama–Shimura–Weil conjecture for the kinds of elliptic curves Frey had identified, together with Ribet's theorem, would also prove Fermat's Last Theorem.
In mathematical terms, Ribet's theorem showed that if the Galois representation associated with an elliptic curve has certain properties (which Frey's curve has), then that curve cannot be modular, in the sense that there cannot exist a modular form which gives rise to the same Galois representation.[10]
Situation prior to Wiles's proof
Following the developments related to the Frey curve, and its link to both Fermat and Taniyama, a proof of Fermat's Last Theorem would follow from a proof of the Taniyama–Shimura–Weil conjecture—or at least a proof of the conjecture for the kinds of elliptic curves that included Frey's equation (known as semistable elliptic curves).
• From Ribet's Theorem and the Frey curve, any 4 numbers able to be used to disprove Fermat's Last Theorem could also be used to make a semistable elliptic curve ("Frey's curve") that could never be modular;
• But if the Taniyama–Shimura–Weil conjecture were also true for semistable elliptic curves, then by definition every Frey's curve that existed must be modular.
• The contradiction could have only one answer: if Ribet's theorem and the Taniyama–Shimura–Weil conjecture for semistable curves were both true, then it would mean there could not be any solutions to Fermat's equation—because then there would be no Frey curves at all, meaning no contradictions would exist. This would finally prove Fermat's Last Theorem.
However, despite the progress made by Serre and Ribet, this approach to Fermat was widely considered unusable as well, since almost all mathematicians saw the Taniyama–Shimura–Weil conjecture itself as completely inaccessible to proof with current knowledge.[1]: 203–205, 223, 226 For example, Wiles's ex-supervisor John Coates stated that it seemed "impossible to actually prove",[1]: 226 and Ken Ribet considered himself "one of the vast majority of people who believed [it] was completely inaccessible".[1]: 223
Andrew Wiles
Hearing of Ribet's 1986 proof of the epsilon conjecture, English mathematician Andrew Wiles, who had studied elliptic curves and had a childhood fascination with Fermat, decided to begin working in secret towards a proof of the Taniyama–Shimura–Weil conjecture, since it was now professionally justifiable,[11] as well as because of the enticing goal of proving such a long-standing problem.
Ribet later commented that "Andrew Wiles was probably one of the few people on earth who had the audacity to dream that you can actually go and prove [it]."[1]: 223
Announcement and subsequent developments
Wiles initially presented his proof in 1993. It was finally accepted as correct, and published, in 1995, following the correction of a subtle error in one part of his original paper. His work was extended to a full proof of the modularity theorem over the following six years by others, who built on Wiles's work.
Announcement and final proof (1993–1995)
During 21–23 June 1993, Wiles announced and presented his proof of the Taniyama–Shimura conjecture for semistable elliptic curves, and hence of Fermat's Last Theorem, over the course of three lectures delivered at the Isaac Newton Institute for Mathematical Sciences in Cambridge, England.[2] There was a relatively large amount of press coverage afterwards.[12]
After the announcement, Nick Katz was appointed as one of the referees to review Wiles's manuscript. In the course of his review, he asked Wiles a series of clarifying questions that led Wiles to recognise that the proof contained a gap. There was an error in one critical portion of the proof which gave a bound for the order of a particular group: the Euler system used to extend Kolyvagin and Flach's method was incomplete. The error would not have rendered his work worthless—each part of Wiles's work was highly significant and innovative by itself, as were the many developments and techniques he had created in the course of his work, and only one part was affected.[1]: 289, 296–297 Without this part proved, however, there was no actual proof of Fermat's Last Theorem.
Wiles spent almost a year trying to repair his proof, initially by himself and then in collaboration with his former student Richard Taylor, without success.[13][14][15] By the end of 1993, rumours had spread that under scrutiny, Wiles's proof had failed, but how seriously was not known. Mathematicians were beginning to pressure Wiles to disclose his work whether or not complete, so that the wider community could explore and use whatever he had managed to accomplish. Instead of being fixed, the problem, which had originally seemed minor, now seemed very significant, far more serious, and less easy to resolve.[16]
Wiles states that on the morning of 19 September 1994, he was on the verge of giving up and was almost resigned to accepting that he had failed, and to publishing his work so that others could build on it and find the error. He states that he was having a final look to try to understand the fundamental reasons why his approach could not be made to work, when he had a sudden insight that the specific reason why the Kolyvagin–Flach approach would not work directly also meant that his original attempt using Iwasawa theory could be made to work if he strengthened it using experience gained from the Kolyvagin–Flach approach since then. Each was inadequate by itself, but fixing one approach with tools from the other would resolve the issue and produce a class number formula (CNF) valid for all cases that were not already proven by his refereed paper:[13][17]
I was sitting at my desk examining the Kolyvagin–Flach method. It wasn't that I believed I could make it work, but I thought that at least I could explain why it didn't work. Suddenly I had this incredible revelation. I realised that, the Kolyvagin–Flach method wasn't working, but it was all I needed to make my original Iwasawa theory work from three years earlier. So out of the ashes of Kolyvagin–Flach seemed to rise the true answer to the problem. It was so indescribably beautiful; it was so simple and so elegant. I couldn't understand how I'd missed it and I just stared at it in disbelief for twenty minutes. Then during the day I walked around the department, and I'd keep coming back to my desk looking to see if it was still there. It was still there. I couldn't contain myself, I was so excited. It was the most important moment of my working life. Nothing I ever do again will mean as much.
— Andrew Wiles, quoted by Simon Singh[18]
On 6 October Wiles asked three colleagues (including Faltings) to review his new proof,[19] and on 24 October 1994 Wiles submitted two manuscripts, "Modular elliptic curves and Fermat's Last Theorem"[4] and "Ring theoretic properties of certain Hecke algebras",[5] the second of which Wiles had written with Taylor and proved that certain conditions were met which were needed to justify the corrected step in the main paper.
The two papers were vetted and finally published as the entirety of the May 1995 issue of the Annals of Mathematics. The new proof was widely analysed, and became accepted as likely correct in its major components.[6][10][11] These papers established the modularity theorem for semistable elliptic curves, the last step in proving Fermat's Last Theorem, 358 years after it was conjectured.
Subsequent developments
Fermat claimed to "... have discovered a truly marvelous proof of this, which this margin is too narrow to contain".[20][21] Wiles's proof is very complex, and incorporates the work of so many other specialists that it was suggested in 1994 that only a small number of people were capable of fully understanding at that time all the details of what he had done.[2][22] The complexity of Wiles's proof motivated a 10-day conference at Boston University; the resulting book of conference proceedings aimed to make the full range of required topics accessible to graduate students in number theory.[9]
As noted above, Wiles proved the Taniyama–Shimura–Weil conjecture for the special case of semistable elliptic curves, rather than for all elliptic curves. Over the following years, Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor (sometimes abbreviated as "BCDT") carried the work further, ultimately proving the Taniyama–Shimura–Weil conjecture for all elliptic curves in a 2001 paper.[23] Now proved, the conjecture became known as the modularity theorem.
In 2005, Dutch computer scientist Jan Bergstra posed the problem of formalizing Wiles's proof in such a way that it could be verified by computer.[24]
Summary of Wiles's proof
Wiles proved the modularity theorem for semistable elliptic curves, from which Fermat’s last theorem follows using proof by contradiction. In this proof method, one assumes the opposite of what is to be proved, and shows if that were true, it would create a contradiction. The contradiction shows that the assumption (that the conclusion is wrong) must have been incorrect, requiring the conclusion to hold.
The proof falls roughly in two parts. In the first part, Wiles proves a general result about "lifts", known as the "modularity lifting theorem". This first part allows him to prove results about elliptic curves by converting them to problems about Galois representations of elliptic curves. He then uses this result to prove that all semistable curves are modular, by proving that the Galois representations of these curves are modular.
Outline proof Comment
Part 1: setting up the proof
1 We start by assuming (for the sake of contradiction) that Fermat's Last Theorem is incorrect. That would mean there is at least one non-zero solution (a, b, c, n) (with all numbers rational and n > 2 and prime) to an + bn = cn.
2 Ribet's theorem (using Frey and Serre's work) shows that we can create a semistable elliptic curve E using the numbers (a, b, c, and n), which is never modular. If we can prove that all such elliptic curves will be modular (meaning that they match a modular form), then we have our contradiction and have proved our assumption (that such a set of numbers exists) was wrong. If the assumption is wrong, that means no such numbers exist, which proves Fermat's Last Theorem is correct.
3 Suppose that Fermat's Last Theorem is incorrect. This means a set of numbers (a, b, c, n) must exist that is a solution of Fermat's equation, and we can use the solution to create a Frey curve which is semistable and elliptic. So we assume that (somehow) we have found a solution and created such a curve (which we will call "E"), and see what happens.
Part 2: the modularity lifting theorem
4 Galois representations of elliptic curves ρ(E, p) for any prime p > 3 have been studied by many mathematicians.
Wiles aims first of all to prove a result about these representations, that he will use later: that if a semistable elliptic curve E has a Galois representation ρ(E, p) that is modular, the elliptic curve itself must be modular.
Proving this is helpful in two ways: it makes counting and matching easier, and, significantly, to prove the representation is modular, we would only have to prove it for one single prime number p, and we can do this using any prime that makes our work easy – it does not matter which prime we use.
This is the most difficult part of the problem – technically it means proving that if the Galois representation ρ(E, p) is a modular form, so are all the other related Galois representations ρ(E, p∞) for all powers of p.[3] This is the so-called "modular lifting problem", and Wiles approached it using deformations.
Any elliptic curve (or a representation of an elliptic curve) can be categorized as either reducible or irreducible. The proof will be slightly different depending whether or not the elliptic curve's representation is reducible.
To compare elliptic curves and modular forms directly is difficult; past efforts to count and match elliptic curves and modular forms had all failed. But elliptic curves can be represented within Galois theory. Wiles realized that working with the representations of elliptic curves instead of the curves themselves would make counting and matching them to modular forms far easier. From this point on, the proof primarily aims to prove:
(1) if the geometric Galois representation of a semistable elliptic curve is modular, so is the curve itself; and
(2) the geometric Galois representations of all semistable elliptic curves are modular.
Together, these allow us to work with representations of curves rather than directly with elliptic curves themselves. Our original goal will have been transformed into proving the modularity of geometric Galois representations of semistable elliptic curves, instead. Wiles described this realization as a "key breakthrough".
A Galois representation of an elliptic curve is G → GL(Zp). To show that a geometric Galois representation of an elliptic curve is a modular form, we need to find a normalized eigenform whose eigenvalues (which are also its Fourier series coefficients) satisfy a congruence relationship for all but a finite number of primes.
5 Wiles's initial strategy is to count and match using proof by induction and a class number formula ("CNF"): an approach in which, once the hypothesis is proved for one elliptic curve, it can automatically be extended to be proven for all subsequent elliptic curves. It was in this area that Wiles found difficulties, first with horizontal Iwasawa theory and later with his extension of Kolyvagin–Flach. Wiles's work extending Kolyvagin–Flach was mainly related to making Kolyvagin–Flach strong enough to prove the full CNF he would use. It later turned out that neither of these approaches by itself could produce a CNF able to cover all types of semistable elliptic curves, and the final piece of his proof in 1995 was to realize that he could succeed by strengthening Iwasawa theory with the techniques from Kolyvagin–Flach.
6 At this point, the proof has shown a key point about Galois representations:
If the geometric Galois representation ρ(E, p) of a semistable elliptic curve E is irreducible and modular (for some prime number p > 2), then subject to some technical conditions, E is modular.
This is Wiles's lifting theorem (or modularity lifting theorem), a major and revolutionary accomplishment at the time.
Crucially, this result does not just show that modular irreducible representations imply modular curves. It also means we can prove a representation is modular by using any prime number > 2 that we find easiest to use (because proving it for just one prime > 2 proves it for all primes > 2).
So we can try to prove all of our elliptic curves are modular by using one prime number as p - but if we do not succeed in proving this for all elliptic curves, perhaps we can prove the rest by choosing different prime numbers as 'p' for the difficult cases.
The proof must cover the Galois representations of all semistable elliptic curves E, but for each individual curve, we only need to prove it is modular using one prime number p.)
Part 3: Proving that all semistable elliptic curves are modular
7 With the lifting theorem proved, we return to the original problem. We will categorize all semistable elliptic curves based on the reducibility of their Galois representations, and use the powerful lifting theorem on the results.
From above, it does not matter which prime is chosen for the representations. We can use any one prime number that is easiest. 3 is the smallest prime number more than 2, and some work has already been done on representations of elliptic curves using ρ(E, 3), so choosing 3 as our prime number is a helpful starting point.
Wiles found that it was easier to prove the representation was modular by choosing a prime p = 3 in the cases where the representation ρ(E, 3) is irreducible, but the proof when ρ(E, 3) is reducible was easier to prove by choosing p = 5. So the proof splits in two at this point.
The proof's use of both p = 3 and p = 5 below, is the so-called "3/5 switch" referred to in some descriptions of the proof, which Wiles noticed in a paper of Mazur's in 1993, although the trick itself dates back to the 19th century.
The switch between p = 3 and p = 5 has since opened a significant area of study in its own right (see Serre's modularity conjecture).
8 If the Galois representation ρ(E, 3) (i.e., using p = 3) is irreducible, then it was known from around 1980 that its Galois representation is also always modular.
Wiles uses his modularity lifting theorem to make short work of this case:
• If the representation ρ(E, 3) is irreducible, then we know the representation is also modular (Langlands and Tunnell), but...
• ... if the representation is both irreducible and modular then E itself is modular (modularity lifting theorem).
Langlands and Tunnell proved this in two papers in the early 1980s. The proof is based on the fact that ρ(E, 3) has the same symmetry group as the general quartic equation in one variable, which was one of the few general classes of diophantine equation known at that time to be modular.
This existing result for p = 3 is crucial to Wiles's approach and is one reason for initially using p = 3.
9 So we now consider what happens if ρ(E, 3) is reducible.
Wiles found that when the representation of an elliptic curve using p = 3 is reducible, it was easier to work with p = 5 and use his new lifting theorem to prove that ρ(E, 5) will always be modular, than to try and prove directly that ρ(E, 3) itself is modular (remembering that we only need to prove it for one prime).
5 is the next prime number after 3, and any prime number can be used, perhaps 5 will be an easier prime number to work with than 3? But it looks hopeless initially to prove that ρ(E, 5) is always modular, for much the same reason that the general quintic equation cannot be solved by radicals. So Wiles has to find a way around this.
9.1 If ρ(E, 3) and ρ(E, 5) are both reducible, Wiles proved directly that ρ(E, 5) must be modular.
9.2 The last case is if ρ(E, 3) is reducible and ρ(E, 5) is irreducible.
Wiles showed that in this case, one could always find another semistable elliptic curve F such that the representation ρ(F, 3) is irreducible and also the representations ρ(E, 5) and ρ(F, 5) are isomorphic (they have identical structures).
- The first of these properties shows that F must be modular (Langlands and Tunnell again: all irreducible representations with p = 3 are modular).
- If F is modular then we know ρ(F, 5) must be modular as well.
- But because the representations of E and F with p = 5 have exactly the same structure, and we know that ρ(F, 5) is modular, ρ(E, 5) must be modular as well.
9.3 Therefore, if ρ(E, 3) is reducible, we have proved that ρ(E, 5) will always be modular. But if ρ(E, 5) is modular, then the modularity lifting theorem shows that E itself is modular. This step shows the real power of the modularity lifting theorem.
Results
10 We have now proved that whether or not ρ(E, 3) is irreducible, E (which could be any semistable elliptic curve) will always be modular. This means that all semistable elliptic curves must be modular.
This proves:
(a) The Taniyama–Shimura–Weil conjecture for semistable elliptic curves; and also
(b) Because there cannot be a contradiction, it also proves that the kinds of elliptic curves described by Frey cannot actually exist. Therefore no solutions to Fermat's equation can exist either, so Fermat's Last Theorem is also true.
We have our proof by contradiction, because we have proven that if Fermat's Last Theorem is incorrect, we could create a semistable elliptic curve that cannot be modular (Ribet's Theorem) and must be modular (Wiles). As it cannot be both, the only answer is that no such curve exists.
Mathematical detail of Wiles's proof
Overview
Wiles opted to attempt to match elliptic curves to a countable set of modular forms. He found that this direct approach was not working, so he transformed the problem by instead matching the Galois representations of the elliptic curves to modular forms. Wiles denotes this matching (or mapping) that, more specifically, is a ring homomorphism:
$R_{n}\rightarrow \mathbf {T} _{n}.$
$R$ is a deformation ring and $\mathbf {T} $ is a Hecke ring.
Wiles had the insight that in many cases this ring homomorphism could be a ring isomorphism (Conjecture 2.16 in Chapter 2, §3 of the 1995 paper[4]). He realised that the map between $R$ and $\mathbf {T} $ is an isomorphism if and only if two abelian groups occurring in the theory are finite and have the same cardinality. This is sometimes referred to as the "numerical criterion". Given this result, Fermat's Last Theorem is reduced to the statement that two groups have the same order. Much of the text of the proof leads into topics and theorems related to ring theory and commutation theory. Wiles's goal was to verify that the map $R\rightarrow \mathbf {T} $ is an isomorphism and ultimately that $R=\mathbf {T} $. In treating deformations, Wiles defined four cases, with the flat deformation case requiring more effort to prove and treated in a separate article in the same volume entitled "Ring-theoretic properties of certain Hecke algebras".
Gerd Faltings, in his bulletin, gives the following commutative diagram (p. 745):
or ultimately that $R=\mathbf {T} $, indicating a complete intersection. Since Wiles could not show that $R=\mathbf {T} $ directly, he did so through $\mathbf {Z} _{3},\mathbf {F} _{3}$ and $\mathbf {T} /{\mathfrak {m}}$ via lifts.
In order to perform this matching, Wiles had to create a class number formula (CNF). He first attempted to use horizontal Iwasawa theory but that part of his work had an unresolved issue such that he could not create a CNF. At the end of the summer of 1991, he learned about an Euler system recently developed by Victor Kolyvagin and Matthias Flach that seemed "tailor made" for the inductive part of his proof, which could be used to create a CNF, and so Wiles set his Iwasawa work aside and began working to extend Kolyvagin and Flach's work instead, in order to create the CNF his proof would require.[25] By the spring of 1993, his work had covered all but a few families of elliptic curves, and in early 1993, Wiles was confident enough of his nearing success to let one trusted colleague into his secret. Since his work relied extensively on using the Kolyvagin–Flach approach, which was new to mathematics and to Wiles, and which he had also extended, in January 1993 he asked his Princeton colleague, Nick Katz, to help him review his work for subtle errors. Their conclusion at the time was that the techniques Wiles used seemed to work correctly.[1]: 261–265 [26]
Wiles's use of Kolyvagin–Flach would later be found to be the point of failure in the original proof submission, and he eventually had to revert to Iwasawa theory and a collaboration with Richard Taylor to fix it. In May 1993, while reading a paper by Mazur, Wiles had the insight that the 3/5 switch would resolve the final issues and would then cover all elliptic curves.
General approach and strategy
Given an elliptic curve E over the field Q of rational numbers $E({\bar {\mathbf {Q} }})$, for every prime power $\ell ^{n}$, there exists a homomorphism from the absolute Galois group
$\operatorname {Gal} ({\bar {\mathbf {Q} }}/\mathbf {Q} )$
to
$\operatorname {GL} _{2}(\mathbf {Z} /l^{n}\mathbf {Z} ),$
the group of invertible 2 by 2 matrices whose entries are integers modulo $\ell ^{n}$. This is because $E({\bar {\mathbf {Q} }})$, the points of E over ${\bar {\mathbf {Q} }}$, form an abelian group, on which $\operatorname {Gal} ({\bar {\mathbf {Q} }}/\mathbf {Q} )$ acts; the subgroup of elements x such that $\ell ^{n}x=0$ is just $(\mathbf {Z} /\ell ^{n}\mathbf {Z} )^{2}$, and an automorphism of this group is a matrix of the type described.
Less obvious is that given a modular form of a certain special type, a Hecke eigenform with eigenvalues in Q, one also gets a homomorphism from the absolute Galois group
$\operatorname {Gal} ({\bar {\mathbf {Q} }}/\mathbf {Q} )\rightarrow \operatorname {GL} _{2}(\mathbf {Z} /l^{n}\mathbf {Z} ).$
This goes back to Eichler and Shimura. The idea is that the Galois group acts first on the modular curve on which the modular form is defined, thence on the Jacobian variety of the curve, and finally on the points of $\ell ^{n}$ power order on that Jacobian. The resulting representation is not usually 2-dimensional, but the Hecke operators cut out a 2-dimensional piece. It is easy to demonstrate that these representations come from some elliptic curve but the converse is the difficult part to prove.
Instead of trying to go directly from the elliptic curve to the modular form, one can first pass to the $(\mathrm {mod} \,\ell ^{n})$ representation for some ℓ and n, and from that to the modular form. In the case ℓ = 3 and n= 1, results of the Langlands–Tunnell theorem show that the $(\mathrm {mod} \,3)$ representation of any elliptic curve over Q comes from a modular form. The basic strategy is to use induction on n to show that this is true for ℓ = 3 and any n, that ultimately there is a single modular form that works for all n. To do this, one uses a counting argument, comparing the number of ways in which one can lift a $(\mathrm {mod} \,\ell ^{n})$ Galois representation to $(\mathrm {mod} \,\ell ^{n+1})$ and the number of ways in which one can lift a $(\mathrm {mod} \,\ell ^{n})$ modular form. An essential point is to impose a sufficient set of conditions on the Galois representation; otherwise, there will be too many lifts and most will not be modular. These conditions should be satisfied for the representations coming from modular forms and those coming from elliptic curves.
3–5 trick
If the original $(\mathrm {mod} \,3)$ representation has an image which is too small, one runs into trouble with the lifting argument, and in this case, there is a final trick which has since been studied in greater generality in the subsequent work on the Serre modularity conjecture. The idea involves the interplay between the $(\mathrm {mod} \,3)$ and $(\mathrm {mod} \,5)$ representations. In particular, if the mod-5 Galois representation ${\overline {\rho }}_{E,5}$ associated to an semistable elliptic curve E over Q is irreducible, then there is another semistable elliptic curve E' over Q such that its associated mod-5 Galois representation ${\overline {\rho }}_{E',5}$ is isomorphic to ${\overline {\rho }}_{E,5}$ and such that its associated mod-3 Galois representation ${\overline {\rho }}_{E,3}$ is irreducible (and therefore modular by Langlands–Tunnell).[27]
Structure of Wiles's proof
In his 108-page article published in 1995, Wiles divides the subject matter up into the following chapters (preceded here by page numbers):
Introduction
443
Chapter 1
455 1. Deformations of Galois representations
472 2. Some computations of cohomology groups
475 3. Some results on subgroups of GL2(k)
Chapter 2
479 1. The Gorenstein property
489 2. Congruences between Hecke rings
503 3. The main conjectures
Chapter 3
517 Estimates for the Selmer group
Chapter 4
525 1. The ordinary CM case
533 2. Calculation of η
Chapter 5
541 Application to elliptic curves
Appendix
545 Gorenstein rings and local complete intersections
Gerd Faltings subsequently provided some simplifications to the 1995 proof, primarily in switching from geometric constructions to rather simpler algebraic ones.[19][28] The book of the Cornell conference also contained simplifications to the original proof.[9]
Overviews available in the literature
Wiles's paper is over 100 pages long and often uses the specialised symbols and notations of group theory, algebraic geometry, commutative algebra, and Galois theory. The mathematicians who helped to lay the groundwork for Wiles often created new specialised concepts and technical jargon.
Among the introductory presentations are an email which Ribet sent in 1993;[29][30] Hesselink's quick review of top-level issues, which gives just the elementary algebra and avoids abstract algebra;[24] or Daney's web page, which provides a set of his own notes and lists the current books available on the subject. Weston attempts to provide a handy map of some of the relationships between the subjects.[31] F. Q. Gouvêa's 1994 article "A Marvelous Proof", which reviews some of the required topics, won a Lester R. Ford award from the Mathematical Association of America.[32][33] Faltings' 5-page technical bulletin on the matter is a quick and technical review of the proof for the non-specialist.[34] For those in search of a commercially available book to guide them, he recommended that those familiar with abstract algebra read Hellegouarch, then read the Cornell book,[9] which is claimed to be accessible to "a graduate student in number theory". The Cornell book does not cover the entirety of the Wiles proof.[12]
See also
• Abstract algebra
• p-adic number
• Semistable curves
References
1. Fermat's Last Theorem, Simon Singh, 1997, ISBN 1-85702-521-0
2. Kolata, Gina (24 June 1993). "At Last, Shout of 'Eureka!' In Age-Old Math Mystery". The New York Times. Retrieved 21 January 2013.
3. "The Abel Prize 2016". Norwegian Academy of Science and Letters. 2016. Retrieved 29 June 2017.
4. Wiles, Andrew (1995). "Modular elliptic curves and Fermat's Last Theorem". Annals of Mathematics. 141 (3): 443–551. CiteSeerX 10.1.1.169.9076. doi:10.2307/2118559. JSTOR 2118559. OCLC 37032255.
5. Taylor R, Wiles A (1995). "Ring theoretic properties of certain Hecke algebras". Annals of Mathematics. 141 (3): 553–572. CiteSeerX 10.1.1.128.531. doi:10.2307/2118560. JSTOR 2118560. OCLC 37032255. Archived from the original on 27 November 2001.
6. "NOVA – Transcripts – The Proof – PBS". PBS. September 2006. Retrieved 29 June 2017.
7. Hellegouarch, Yves (2001). Invitation to the Mathematics of Fermat–Wiles. Academic Press. ISBN 978-0-12-339251-0.
8. Singh, pp. 194–198; Aczel, pp. 109–114.
9. G. Cornell, J. H. Silverman and G. Stevens, Modular forms and Fermat's Last Theorem, ISBN 0-387-94609-8
10. Daney, Charles (13 March 1996). "The Proof of Fermat's Last Theorem". Archived from the original on 10 December 2008. Retrieved 29 June 2017.
11. "Andrew Wiles on Solving Fermat". PBS. 1 November 2000. Retrieved 29 June 2017.
12. Buzzard, Kevin (22 February 1999). "Review of Modular forms and Fermat's Last Theorem, by G. Cornell, J. H. Silverman, and G. Stevens" (PDF). Bulletin of the American Mathematical Society. 36 (2): 261–266. doi:10.1090/S0273-0979-99-00778-8.
13. Singh, pp. 269–277.
14. Kolata, Gina (28 June 1994). "A Year Later, Snag Persists In Math Proof". The New York Times. ISSN 0362-4331. Retrieved 29 June 2017.
15. Kolata, Gina (3 July 1994). "June 26-July 2; A Year Later Fermat's Puzzle Is Still Not Quite Q.E.D." The New York Times. ISSN 0362-4331. Retrieved 29 June 2017.
16. Singh, pp. 175–185.
17. Aczel, pp. 132–134.
18. Singh pp. 186–187 (text condensed).
19. "Fermat's last theorem". MacTutor History of Mathematics. February 1996. Retrieved 29 June 2017.
20. Cornell, Gary; Silverman, Joseph H.; Stevens, Glenn (2013). Modular Forms and Fermat's Last Theorem (illustrated ed.). Springer Science & Business Media. p. 549. ISBN 978-1-4612-1974-3. Extract of page 549
21. O'Carroll, Eoin (17 August 2011). "Why Pierre de Fermat is the patron saint of unfinished business". The Christian Science Monitor. ISSN 0882-7729. Retrieved 29 June 2017.
22. Granville, Andrew. "History of Fermat's Last Theorem". Retrieved 29 June 2017.
23. Breuil, Christophe; Conrad, Brian; Diamond, Fred; Taylor, Richard (2001). "On the modularity of elliptic curves over 𝐐: Wild 3-adic exercises". Journal of the American Mathematical Society. 14 (4): 843–939. doi:10.1090/S0894-0347-01-00370-8. ISSN 0894-0347.
24. Hesselink, Wim H. (3 April 2008). "Computer verification of Wiles' proof of Fermat's Last Theorem". www.cs.rug.nl. Retrieved 29 June 2017.
25. Singh p.259-262
26. Singh, pp. 239–243; Aczel, pp. 122–125.
27. Chapter 5 of Wiles, Andrew (1995). "Modular elliptic curves and Fermat's Last Theorem" (PDF). Annals of Mathematics. 141 (3): 443–551. CiteSeerX 10.1.1.169.9076. doi:10.2307/2118559. JSTOR 2118559. OCLC 37032255.
28. Malek, Massoud (6 January 1996). "Fermat's Last Theorem". Retrieved 29 June 2017.
29. "sci.math FAQ: Wiles attack". www.faqs.org. Retrieved 29 June 2017.
30. "Fermat's Last Theorem, a Theorem at Last" (PDF). FOCUS. August 1993. Retrieved 29 June 2017.
31. Weston, Tom. "Research Summary Topics". people.math.umass.edu. Retrieved 29 June 2017.
32. Gouvêa, Fernando (1994). "A Marvelous Proof". American Mathematical Monthly. 101 (3): 203–222. doi:10.2307/2975598. JSTOR 2975598. Retrieved 29 June 2017.
33. "The Mathematical Association of America's Lester R. Ford Award". Retrieved 29 June 2017.
34. Faltings, Gerd (July 1995). "The Proof of Fermat's Last Theorem by R. Taylor and A. Wiles" (PDF). Notices of the American Mathematical Society. 42 (7): 743–746.
Bibliography
• Aczel, Amir (1 January 1997). Fermat's Last Theorem: Unlocking the Secret of an Ancient Mathematical Problem. ISBN 978-1-56858-077-7. Zbl 0878.11003.
• Coates, John (July 1996). "Wiles Receives NAS Award in Mathematics" (PDF). Notices of the AMS. 43 (7): 760–763. Zbl 1029.01513.
• Cornell, Gary (1 January 1998). Modular Forms and Fermat's Last Theorem. ISBN 978-0-387-94609-2. Zbl 0878.11004. (Cornell, et al.)
• Daney, Charles (2003). "The Mathematics of Fermat's Last Theorem". Archived from the original on 3 August 2004. Retrieved 5 August 2004.
• Darmon, H. (9 September 2007). "Wiles' theorem and the arithmetic of elliptic curves" (PDF).
• Faltings, Gerd (July 1995). "The Proof of Fermat's Last Theorem by R. Taylor and A. Wiles" (PDF). Notices of the AMS. 42 (7): 743–746. ISSN 0002-9920. Zbl 1047.11510.
• Frey, Gerhard (1986). "Links between stable elliptic curves and certain diophantine equations". Ann. Univ. Sarav. Ser. Math. 1: 1–40. Zbl 0586.10010.
• Hellegouarch, Yves (1 January 2001). Invitation to the Mathematics of Fermat–Wiles. ISBN 978-0-12-339251-0. Zbl 0887.11003. See review
• Mozzochi, Charles (7 December 2000). The Fermat Diary. American Mathematical Society. ISBN 978-0-8218-2670-6. Zbl 0955.11002. See also Gouvêa, Fernando Q. (2001). "Review: Wiles's Proof, 1993–1995: The Fermat Diary by C. J. Mozzochi". American Scientist. 89 (3): 281–282. JSTOR 27857485.
• Mozzochi, Charles (6 July 2006). The Fermat Proof. Trafford Publishing. ISBN 978-1-4120-2203-3. Zbl 1104.11001.
• O'Connor, J. J.; Robertson, E. F. (1996). "Fermat's last theorem". Retrieved 5 August 2004.
• van der Poorten, Alfred (1 January 1996). Notes on Fermat's Last Theorem. ISBN 978-0-471-06261-5. Zbl 0882.11001.
• Ribenboim, Paulo (1 January 2000). Fermat's Last Theorem for Amateurs. ISBN 978-0-387-98508-4. Zbl 0920.11016.
• Singh, Simon (October 1998). Fermat's Enigma. New York: Anchor Books. ISBN 978-0-385-49362-8. Zbl 0930.00002.
• Simon Singh "The Whole Story". Archived from the original on 10 May 2011. Edited version of ~2,000-word essay published in Prometheus magazine, describing Andrew Wiles's successful journey.
• Richard Taylor and Andrew Wiles (May 1995). "Ring-theoretic properties of certain Hecke algebras". Annals of Mathematics. 141 (3): 553–572. CiteSeerX 10.1.1.128.531. doi:10.2307/2118560. ISSN 0003-486X. JSTOR 2118560. OCLC 37032255. Zbl 0823.11030.
• Wiles, Andrew (1995). "Modular elliptic curves and Fermat's Last Theorem". Annals of Mathematics. 141 (3): 443–551. CiteSeerX 10.1.1.169.9076. doi:10.2307/2118559. ISSN 0003-486X. JSTOR 2118559. OCLC 37032255. Zbl 0823.11029.
External links
• Weisstein, Eric W. "Fermat's Last Theorem". MathWorld.
• "The Proof". PBS. The title of one edition of the PBS television series NOVA discusses Andrew Wiles's effort to prove Fermat's Last Theorem that broadcast on BBC Horizon and UTV/Documentary as Fermat's Last Theorem (Adobe Flash) (subscription required)
• Wiles, Ribet, Shimura–Taniyama–Weil and Fermat's Last Theorem
• Are mathematicians finally satisfied with Andrew Wiles's proof of Fermat's Last Theorem? Why has this theorem been so difficult to prove?, Scientific American, 21 October 1999
Explanations of the proof (varying levels)
• Overview of Wiles proof, accessible to non-experts, by Henri Darmon
• Very short summary of the proof by Charles Daney
• 140 page students work-through of the proof, with exercises, by Nigel Boston
Pierre de Fermat
Work
• Fermat's Last Theorem
• Fermat number
• Fermat's principle
• Fermat's little theorem
• Fermat polygonal number theorem
• Fermat pseudoprime
• Fermat point
• Fermat's theorem (stationary points)
• Fermat's theorem on sums of two squares
• Fermat's spiral
• Fermat's right triangle theorem
Related
• List of things named after Pierre de Fermat
• Wiles's proof of Fermat's Last Theorem
• Fermat's Last Theorem in fiction
• Fermat Prize
• Fermat's Last Tango (2000 musical)
• Fermat's Last Theorem (popular science book)
| Wikipedia |
Wilf–Zeilberger pair
In mathematics, specifically combinatorics, a Wilf–Zeilberger pair, or WZ pair, is a pair of functions that can be used to certify certain combinatorial identities. WZ pairs are named after Herbert S. Wilf and Doron Zeilberger, and are instrumental in the evaluation of many sums involving binomial coefficients, factorials, and in general any hypergeometric series. A function's WZ counterpart may be used to find an equivalent and much simpler sum. Although finding WZ pairs by hand is impractical in most cases, Gosper's algorithm provides a sure method to find a function's WZ counterpart, and can be implemented in a symbolic manipulation program.
Definition
Two functions F and G form a WZ pair if and only if the following two conditions hold:
$F(n+1,k)-F(n,k)=G(n,k+1)-G(n,k),$
$\lim _{M\to \pm \infty }G(n,M)=0.$
Together, these conditions ensure that
$\sum _{k=-\infty }^{\infty }[F(n+1,k)-F(n,k)]=0$
because the function G telescopes:
${\begin{aligned}\sum _{k=-\infty }^{\infty }[F(n+1,k)-F(n,k)]&{}=\lim _{M\to \infty }\sum _{k=-M}^{M}[F(n+1,k)-F(n,k)]\\&{}=\lim _{M\to \infty }\sum _{k=-M}^{M}[G(n,k+1)-G(n,k)]\\&{}=\lim _{M\to \infty }[G(n,M+1)-G(n,-M)]\\&{}=0-0\\&{}=0.\end{aligned}}$
Therefore,
$\sum _{k=-\infty }^{\infty }F(n+1,k)=\sum _{k=-\infty }^{\infty }F(n,k),$
that is
$\sum _{k=-\infty }^{\infty }F(n,k)={\text{const}}.$
The constant does not depend on n. Its value can be found by substituting n = n0 for a particular n0.
If F and G form a WZ pair, then they satisfy the relation
$G(n,k)=R(n,k)F(n,k-1),$
where $R(n,k)$ is a rational function of n and k and is called the WZ proof certificate.
Example
A Wilf–Zeilberger pair can be used to verify the identity
$\sum _{k=0}^{\infty }(-1)^{k}{n \choose k}{2k \choose k}4^{n-k}={2n \choose n}.$
Divide the identity by its right-hand side:
$\sum _{k=0}^{\infty }{\frac {(-1)^{k}{n \choose k}{2k \choose k}4^{n-k}}{2n \choose n}}=1.$
Use the proof certificate
$R(n,k)={\frac {2k-1}{2n+1}}$
to verify that the left-hand side does not depend on n, where
${\begin{aligned}F(n,k)&={\frac {(-1)^{k}{n \choose k}{2k \choose k}4^{n-k}}{2n \choose n}},\\G(n,k)&=R(n,k)F(n,k-1).\end{aligned}}$
Now F and G form a Wilf–Zeilberger pair.
To prove that the constant in the right-hand side of the identity is 1, substitute n = 0, for instance.
References
• Marko Petkovsek; Herbert Wilf and Doron Zeilberger (1996). A=B. AK Peters. ISBN 1-56881-063-6.
• Tefera, Akalu (2010), "What Is . . . a Wilf-Zeilberger Pair?" (PDF), AMS Notices, 57 (4): 508–509.
See also
• Almkvist–Zeilberger method, an analog of WZ method for evaluating definite integrals.
• List of mathematical identities
External links
• Gosper's algorithm gives a method for generating WZ pairs when they exist.
• Generatingfunctionology provides details on the WZ method of identity certification.
| Wikipedia |
Wilf equivalence
In the study of permutations and permutation patterns, Wilf equivalence is an equivalence relation on permutation classes. Two permutation classes are Wilf equivalent when they have the same numbers of permutations of each possible length, or equivalently if they have the same generating functions.[1] The equivalence classes for Wilf equivalence are called Wilf classes;[2] they are the combinatorial classes of permutation classes. The counting functions and Wilf equivalences among many specific permutation classes are known.
Wilf equivalence may also be described for individual permutations rather than permutation classes. In this context, two permutations are said to be Wilf equivalent if the principal permutation classes formed by forbidding them are Wilf equivalent.[1]
References
1. Bevan, David (2015), Permutation patterns: basic definitions and notation, arXiv:1506.06673, Bibcode:2015arXiv150606673B
2. Steingrímsson, Einar (2013), "Some open problems on permutation patterns", Surveys in combinatorics 2013, London Math. Soc. Lecture Note Ser., vol. 409, Cambridge Univ. Press, Cambridge, pp. 239–263, MR 3156932
| Wikipedia |
Wilfred Kaplan
Wilfred Kaplan (November 28, 1915 – December 26, 2007) was a professor of mathematics at the University of Michigan for 46 years, from 1940 through 1986. His research focused on dynamical systems, the topology of curve families, complex function theory, and differential equations. In total, he authored over 30 research papers and 11 textbooks.
Wilfred Kaplan
Wilfred Kaplan, circa 1960
Born(1915-11-28)November 28, 1915
Boston, Massachusetts, US
DiedDecember 26, 2007(2007-12-26) (aged 92)
Ann Arbor, Michigan, US
NationalityAmerican
SpouseIda Roetting (married 1938)
Children
• Roland Kaplan
• Muriel Kaplan
Parents
• Jacob Kaplan
• Anne Kaplan
Academic background
Alma materHarvard University
ThesisRegular Curve-Families Filling the Plane (1939)
Doctoral advisorHassler Whitney
Academic work
DisciplineMathematics
Doctoral students
• William M. Boothby
• Helen F. Cullen
• George R. Sell
For over thirty years Kaplan was an active member of the American Association of University Professors (AAUP) and served as president of the University of Michigan chapter from 1978 to 1985.
Early life
Education
Wilfred Kaplan was born in Boston, Massachusetts to Jacob and Anne Kaplan.[1] He attended Boston Latin School[1] and furthered his education at Harvard University, where he was granted his A.B. in mathematics in 1936 and graduated summa cum laude.[2] Later that same year he received his master's degree at Harvard.[3] Kaplan received a Rogers Fellow scholarship to study in Europe from 1936-1937, during his second year of graduate school.[3] He was based out of Zürich, Switzerland where many of the mathematicians working on the applications of topology to differential equations were located.[3] He also spent a month in Rome to work with famous mathematician Tullio Levi-Civita.[3] Upon returning to the United States, Kaplan accepted a yearlong teaching fellowship at Rice Institute for the 1938-1939 school year, thus completing his graduate program.[2] He received his Ph.D. from Harvard in 1939 under the advisement of Hassler Whitney. His dissertation covered regular curve families filling the plane.[4]
Personal life
While attending lectures at the Eidgenössische Technische Hochschule (ETH) Zürich he met a fellow mathematician, Ida Roetting, whom he nicknamed Heidi and would eventually marry in 1938.[3] The couple lived in Houston for a year after their wedding while Kaplan taught at the Rice Institute. The Kaplans had two children, Roland and Muriel.[1] Wilfred Kaplan died at the age of 92 after a short illness.[2]
Work
Teaching and research
After Kaplan's short teaching position at Rice Institute, he went on to teach at the College of William and Mary in Virginia for one year.[3] In 1940 he was invited by T. H. Hildebrandt to join the faculty at the University of Michigan, after he had previously attended the Topology Congress.[5] The mathematics department at this time was diminishing due to the effects of World War II. Enrollment was down and some of the faculty had been granted leaves to do military research.[5] When asked to record his contribution to the war effort, Kaplan mentioned teaching math exclusively to Air Force pre-meteorology students in the spring and summer of 1943, as well as teaching Navy V-12 and Army ASTP students for the majority of the academic year 1943-44. In June 1944, Kaplan worked at Brown University as a researcher in an Applied Mathematics Group for the Taylor Model Basin, the Watertown Arsenal and the Bureau of Ordnance of the Navy Department. He continued his research at Brown for 17 months. In May 1947 he outlined a curriculum for a new Lectures on Mathematics Project sponsored by the Office of Naval Research.[3]
His early research focused primarily on dynamical systems, and the topology of curve families. In 1955, he became especially interested in complex function theory and made a significant contribution to mathematics in his study of a special class of Schlicht functions, for which he showed that the Bieberbach Conjecture held.[2][6] His later research took on a more applied approach as he returned once more to differential equations, this time engaging in a more global analysis. In total, Wilfred Kaplan authored about 30 research papers.[2][3]
Kaplan was named assistant professor in 1944, associate professor in 1950 and full professor in 1956.[3] His lectures were characterized by clarity and directness, a skill which allowed him to write several popular mathematics textbooks.[2] An updated version of his Advanced Calculus textbook is still used widely today. A selection of Kaplan's books can be found in the bibliography.
Kaplan taught a variety of undergraduate and graduate courses during his time at Michigan, and further advised eight doctoral students.[4] Kaplan's skillful teaching won him respect among students and coworkers. Donald Lewis, chair of the mathematics department and co-author said,[2] "First and foremost, Wilfred Kaplan was a teacher. He enjoyed conveying the beauty and usefulness of mathematics, and his students responded enthusiastically. He was a superb expositor, and his ability to elegantly convey mathematical ideas explains the enormous impact of his textbooks. When we were writing our joint texts, he never came to a meeting without a new idea to be incorporated."
One of Kaplan's primary goals as an educator was bridging the gap between pure and applied mathematics. He sympathized with the plight of engineers who felt the pressure to master more and more math concepts and then master the additional skill of applying it to their field. Kaplan wrote math textbooks specifically for engineers, such as Advanced Mathematics for Engineers (1981), because he believed teachers needed to work on presenting mathematical knowledge more efficiently to this group.[3] Furthermore, he argued that science students in general, but specifically engineers, needed to be given other resources such as textbooks and articles to further their study outside of lectures along with the tools to employ those resources appropriately. He urged other textbook authors to use clear and simple language whenever possible, in order to “make the more advanced material accessible to those with limited background.[3]”
He also taught a class called "Mathematical Ideas in Science and the Humanities," which focused on the use of math as an instrument to organize thinking about complex problems.[3] More than just learning specific math content, Kaplan believed math was a medium through which to teach conciseness and how to recognize analogies, determine logical consequences of assumptions, and learn what questions need to be asked to tackle a given problem in any field.[3]
AAUP
Wilfred Kaplan became a member of the national American Association of University Professors (AAUP) in 1946.[6] He served as the vice president of the Michigan Conference AAUP from 1966 to 1968 and as president from 1969 to 1970.[7] From 1973 to 1978, Kaplan served on the executive committee of the University of Michigan Chapter of the AAUP and took over as president in the years from 1978 through 1985. He continued being an active member even in his retirement, serving as executive secretary from 1985 to 2002.[3] Kaplan received grievances from faculty members and supported collective bargaining (although the University faculty was never unionized). One of his primary concerns was retired professors on fixed incomes who were suffering under rising inflation. He sought to obtain a grant to provide the required financial aid. He also argued that retirees should receive more information about the health care options available to them, and he secured increases in maximum coverage and the annual reinstatement amounts allowed to retirees under the university health plan.[3]
In the 1980s, Kaplan wrote an extensive proposal for a study of higher education in the United States. He argued that there should be more research of the inevitable challenges that would arise and that the University of Michigan would be a great case study whose results would be relevant for many public universities. In the proposal he outlined a variety of topics to be explored in the study: the historical background of higher education including tuition rates, enrollment rates, and changes in social customs; a study of changing demographics; the economic need for college-trained people for the betterment of society; and a thorough account of the present resources available to higher education and how these could be modified for greater efficiency.
In the 1990s, Kaplan's correspondence and reports focused heavily on the "grave difficulties" between faculty government leaders (members of the Senate Assembly) and the top administration offices, specifically President Duderstadt and Provost Whitaker. Many faculty grievances were concerned with the many decisions being made without faculty input and included complaints that President Duderstadt had only “modest interest in the views of others within the faculty.” [3]
Additional clubs/Memberships
Kaplan was involved in a wide variety of other clubs as well. His interest in art lead him to become the president of the Washtenaw County Council for Arts.[3] Kaplan also made a significant financial contribution to the ONCE Group, a group of artists, musicians, and film-makers known for their annual ONCE music festival in Ann Arbor; the group also spearheaded a film festival and a theatre ensemble in the 1960s. He was also a member of the American Mathematical Society (AMS), the American Physical Society (APS), and the Mathematical Association of America (MAA). He was on the State's Higher Education Capital Investment Advisory Committee although little is known about his specific role there.[3] He was also Vice Chair of the University's Senate Advisory Committee on Academic Affairs, a subcommittee of the Senate Assembly. Additionally, Kaplan served as the treasurer of the Ann Arbor Unitarian Fellowship from 1972 to 2002.[3] In his later years he was president of the University of Michigan Retirees Association.[3]
Awards
Kaplan was named collegiate professor by the Board of Regents of the university from 1973-1975 for his accomplishments as a teacher.[3] In 1984, he received the Good Teaching Award, from what was then called the AMOCO foundation, now BP Amoco.[8] The award recognized excellence in undergraduate instruction and sought to incentive great teaching. While serving on the Senate Advisory Committee on University Affairs, Kaplan received their Distinguished Faculty Governance Award in 1986.[9]
Retirement
Wilfred Kaplan retired in May 1986 after 46 years of service to the university. It was then that he received the emeritus distinction. Despite his retirement, Kaplan was still involved with the university. In the years to come he would receive several requests from the University Regents Commission to return to active duty to teach specific classes in the mathematics department.[3] In 1990 he helped establish the Academic Freedom Lecture Fund (AFLF), allowing professors that were suspended or fired during the McCarthy era to hold lectures on campus.[2][6] The film Keeping in Mind, an account of the mistreatment of the three professors who were suspended for their unpopular views during the McCarthy era, was played in the spring of 1989. After an audience member suggested the University make amends for its mistreatment of the three professors, the AAUP pursued this goal. First, university officials were contacted and a proposal was sent to the Senate Advisory Committee on University Affairs in October 1989. The Senate Assembly established the Academic Lecture Freedom Fund which was funded, in part, by the national AAUP.[10] Kaplan was on the AFLF's board of directors until his death.[6]
After his wife's death in 2005, Kaplan wrote a book titled Bill and Heidi: Beginning of our Lives Together, which was a translated composition of all their early correspondence before their wedding.[3] Kaplan died on December 26, 2007, after a short illness.[2] After his death, Walter Dublin, Professor Emeritus of Mechanical Engineering, wrote about Kaplan in a letter to the editor of the Ann Arbor News that Kaplan "worked tirelessly to improve the faculty--and, de facto, the university--by his work on many committees, work that spanned multiple decades. While he was a leader, he was never domineering, but always logical. Often, he would quietly remind his associates when they had strayed from their stated purpose or point out a legal or historical obstacle to what was being considered. He was always up to date and on the mark until he died, many years after he had retired."[9]
Selected bibliography
• Kaplan, Wilfred. Advanced Calculus. Reading, Mass.: Addison-Wesley, 1952. Print.
• Kaplan, Wilfred. Ordinary Differential Equations. Reading, Mass.: Addison-Wesley Pub., 1958. Print.
• Kaplan, Wilfred, and Donald J. Lewis. Calculus and Linear Algebra. New York: Wiley, 1970. Print.
• Kaplan, Wilfred. Advanced Mathematics for Engineers. Reading, Mass.: Addison-Wesley Pub., 1981. Print.
• Kaplan, Wilfred. Operational Methods for Linear Systems. Reading, Mass.: Addison-Wesley Pub., 1962. Print.
• Kaplan, Wilfred. "Regular curve-families filling the plane, I." Duke Mathematical Journal 7.1 (1940): 154–185. Web.
• Kaplan, Wilfred. "Regular Curve-families Filling the Plane, II." Duke Mathematical Journal 8.1 (1941): 11-46. Web.
• Kaplan, Wilfred. "Topology of Level Curves of Harmonic Functions." Transactions of the American Mathematical Society 63.3 (1948): 514. Web.
• Kaplan, Wilfred. "Dynamics of Linear Systems (Vaclav Dolez Al)." SIAM Review 8.2 (1966): 246-2. ProQuest. Web. 21 Mar. 2015.
• Kaplan, Wilfred. "Applications of Undergraduate Mathematics in Engineering (Ben Noble)." SIAM Review 10.3 (1968): 383-2. ProQuest. Web. 21 Mar. 2015.
• Kaplan, Wilfred. "Topics in Mathematical System Theory (Rudolf E. Kalman, Peter L. Falb and Michael A. Arbib)." SIAM Review 12.1 (1970): 157-2. ProQuest. Web. 21 Mar. 2015.
• Kaplan, Wilfred. "Topics in Ordinary Differential Equations: A Potpourri (William D. Lakin and David A. Sanchez)." SIAM Review 14.3 (1972): 508-2. ProQuest. Web. 21 Mar. 2015.
• Kaplan, Wilfred. "Ordinary Differential Equations in the Complex Domain (Einar Hills)." SIAM Review 19.4 (1977): 749-1. ProQuest. Web. 21 Mar. 2015.
• Kaplan, Wilfred. "Green's Functions and Boundary Value Problems (Ivar Stackgold)." SIAM Review 23.1 (1981): 117-2. ProQuest. Web. 21 Mar. 2015.
References
1. Kalte, Pamela M.; Nemeh, Katherine H.; editors, Noah Schusterbauer, project (2005). American men & women of science a biographical directory of today's leaders in physical, biological, and related sciences (22nd ed.). Detroit: Thomson Gale. p. 211. ISBN 9781414404622. Retrieved 21 March 2015. {{cite book}}: |last3= has generic name (help)CS1 maint: multiple names: authors list (link)
2. Duren, Peter. "Obituaries". The University Record. University of Michigan. Retrieved 21 March 2015.
3. "Finding Aid". Bentley Historical Library. Retrieved 30 January 2015.
4. Wilfred Kaplan at the Mathematics Genealogy Project
5. Kaplan, Wilfred. "Mathematics at the University of Michigan" (PDF). University of Michigan. Retrieved 21 March 2015.
6. "Faculty History Project: Wilfred Kaplan". um2017. University of Michigan. Retrieved 24 March 2015.
7. "Past Officers/Boards". Michigan Conference American Association of University Professors. Retrieved 24 March 2015.
8. "GOOD TEACHING AWARD (formerly AMOCO)" (PDF). University of Oklahoma Health Sciences Center.
9. Maloney, Wendi (May 2008). "Wilfred Kaplan, 1915-2007". Academe. 94 (3): 9. ProQuest 232312313.
10. "Academic Freedom Lecture Fund". University of Michigan. Retrieved 21 March 2015.
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| Wikipedia |
Wilfrid Dixon
Wilfrid Joseph Dixon (December 13, 1915 – September 20, 2008) was an American mathematician and statistician. He made notable contributions to nonparametric statistics, statistical education and experimental design.
Wilfrid Dixon
Born(1915-12-13)December 13, 1915
Portland, Oregon
DiedSeptember 20, 2008(2008-09-20) (aged 92)
NationalityAmerican
Alma materPrinceton University
University of Wisconsin–Madison
Known forBMDP
Scientific career
FieldsMathematical statistics
InstitutionsUniversity of California, Los Angeles
University of Oregon
Doctoral advisorSamuel S. Wilks
Doctoral studentsPaula Diehr
A native of Portland, Oregon, Dixon received a bachelor's degree in mathematics from Oregon State College in 1938. He continued his graduate studies at the University of Wisconsin–Madison, where he earned a master's degree in 1939. Under supervision of Samuel S. Wilks, he then earned a Ph.D. in mathematical statistics from Princeton in 1944.[1] During World War II, he was an operations analyst on Guam.
Dixon was on the faculties at Oklahoma (1942–1943), Oregon (1946–1955), and UCLA (1955–1986, then emeritus). While at Oregon, Dixon (together with A.M. Mood) described and provided theory and estimation methods for the adaptive Up-and-Down experimental design, which was new and poorly documented at the time.[2] This article became the cornerstone publication for up-and-down, a family of designs used in many scientific, engineering and medical fields, and to which Dixon continued to contribute in later years. In 1951 Dixon, together with Frank Massey, published a statistics textbook - the first such textbook intended to a non-mathematical audience.[3] In 1955 he was elected as a Fellow of the American Statistical Association.[4]
In the 1960s at UCLA, Dixon developed BMDP, a statistical software package for biomedical analyses.[5]
His daughter, Janet D. Elashoff, is also a statistician who became a UCLA faculty member, and an ASA fellow in 1978.[6] In December 2008 she funded the W. J. Dixon Award for Excellence in Statistical Consulting of the American Statistical Association in his honor.[7]
References
1. "Wilfrid J. Dixon *44". Princeton Alumni Weekly. September 23, 2009.
2. Dixon, WJ; Mood, AM (1948). "A method for obtaining and analyzing sensitivity data". Journal of the American Statistical Association. 43 (241): 109–126. doi:10.1080/01621459.1948.10483254.
3. Dixon, WJ; Massey, FJ (1951). Introduction to statistical analysis. McGraw-Hill.
4. View/Search Fellows of the ASA, accessed 2016-07-23.
5. "Oral History of Wilfrid J. (Wil) Dixon and Linda Glassner: Interviewed by Luanne Johnson" (PDF). March 27, 1986.
6. ASA Fellows, Caucus for Women in Statistics, 29 March 2016, retrieved 2017-10-24
7. W. J. Dixon Award for Excellence in Statistical Consulting, American Statistical Association, retrieved 2017-10-24
External links
• Wilfrid Dixon at the Mathematics Genealogy Project
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Wilfrid Hodges
Wilfrid Augustine Hodges, FBA (born 27 May 1941) is a British mathematician and logician known for his work in model theory.
Wilfrid Hodges
Wilfrid Hodges, 1988 at the MFO
Born (1941-05-27) 27 May 1941
Alma materNew College, Oxford
Parent(s)H. A. Hodges, Vera Joan Willis
Scientific career
FieldsModel theory
Doctoral advisorJohn Crossley
Doctoral studentsAlex Wilkie
President of the DLMPST/IUHPST
In office
2008–2011
Preceded byAdolf Grünbaum
Succeeded byElliott Sober
Life
Hodges attended New College, Oxford (1959–65), where he received degrees in both Literae Humaniores and (Christianic) Theology. In 1970 he was awarded a doctorate for a thesis in Logic. He lectured in both Philosophy and Mathematics at Bedford College, University of London. He has held visiting appointments in the department of philosophy at the University of California and in the department of mathematics at University of Colorado. Hodges was Professor of Mathematics at Queen Mary College, University of London from 1987 to 2006 and is the author of books on logic.
Honors and awards
Hodges was President of the British Logic Colloquium, of the European Association for Logic, Language and Information and of the Division of Logic, Methodology, and Philosophy of Science. In 2009 he was elected a Fellow of the British Academy.
Writing style
Hodges' books are written in an informal style. The "Notes on Notation" in his book "Model theory" end with the following characteristic sentence:
'I' means I, 'we' means we.
When this 780-page book appeared in 1993, it became one of the standard textbooks on model theory. Due to its success an abbreviated version (but with a new chapter on stability theory) was published as a paperback.
Bibliography
Only first editions are listed.
• Hodges, Wilfrid (1977). Logic – An Introduction to Elementary Logic. Penguin Books.[1]
• Hodges, Wilfrid (1985). Building Models by Games. London Mathematical Society Student Texts. Cambridge University Press. ISBN 9780521268974.
• Hodges, Wilfrid (1993). Model Theory. Encyclopedia of Mathematics. Cambridge University Press. ISBN 0-521-30442-3.[2]
• Hodges, Wilfrid (1997). A Shorter Model Theory. Cambridge University Press. ISBN 0-521-58713-1.
• Chiswell, Ian; Hodges, Wilfrid (2007). Mathematical Logic. Oxford University Press. ISBN 978-0-19-921562-1.
References
1. Leeds, Stephen (1980). "Review of Logic by Wilfrid Hodges". Journal of Symbolic Logic. 45 (2): 382–383. doi:10.2307/2273212. JSTOR 2273212. S2CID 117796294.
2. Baldwin, John T. (1995). "Review: Model Theory by Wilfrid Hodges" (PDF). Bull. Amer. Math. Soc. (N.S.). 32 (2): 280–285. doi:10.1090/s0273-0979-1995-00578-1.
External links
• Home page of Wilfrid Hodges
• Wilfrid Hodges at the Mathematics Genealogy Project
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Wilhelm Ahrens
Wilhelm Ahrens (3 March 1872 – 23 May 1927) was a German mathematician and writer on recreational mathematics.[1]
Biography
Ahrens was born in Lübz at the Elde in Mecklenburg and studied from 1890 to 1897 at the University of Rostock, Humboldt University of Berlin, and the University of Freiburg. In 1895 at the University of Rostock he received his Promotion (Ph.D.), summa cum laude, under the supervision of Otto Staude[2] with dissertation entitled Über eine Gattung n-fach periodischer Functionen von n reellen Veränderlichen.[3] From 1895 to 1896 he taught at the German school in Antwerp and then studied another semester under Sophus Lie in Leipzig. In 1897 Ahrens was a teacher in Magdeburg at the Baugewerkeschule, from 1901 at the engineering school. Inspired by Sophus Lie, he wrote "On transformation groups, all of whose subgroups are invariant" (Hamburger Math Society Vol 4, 1902).
He worked a lot on the history of mathematics and mathematical games (recreational mathematics), about which he wrote a great work and also contributed to the Encyclopedia of mathematical sciences His predecessors were the great Jacques Ozanam in France, where the number theorist Édouard Lucas (1842–1891) in the 19th century wrote similar books, and Walter William Rouse Ball (1850–1925) in England (Mathematical recreations and essays 1892), Sam Loyd (1841–1901) in the U.S. and Henry Dudeney (1857–1930) in England. In this sense Martin Gardner (1914-2010) and Ian Stewart, the editor of the math column in Scientific American, might be regarded as his successors. He also wrote a book of quotations and anecdotes about mathematicians. He was the author of numerous journal articles.
Scherz und Ernst in der Mathematik
According to R. C. Archibald:
Ahrens's Scherz und Ernst in der Mathematik ... is strictly a book of quotations; secondly, each quotation is invariably given in the original language, spoken or written; thirdly, exact bibliographical data are provided for all quotations; fourthly, the quotations follow one another consecutively from pages 1 to 495 without grouping under subject headings. A 24-page detailed index of subjects and authors provides the means for rapid orientation. Names of living mathematicians are rarely met with, but references to the "old masters" such as Abel, Euclid, Euler, Gauss, Helmholtz, Lagrange, Laplace, Steiner, and Weierstrass, are very numerous.
The whole constitutes a most admirable piece of work and must long serve as a desirable model for works of like nature.[4]
Bibliography
• Mathematische Unterhaltungen und Spiele [Mathematical Recreations and Games], 1901
• Mathematische Spiele [Mathematical Games], 1902
• Scherz und Ernst in der Mathematik; geflügelte und ungeflügelte Worte [Fun and seriousness in mathematics: well-known and less well-known words], 1904. Scherz und Ernst in der Mathematik is available for free viewing and download at the Internet Archive; 2002 Auflage
• Gelehrten-Anekdoten [Scholarly anecdotes], 1911
• Mathematiker-Anekdoten [Anecdotes of Mathematicians], 1916;[5] Zweite, stark veränderte Auflage (2nd revised edition) 1920[6]
References
1. britannica.com
2. See entries of Wilhelm Ahrens in Rostock Matrikelportal
3. Wilhelm Ahrens at the Mathematics Genealogy Project
4. Archibald, R. C. (1916). "Review of Memorabilia Mathematica or the Philomath's Quotation-Book by Robert Edouard Moritz". Bulletin of the American Mathematical Society. 22 (4): 188–192. doi:10.1090/S0002-9904-1916-02751-0. MR 1559751.
5. Smith, David Eugene (1916). "Book Review: Mathematiker-Anekdoten". Bulletin of the American Mathematical Society. 23: 44–47. doi:10.1090/S0002-9904-1916-02873-4. MR 1559860.
6. Carmichael, R. D. (1921). "Book Review: Mathematiker Anekdoten. Zweite, stark veränderte Auflage". Bulletin of the American Mathematical Society. 27 (5): 230–232. doi:10.1090/S0002-9904-1921-03410-0. MR 1560405.
External links
• O'Connor, John J.; Robertson, Edmund F., "Wilhelm Ernst Martin Georg Ahrens", MacTutor History of Mathematics Archive, University of St Andrews
• Wilhelm Ernst Martin Georg Ahrens at the Mathematics Genealogy Project
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Karl Wilhelm Feuerbach
Karl Wilhelm Feuerbach (30 May 1800 – 12 March 1834) was a German geometer and the son of legal scholar Paul Johann Anselm Ritter von Feuerbach, and the brother of philosopher Ludwig Feuerbach. After receiving his doctorate at age 22, he became a professor of mathematics at the Gymnasium at Erlangen. In 1822 he wrote a small book on mathematics noted mainly for a theorem on the nine-point circle, which is now known as Feuerbach's theorem. In 1827 he introduced homogeneous coordinates, independently of Möbius.[1]
Karl Wilhelm Feuerbach
Born30 May 1800 (1800-05-30)
Jena, Saxe-Weimar, Holy Roman Empire
Died12 March 1834 (1834-03-13) (aged 33)
Erlangen, Germany
NationalityGerman
Alma materAlbert Ludwigs University of Freiburg
Known forFeuerbach's theorem
Scientific career
FieldsMathematician
InstitutionsUniversity of Basel
Notes
Brother of Ludwig Andreas Feuerbach
Works
• Eigenschaften einiger merkwürdigen Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten Linien und Figuren. Eine analytisch-trigonometrische Abhandlung (Monograph ed.), Nürnberg: Wiessner, 1822. online book at Google Books ("Properties of some special points in the plane of a triangle, and various lines and figures determined by these points: an analytic-trigonometric treatment")
• Grundriss zu analytischen Untersuchungen der dreyeckigen Pyramide ("Foundations of the analytic theory of the triangular pyramid")
References
1. "Feuerbach". sfabel.tripod.com. Retrieved 2020-07-21.
External links
• Works by or about Karl Wilhelm Feuerbach at Internet Archive
• O'Connor, John J.; Robertson, Edmund F., "Karl Wilhelm Feuerbach", MacTutor History of Mathematics Archive, University of St Andrews
• Feuerbach's Theorem: a Proof
• Karl Wilhelm Feuerbach: Geometer
• Media related to Karl Wilhelm Feuerbach (geometer) at Wikimedia Commons
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Wilhelm Fuhrmann
Wilhelm Ferdinand Fuhrmann (28 February 1833 - 11 June 1904) was a German mathematician. The Fuhrmann circle and the Fuhrmann triangle are named after him.[1]
Biography
Fuhrmann was born on 28 February 1833 in Burg bei Magdeburg. Fuhrmann had shortly worked as sailor before he returned to school and attended the Altstadt Gymnasium in Königsberg, where his teachers noticed his interest and talent in mathematics and geography. He graduated in 1853 and went on to study mathematics and physics at the University of Königsberg. One of his peers later remembered him as the most talented and diligent student of his class. Fuhrmann however despite his talent did not pursue a career at the university, instead he became a math and science teacher at the Burgschule in Königsberg after his graduation. He joined the school in 1860 and remained there until his death in 1904.[2]
Fuhrmann authored several books and a number of papers on different mathematical subjects. Today he is best remembered for his interest in and contribution to elementary geometry. With Synthetische Beweise planimetrischer Sätze he wrote an influential book on the subject and in 1890 he published an article entitled Sur un nouveau cercle associé à un triangle in the Belgian math journal Mathesis. In this article Fuhrmann described the circle and the triangle that now carry his name.[2][3]
Publications
Papers
• Transformationen der Theta-Funktionen (1864)
• Einige Untersuchungen über die Abhängigkeit geometrischer Gebilde (1869)
• Einige Anmerkungen der projektiven Eigenschaften der Figuren (1875)
• Aufgaben über Kegelschnitte (1879)
• Aufgaben aus der niederen Analysis (1886)
• Der Brocardsche Winkel (1889)
• "Sur un nouveau cercle associé à un triangle". In: Mathesis, 1890 (English translation)
• Sätze und Aufgaben aus der sphärischen Trigonometrie (1894)
• Beiträge zur Transformation algebraisch-trigonometrischer Figuren Teil 1 (1898)
• Beiträge zur Transformation algebraisch-trigonometrischer Figuren Teil 2 (1899)
• Kollineare und orthologische Dreiecke (1902)
• Aufgaben aus der analytischen Geometrie (1904, post mortem)
Books
• Synthetische Beweise planimetrischer Sätze. Berlin: L. Simion, 1890. Heute: Wentworth Press, 2018, ISBN 9780270116830 (online copy in the Internet Archive)
• Kollineare und orthologische Dreiecke. Königsberg: Hartung, 1902.
• Wegweiser in der Arithmetik, Algebra und niedern Analysis. Leipzig: Teubner, 1886.
References
1. Roger A. Johnson: Advanced Euclidean Geometry. Dover 2007, ISBN 978-0-486-46237-0, pp. 228–229, 300 (originally published 1929 with Houghton Mifflin Company (Boston) as Modern Geometry).
2. L. Saalschütz: "Zur Erinnerung an Wilhelm Fuhrmann". In: Jahresbericht der Deutschen Mathematiker-Vereinigung, Volume 14, 1905, pp. 56–60. (online copy)
3. Jan Vonk, J. Chris Fisher: "Translation of Fuhrmann’s “Sur un nouveau cercle associe´ a un triangle”. In: Forum Geometricorum, Volume 11 (2011), pp. 13–26
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Wilhelm Grunwald
Wilhelm Grunwald (15 July 1909 – 7 June 1989) was a German mathematician who introduced the Grunwald–Wang theorem, though his original statement and proof of this contained a small error that was corrected by Shianghao Wang. He later left mathematics to become a science librarian, and was director of the Göttingen university library (Roquette 2005, p.29).
References
• Grunwald, W. (1933), "Ein allgemeiner Existenzsatz für algebraische Zahlkörper", Journal für die reine und angewandte Mathematik, 169: 103–107
• Roquette, Peter (2005), The Brauer–Hasse–Noether theorem in historical perspective (PDF), Schriften der Mathematisch-Naturwissenschaftlichen Klasse der Heidelberger Akademie der Wissenschaften [Publications of the Mathematics and Natural Sciences Section of Heidelberg Academy of Sciences], vol. 15, Berlin, New York: Springer-Verlag, ISBN 978-3-540-23005-2
• Wilhelm Grunwald at the Mathematics Genealogy Project
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Wilhelm Lexis
Wilhelm Lexis (17 July 1837, Eschweiler, Germany – 24 August 1914, Göttingen, Germany), full name Wilhelm Hector Richard Albrecht Lexis,[1] was a German statistician, economist, and social scientist. The Oxford Dictionary of Statistics cites him as a "pioneer of the analysis of demographic time series".[2] Lexis is largely remembered for two items that bear his name—the Lexis ratio and the Lexis diagram.
Wilhelm Lexis
Born
Wilhelm Hector Richard Albrecht Lexis
(1837-07-17)17 July 1837
Eschweiler
Died24 August 1914(1914-08-24) (aged 77)
Göttingen
CitizenshipGerman
Scientific career
FieldsSocial scientist
Doctoral advisorAugust Beer[1]
Doctoral studentsLadislaus Bortkiewicz[1]
Life
Lexis graduated in 1859 from the University of Bonn, where he studied science and mathematics. He spent some time afterwards in various occupations and, in 1861, went to Paris to study social science. It was there that Lexis became acquainted with the work of Adolphe Quetelet, whose quantitative approach to the social sciences was to guide much of Lexis' work. He spent about ten years in Paris, after which he took a teaching position in Strasbourg (France). At some point during this period, Lexis wrote his first book (Introduction to the Theory of Population Statistics) and had it published in 1875, by which time he was teaching at the Imperial University of Dorpat (today the University of Tartu) in what is today Tartu, Estonia.
Starting in 1876, Lexis was the chair of the Economics Department at the University of Freiburg. The various papers written by him during his eight-year tenure at Freiburg were, in the eyes of statistics historian Stephen Stigler, "his most important statistical work". Foremost among them was the 1879 paper "On the Theory of the Stability of Statistical Series", which introduced the quantity now often called the Lexis ratio.
Lexis moved on from Freiburg to the University of Breslau but stayed there only a few years (from 1884 to 1887). He then settled in Göttingen, taking a position at that city's University. In 1895, he established a course in actuarial science at the university, the first ever in Germany. In 1901, Lexis became a member of the Insurance Advisory Council for Germany's Federal Insurance Supervisory Office. He remained a member of the Council until his death in 1914. During this final period of his life, Lexis published two more books: Treatises on Population and Social Statistics (Jena: Gustav Fischer, 1903) and General Economics (Leipzig: Teubner, 1910). He was also the editor of a book on the German education system.[3][4]
Work
Throughout his professional career, Lexis published books and articles on a wide variety of topics, including demography, economics and mathematical statistics. However, little of that work proved to have lasting significance. Today, Lexis is largely remembered for two items that bear his name—the Lexis ratio and the Lexis diagram. His theory of mortality has also enjoyed a recent revival of interest.
Lexis ratio
To Lexis, a time series was "stable" if the underlying probability giving rise to the observed rates remained constant from year to year (or, more generally, from one measurement period to the next). Using modern terminology, such a time series would be called a zero-order moving-average series (also known as a white noise process). Lexis was aware that many series were not stable. For non-stable series, he imagined that the underlying probabilities varied over time, being affected by what he called "physical" forces (as opposed to the random "non-essential" forces that would cause an observed rate to be different than the underlying probability). In his 1879 paper "On the Theory of the Stability of Statistical Series",[5] Lexis set himself the task of devising a method for distinguishing between stable and non-stable time series.
To this end, Lexis created a test statistic equal to the ratio between (i) the probable error of the observed rates and (ii) the probable error that would be expected if the underlying probabilities for each of the observed rates were all equal to the average rate observed across all of the observations. He called this ratio Q. Lexis then reasoned that if Q was sufficiently close to 1, then the time series was exhibiting what he called "normal dispersion" and one could assume that it was stable. If Q was substantially greater than 1, then the series was exhibiting "supernomal dispersion" and one must conclude that physical forces were having a discernible effect on the variability of the observations. Lexis used a Q value of 1.41 (i.e., the square root of 2) as the dividing line between "normal" and "supernormal" dispersion.
"Stability of Statistical Series" is the only one of Lexis' works cited in his entry in the Oxford Dictionary of Statistics. It is also the only one that receives an extended discussion in Stigler's A History of Statistics. And yet, Stigler ends his discussion by labeling the work a failure. To Stigler, its chief value was the discussion that it generated from other researchers in the field. It was those other researchers, and not Lexis, who created the modern science of time-series analysis.[6]
Lexis diagram
Although it can take various forms, the typical Lexis diagram is a graphical illustration of the lifetime of either an individual or a cohort of same-aged individuals. On the diagram, each such lifetime appears as a straight line in a two-dimensional plane, with one dimension representing time and the other representing age. The use of Lexis diagrams is very common amongst demographers, so much so that they often are used without being identified as Lexis diagrams.[7]
Lexis introduced his diagram in his first book, Introduction to the Theory of Population Statistics (Strasbourg: Trubner, 1875). However, the notion of using a time vs. age diagram appears to have been developed more or less simultaneously by other authors. See the paper by Vandeschrick (2001) for more detail.
Theory of mortality
In his 1877 book On the Theory of Mass Phenomena in Human Society (Freiburg: Wagnersche Buchhandlung), Lexis proposed that all human deaths could be classified into one of three types: (i) normal deaths, (ii) infant deaths and (iii) premature adult deaths. He also proposed that the normal deaths were subject to random forces such that, if all infant and other premature deaths were eliminated, the ages at which people died would exhibit a normal (i.e., Gaussian) distribution. Furthermore, the average of those ages would be equal to the age at which most adults are actually observed to die (i.e., the modal age at death), even though the actual observations are taking place in the presence of infant and other premature deaths.[8]
In the adjacent diagram, the normal deaths are represented by the vertically-shaded bell-shaped area centered slightly above age 70; the infant deaths are represented by the unshaded area starting at age 0; the premature deaths are represented by the horizontally-shaded area bridging the infant and normal deaths.
Although Lexis' theory did generate some contemporaneous discussion, it never supplanted the traditional demographic measures of life expectancy and age-adjusted mortality rates. However, recent research suggests that the modal age at death might be a useful statistic for tracking changes in the lifespans of the elderly. For a survey of the contemporaneous response to Lexis' theory, see section IV ("Reception of Lexis' hypothesis in the late 19th century") of Véron and Rohrbasser (2003). For a discussion of the modern-day use of the modal age at death, see Horiuchi et al. (2013).
Further reading
• Horiuchi, Shiro; Ouelette, Nadine; Cheung, Siu Lan Karen; Robine, Jean-Marie (2013). "Modal Age at Death: Lifespan Indicator in the Era of Longevity Extension" (PDF). Vienna Yearbook of Population Research. 11: 37–69. doi:10.1553/populationyearbook2013s37.
• Vandeschrick, Christophe (2001). "The Lexis Diagram, a Misnomer" (PDF). Demographic Research. 4: 97–124. doi:10.4054/DemRes.2001.4.3.
• Véron, Jacques; Rohrbasser, Jean-Marc (2003). "Wilhelm Lexis: The Normal Length of Life as an Expression of the 'Nature of Things'". Population. 53 (3): 303–322.
Two biographies of Lexis are:
• Heiss, Klaus-Peter (1978) "Wilhelm Lexis", in Kruskal, William H. and Tanur, Judith M. (eds.) International Encyclopedia of Statistics (New York: Free Press), Volume 1, pages 507-512
• Klein, Felix (1914) "Wilhelm Lexis" in Jahresbericht der Deutschen Mathematiker-Vereinigung, Volume 23, pages 314-317 (obituary, in German)
References
• Koch, Peter (1985) "Wilhem Lexis" in Neue Deutsche Biographie (Berlin: Duncker & Humblot) Volume 14, pages 421-422 (in German)
• Stigler, Stephen M. (1986) The History of Statistics: The Measurement of Uncertainty before 1900 (Cambridge, Massachusetts: Belknap Press, ISBN 0-674-40340-1), chapter 6 ("Attempts to Revive the Binomial"), pages 221-238
• Upton, Graham and Cook, Ian (2006) A Dictionary of Statistics, Second edition (Oxford: Oxford University Press, ISBN 0-19-861431-4), pages 237-238
1. Lexis' page at the Mathematics Genealogy Project Note that the date of death given in the MacTutor biography does not agree with the German sources, including the 1914 obituary by Felix Klein. These other sources give the date as shown above.
2. Upton and Cook (2006), page 238
3. A General Overview of the History and Organisation of Public Education in the German Empire (Berlin: A. Asher, 1904)
4. Factual details in this section are taken from the Lexis entry in the Neue Deutsche Biographie. The relationship of Lexis' work to that of Quetelet is from page 223 of Stigler (1986), as is the direct quote concerning Lexis' Freiburg-period work.
5. Jahrbücher für National Ökonomie und Statistik, Volume 32, 1879, pages 60-98. The distinction between "physical" and "non-essential" forces is made on page 66.
6. Stigler's discussion of the Lexis ratio is at pages 229 through 234 of Stigler (1986) (i.e., the sections titled "The Dispersion of Series" and "Lexis's Analysis and Interpretation"). His finding that Lexis' work was a failure is at pages 234 through 236 ("Why Lexis Failed"). The effect on other researchers is at pages 237-238 ("Lexian Dispersion after Lexis").
7. For example, see Dick London's discussion of U.S. Census Bureau techniques in chapter 9 of his Survival Models and their Estimation (Winsted, Connecticut: Actex, 1988 ISBN 0-936031-01-8). Also see Kenneth P. Veit's "Stationary Population Methods" in the Transactions of the Society of Actuaries, Volume XVI (1964), page 233 ff. (available here).
8. The discussion of normal vs. premature deaths starts at page 45 of Mass Phenomena. Note that Lexis uses the word jugendlichen to describe the infant deaths. Although Lexis' word might equally well be translated as "youth", his calculations later in the text indicate that no jugendlichen deaths are assumed to take place after age 15. Véron and Rohrbasser (2003) and Horiuchi et al. (2013) both translate Lexis' word as "infant".
External links
Wikisource has the text of a 1905 New International Encyclopedia article about "Wilhelm Lexis".
Wikimedia Commons has media related to Wilhelm Lexis.
Works by Lexis
• Lexis' Ph.D. dissertation (in Latin)
• Introduction to the Theory of Population Statistics (1875) (in German)
• On the Theory of Mass Phenomena in Human Society (1877) (in German)
• On the Theory of the Stability of Statistical Series (1879) (in German)
• Treatises on Population and Social Statistics (1903) (in German)
• 1904 text on the German education system (in English translation)
Biographies of Lexis
• Entry in the Neue Deutsche Biographie (in German)
• Entry in the Oxford Dictionary of Statistics
• Obituary by Felix Klein (in German)
• O'Connor, John J.; Robertson, Edmund F., "Wilhelm Lexis", MacTutor History of Mathematics Archive, University of St Andrews
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Wilhelm Ackermann
Wilhelm Friedrich Ackermann (/ˈækərmən/; German: [ˈakɐˌman]; 29 March 1896 – 24 December 1962) was a German mathematician and logician best known for his work in mathematical logic[1] and the Ackermann function, an important example in the theory of computation.
Wilhelm Ackermann
Wilhelm Ackermann in c. 1935
Born(1896-03-29)29 March 1896
Herscheid, German Empire
Died24 December 1962(1962-12-24) (aged 66)
Lüdenscheid, West Germany
NationalityGerman
Alma materUniversity of Göttingen
Known for
• Ackermann coding
Ackermann function
Ackermann set theory
Scientific career
FieldsMathematics
Doctoral advisorDavid Hilbert
Biography
Ackermann was born in Herscheid, Germany, and was awarded a Ph.D. by the University of Göttingen in 1925 for his thesis Begründung des "tertium non datur" mittels der Hilbertschen Theorie der Widerspruchsfreiheit, which was a consistency proof of arithmetic apparently without Peano induction (although it did use e.g. induction over the length of proofs). This was one of two major works in proof theory in the 1920s and the only one following Hilbert's school of thought.[1] From 1929 until 1948, he taught at the Arnoldinum Gymnasium in Burgsteinfurt, and then at Lüdenscheid until 1961. He was also a corresponding member of the Akademie der Wissenschaften (Academy of Sciences) in Göttingen, and was an honorary professor at the University of Münster.
In 1928, Ackermann helped David Hilbert turn his 1917 – 22 lectures on introductory mathematical logic into a text, Principles of Mathematical Logic. This text contained the first exposition ever of first-order logic, and posed the problem of its completeness and decidability (Entscheidungsproblem). Ackermann went on to construct consistency proofs for set theory (1937), full arithmetic (1940), type-free logic (1952), and a new axiomatization of set theory (1956).
Later in life, Ackermann continued working as a high school teacher. Still, he kept continually engaged in the field of research and published many contributions to the foundations of mathematics until the end of his life. He died in Lüdenscheid, West Germany in December 1962.
See also
• Ackermann's bijection
• Ackermann coding
• Ackermann function
• Ackermann ordinal
• Ackermann set theory
• Hilbert–Ackermann system
• Entscheidungsproblem
• Ordinal notation
• Inverse Ackermann function
Bibliography
• 1928. "On Hilbert's construction of the real numbers" in Jean van Heijenoort, ed., 1967. From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931. Harvard Univ. Press: 493–507.
• 1940. "Zur Widerspruchsfreiheit der Zahlentheorie", Mathematische Annalen, vol. 117, pp 162–194.
• 1950 (1928). (with David Hilbert) Principles of Mathematical Logic. Chelsea. Translation of 1938 German edition.
• 1954. Solvable cases of the decision problem. North Holland.
References
1. O'Connor, J J; Robertson, E F; Felscher, Walter. "Wilhelm Ackermann". MacTutor History of Mathematics. Retrieved 18 August 2021.
External links
• O'Connor, John J.; Robertson, Edmund F., "Wilhelm Ackermann", MacTutor History of Mathematics Archive, University of St Andrews
• Wilhelm Ackermann at the Mathematics Genealogy Project
• Erich Friedman's page on Ackermann at Stetson University
• Hermes, In memoriam WILHELM ACKERMANN 1896-1962 (PDF, 945 KB)
• Author profile in the database zbMATH
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Wilhelm Blaschke
Wilhelm Johann Eugen Blaschke (13 September 1885 – 17 March 1962) was an Austrian mathematician working in the fields of differential and integral geometry.
Wilhelm Blaschke
Born(1885-09-13)13 September 1885
Graz
Died17 March 1962(1962-03-17) (aged 76)
Hamburg
NationalityAustrian
Alma materUniversity of Vienna
Known forBlaschke product
Blaschke selection theorem
Blaschke–Santaló inequality
Scientific career
FieldsMathematics
InstitutionsUniversity of Hamburg
Doctoral advisorWilhelm Wirtinger
Doctoral studentsShiing-Shen Chern
Luis Santaló
Emanuel Sperner
Other notable studentsAlberto Dou Mas de Xaxàs
Education and career
Blaschke was the son of mathematician Josef Blaschke, who taught geometry at the Landes Oberrealschule in Graz. After studying for two years at the Technische Hochschule in Graz, he went to the University of Vienna, and completed a doctorate in 1908 under the supervision of Wilhelm Wirtinger.[1] His dissertation was Über eine besondere Art von Kurven vierter Klasse.[2]
After completing his doctorate he spent several years visiting mathematicians at the major universities in Italy and Germany. He spent two years each in positions in Prague, Leipzig, Göttingen, and Tübingen until, in 1919, he took the professorship at the University of Hamburg that he would keep for the rest of his career.[1] His students at Hamburg included Shiing-Shen Chern, Luis Santaló, and Emanuel Sperner.[2]
In 1933 Blaschke signed the Vow of allegiance of the Professors of the German Universities and High-Schools to Adolf Hitler and the National Socialistic State.[3][4] However, he defended Kurt Reidemeister against the Nazis and, in the early 1930s, campaigned against Ludwig Bieberbach for leadership of the German Mathematical Society, arguing that the society should remain international and apolitical in opposition to Bieberbach's wish to "enforce Nazi policies on German mathematics and race". However, by 1936 he was supporting Nazi policies, called himself "a Nazi at Heart", and was described by colleagues as "Mussolinetto" for his fascist beliefs.[1] He officially joined the Nazi Party in 1937.[3]
After the war, Blaschke was removed from his position at the University of Hamburg for his Nazi affiliation, but after an appeal his professorship was restored in 1946.[1] He remained at the university until his retirement in 1953.[1]
Publications
In 1916 Blaschke published one of the first books devoted to convex sets: Circle and Sphere (Kreis und Kugel). Drawing on dozens of sources, Blaschke made a thorough review of the subject with citations within the text to attribute credit in a classical area of mathematics.
• Kreis und Kugel, Leipzig, Veit 1916; 3rd edn. Berlin, de Gruyter 1956
• Vorlesungen über Differentialgeometrie, 3 vols., Springer, Grundlehren der mathematischen Wissenschaften 1921-1929 (vol. 1, Elementare Differentialgeometrie;[5] vol. 2, Affine Differentialgeometrie; vol. 3, Differentialgeometrie der Kreise und Kugeln, 1929)
• with G. Bol: Geometrie der Gewebe. Berlin: Springer 1938[6]
• Ebene Kinematik. Leipzig: B.G. Teubner 1938,[7] 2nd expanded edn. with Hans Robert Müller, Oldenbourg, München 1956
• Nicht-Euklidische Geometrie und Mechanik I, II, III. Leipzig: B.G.Teubner (1942)
• Zur Bewegungsgeometrie auf der Kugel. In: Sitzungsberichte der Heidelberger Akademie der Wissenschaften (1948)
• Einführung in die Differentialgeometrie. Springer 1950,[8] 2nd expanded edn. with H. Reichardt 1960
• with Kurt Leichtweiß: Elementare Differentialgeometrie. Berlin: Springer (5th edn. 1973)
• Reden und Reisen eines Geometers. Berlin : VEB Deutscher Verlag der Wissenschaften (1961; 2nd expanded edn.)
• Mathematik und Leben, Wiesbaden, Steiner 1951
• Griechische und anschauliche Geometrie, Oldenbourg 1953
• Projektive Geometrie, 3rd edn, Birkhäuser 1954
• Analytische Geometrie, 2nd edn., Birkhäuser 1954
• Einführung in die Geometrie der Waben, Birkhäuser 1955[9]
• Vorlesungen über Integralgeometrie, VEB, Berlin 1955
• Reden und Reisen eines Geometers, 1957
• Kinematik und Quaternionen. Berlin: VEB Deutscher Verlag der Wissenschaften (1960)
• Gesammelte Werke, Thales, Essen 1985
Namesake
A number of mathematical theorems and concepts is associated with the name of Blaschke.
• Blaschke selection theorem
• Blaschke–Lebesgue theorem
• Blaschke product
• Blaschke sum
• Blaschke condition
• Blaschke–Busemann measure
• Blaschke–Santaló inequality
• Blaschke conjecture: "The only Wiedersehen manifolds in any dimension are the standard Euclidean spheres."
See also
• Affine differential geometry
• Affine geometry of curves
• Body of constant brightness
• Web (differential geometry)
• Pestov–Ionin theorem
References
1. O'Connor, John J.; Robertson, Edmund F. "Wilhelm Blaschke". MacTutor History of Mathematics Archive. University of St Andrews.
2. Wilhelm Blaschke at the Mathematics Genealogy Project
3. Heinrich Behnke (1898-1979): zwischen Mathematik und deren Didaktik, Uta Hartmann, 2009 ISBN 9783631588604
4. Ernst Klee: Das Personenlexikon zum Dritten Reich. Wer war was vor und nach 1945. Fischer Taschenbuch Verlag, Zweite aktualisierte Auflage, Frankfurt am Main 2005, S. 52.
5. Bliss, G. A. (1923). "Blaschke on Differential Geometry". Bull. Amer. Math. Soc. 29 (7): 322–325. doi:10.1090/S0002-9904-1923-03737-3.
6. Walker, R. J. (1939). "Review: W. Blaschke and G. Bol, Geometrie der Gewebe". Bull. Amer. Math. Soc. 45 (9): 652–653. doi:10.1090/s0002-9904-1939-07052-3.
7. Synge, J. L. (1939). "Review: W. Blaschke, Eben Kinematik". Bull. Amer. Math. Soc. 45 (11): 814–815. doi:10.1090/s0002-9904-1939-07071-7.
8. Allendoerfer, Carl B. (1951). "Review, W. Blaschke, Einführung in die Differentialgeometrie". Bull. Amer. Math. Soc. 57 (1, Part 1): 84–85. doi:10.1090/s0002-9904-1951-09454-9.
9. Hsiung, C. C. (1957). "Review: W. Blaschke, Einführung in die Geometrie der Waben". Bull. Amer. Math. Soc. 63 (3): 203–204. doi:10.1090/s0002-9904-1957-10104-9.
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• Newspaper clippings about Wilhelm Blaschke in the 20th Century Press Archives of the ZBW
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| Wikipedia |
Wilkie's theorem
In mathematics, Wilkie's theorem is a result by Alex Wilkie about the theory of ordered fields with an exponential function, or equivalently about the geometric nature of exponential varieties.
Formulations
In terms of model theory, Wilkie's theorem deals with the language Lexp = (+, −, ·, <, 0, 1, ex), the language of ordered rings with an exponential function ex. Suppose φ(x1, ..., xm) is a formula in this language. Then Wilkie's theorem states that there is an integer n ≥ m and polynomials f1, ..., fr ∈ Z[x1, ..., xn, ex1, ..., exn] such that φ(x1, ..., xm) is equivalent to the existential formula
$\exists x_{m+1}\ldots \exists x_{n}\,f_{1}(x_{1},\ldots ,x_{n},e^{x_{1}},\ldots ,e^{x_{n}})=\cdots =f_{r}(x_{1},\ldots ,x_{n},e^{x_{1}},\ldots ,e^{x_{n}})=0.$
Thus, while this theory does not have full quantifier elimination, formulae can be put in a particularly simple form. This result proves that the theory of the structure Rexp, that is the real ordered field with the exponential function, is model complete.[1]
In terms of analytic geometry, the theorem states that any definable set in the above language — in particular the complement of an exponential variety — is in fact a projection of an exponential variety. An exponential variety over a field K is the set of points in Kn where a finite collection of exponential polynomials simultaneously vanish. Wilkie's theorem states that if we have any definable set in an Lexp structure K = (K, +, −, ·, 0, 1, ex), say X ⊂ Km, then there will be an exponential variety in some higher dimension Kn such that the projection of this variety down onto Km will be precisely X.
Gabrielov's theorem
The result can be considered as a variation of Gabrielov's theorem. This earlier theorem of Andrei Gabrielov dealt with sub-analytic sets, or the language Lan of ordered rings with a function symbol for each proper analytic function on Rm restricted to the closed unit cube [0, 1]m. Gabrielov's theorem states that any formula in this language is equivalent to an existential one, as above.[2] Hence the theory of the real ordered field with restricted analytic functions is model complete.
Intermediate results
Gabrielov's theorem applies to the real field with all restricted analytic functions adjoined, whereas Wilkie's theorem removes the need to restrict the function, but only allows one to add the exponential function. As an intermediate result Wilkie asked when the complement of a sub-analytic set could be defined using the same analytic functions that described the original set. It turns out the required functions are the Pfaffian functions.[1] In particular the theory of the real ordered field with restricted, totally defined Pfaffian functions is model complete.[3] Wilkie's approach for this latter result is somewhat different from his proof of Wilkie's theorem, and the result that allowed him to show that the Pfaffian structure is model complete is sometimes known as Wilkie's theorem of the complement. See also.[4]
References
1. A.J. Wilkie, Model completeness results for expansions of the ordered field of real numbers by restricted pfaffian functions and the exponential functions, J. Amer. Math. Soc. 9 (1996), pp. 1051–1094.
2. A. Gabrielov, Projections of semi-analytic sets, Functional Anal. Appl. 2 (1968), pp.282–291.
3. A.J. Wilkie, A theorem of the complement and some new o-minimal structures, Sel. Math. 5 (1999), pp.397–421.
4. M. Karpinski and A. Macintyre, A generalization of Wilkie's theorem of the complement, and an application to Pfaffian closure, Sel. math., New ser. 5 (1999), pp.507-516
| Wikipedia |
Wilkie investment model
The Wilkie investment model, often just called Wilkie model, is a stochastic asset model developed by A. D. Wilkie that describes the behavior of various economics factors as stochastic time series. These time series are generated by autoregressive models. The main factor of the model which influences all asset prices is the consumer price index. The model is mainly in use for actuarial work and asset liability management. Because of the stochastic properties of that model it is mainly combined with Monte Carlo methods.
Wilkie first proposed the model in 1986, in a paper published in the Transactions of the Faculty of Actuaries.[1] It has since been the subject of extensive study and debate.[2][3] Wilkie himself updated and expanded the model in a second paper published in 1995.[4] He advises to use that model to determine the "funnel of doubt", which can be seen as an interval of minimum and maximum development of a corresponding economic factor.
Components
• price inflation
• wage inflation
• share yield
• share dividend
• consols yield (long-term interest rate)
• bank rate (short-term interest rate)
References
1. Wilkie, A.D. (1986). "A stochastic investment model for Actuarial Use" (PDF). Transactions of the Faculty of Actuaries. 39: 341–403. doi:10.1017/S0071368600009009.
2. Geoghegan, T J; Clarkson, R S; Feldman, K S; Green, S J; Kitts, A; Lavecky, J P; Ross, F J M; Smith, W J; Toutounchi, A (27 January 1992). "Report on the Wilkie investment model". Journal of the Institute of Actuaries. 119: 173–228. doi:10.1017/S0020268100019879.
3. Şahin, Şule; Cairns, Andrew; Kleinow, Torsten; Wilkie, A. D. (12 June 2008). Revisiting the Wilkie Investment Model (PDF). International Actuarial Association, AFIR/ERM Sectional Colloquium, Rome, 2008.
4. Wilkie, A.D. (1995). "More on a stochastic asset model for actuarial use". British Actuarial Journal. 1 (5): 777–964. doi:10.1017/S1357321700001331. S2CID 153338215.
| Wikipedia |
Wilkinson matrix
In linear algebra, Wilkinson matrices are symmetric, tridiagonal, order-N matrices with pairs of nearly, but not exactly, equal eigenvalues.[1] It is named after the British mathematician James H. Wilkinson. For N = 7, the Wilkinson matrix is given by
${\begin{bmatrix}3&1&0&0&0&0&0\\1&2&1&0&0&0&0\\0&1&1&1&0&0&0\\0&0&1&0&1&0&0\\0&0&0&1&1&1&0\\0&0&0&0&1&2&1\\0&0&0&0&0&1&3\\\end{bmatrix}}.$
Wilkinson matrices have applications in many fields, including scientific computing, numerical linear algebra, and signal processing.
References
1. Wilkinson (1965). The Algebraic Eigenvalue Problem. Oxford University Press. ISBN 0-19-853418-3.
Numerical linear algebra
Key concepts
• Floating point
• Numerical stability
Problems
• System of linear equations
• Matrix decompositions
• Matrix multiplication (algorithms)
• Matrix splitting
• Sparse problems
Hardware
• CPU cache
• TLB
• Cache-oblivious algorithm
• SIMD
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• MATLAB
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| Wikipedia |
Wilks' theorem
In statistics Wilks' theorem offers an asymptotic distribution of the log-likelihood ratio statistic, which can be used to produce confidence intervals for maximum-likelihood estimates or as a test statistic for performing the likelihood-ratio test.
Statistical tests (such as hypothesis testing) generally require knowledge of the probability distribution of the test statistic. This is often a problem for likelihood ratios, where the probability distribution can be very difficult to determine.
A convenient result by Samuel S. Wilks says that as the sample size approaches $\infty $, the distribution of the test statistic $-2\log(\Lambda )$ asymptotically approaches the chi-squared ($\chi ^{2}$) distribution under the null hypothesis $H_{0}$.[1] Here, $\Lambda $ denotes the likelihood ratio, and the $\chi ^{2}$ distribution has degrees of freedom equal to the difference in dimensionality of $\Theta $ and $\Theta _{0}$, where $\Theta $ is the full parameter space and $\Theta _{0}$ is the subset of the parameter space associated with $H_{0}$. This result means that for large samples and a great variety of hypotheses, a practitioner can compute the likelihood ratio $\Lambda $ for the data and compare $-2\log(\Lambda )$ to the $\chi ^{2}$ value corresponding to a desired statistical significance as an approximate statistical test.
The theorem no longer applies when the true value of the parameter is on the boundary of the parameter space: Wilks’ theorem assumes that the ‘true’ but unknown values of the estimated parameters lie within the interior of the supported parameter space. In practice, one will notice the problem if the estimate lies on that boundary. In that event, the likelihood test is still a sensible test statistic and even possess some asymptotic optimality properties, but the significance (the p-value) can not be reliably estimated using the chi-squared distribution with the number of degrees of freedom prescribed by Wilks. In some cases, the asymptotic null-hypothesis distribution of the statistic is a mixture of chi-square distributions with different numbers of degrees of freedom.
Use
Each of the two competing models, the null model and the alternative model, is separately fitted to the data and the log-likelihood recorded. The test statistic (often denoted by D) is twice the log of the likelihoods ratio, i.e., it is twice the difference in the log-likelihoods:
${\begin{aligned}D&=-2\ln \left({\frac {\text{likelihood for null model}}{\text{likelihood for alternative model}}}\right)\\[5pt]&=2\ln \left({\frac {\text{likelihood for alternative model}}{\text{likelihood for null model}}}\right)\\[5pt]&=2\times [\ln({\text{likelihood for alternative model}})-\ln({\text{likelihood for null model}})]\\[5pt]\end{aligned}}$
The model with more parameters (here alternative) will always fit at least as well — i.e., have the same or greater log-likelihood — than the model with fewer parameters (here null). Whether the fit is significantly better and should thus be preferred is determined by deriving how likely (p-value) it is to observe such a difference D by chance alone, if the model with fewer parameters were true. Where the null hypothesis represents a special case of the alternative hypothesis, the probability distribution of the test statistic is approximately a chi-squared distribution with degrees of freedom equal to $\,df_{\text{alt}}-df_{\text{null}}\,$,[2] respectively the number of free parameters of models alternative and null.
For example: If the null model has 1 parameter and a log-likelihood of −8024 and the alternative model has 3 parameters and a log-likelihood of −8012, then the probability of this difference is that of chi-squared value of $2\times (-8012-(-8024))=24$ with $3-1=2$ degrees of freedom, and is equal to $6\times 10^{-6}$. Certain assumptions[1] must be met for the statistic to follow a chi-squared distribution, but empirical p-values may also be computed if those conditions are not met.
Examples
Coin tossing
An example of Pearson's test is a comparison of two coins to determine whether they have the same probability of coming up heads. The observations can be put into a contingency table with rows corresponding to the coin and columns corresponding to heads or tails. The elements of the contingency table will be the number of times each coin came up heads or tails. The contents of this table are our observations X.
${\begin{array}{c|cc}X&{\text{Heads}}&{\text{Tails}}\\\hline {\text{Coin 1}}&k_{\mathrm {1H} }&k_{\mathrm {1T} }\\{\text{Coin 2}}&k_{\mathrm {2H} }&k_{\mathrm {2T} }\end{array}}$
Here Θ consists of the possible combinations of values of the parameters $p_{\mathrm {1H} }$, $p_{\mathrm {1T} }$, $p_{\mathrm {2H} }$, and $p_{\mathrm {2T} }$, which are the probability that coins 1 and 2 come up heads or tails. In what follows, $i=1,2$ and $j=\mathrm {H,T} $. The hypothesis space H is constrained by the usual constraints on a probability distribution, $0\leq p_{ij}\leq 1$, and $p_{i\mathrm {H} }+p_{i\mathrm {T} }=1$. The space of the null hypothesis $H_{0}$ is the subspace where $p_{1j}=p_{2j}$. The dimensionality of the full parameter space Θ is 2 (either of the $p_{1j}$ and either of the $p_{2j}$ may be treated as free parameters under the hypothesis $H$), and the dimensionality of $\Theta _{0}$ is 1 (only one of the $p_{ij}$ may be considered a free parameter under the null hypothesis $H_{0}$).
Writing $n_{ij}$ for the best estimates of $p_{ij}$ under the hypothesis H, the maximum likelihood estimate is given by
$n_{ij}={\frac {k_{ij}}{k_{i\mathrm {H} }+k_{i\mathrm {T} }}}\,.$
Similarly, the maximum likelihood estimates of $p_{ij}$ under the null hypothesis $H_{0}$ are given by
$m_{ij}={\frac {k_{1j}+k_{2j}}{k_{\mathrm {1H} }+k_{\mathrm {2H} }+k_{\mathrm {1T} }+k_{\mathrm {2T} }}}\,,$
which does not depend on the coin i.
The hypothesis and null hypothesis can be rewritten slightly so that they satisfy the constraints for the logarithm of the likelihood ratio to have the desired distribution. Since the constraint causes the two-dimensional H to be reduced to the one-dimensional $H_{0}$, the asymptotic distribution for the test will be $\chi ^{2}(1)$, the $\chi ^{2}$ distribution with one degree of freedom.
For the general contingency table, we can write the log-likelihood ratio statistic as
$-2\log \Lambda =2\sum _{i,j}k_{ij}\log {\frac {n_{ij}}{m_{ij}}}\,.$
Invalidity for random or mixed effects models
Wilks’ theorem assumes that the true but unknown values of the estimated parameters are in the interior of the parameter space. This is commonly violated in random or mixed effects models, for example, when one of the variance components is negligible relative to the others. In some such cases, one variance component can be effectively zero, relative to the others, or in other cases the models can be improperly nested.
To be clear: These limitations on Wilks’ theorem do not negate any power properties of a particular likelihood ratio test.[3] The only issue is that a $\chi ^{2}$ distribution is sometimes a poor choice for estimating the statistical significance of the result.
Bad examples
Pinheiro and Bates (2000) showed that the true distribution of this likelihood ratio chi-square statistic could be substantially different from the naïve $\chi ^{2}$ – often dramatically so.[4] The naïve assumptions could give significance probabilities (p-values) that are, on average, far too large in some cases and far too small in others.
In general, to test random effects, they recommend using Restricted maximum likelihood (REML). For fixed-effects testing, they say, “a likelihood ratio test for REML fits is not feasible”, because changing the fixed effects specification changes the meaning of the mixed effects, and the restricted model is therefore not nested within the larger model.[4] As a demonstration, they set either one or two random effects variances to zero in simulated tests. In those particular examples, the simulated p-values with k restrictions most closely matched a 50–50 mixture of $\chi ^{2}(k)$ and $\chi ^{2}(k-1)$. (With k = 1 , $\chi ^{2}(0)$ is 0 with probability 1. This means that a good approximation was $\,0.5\,\chi ^{2}(1)\,.$)[4]
Pinheiro and Bates also simulated tests of different fixed effects. In one test of a factor with 4 levels (degrees of freedom = 3), they found that a 50–50 mixture of $\chi ^{2}(3)$ and $\chi ^{2}(4)$ was a good match for actual p-values obtained by simulation – and the error in using the naïve $\chi ^{2}(3)$ “may not be too alarming.”[4]
However, in another test of a factor with 15 levels, they found a reasonable match to $\chi ^{2}(18)$ – 4 more degrees of freedom than the 14 that one would get from a naïve (inappropriate) application of Wilks’ theorem, and the simulated p-value was several times the naïve $\chi ^{2}(14)$. They conclude that for testing fixed effects, “it's wise to use simulation.”[lower-alpha 1]
See also
• Bayes factor
• Model selection
• Sup-LR test
Notes
1. Pinheiro and Bates (2000)[4] provided a simulate.lme function in their nlme package for S-PLUS and R to support REML simulation; see ref.[5]
References
1. Wilks, Samuel S. (1938). "The large-sample distribution of the likelihood ratio for testing composite hypotheses". The Annals of Mathematical Statistics. 9 (1): 60–62. doi:10.1214/aoms/1177732360.
2. Huelsenbeck, J.P.; Crandall, K.A. (1997). "Phylogeny Estimation and Hypothesis Testing Using Maximum Likelihood". Annual Review of Ecology and Systematics. 28: 437–466. doi:10.1146/annurev.ecolsys.28.1.437.
3. Neyman, Jerzy; Pearson, Egon S. (1933). "On the problem of the most efficient tests of statistical hypotheses" (PDF). Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences. 231 (694–706): 289–337. Bibcode:1933RSPTA.231..289N. doi:10.1098/rsta.1933.0009. JSTOR 91247.
4. Pinheiro, José C.; Bates, Douglas M. (2000). Mixed-Effects Models in S and S-PLUS. Springer-Verlag. pp. 82–93. ISBN 0-387-98957-9.
5. "Simulate results from lme models" (PDF). R-project.org (software documentation). Package nlme. 12 May 2019. pp. 281–282. Retrieved 8 June 2019.
Other sources
• Casella, George; Berger, Roger L. (2001). Statistical Inference (Second ed.). ISBN 0-534-24312-6.
• Mood, A.M.; Graybill, F.A. (1963). Introduction to the Theory of Statistics (2nd ed.). McGraw-Hill. ISBN 978-0070428638.
• Cox, D.R.; Hinkley, D.V. (1974). Theoretical Statistics. Chapman and Hall. ISBN 0-412-12420-3.
• Stuart, A.; Ord, K.; Arnold, S. (1999). Kendall's Advanced Theory of Statistics. Vol. 2A. London: Arnold. ISBN 978-0-340-66230-4.
External links
• "Likelihood Ratio: Wilks's Theorem".
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| Wikipedia |
Wilks Memorial Award
The Wilks Memorial Award is awarded by the American Statistical Association to recognize outstanding contributions to statistics. It was established in 1964 and is awarded yearly. It is named in memory of the statistician Samuel S. Wilks. The award consists of a medal, a citation and a cash honorarium of US$1500 (as of 2008).[1]
Wilks Memorial Award
Awarded forOutstanding contributions to statistics
CountryUSA
Presented byAmerican Statistical Association
First awarded1964
Websitehttps://www.amstat.org/ASA/Your-Career/Awards/Samuel-S-Wilks-Memorial-Award.aspx
Recipients
• 1964 Frank E. Grubbs
• 1965 John W. Tukey
• 1966 Leslie E. Simon
• 1967 William G. Cochran
• 1968 Jerzy Neyman
• 1969 W. J. Youden
• 1970 George W. Snedecor
• 1971 Harold F. Dodge
• 1972 George E.P. Box
• 1973 Herman Otto Hartley
• 1974 Cuthbert Daniel
• 1975 Herbert Solomon
• 1976 Solomon Kullback
• 1977 Churchill Eisenhart
• 1978 William Kruskal
• 1979 Alexander M. Mood
• 1980 W. Allen Wallis
• 1981 Holbrook Working
• 1982 Frank Proschan
• 1983 W. Edwards Deming
• 1984 Z. W. Birnbaum
• 1985 Leo A. Goodman
• 1986 Frederick Mosteller
• 1987 Herman Chernoff
• 1988 Theodore W. Anderson
• 1989 C. R. Rao
• 1990 Bradley Efron
• 1991 Ingram Olkin
• 1992 Wilfrid Dixon
• 1993 Norman L. Johnson
• 1994 Emanuel Parzen
• 1995 Donald Rubin
• 1996 Erich L. Lehmann
• 1997 Leslie Kish
• 1998 David O. Siegmund
• 1999 Lynne Billard
• 2000 Stephen Fienberg
• 2001 George C. Tiao
• 2002 Lawrence D. Brown
• 2003 David L. Wallace
• 2004 Paul Meier
• 2005 Roderick J. A. Little
• 2006 Marvin Zelen
• 2007 Colin L. Mallows
• 2008 Scott Zeger
• 2009 Lee-Jen Wei
• 2010 Pranab K. Sen
• 2011 Nan Laird
• 2012 Peter Gavin Hall
• 2013 Kanti Mardia
• 2014 Madan L. Puri
• 2015 James O. Berger
• 2016 David Donoho
• 2017 Wayne Fuller
• 2018 Peter J. Bickel
• 2019 Alan E. Gelfand
• 2020 Malay Ghosh
• 2021 Sallie Ann Keller
• 2022 Jessica Utts[2]
References
1. "Wilks Memorial Award". American Statistical Association.
2. "Professors Utts and Stern Honored with American Statistical Association Awards". www.stat.uci.edu. Retrieved 2023-05-05.
| Wikipedia |
Shapiro–Wilk test
The Shapiro–Wilk test is a test of normality. It was published in 1965 by Samuel Sanford Shapiro and Martin Wilk.[1]
Not to be confused with the likelihood-ratio test, which is sometimes referred to as Wilks test.
Theory
The Shapiro–Wilk test tests the null hypothesis that a sample x1, ..., xn came from a normally distributed population. The test statistic is
$W={\left(\sum _{i=1}^{n}a_{i}x_{(i)}\right)^{2} \over \sum _{i=1}^{n}(x_{i}-{\overline {x}})^{2}},$
where
• $x_{(i)}$ with parentheses enclosing the subscript index i is the ith order statistic, i.e., the ith-smallest number in the sample (not to be confused with $x_{i}$).
• ${\overline {x}}=\left(x_{1}+\cdots +x_{n}\right)/n$ is the sample mean.
The coefficients $a_{i}$ are given by:[1]
$(a_{1},\dots ,a_{n})={m^{\mathsf {T}}V^{-1} \over C},$
where C is a vector norm:[2]
$C=\|V^{-1}m\|=(m^{\mathsf {T}}V^{-1}V^{-1}m)^{1/2}$
and the vector m,
$m=(m_{1},\dots ,m_{n})^{\mathsf {T}}\,$
is made of the expected values of the order statistics of independent and identically distributed random variables sampled from the standard normal distribution; finally, $V$ is the covariance matrix of those normal order statistics.[3]
There is no name for the distribution of $W$. The cutoff values for the statistics are calculated through Monte Carlo simulations.[2]
Interpretation
The null-hypothesis of this test is that the population is normally distributed. Thus, if the p value is less than the chosen alpha level, then the null hypothesis is rejected and there is evidence that the data tested are not normally distributed. On the other hand, if the p value is greater than the chosen alpha level, then the null hypothesis (that the data came from a normally distributed population) can not be rejected (e.g., for an alpha level of .05, a data set with a p value of less than .05 rejects the null hypothesis that the data are from a normally distributed population – consequently, a data set with a p value more than the .05 alpha value fails to reject the null hypothesis that the data is from a normally distributed population).[4]
Like most statistical significance tests, if the sample size is sufficiently large this test may detect even trivial departures from the null hypothesis (i.e., although there may be some statistically significant effect, it may be too small to be of any practical significance); thus, additional investigation of the effect size is typically advisable, e.g., a Q–Q plot in this case.[5]
Power analysis
Monte Carlo simulation has found that Shapiro–Wilk has the best power for a given significance, followed closely by Anderson–Darling when comparing the Shapiro–Wilk, Kolmogorov–Smirnov, and Lilliefors.[6]
Approximation
Royston proposed an alternative method of calculating the coefficients vector by providing an algorithm for calculating values that extended the sample size from 50 to 2,000.[7] This technique is used in several software packages including GraphPad Prism, Stata,[8][9] SPSS and SAS.[10] Rahman and Govidarajulu extended the sample size further up to 5,000.[11]
See also
• Anderson–Darling test
• Cramér–von Mises criterion
• D'Agostino's K-squared test
• Kolmogorov–Smirnov test
• Lilliefors test
• Normal probability plot
• Shapiro–Francia test
References
1. Shapiro, S. S.; Wilk, M. B. (1965). "An analysis of variance test for normality (complete samples)". Biometrika. 52 (3–4): 591–611. doi:10.1093/biomet/52.3-4.591. JSTOR 2333709. MR 0205384. p. 593
2. RMD (2022). "The Shapiro-Wilk and related tests for normality" (PDF). Retrieved 2022-06-16.
3. Davis, C. S.; Stephens, M. A. (1978). The covariance matrix of normal order statistics (PDF) (Technical report). Department of Statistics, Stanford University, Stanford, California. Technical Report No. 14. Retrieved 2022-06-17.
4. "How do I interpret the Shapiro–Wilk test for normality?". JMP. 2004. Retrieved March 24, 2012.
5. Field, Andy (2009). Discovering statistics using SPSS (3rd ed.). Los Angeles [i.e. Thousand Oaks, Calif.]: SAGE Publications. p. 143. ISBN 978-1-84787-906-6.
6. Razali, Nornadiah; Wah, Yap Bee (2011). "Power comparisons of Shapiro–Wilk, Kolmogorov–Smirnov, Lilliefors and Anderson–Darling tests". Journal of Statistical Modeling and Analytics. 2 (1): 21–33. Retrieved 30 March 2017.
7. Royston, Patrick (September 1992). "Approximating the Shapiro–Wilk W-test for non-normality". Statistics and Computing. 2 (3): 117–119. doi:10.1007/BF01891203. S2CID 122446146.
8. Royston, Patrick. "Shapiro–Wilk and Shapiro–Francia Tests". Stata Technical Bulletin, StataCorp LP. 1 (3).
9. Shapiro–Wilk and Shapiro–Francia tests for normality
10. Park, Hun Myoung (2002–2008). "Univariate Analysis and Normality Test Using SAS, Stata, and SPSS". [working paper]. Retrieved 29 July 2023.
11. Rahman und Govidarajulu (1997). "A modification of the test of Shapiro and Wilk for normality". Journal of Applied Statistics. 24 (2): 219–236. doi:10.1080/02664769723828.
External links
• Worked example using Excel
• Algorithm AS R94 (Shapiro Wilk) FORTRAN code
• Exploratory analysis using the Shapiro–Wilk normality test in R
• Real Statistics Using Excel: the Shapiro-Wilk Expanded Test
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| Wikipedia |
Willard L. Miranker
Willard L. Miranker (March 8, 1932 – April 28, 2011) was an American mathematician and computer scientist, known for his contributions to applied mathematics and numerical mathematics.[1]
Raised in Brooklyn, New York, he earned B.A. (1952), M.S. (1953) and Ph.D. (1956) from the Courant Institute at New York University, the latter on the thesis The Asymptotic Theory of Solutions of U + (K2)U = 0 advised by Joseph Keller. He then worked for the mathematics department at Bell Labs (1956–1958) before joining IBM Research (1961). After retirement from IBM, he joined the computer science faculty at Yale University (1989) as research faculty.
He also held professor affiliations at California Institute of Technology (1963), Hebrew University of Jerusalem (1968), Yale University (1973), University of Paris-Sud (1974), City University of New York (1966–) and New York University (1970–1973).
Miranker's work[2] includes articles and books on stiff differential equations,[3] interval arithmetic,[4] analog computing, and neural networks and the modeling of consciousness.
Miranker was also an accomplished and prolific painter. Over the course of his life, Willard Miranker painted ~4000 watercolors/aquarelles and ~200 oil paintings, many of which are displayed online. He exhibited internationally in New York City, Paris and Bonn.[5]
Awards
• Fellow of the American Association for the Advancement of Science
References
1. "Willard L. Miranker". Findagrave. Retrieved 2015-09-21.
2. Willard L. Miranker, 60 years, Computing 48:1-3, 1992 fulltext
3. Miranker, Willard L., Numerical Methods for Stiff Equations And Singular Perturbation Problems, D. Reidel Publishing Co., Dordrecht-Boston, Mass., 1981. ISBN 90-277-1107-0
4. Kulisch, Ulrich W.; Miranker, Willard L. (1981). Rheinboldt, Werner (ed.). Computer arithmetic in theory and practice. Computer Science and Applied Mathematics (1 ed.). New York, USA: Academic Press, Inc. ISBN 978-0-12-428650-4.
5. The Guide from New York Times (August 29, 1993).
External links
• Works by Willard L. Miranker at Open Library
• Paintings by Will Miranker
• Willard L. Miranker at the Mathematics Genealogy Project
• short page at Yale University
• FindaGrave entry
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| Wikipedia |
Willem van Zwet
Willem Rutger van Zwet (31 March 1934 – 2 July 2020) was a Dutch mathematical statistician.[2] He was a professor at Leiden University between 1968 and 1999.
Willem van Zwet
Born31 March 1934
Leiden, Netherlands
Died2 July 2020(2020-07-02) (aged 86)
Oegstgeest, Netherlands
NationalityDutch
OccupationMathematician
Scientific career
ThesisConvex Transformations of Random Variables[1] (1964)
Doctoral advisorJan Hemelrijk
Doctoral studentsSara van de Geer
Aad van der Vaart
Biography
Van Zwet was born on 31 March 1934 in Leiden.[3] Van Zwet obtained his doctoral degree in 1964 under the supervision of Jan Hemelrijk at the University of Amsterdam with a thesis titled "Convex Transformations of Random Variables".[4] After that, he worked at the Centrum Wiskunde & Informatica in Amsterdam, and became a lector of statistics at Leiden University in 1964 and was named professor in 1968. He retired in 1999.[3]
From 1992 to 1999, van Zwet was the Director of the Thomas Stieltjes Institute of Mathematics. He co-founded Eurandom in 1997, and served as its director until 2000. From 1997 to 1999, he was also the President of the International Statistical Institute.
Van Zwet was a Fellow of the Institute of Mathematical Statistics and a member of the Academia Europaea since 1990.[5] He received the Humboldt Prize in 2006. He won the Adolphe Quetelet Medal in 1993, and had been a Fellow of the Royal Statistical Society since 1978. In 1979, he became a member of the Royal Netherlands Academy of Arts and Sciences.[6] In 1996, he was made Knight of the Order of the Netherlands Lion, and was named Doctoris Honoris causa of Charles University the following year. He died on 2 July 2020 in Oegstgeest.[3]
References
1. Willem van Zwet at the Mathematics Genealogy Project
2. "In Memoriarium: William van Zwet". International Statistical Institute. 2 July 2020.
3. "Willem Rutger van Zwet". Leiden University. Archived from the original on 9 July 2020.
4. "Willem Rutger van Zwet". North Dakota State University.
5. "Willem van Zwet". Academia Europaea. Archived from the original on 28 March 2019.
6. "Prof. dr. w.r. Willem van Zwet" (in Dutch). KNAW genootschap. Archived from the original on 8 July 2020.
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| Wikipedia |
Willerton's fish
In knot theory, Willerton's fish is an unexplained relationship between the first two Vassiliev invariants of a knot. These invariants are c2, the quadratic coefficient of the Alexander–Conway polynomial, and j3, an order-three invariant derived from the Jones polynomial.[1][2]
When the values of c2 and j3, for knots of a given fixed crossing number, are used as the x and y coordinates of a scatter plot, the points of the plot appear to fill a fish-shaped region of the plane, with a lobed body and two sharp tail fins. The region appears to be bounded by cubic curves,[2] suggesting that the crossing number, c2, and j3 may be related to each other by not-yet-proven inequalities.[1]
This shape is named after Simon Willerton,[1] who first observed this phenomenon and described the shape of the scatterplots as "fish-like".[3]
References
1. Chmutov, S.; Duzhin, S.; Mostovoy, J. (2012), "14.3 Willerton's fish and bounds for c2 and j3", Introduction to Vassiliev knot invariants (PDF), Cambridge University Press, Cambridge, pp. 419–420, arXiv:1103.5628, doi:10.1017/CBO9781139107846, ISBN 978-1-107-02083-2, MR 2962302.
2. Dunin-Barkowski, P.; Sleptsov, A.; Smirnov, A. (2013), "Kontsevich integral for knots and Vassiliev invariants", International Journal of Modern Physics A, 28 (17): 1330025, arXiv:1112.5406, Bibcode:2013IJMPA..2830025D, doi:10.1142/S0217751X13300251, MR 3081407. See in particular Section 4.2.1, "Willerton's fish and families of knots".
3. Willerton, Simon (2002), "On the first two Vassiliev invariants", Experimental Mathematics, 11 (2): 289–296, MR 1959269.
| Wikipedia |
Willi Rinow
Willi Ludwig August Rinow (February 28th, 1907 in Berlin – March 29th, 1979 in Greifswald) was a German mathematician who specialized in differential geometry and topology. Rinow was the son of a schoolteacher. In 1926, he attended the Humboldt University of Berlin, studying mathematics and physics under professors such as Max Planck, Ludwig Bieberbach, and Heinz Hopf. There, he received his doctorate in 1931 (Über Zusammenhänge zwischen der Differentialgeometrie im Großen und im Kleinen, Math. Zeitschrift volume 35, 1932, page 512). In 1933, he worked at the Jahrbuch über die Fortschritte der Mathematik in Berlin. In 1937, he joined the Nazi Party.[1] During 1937—1940, he was an editor of the journal Deutsche Mathematik. In 1937, he became a professor in Berlin and lectured there until 1950. His lecturing was interrupted because of his work as a mathematician at the Oberspreewerk in Berlin (a producer of radio and telecommunications technology) from 1946 to 1949.
In 1950, he became a professor at the University of Greifswald. He retired in 1972.
The Hopf–Rinow theorem is named after Hopf and Rinow.
In 1959, he became the director of the Institute for Pure Mathematics at the German Academy of Sciences at Berlin and president of the German Mathematical Society.
War work
During World War II, Rinow worked as a cryptanalyst in Subsection F of Referat I of Group IV of the Inspectorate 7/VI, that was later called the General der Nachrichtenaufklärung (GdNA), achieving the rank of Colonel. Rinow worked on researching methods to solve foreign ciphers. Rinow was subordinated to Herbert von Denffer who was Director of the section and subordinated to Hans Pietsch who was Director of the section. Major Rudolf Hentze was head of the group. Otto Buggisch who was a cryptanalyst who working in the GdNA and earlier the OKW/Chi and was interrogated by TICOM agents after the war, stated that Rinow was one of the most capable people in the unit.[2]
Publications
• Die innere Geometrie der metrischen Räume, Springer 1961[3]
• Lehrbuch der Topologie, Berlin, Deutscher Verlag der Wissenschaften 1975
• Rinow Über Zusammenhänge der Differentialgeometrie im Großen und Kleinen, Mathematische Zeitschrift, volume 32, 1932, pages 512-528, Dissertation
Further reading
• Renate Tobies: Biographisches Lexikon in Mathematik promovierter Personen, 2006
References
1. Harry Waibel: Diener vieler Herren : Ehemalige NS-Funktionäre in der SBZ/DDR. Peter Lang, Frankfurt 2011 ISBN 978-3-631-63542-1 page 270
2. "Volume 4 – Signal Intelligence Service of the Army High Command" (PDF). NSA. p. 182. Retrieved 12 November 2016. This article incorporates text from this source, which is in the public domain.
3. Green, Leon W. (1963). "Review: Die innere Geometrie der metrischen Räume by Willi Rinow" (PDF). Bull. Amer. Math. Soc. 69 (2): 210–212. doi:10.1090/s0002-9904-1963-10916-7.
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| Wikipedia |
William A. Massey (mathematician)
William Alfred Massey is an American mathematician and operations researcher, the Edwin S. Wilsey Professor of Operations Research and Financial Engineering at Princeton University. He is an expert in queueing theory.
William A Massey
Born1956 (age 66–67)
Jefferson City, Missouri
OccupationEdwin S Wilsey Professor of Princeton University
Parent(s)Juliete and Richard Massey
Biography
Massey was born in Jefferson City, Missouri in 1956,[1] the son of Juliette and Richard Massey Sr., both educators. His family moved to St. Louis, Missouri when he was four.[2] He went to college at Princeton University, graduating in 1977.[2] Massey obtained his Ph.D. from Stanford University in 1981, with a thesis on queueing theory supervised by Joseph Keller.[3] His first research publication was developed during a summer program at Bell Laboratories while he was a graduate student and was published in 1978. After earning his doctorate, he became a permanent staff member at Bell Labs.[2] In 2001, Massey moved to his current position at Princeton, becoming the first African-American Princeton undergraduate alumnus to return as a faculty member.[2]
Contributions
Massey has made many original contributions as a mathematician by developing a theory of "dynamical queueing systems". Classical queueing models assumed that calling rates were constant so they could use the static, equilibrium analysis of time homogeneous Markov chains. However, real communication systems call for the large scale analysis of queueing models with time-varying rates. His thesis at Stanford University created a dynamic, asymptotic method for time inhomogeneous Markov chains called "uniform acceleration" to deal with such problems. Moreover, his research on queueing networks led to new methods of comparing multi-dimensional, Markov processes by viewing them as "stochastic orderings" on "partially ordered spaces". Finally, one of his most cited papers develops an algorithm to find a dynamic, optimal server staffing schedule for telephone call centers with time varying demand, which led to a patent. Another highly cited paper creates a temporally and spatially dynamic model for the offered load traffic of wireless communication networks.
Awards and honors
In 2006, Massey won the Blackwell–Tapia Prize of the Institute for Mathematics and its Applications for his "outstanding record of achievement in mathematical research and his mentoring of minorities and women in the field of mathematics".[2] He was elected to the 2006 class of Fellows of the Institute for Operations Research and the Management Sciences.[4] In 2012 he became a fellow of the American Mathematical Society.[5] Massey's accomplishments have earned him recognition by Mathematically Gifted & Black as a Black History Month 2018 Honoree.[6]
References
1. "William A. Massey". Mathematicians of the African diaspora. State University of New York at Buffalo. Retrieved 16 April 2017.
2. "Massey's mentorship creates network of mathematicians", Princeton Weekly Bulletin, October 23, 2006.
3. William Alfred Massey at the Mathematics Genealogy Project.
4. Fellows: Alphabetical List, Institute for Operations Research and the Management Sciences, retrieved 2019-10-09
5. List of Fellows of the American Mathematical Society, retrieved 2013-02-02.
6. "William A. Massey". Mathematically Gifted & Black.
External links
• http://www.princeton.edu/~wmassey/ William A Massey Profile
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William A. Stein
William Arthur Stein (born February 21, 1974 in Santa Barbara, California) is a software developer and previously a professor of mathematics at the University of Washington.
William A. Stein
Born
William Arthur Stein
(1974-02-21) 21 February 1974
Santa Barbara, California
Occupation(s)Software Developer, Professor of Mathematics
Known forLead developer of SageMath and founder of CoCalc.
Websitewww.wstein.org
He is the lead developer of SageMath and founder of CoCalc. Stein does computational and theoretical research into the problem of computing with modular forms and the Birch and Swinnerton-Dyer conjecture.[1] He is considered "a leading expert in the field of computational arithmetic".[2]
References
1. "NSF Award Search: Award # 0555776 - Explicit Approaches to Modular Forms and Modular Abelian Varieties".
2. Kleinert, Werner. "Zbl 1110.11015". Zentralblatt MATH. Retrieved 18 April 2012.
External links
• Official website
• William A. Stein at the Mathematics Genealogy Project
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| Wikipedia |
William Allen Whitworth
William Allen Whitworth (1 February 1840 – 12 March 1905) was an English mathematician and a priest in the Church of England.[1][2]
Education and mathematical career
Whitworth was born in Runcorn; his father, William Whitworth, was a school headmaster, and he was the oldest of six siblings. He was schooled at the Sandicroft School in Northwich and then at St John's College, Cambridge, earning a B.A. in 1862 as 16th Wrangler. He taught mathematics at the Portarlington School and the Rossall School, and was a professor of mathematics at Queen's College in Liverpool from 1862 to 1864. He returned to Cambridge to earn a master's degree in 1865, and was a fellow there from 1867 to 1882.[1]
Mathematical contributions
As an undergraduate, Whitworth became the founding editor in chief of the Messenger of Mathematics, and he continued as its editor until 1880.[1] He published works about the logarithmic spiral and about trilinear coordinates, but his most famous mathematical publication is the book Choice and Chance: An Elementary Treatise on Permutations, Combinations, and Probability (first published in 1867 and extended over several later editions).[1] The first edition of the book treated the subject primarily from the point of view of arithmetic calculations, but had an appendix on algebra, and was based on lectures he had given at Queen's College.[2] Later editions added material on enumerative combinatorics (the numbers of ways of arranging items into groups with various constraints), derangements, frequentist probability, life expectancy, and the fairness of bets, among other topics.[2]
Among the other contributions in this book, Whitworth was the first to use ordered Bell numbers to count the number of weak orderings of a set, in the 1886 edition. These numbers had been studied earlier by Arthur Cayley, but for a different problem.[3] He was the first to publish Bertrand's ballot theorem, in 1878; the theorem is misnamed after Joseph Louis François Bertrand, who rediscovered the same result in 1887.[4] He is the inventor of the E[X] notation for the expected value of a random variable X, still commonly in use,[5] and he coined the name "subfactorial" for the number of derangements of n items.[6]
Another of Whitworth's contributions, in geometry, concerns equable shapes, shapes whose area has the same numerical value (with a different set of units) as their perimeter. As Whitworth showed with D. Biddle in 1904, there are exactly five equable triangles with integer sides: the two right triangles with side lengths (5,12,13) and (6,8,10), and the three triangles with side lengths (6,25,29), (7,15,20), and (9,10,17).[7]
Religious career
Whitworth was ordained as a deacon in 1865, and became a priest in 1866. He served as the curate of St Anne's Church in Birkenhead in 1865, of the Church of St Luke, Liverpool from 1866 to 1870 and of Christ Church in Liverpool from 1870 to 1875. He was then a vicar in London at St John the Evangelist's in Hammersmith. From 1886 to 1905 he was vicar of All Saints, Margaret Street.[1]
He was the Hulsean Lecturer in 1903.[1]
References
1. Lee, Sidney, ed. (1912). "Whitworth, William Allen" . Dictionary of National Biography (2nd supplement). Vol. 2. London: Smith, Elder & Co.
2. Irwin, J. O. (1967). "William Allen Whitworth and a Hundred Years of Probability". Journal of the Royal Statistical Society. Series A. 130 (2): 147–176. doi:10.2307/2343399. JSTOR 2343399..
3. Pippenger, Nicholas (2010), "The hypercube of resistors, asymptotic expansions, and preferential arrangements", Mathematics Magazine, 83 (5): 331–346, arXiv:0904.1757, doi:10.4169/002557010X529752, MR 2762645, S2CID 17260512.
4. Feller, William (1968). An Introduction to Probability Theory and its Applications, Volume I (3rd ed.). Wiley. p. 69..
5. Aldrich, John (2007). "Earliest Uses of Symbols in Probability and Statistics". Retrieved 13 March 2013..
6. Cajori, Florian (2011), A History of Mathematical Notations: Two Volumes in One, Cosimo, Inc., p. 77, ISBN 9781616405717.
7. Dickson, Leonard Eugene (2005), History of the Theory of Numbers, Volume Il: Diophantine Analysis, Courier Dover Publications, p. 199, ISBN 9780486442334.
External links
• Works by or about William Allen Whitworth at Internet Archive
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W. B. R. Lickorish
William Bernard Raymond Lickorish (born 19 February 1938) is a mathematician. He is emeritus professor of geometric topology in the Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, and also an emeritus fellow of Pembroke College, Cambridge. His research interests include topology and knot theory. He was one of the discoverers of the HOMFLY polynomial invariant of links, and proved the Lickorish-Wallace theorem which states that all closed orientable 3-manifolds can be obtained by Dehn surgery on a link.
W. B. R. Lickorish
Lickorish in 1974
Born19 February 1938 (1938-02-19) (age 85)
NationalityBritish
Alma materUniversity of Cambridge
Known forTopology
AwardsChauvenet Prize (1991)
Senior Whitehead Prize (1991)
Scientific career
FieldsMathematician
Doctoral advisorErik Christopher Zeeman
Doctoral studentsMarc Lackenby
Education
Lickorish received his Ph.D from Cambridge in 1964; his thesis was written under the supervision of Christopher Zeeman.[1]
Recognition and awards
In 1991, Lickorish received the Senior Whitehead Prize from the London Mathematical Society.[2] Lickorish and Kenneth Millett won the 1991 Chauvenet Prize for their paper "The New Polynomial Invariants of Knots and Links".[3] Lickorish was included in the 2019 class of fellows of the American Mathematical Society "for contributions to knot theory and low-dimensional topology".[4]
Selected publications
• Lickorish, W. B. R. (November 1962). "A Representation of Orientable Combinatorial 3-Manifolds". Annals of Mathematics. 76 (3): 531–540. doi:10.2307/1970373. JSTOR 1970373.
• Freyd, Peter; Yetter, David; Hoste, Jim; Lickorish, W.B.R.; Millett, Kenneth; Ocneanu, Adrian (1985). "A New Polynomial Invariant of Knots and Links". Bulletin of the American Mathematical Society. 12 (2): 239–246. doi:10.1090/S0273-0979-1985-15361-3.
• Lickorish, W. B. R. (1997). An Introduction to Knot Theory. Graduate Texts in Mathematics 175. Springer. ISBN 0-387-98254-X.
See also
• Lickorish twist theorem
• Lickorish–Wallace theorem
References
1. W. B. R. Lickorish at the Mathematics Genealogy Project
2. London Mathematical Society. "List of Prizewinners". Retrieved 1 April 2015.
3. Lickorish, W. B. R.; Millett, K. C. (1988). "The New Polynomial Invariants of Knots and Links". Mathematics Magazine. Taylor & Francis. 61 (1): 3–23. doi:10.1080/0025570x.1988.11977338. ISSN 0025-570X.
4. "2019 Class of the Fellows of the AMS". American Mathematical Society. Retrieved 7 November 2018.
Chauvenet Prize recipients
• 1925 G. A. Bliss
• 1929 T. H. Hildebrandt
• 1932 G. H. Hardy
• 1935 Dunham Jackson
• 1938 G. T. Whyburn
• 1941 Saunders Mac Lane
• 1944 R. H. Cameron
• 1947 Paul Halmos
• 1950 Mark Kac
• 1953 E. J. McShane
• 1956 Richard H. Bruck
• 1960 Cornelius Lanczos
• 1963 Philip J. Davis
• 1964 Leon Henkin
• 1965 Jack K. Hale and Joseph P. LaSalle
• 1967 Guido Weiss
• 1968 Mark Kac
• 1970 Shiing-Shen Chern
• 1971 Norman Levinson
• 1972 François Trèves
• 1973 Carl D. Olds
• 1974 Peter D. Lax
• 1975 Martin Davis and Reuben Hersh
• 1976 Lawrence Zalcman
• 1977 W. Gilbert Strang
• 1978 Shreeram S. Abhyankar
• 1979 Neil J. A. Sloane
• 1980 Heinz Bauer
• 1981 Kenneth I. Gross
• 1982 No award given.
• 1983 No award given.
• 1984 R. Arthur Knoebel
• 1985 Carl Pomerance
• 1986 George Miel
• 1987 James H. Wilkinson
• 1988 Stephen Smale
• 1989 Jacob Korevaar
• 1990 David Allen Hoffman
• 1991 W. B. Raymond Lickorish and Kenneth C. Millett
• 1992 Steven G. Krantz
• 1993 David H. Bailey, Jonathan M. Borwein and Peter B. Borwein
• 1994 Barry Mazur
• 1995 Donald G. Saari
• 1996 Joan Birman
• 1997 Tom Hawkins
• 1998 Alan Edelman and Eric Kostlan
• 1999 Michael I. Rosen
• 2000 Don Zagier
• 2001 Carolyn S. Gordon and David L. Webb
• 2002 Ellen Gethner, Stan Wagon, and Brian Wick
• 2003 Thomas C. Hales
• 2004 Edward B. Burger
• 2005 John Stillwell
• 2006 Florian Pfender & Günter M. Ziegler
• 2007 Andrew J. Simoson
• 2008 Andrew Granville
• 2009 Harold P. Boas
• 2010 Brian J. McCartin
• 2011 Bjorn Poonen
• 2012 Dennis DeTurck, Herman Gluck, Daniel Pomerleano & David Shea Vela-Vick
• 2013 Robert Ghrist
• 2014 Ravi Vakil
• 2015 Dana Mackenzie
• 2016 Susan H. Marshall & Donald R. Smith
• 2017 Mark Schilling
• 2018 Daniel J. Velleman
• 2019 Tom Leinster
• 2020 Vladimir Pozdnyakov & J. Michael Steele
• 2021 Travis Kowalski
• 2022 William Dunham, Ezra Brown & Matthew Crawford
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| Wikipedia |
William Bigelow Easton
William Bigelow Easton was an American mathematician who proved Easton's theorem about the possible values of the continuum function. His advisor at Princeton was the famed mathematician and computer scientist Alonzo Church.[1]
Publications
• Easton, W. (1970), "Powers of regular cardinals", Ann. Math. Logic, 1 (2): 139–178, doi:10.1016/0003-4843(70)90012-4
References
1. "William Bigelow Easton". Mathematics Genealogy Project. Retrieved 2 November 2022.
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William Binney (intelligence official)
William "Bill" Edward Binney (born September 1943) [4] is a former intelligence official with the United States National Security Agency (NSA)[5] and whistleblower. He retired on October 31, 2001, after more than 30 years with the agency.
William Binney
Binney at the Congress on Privacy & Surveillance (2013) of the École polytechnique fédérale de Lausanne (EPFL)
Born
William Edward Binney
September 1943 (age 79)
Pennsylvania, U.S.
EducationPennsylvania State University (B.S., 1970)
OccupationCryptanalyst-mathematician
EmployerNational Security Agency (NSA)
Known forCryptography, SIGINT analysis, whistleblowing
Awards
• Joe A. Callaway Award for Civic Courage (2012)[1]
• Sam Adams Award (2015)[2]
• Allard Prize for International Integrity (2019)[3]
Signature
He was a critic of his former employers during the George W. Bush administration, and later criticized the NSA's data-collection policies during the Barack Obama administration. He dissented from the view that Russia interfered with the 2016 US election. More specifically, he was critical of the view that Russia hacked the DNC server.[6]
Biography
Binney grew up in rural Pennsylvania and graduated with a Bachelor of Science degree in mathematics from the Pennsylvania State University in 1970. He said that he volunteered for the Army during the Vietnam era in order to select work that would interest him rather than be drafted and have no input. He was found to have strong aptitudes for mathematics, analysis, and code breaking,[7] and served from 1965 to 1969 in the Army Security Agency before going to the NSA in 1970.
Binney was a Russia specialist and worked in the operations side of intelligence, starting as an analyst and ending as a Technical Director prior to becoming a geopolitical world Technical Director. In the 1990s, he co-founded a unit on automating signals intelligence with NSA research chief John Taggart.[8] Binney's NSA career culminated as Technical Leader for intelligence in 2001. He has expertise in intelligence analysis, traffic analysis, systems analysis, knowledge management, and mathematics (including set theory, number theory, and probability).[9][10]
After retiring from the NSA, he founded, together with fellow NSA whistleblower J. Kirk Wiebe, Entity Mapping, LLC, a private intelligence agency to market their analysis program to government agencies.[11]
Whistleblowing
National Security Agency surveillance
Map of global NSA data collection as of 2007, with countries subject to the most data collection shown in red
Programs
Pre-1978
• ECHELON
• MINARET
• SHAMROCK
• PROMIS
Since 1978
• Upstream collection
• BLARNEY
• FAIRVIEW
• Main Core
• ThinThread
• Genoa
Since 1990
• RAMPART-A
Since 1998
• Tailored Access Operations
Since 2001
• OAKSTAR
• STORMBREW
• Trailblazer
• Turbulence
• Genoa II
• Total Information Awareness
• President's Surveillance Program
• Terrorist Surveillance Program
Since 2007
• PRISM
• Dropmire
• Stateroom
• Bullrun
• MYSTIC
Databases, tools etc.
• PINWALE
• MARINA
• Main Core
• MAINWAY
• TRAFFICTHIEF
• DISHFIRE
• XKeyscore
• ICREACH
• BOUNDLESSINFORMANT
GCHQ collaboration
• MUSCULAR
• Tempora
Legislation
• Safe Streets Act
• Privacy Act of 1974
• FISA
• ECPA
• Patriot Act
• Homeland Security Act
• Protect America Act of 2007
• FISA Amendments Act of 2008
Institutions
• FISC
• Senate Intelligence Committee
• National Security Council
Lawsuits
• ACLU v. NSA
• Hepting v. AT&T
• Jewel v. NSA
• Clapper v. Amnesty
• Klayman v. Obama
• ACLU v. Clapper
• Wikimedia v. NSA
• US v. Moalin
Whistleblowers
• William Binney
• Thomas Drake
• Mark Klein
• Thomas Tamm
• Russ Tice
Publication
• 2005 warrantless surveillance scandal
• Global surveillance disclosures (2013–present)
Related
• Cablegate
• Surveillance of reporters
• Mail tracking
• UN diplomatic spying
• Insider Threat Program
• Mass surveillance in the United States
• Mass surveillance in the United Kingdom
Concepts
• SIGINT
• Metadata
Collaboration
United States
• CSS
• CYBERCOM
• DOJ
• FBI
• CIA
• DHS
• IAO
Five Eyes
• CSEC
• GCHQ
• ASD
• GCSB
Other
• DGSE
• BND
In September 2002, he, along with J. Kirk Wiebe and Edward Loomis, asked the U.S. Defense Department Inspector General (DoD IG) to investigate the NSA for allegedly wasting "millions and millions of dollars" on Trailblazer, a system intended to analyze mass collection of data carried on communications networks such as the Internet. Binney had been one of the inventors of an alternative system, ThinThread, which was shelved when Trailblazer was chosen instead. Binney has also been publicly critical of the NSA for spying on U.S. citizens, saying of its expanded surveillance after the September 11, 2001 attacks that "it's better than anything that the KGB, the Stasi, or the Gestapo and SS ever had"[12] as well as noting Trailblazer's ineffectiveness and unjustified high cost compared to the far less intrusive ThinThread.[13] He was furious that the NSA hadn't uncovered the 9/11 plot and stated that intercepts it had collected but not analyzed likely would have garnered timely attention with his leaner more focused system.[10]
Post-NSA career
After he left the NSA in 2001, Binney was one of several people investigated as part of an inquiry into a 2005 exposé by The New York Times on the agency's warrantless eavesdropping program.[14] Binney was cleared of wrongdoing after three interviews with FBI agents beginning in March 2007, but in early July 2007, in an unannounced early morning raid, a dozen agents armed with rifles appeared at his house, one of whom entered the bathroom and pointed his gun at Binney, who was taking a shower. The FBI confiscated a desktop computer, disks, and personal and business records.[15] The NSA revoked his security clearance, forcing him to close a business he ran with former colleagues at a loss of a reported $300,000 in annual income. The FBI raided the homes of Wiebe and Loomis, as well as House Intelligence Committee staffer Diane Roark, the same morning. Several months later the FBI raided the home of then still active NSA executive Thomas Andrews Drake who had also contacted DoD IG, but anonymously with confidentiality assured. The Assistant Inspector General, John Crane, in charge of the Whistleblower Program, suspecting his superiors provided confidential information to the United States Department of Justice (DOJ), challenged them, was eventually forced from his position, and subsequently himself became a public whistleblower. The punitive treatment of Binney, Drake, and the other whistleblowers also led Edward Snowden to go public with his revelations rather than report through the internal whistleblower program.[16] In 2012, Binney and his co-plaintiffs went to federal court to retrieve the confiscated items.[17]
Allegations on intercepts
Binney is known for making the claim that the NSA collects and stores information about every U.S. communication.[18] Binney was invited as a witness by the NSA commission of the German Bundestag. On July 3, 2014 Der Spiegel wrote, he said that the NSA wanted to have information about everything. In Binney's view this is a totalitarian approach, which had previously been seen only in dictatorships.[19] Binney stated that the goal was to control people. Meanwhile, he said that it is possible in principle to monitor the whole population, abroad and in the U.S., which in his view contradicts the United States Constitution.[19]
In August 2014, Binney was among the signatories of an open letter by the group Veteran Intelligence Professionals for Sanity to German chancellor Angela Merkel in which they urged the Chancellor to be suspicious of U.S. intelligence regarding the alleged invasion by Russia in Eastern Ukraine.[20][21][22] In the open letter, the group said:
[A]ccusations of a major Russian "invasion" of Ukraine appear not to be supported by reliable intelligence. Rather, the "intelligence" seems to be of the same dubious, politically "fixed" kind used 12 years ago to "justify" the U.S.-led attack on Iraq.[21]
Russian Interference in the 2016 election
Binney has said he voted for Trump in the 2016 presidential election, calling Hillary Clinton a "war monger".[23]
Binney has asserted that the U.S. intelligence community's assessment that Russia interfered in the 2016 presidential election is false, and that the Democratic National Committee e-mails were leaked by an insider instead.[24][25][26] An investigation by Duncan Campbell later detailed how Binney had been persuaded by a pro-Kremlin disinformant that the theft of the DNC emails was an inside job, and not the work of Russian agents (contrary to the findings of the US intelligence community).[27] The disinformation agent altered metadata in the files released by Guccifer 2.0 (whom the US intelligence community identifies as a Russian military intelligence operation) to make it appear as if the documents came from a computer in the Eastern United States, not Russia. (Specifically, the local time zone of the computer's system clock was changed to UTC−05:00.)[27] Binney appeared on Fox News at least ten times between September 2016 and November 2017 to promote this theory.[18][24][25] Binney said that the "intelligence community wasn't being honest here".[24] He has been a frequent guest on RT and Fox News and has been frequently cited on Breitbart News.[18] In October 2017, Binney met with CIA Director Mike Pompeo at the behest of President Trump to discuss his theory.[24] However, on meeting Campbell and analysing the material again, Binney changed his position: he said there was “no evidence to prove where the download/copy was done”, and that the files he had based his previous assessment were “manipulated” and a “fabrication”.[27]
Role in apparent release of the Nunes Memo
On January 23, 2018, Binney made an appearance on InfoWars[28] in connection with the Nunes memo, a Congressional document alleging irregularities in the application of the FISA Act, which at that time was not publicly available although its potential release was a topic of public debate.[29] During the show, host Alex Jones announced that Binney had been able to provide him with the actual memo, and the purported leaked document was shown on air.[30] However, this was in fact a public document that had been available on the website of the Office of the Director of National Intelligence since at least May 2017.[31][32] The actual Nunes memo was released February 2, 2018.[33]
Claims of fraud in the 2020 election
After Joe Biden won the 2020 United States presidential election and Donald Trump refused to concede, Binney doubted the official results and claimed that there had been large-scale voter fraud. One of Binney's tweets alleging missing votes was based on a mistaken conflation between eligible voters and an outdated number of registered voters; this was cited in an article by The Gateway Pundit, which in turn was promoted by Trump.[34][35][36][37]
Documentary film
Binney's story is recounted in A Good American, a documentary film.[38]
See also
• MAINWAY
• PRISM (surveillance program)
• Mark Klein
• Thomas Tamm
• Russ Tice
• Perry Fellwock
• Targeted surveillance
• Citizenfour – a 2014 documentary
• A Good American – a 2015 documentary
References
1. "For Immediate Release: Callaway Awards Tuesday November 13, 2012". The Joe A. Callaway Award for Civic Justice. November 13, 2012. Retrieved July 1, 2013.
2. Hannah Borno (January 23, 2015). "NSA whistleblower William Binney wins 2015 Sam Adams award". International Business Times. Retrieved August 31, 2015.
3. "Past Winners and Honourees of the Allard Prize". Allard Prize For International Integrity. Allard Prize Foundation. Retrieved October 24, 2020.
4. Video-Interview by Thomas Drake (October 26, 2011). "William Edward Binney Collection" (Video; 25 Min). Veterans History Project. American Folklife Center of the Library of Congress. Retrieved June 29, 2013.
5. "Three NSA Whistleblowers Back EFF's Lawsuit Over Government's Massive Spying Program". Electronic Frontier Foundation. July 2, 2012. Retrieved May 11, 2013.
6. Kopan, Tal (June 21, 2016). "DNC hack: What you need to know". CNN. Retrieved March 16, 2023.
7. "Keynote Address: William Binney". Schedule – HOPE Number Nine. 2600 Enterprises. 2012. Retrieved May 11, 2013.
8. "The Government Is Profiling You". MIT Center for Internet and Society. November 12, 2012. Retrieved June 8, 2013.
9. "Sworn Declaration of Whistleblower William Binney on NSA Domestic Surveillance Capabilities". Public Intelligence. July 16, 2012. Retrieved May 11, 2013.
10. Mayer, Jane (May 23, 2011). "The Secret Sharer: Is Thomas Drake an enemy of the state?". The New Yorker.
11. "NSA Whistleblowers William (Bill) Binney and J. Kirk Wiebe". Government Accountability Project website. Archived from the original on December 12, 2013. Retrieved June 9, 2013.
12. Shorrock, Tim (April 15, 2013). "The Untold Story: Obama's Crackdown on Whistleblowers: The NSA Four reveal how a toxic mix of cronyism and fraud blinded the agency before 9/11". The Nation.
13. "NSA Whistleblowers William (Bill) Binney and J. Kirk Wiebe". Government Accountability Project. Archived from the original on December 12, 2013. Retrieved May 11, 2013.
14. "The FRONTLINE Interview: William Binney – United States of Secrets". FRONTLINE. Retrieved November 10, 2020.
15. "Exclusive: National Security Agency Whistleblower William Binney on Growing State Surveillance". Democracy Now!. Retrieved September 23, 2016.
16. Hertsgaard, Mark; Kasten, Felix; Rosenbach, Marcel; Stark, Holger (May 22, 2016). "Blowing the Whistle: Former US Official Reveals Risks Faced by Internal Critics". Der Spiegel. Retrieved June 16, 2016.
17. Bronner, Ethan; Charlie Savage; Scott Shane (May 25, 2013). "Leak Inquiries Show How Wide A Net U.S. Cast". The New York Times.
18. "Was Donald Trump behind meeting of CIA chief and conspiracy theorist?". NBC News. Retrieved November 9, 2017.
19. "Untersuchungsausschuss im Bundestag: US-Informant vergleicht NSA mit einer Diktatur". Spiegel. Spiegelonline GmbH. July 3, 2014. Retrieved July 4, 2014.
20. "US-Geheimdienst-Pensionäre warnen Merkel vor Fehlinformationen" (in German). Der Tagesspiegel. September 4, 2014.
21. "The State Department Says Russia Is Invading Ukraine—Should We Believe It?". The Nation. September 2, 2014. Retrieved September 28, 2015.
22. "US-Geheimdienst-Pensionäre warnen Merkel vor Fehlinformationen". Der Tagesspiegel Online. September 4, 2014.
23. "Was Donald Trump behind meeting of CIA chief and conspiracy theorist?". NBC News. Retrieved August 4, 2019.
24. Campbell, Duncan; Risen, James (November 7, 2017). "CIA Director Met Advocate of Disputed DNC Hack Theory — at Trump's Request". The Intercept. Retrieved November 7, 2017.
25. "Conservative media figures are embracing a wild WikiLeaks conspiracy theory that the CIA hacked the DNC, and then framed Russia". Business Insider. Retrieved November 7, 2017.
26. "Why Some U.S. Ex-Spies Don't Buy the Russia Story". Bloomberg.com. August 10, 2017. Retrieved November 7, 2017.
27. "Briton ran pro-Kremlin disinformation campaign that helped Trump deny Russian links". ComputerWeekly.com. Retrieved August 1, 2018.
28. "Alex Jones tries and fails to pass off a publicly available document as the House GOP's secret Russia memo". Media Matters for America.
29. Mckew, Molly K. "How Twitter Bots and Trump Fans Made #ReleaseTheMemo Go Viral". POLITICO Magazine.
30. "Read the disputed memo here". CNN. February 2, 2018.
31. "Fact Check: Did Infowars Release the 'Secret FISA Memo'?". The Weekly Standard. January 23, 2018.
32. Matsakis, Louise (January 24, 2018). "The Cynical Misdirection Behind #ReleaseTheMemo". Wired – via wired.com.
33. McCarthy, Tom; Yuhas, Alan (February 2, 2018). "'Nunes memo' published after Trump declassifies controversial document". The Guardian – via theguardian.com.
34. "PolitiFact - Bogus analysis leads to ridiculous claim about Biden votes". PolitiFact. 2020. Retrieved December 28, 2020.{{cite web}}: CS1 maint: url-status (link)
35. "Fact check: Claim that turnout numbers prove U.S. election fraud uses wrong figures". Reuters. January 4, 2021. Retrieved February 19, 2021.
36. Bump, Philip (December 21, 2020). "Analysis - Another good example of how laughably flimsy Trump's electoral fraud claims are". Washington Post. Retrieved February 19, 2021.
37. "Flawed Calculation Behind False Claim of Fraudulent Votes". FactCheck.org. December 21, 2020. Retrieved February 19, 2021.
38. Kenigsberg, Ben (February 2, 2017). "Review: In 'A Good American,' Examining Sept. 11 and Data Collection". The New York Times. ISSN 0362-4331. Retrieved February 7, 2017.
External links
Wikiquote has quotations related to William Binney.
Wikimedia Commons has media related to William Binney.
• The Future of Freedom: A Feature Interview with NSA Whistleblower William Binney, February 2015
• William.Binney.HOPE.9.KEYNOTE.Part1, related to ThinThread development
• William.Binney.HOPE.9.KEYNOTE.Part2, related to ThinThread development
• "Who's Watching the N.S.A. Watchers?: Giving In to the Surveillance State", Shane Harris op-ed in The New York Times, 22 August 2012
• "The National Security Agency's Domestic Spying Program", Laura Poitras opinion piece in The New York Times, 22 August 2012
• 'The Program' – a video by Laura Poitras for The New York Times, 22 August 2012.
• William Binney at IMDb
• O'Brien, Alexa. "Retired NSA Technical Director Explains Snowden Docs". Archived from the original on October 15, 2014. Retrieved October 7, 2014.
• Web site for documentary A Good American
Laureates of the Sam Adams Award
• 2002: Coleen Rowley
• 2003: Katharine Gun
• 2004: Sibel Edmonds
• 2005: Craig Murray
• 2006: Samuel Provance
• 2007: Andrew Wilkie
• 2008: Frank Grevil
• 2009: Larry Wilkerson
• 2010: Julian Assange
• 2011: Thomas A. Drake and Jesselyn Radack
• 2012: Thomas Fingar
• 2013: Edward Snowden
• 2014: Chelsea Manning
• 2015: William Binney
• 2016: John Kiriakou
• 2017: Seymour Hersh
• 2018: Karen Kwiatkowski
• 2019: Jeffrey Sterling
• 2020: Annie Machon
• 2021: Daniel Hale
Authority control
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William Brouncker, 2nd Viscount Brouncker
William Brouncker, 2nd Viscount Brouncker FRS (c. 1620 – 5 April 1684) was an Anglo-Irish peer and mathematician who served as the president of the Royal Society from 1662 to 1677. Best known for introducing Brouncker's formula, he also worked as a civil servant, serving as a commissioner in the Royal Navy. Brouncker was a friend and colleague of Samuel Pepys, and features prominently in the Pepys' diary.
The Right Honourable
The Viscount Brouncker
President of the Royal Society
In office
1662–1677
Preceded byOffice established
Succeeded byJoseph Williamson
Personal details
Bornc. 1620
Castlelyons, Ireland
Died5 April 1684(1684-04-05) (aged 64)
Westminster, London
ResidenceEngland
Alma materUniversity of Oxford
Known forBrouncker's formula, leadership of Royal Society
Scientific career
FieldsMathematician, civil servant
InstitutionsSaint Catherine's Hospital
Academic advisorsJohn Wallis
Biography
Brouncker was born c. 1620 in Castlelyons, County Cork, the elder son of William Brouncker (1585–1649), 1st Viscount Brouncker and Winifred, daughter of Sir William Leigh of Newnham. His family came originally from Melksham in Wiltshire. His grandfather Sir Henry Brouncker (died 1607) had been Lord President of Munster 1603–1607, and settled his family in Ireland. His father was created a viscount in the Peerage of Ireland in 1645 for his services to the Crown. Although the first viscount had fought for the Crown in the Anglo-Scots war of 1639, malicious gossip said that he paid the then enormous sum of £1200 for the title and was almost ruined as a result. He died only a few months afterwards.
William obtained a DM at the University of Oxford in 1647. Until 1660 he played no part in public life: being a staunch Royalist, he felt it best to live quietly and devote himself to his mathematical studies. He was one of the founders and the first president of the Royal Society. In 1662, he became chancellor to Queen Catherine, then head of the Saint Catherine's Hospital. He was appointed one of the commissioners of the Royal Navy in 1664, and his career thereafter can be traced in the Diary of Samuel Pepys; despite their frequent disagreements, Samuel Pepys on the whole respected Brouncker more than most of his other colleagues, writing in 1668 that "in truth he is the best of them".
Although his attendance at the Royal Society had become infrequent, and he had quarrelled with some of his fellow members, he was nonetheless greatly displeased to be deprived of the presidency in 1677. He was commissioner for executing the office of Lord High Admiral of England from 1679.[1]
Abigail Williams
Brouncker never married, but lived for many years with the actress Abigail Williams (much to Pepys' disgust) and left most of his property to her. She was the daughter of Sir Henry Clere (died 1622), first and last of the Clere Baronets, and the estranged wife of John Williams, otherwise Cromwell, second son of Sir Oliver Cromwell, and first cousin to the renowned Oliver Cromwell. She and John had a son and a daughter. The fire of 1673 which destroyed the Royal Navy Office started in her private closet: this is unlikely to have improved her relations with Samuel Pepys, whose private apartments were also destroyed in the blaze.
On Brouncker's death in 1684, his title passed to his brother Henry, one of the most detested men of the era. William left him almost nothing in his will "for reasons I think not fit to mention".
Mathematical works
His mathematical work concerned in particular the calculations of the lengths of the parabola and cycloid, and the quadrature of the hyperbola,[2] which requires approximation of the natural logarithm function by infinite series.[3] He was the first European to solve what is now known as Pell's equation. He was the first in England to take interest in generalized continued fractions and, following the work of John Wallis, he provided development in the generalized continued fraction of pi.
Brouncker's formula
This formula provides a development of π/4 in a generalized continued fraction:
${\frac {\pi }{4}}={\cfrac {1}{1+{\cfrac {1^{2}}{2+{\cfrac {3^{2}}{2+{\cfrac {5^{2}}{2+{\cfrac {7^{2}}{2+{\cfrac {9^{2}}{2+\ddots }}}}}}}}}}}}$
The convergents are related to the Leibniz formula for pi: for instance
${\frac {1}{1+{\frac {1^{2}}{2}}}}={\frac {2}{3}}=1-{\frac {1}{3}}$
and
${\frac {1}{1+{\frac {1^{2}}{2+{\frac {3^{2}}{2}}}}}}={\frac {13}{15}}=1-{\frac {1}{3}}+{\frac {1}{5}}.$
Because of its slow convergence, Brouncker's formula is not useful for practical computations of π.
Brouncker's formula can also be expressed as[4]
${\frac {4}{\pi }}=1+{\cfrac {1^{2}}{2+{\cfrac {3^{2}}{2+{\cfrac {5^{2}}{2+{\cfrac {7^{2}}{2+{\cfrac {9^{2}}{2+\ddots }}}}}}}}}}$
See also
• List of presidents of the Royal Society
References
1. "No. 1485". The London Gazette. 9 February 1679. p. 2.
2. W. Brouncker (1667) The Squaring of the Hyperbola, Philosophical Transactions of the Royal Society of London, abridged edition 1809, v. i, pp 233–6, link form Biodiversity Heritage Library
3. Julian Coolidge Mathematics of Great Amateurs, chapter 11, pp. 136–46
4. John Wallis, Arithmetica Infinitorum, ... (Oxford, England: Leon Lichfield, 1656), page 182. Brouncker expressed, as a continued fraction, the ratio of the area of a circle to the area of the circumscribed square (i.e., 4/π). The continued fraction appears at the top of page 182 (roughly) as: ☐ = 1 1/2 9/2 25/2 49/2 81/2 &c , where the square denotes the ratio that is sought. (Note: On the preceding page, Wallis names Brouncker as: "Dom. Guliel. Vicecon, & Barone Brouncher" (Lord William Viscount and Baron Brouncker).)
External links
• O'Connor, John J.; Robertson, Edmund F., "William Brouncker, 2nd Viscount Brouncker", MacTutor History of Mathematics Archive, University of St Andrews
• "Brouncker, William" . Dictionary of National Biography. London: Smith, Elder & Co. 1885–1900.
Presidents of the Royal Society
17th century
• Viscount Brouncker (1662)
• Joseph Williamson (1677)
• Christopher Wren (1680)
• John Hoskyns (1682)
• Cyril Wyche (1683)
• Samuel Pepys (1684)
• Earl of Carbery (1686)
• Earl of Pembroke (1689)
• Robert Southwell (1690)
• Charles Montagu (1695)
• Lord Somers (1698)
18th century
• Isaac Newton (1703)
• Hans Sloane (1727)
• Martin Folkes (1741)
• Earl of Macclesfield (1752)
• Earl of Morton (1764)
• James Burrow (1768)
• James West (1768)
• James Burrow (1772)
• John Pringle (1772)
• Joseph Banks (1778)
19th century
• William Hyde Wollaston (1820)
• Humphry Davy (1820)
• Davies Gilbert (1827)
• Duke of Sussex (1830)
• Marquess of Northampton (1838)
• Earl of Rosse (1848)
• Lord Wrottesley (1854)
• Benjamin Collins Brodie (1858)
• Edward Sabine (1861)
• George Biddell Airy (1871)
• Joseph Dalton Hooker (1873)
• William Spottiswoode (1878)
• Thomas Henry Huxley (1883)
• George Gabriel Stokes (1885)
• Lord Kelvin (1890)
• Joseph Lister (1895)
20th century
• William Huggins (1900)
• Lord Rayleigh (1905)
• Archibald Geikie (1908)
• William Crookes (1913)
• J. J. Thomson (1915)
• Charles Scott Sherrington (1920)
• Ernest Rutherford (1925)
• Frederick Gowland Hopkins (1930)
• William Henry Bragg (1935)
• Henry Hallett Dale (1940)
• Robert Robinson (1945)
• Lord Adrian (1950)
• Cyril Norman Hinshelwood (1955)
• Howard Florey (1960)
• Patrick Blackett (1965)
• Alan Lloyd Hodgkin (1970)
• Lord Todd (1975)
• Andrew Huxley (1980)
• George Porter (1985)
• Sir Michael Atiyah (1990)
• Sir Aaron Klug (1995)
21st century
• Lord May (2000)
• Lord Rees (2005)
• Sir Paul Nurse (2010)
• Venki Ramakrishnan (2015)
• Adrian Smith (2020)
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William B. Johnson (mathematician)
William Buhmann Johnson (born December 5, 1944) is an American mathematician, one of the namesakes of the Johnson–Lindenstrauss lemma. He is Distinguished Professor and A.G. & M.E. Owen Chair of Mathematics at Texas A&M University. His research specialties include the theory of Banach spaces, nonlinear functional analysis, and probability theory.[1] He was born in Palo Alto, California and raised from an early age in Dallas, Texas.
Johnson graduated from Southern Methodist University in 1966,[2] and earned a doctorate from Iowa State University in 1969 under the supervision of James A. Dyer.[3] After faculty positions at the University of Houston, and Ohio State University, he joined the Texas A&M faculty in 1984.
In 2007, Johnson was awarded the Stefan Banach Medal of the Polish Academy of Sciences.[4][5] In 2012 he became a fellow of the American Mathematical Society.[6] In 2018 he was an Invited Speaker at the International Congress of Mathematicians in Rio de Janeiro.[7] His doctoral students include Edward Odell.
References
1. Faculty directory listing, Texas A&M Mathematics, retrieved 2013-01-26.
2. Faculty web page, retrieved 2013-01-26.
3. William Buhmann Johnson at the Mathematics Genealogy Project
4. Stefan Banach Medal, Polish Academy of Sciences, retrieved 2013-01-26.
5. 2007 Personal News, Texas A&M Mathematics, retrieved 2013-01-26.
6. List of Fellows of the American Mathematical Society, retrieved 2013-01-26.
7. Johnson, William B. "Some 20+ year old problems about Banach spaces and operators on them" (PDF). Proceedings of the ICM – 2018 Rio de Janeiro. Vol. 2. pp. 1669–1686.
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William C. Waterhouse
William Charles Waterhouse (December 31, 1941 – June 26, 2016) was an American mathematician. He was a professor emeritus of Mathematics at Pennsylvania State University,[1] after having taught there for over 35 years.[2] The early part of his career was at Cornell University.[2] His research interests included abstract algebra, number theory, group schemes, and the history of mathematics.[3]
William C. Waterhouse
Born(1941-12-31)December 31, 1941
Galveston, Texas, U.S.
DiedJune 26, 2016(2016-06-26) (aged 74)
State College, Pennsylvania, U.S.
NationalityAmerican
Alma materHarvard University
AwardsLester R. Ford Award (1984, 1995)
Scientific career
FieldsMathematics
InstitutionsPennsylvania State University
Cornell University
ThesisAbelian Varieties over Finite Fields (1968)
Doctoral advisorJohn Tate
Early life and education
Waterhouse was born in Galveston, Texas, on December 31, 1941, the son of William T. Waterhouse and Grace D. Waterhouse,[4] but grew up in Denver, Colorado.[5] His father was an engineer who was employed with the United States Bureau of Reclamation.[6]
He attended East High School in Denver.[7] In a high school mathematics competition spanning the states of Colorado, South Dakota, and Wyoming, he received the highest score in the competition's history and helped his school gain the top mark.[7] As a senior, he took the Scholastic Aptitude Test and received a near-perfect 797 on the verbal portion and a perfect 800 on the math portion.[6] He then received perfect 800 scores on three different college board Achievement Tests, those for English Composition, for Chemistry, and for advanced Mathematics,[6] a feat that the Associated Press filed a story about and that ran in a number of newspapers around the country.[8] Time magazine ran a profile of him as well.[6] Waterhouse received the National Merit Scholarship and the General Motors Scholarship;[6] he graduated from East High in 1959.[7]
Waterhouse attended Harvard College. There he was a standout in the Putnam Competition: As a sophomore in the 1960 competition, he was not part of Harvard's three-person team that finished second overall, but he did achieve a top-ten individual mark;[9] as a junior in the 1961 competitition, he attained the highest individual level – a top-five score – while helping his Harvard team to a fourth-place finish overall;[10] and as a senior in the 1962 competition, he again was a Putnam Fellow with a top-five score and helped his Harvard team to a third-place finish overall.[11]
After graduating from Harvard College with a bachelor's degree summa cum laude and being elected to Phi Beta Kappa, Waterhouse continued at Harvard Graduate School of Arts and Sciences where he received a master's degree.[12] While in the graduate school, he was awarded a National Science Foundation Fellowship.[12] He then received his Ph.D. in 1968 from Harvard for his thesis Abelian Varieties over Finite Fields under the supervision of John Tate.[13]
Career
Waterhouse began teaching at Cornell University in 1968.[4]
He had a career-long interest in the history of mathematics,[3] and while at Cornell wrote a history of the early years of that university's Oliver Club, a discussion forum begun by pioneering Cornell mathematican James Edward Oliver in the 1890s.[14]
Waterhouse remained an assistant professor at Cornell until 1975, at which point he was appointed an associate professor at Penn State.[12] At Penn State, he subsequently became a full professor.[5]
In 1980 he married Betty Ann Senk, a doctoral student and teacher in comparative literature at Penn State.[15] They lived in State College, Pennsylvania.[15]
According to his obituary published in the Centre Daily Times, during his career Waterhouse published over 250 articles in scholarly journals and other publications.[4] He was the author of the 1979 textbook Introduction to Affine Group Schemes for Springer-Verlag.[16] Telegraphic Reviews characterized the work as a "fairly intuitive and accessible" development of the topic, suitable for second-year graduate students.[17] In a 1986 volume for Springer-Verlag, he edited the 1966 translation by Arthur A. Clarke of Gauss's Disquisitiones Arithmeticae.[18] Waterhouse and his wife collaborated on several translations of works by German mathematicians.[19] He was a member of the Mathematical Association of America and the American Mathematical Society. [12]
Waterhouse long had an interest in classical studies;[4] as such, he was a member of the Classical Association of the Middle West and South.[20] He published a number of entries about language- and classics-focused matters in the journal Notes and Queries, [21] as well as, in the journal Classical World, an exegesis of an unusual word form found in Ovid's Amores.[22] Waterhouse also had an interest in investigating quotations, whether via a Usenet newsgroup or publishing with The Skeptics Society.[23]
By 2012, Waterhouse had moved to emeritus status.[1]
Waterhouse died on June 26, 2016, in State College, Pennsylvania.[4]
Awards and honors
Waterhouse twice won the Lester R. Ford Award of the Mathematical Association of America, given to authors of articles of expository excellence. The first was in 1984 for his paper "Do Symmetric Problems Have Symmetric Solutions?"[13] and the second was in 1995 for his paper "A Counterexample for Germain".[24] The latter has been characterized as "a historical and mathematical detective story" that investigated an aspect of the correspondence between Carl Friedrich Gauss and Sophie Germain, a French mathematician who used a pseudonym to disguise the fact that she was a woman.[25] According to his obituary, Waterhouse had a special pride in having won the two Lester R. Ford Awards.[4]
References
1. "Mathematics Department: Emeriti". Pennsylvania State University. Archived from the original on October 30, 2012. Retrieved March 15, 2023.
2. "News from the AMS: William C. 'Bill' Waterhouse (1941–2016)". American Mathematical Society. September 6, 2016. Retrieved March 15, 2023.
3. PSU Mathematics Department - Faculty, retrieved 2010-02-06.
4. "William C. Waterhouse Obituary". Centre Daily Times. June 29, 2016. Retrieved April 21, 2022.. Also available at this CAMWS page.
5. Waterhouse, William C. (1989). "Two Elementary Proofs of an Inequality (and 1½ Better Ones)". The College Mathematics Journal. 20 (3): 201–205. doi:10.1080/07468342.1989.11973231.
6. "Education: The Good Student". Time. April 27, 1959.
7. "uncertain". The Denver Post. March 25, 1959.
8. See for example "Flawless Performance". The Tampa Times. Associated Press. June 13, 1959. p. 2-C – via Newspapers.com. Newspapers.com shows over fifty papers running the AP report.
9. Bush, L. E. (1961). "The 1960 William Lowell Putnam Mathematical Competition". The American Mathematical Monthly. 68 (7): 629–637. doi:10.2307/2311508.
10. Bush, L. E. (1962). "The 1961 William Lowell Putnam Mathematical Competition". The American Mathematical Monthly. 69 (8): 759–767. doi:10.2307/2310772.
11. "Three Math Students Win Third in Contest". The Harvard Crimson. March 16, 1963.
12. "2 Named in Mathematics". Centre Daily Times. State College, Pennsylvania. September 10, 1975. p. 20 – via Newspapers.com.
13. "MAA Writing Awards: Do Symmetric Problems Have Symmetric Solutions?". Mathematical Association of America. Retrieved February 7, 2010.
14. "About the Oliver Club". Department of Mathematics, Cornell University. Retrieved February 8, 2023.
15. "Senk–Waterhouse Marriage". Centre Daily Times. State College, Pennsylvania. October 17, 1980. p. 10 – via Newspapers.com.
16. Graduate Texts in Mathematics 66, Springer-Verlag, 1979, ISBN 978-0-387-90421-4.
17. "Telegraphic Reviews". The American Mathematical Monthly. 87 (3): 233–238. 1980 – via JSTOR.
18. Reprinted in 1986 by Springer-Verlag, ISBN 978-0-387-96254-2.
19. Wynn, James; Reyes, G. Mitchell (2021). "From Division to Multiplication: Uncovering the Relationship Between Mathematics and Rhetoric Through Transdisciplinary Scholarship". In Wynn, James; Reyes, G. Mitchell (eds.). Arguing with Numbers: The Intersections of Rhetoric and Mathematics. Pennsylvania State University Press. p. 23. also Poonen, Bjorn (2017). Rational Points on Varieties. Providence, Rhode Island: American Mathematical Society. p. 317.
20. "CAMWS Necrology". Program: 115th Annual CAMWS Meeting (PDF). Lincoln, Nebraska: Classical Association of the Middle West and South. 2019. p. 69.
21. See search results at Notes and Queries archive, accessed March 10, 2023.
22. Waterhouse, William C. (2008). "Emodulanda in Ovid's Amores 1.1". Classical World. 101 (4): 533–534. doi:10.1353/clw.0.0027 – via Project Muse.
23. Waterhouse, William C. (January 5, 2006). "Did Einstein Praise the Church?". eSkeptic. The Skeptics Society. Retrieved March 15, 2023.
24. "MAA Writing Awards: A Counterexample for Germain". Mathematical Association of America. Retrieved February 7, 2010.
25. "Penn State Briefs: Math Professor Wins Award". Centre Daily Times. August 21, 1995. p. 3 – via Newspapers.com.
External links
• William Charles Waterhouse at the Mathematics Genealogy Project
• Abelian Varieties over Finite Fields dissertation, at French Numdam service
• Early History of the Oliver Club at Cornell University
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Bill Casselman
William Allen Casselman (born November 27, 1941) is an American Canadian mathematician who works in representation theory and automorphic forms. He is a Professor Emeritus at the University of British Columbia.[1] He is closely connected to the Langlands program and has been involved in posting all of the work of Robert Langlands on the internet.[2]
Bill Casselman
Casselman in 1969
Born
William Allen Casselman
(1941-11-27) November 27, 1941
Glen Ridge, New Jersey, U.S.
CitizenshipCanadian
Alma materPrinceton University
Scientific career
FieldsRepresentation theory
Automorphic forms
Geometric combinatorics
Structure of algebraic groups
InstitutionsUniversity of British Columbia
Doctoral advisorGoro Shimura
InfluencesRobert Langlands
Career
Casselman did his undergraduate work at Harvard College where his advisor was Raoul Bott and received his Ph.D from Princeton University in 1966 where his advisor was Goro Shimura. He was a visiting scholar at the Institute for Advanced Study in 1974, 1983, and 2001.[3] He emigrated to Canada in 1971 and is a Professor Emeritus in mathematics at the University of British Columbia.[1]
Research
Casselman specializes in representation theory, automorphic forms, geometric combinatorics, and the structure of algebraic groups. He has an interest in mathematical graphics[4] and has been the graphics editor of the Notices of the American Mathematical Society since January, 2001.[5]
Awards
In 2012, he became one of the inaugural fellows of the American Mathematical Society.[6]
Selected publications
• Casselman, Bill (1973). "On some results of Atkin and Lehner". Mathematische Annalen. 201 (4): 301–314. doi:10.1007/BF01428197. S2CID 121867474.
• Casselman, Bill (1977). "Characters and Jacquet modules". Mathematische Annalen. 230 (2): 101–105. doi:10.1007/BF01370657. ISSN 0025-5831. S2CID 121574262.
• Casselman, Bill (1980). "The unramified principal series of p-adic groups. I. The Spherical function". Compositio Mathematica. 40 (3): 387–406.
• Casselman, Bill; Shalika, Joseph (1980). "The unramified principal series of p-adic groups. II. The Whittaker function". Compositio Mathematica. 41 (2): 207–231.
• Casselman, Bill; Milicic, Dragan (1982). "representations". Duke Mathematical Journal. 49 (4): 869–930. doi:10.1215/S0012-7094-82-04943-2. ISSN 0012-7094.
• Borel, Armand; Casselman, Bill (1983). "L2-cohomology of locally symmetric manifolds of finite volume". Duke Mathematical Journal. 50 (3): 625–647. doi:10.1215/S0012-7094-83-05029-9. S2CID 122723214.
• Casselman, Bill; Shahidi, Freydoon (1998). "On irreducibility of standard modules for generic representations". Annales Scientifiques de l'École Normale Supérieure. 31 (4): 561–589. doi:10.1016/S0012-9593(98)80107-9.
• Casselman, Bill (2005). Mathematical Illustrations: A Manual of Geometry and PostScript. Cambridge University Press. ISBN 0521839211.
References
1. "Emeriti and Retirees". University of British Columbia Mathematics Department. Retrieved September 27, 2020.
2. Institute for Advanced Study: The Work of Robert Langlands
3. Institute for Advanced Study: A Community of Scholars: Casselman, William
4. Mathematical Illustrations: A Manual of Geometry and PostScript reviewed by Denis Roegel in Notices of the AMS
5. Notices of the American Mathematical Society: Editors and Staff
6. List of Fellows of the American Mathematical Society, retrieved 2015-11-18.
External links
• Publications of Bill Casselman
• Bill Casselman's Home Page
• William Allen Casselman at the Mathematics Genealogy Project
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William Chapple (surveyor)
William Chapple (1718–1781) was an English surveyor and mathematician. His mathematical discoveries were mostly in plane geometry and include:
• the first proof of the existence of the orthocentre of a triangle,
• a formula for the distance between the incentre and circumcentre of a triangle,
• the discovery of Poncelet's porism on triangles with a common incircle and circumcircle.
He was also one of the earliest mathematicians to calculate the values of annuities.
Life
Chapple was born in Witheridge on 25 January 1719 [O.S. 14 January 1718], the son of a poor farmer and parish clerk.[1] He was a devoted bibliophile,[2] and gained much of his knowledge of mathematics from Ward's The Young Mathematician's Guide: Being a Plain and Easie Introduction to the Mathematicks, in Five Parts.[3] He became an assistant to the parish priest, and a regular contributor to The Ladies' Diary, especially concerning mathematical problems. He also later contributed work on West Country English to The Gentleman's Magazine.[1]
His correspondence led him to become, in 1738, the clerk for a surveyor in Exeter. He married the surveyor's niece, supervised the construction of a new hospital in Exeter, and became secretary of the hospital.[1] He also worked as the estate steward for William Courtenay, 1st Viscount Courtenay.[4] In 1772 he began work on an update to Tristram Risdon's Survey of the County of Devon, and spent much of the rest of his life working on it; it was published in part throughout his life, and in complete form posthumously in 1785.[1]
He died in early September 1781.[1] A tablet in his memory could be found in the west end of the nave of the Church of St Mary Major, Exeter, prior to that church's demolition in 1971.[5] Chapple Road in Witheridge is named after him.[2]
Contributions to mathematics
Andrea del Centina writes that:
"To illustrate the work of Chapple, whose arguments are often confused and whose logic is very poor, even for the standard of his time, is not easy especially when trying to keep as faithful as possible to his thought."[3]
Nevertheless, Chapple made several significant discoveries in mathematics.
Plane geometry
Euler's theorem in geometry gives a formula for the distance $d$ between the incentre and circumcentre of a circle, as a function of the inradius $r$ and circumradius $R$:
$d={\sqrt {R(R-2r)}}.$
An immediate consequence is the related inequality $R\geq 2r$. Although these results are named for Leonhard Euler, who published them in 1765, they were found earlier by Chapple, in a 1746 essay in The Gentleman's Magazine.[6][7] In the same work he stated that, when two circles are the incircle and circumcircle of a triangle, then there is an infinite family of triangles for which they are the incircle and circumcircle. This is the triangular case of Poncelet's closure theorem, which applies more generally to polygons of any number of sides and to conics other than circles. It is the first known mathematical publication on pairs of inscribed and circumscribed circles of polygons, and significantly predates Poncelet's own 1822 work in this area.[3]
In 1749, Chapple published the first known proof of the existence of the orthocentre of a triangle, the point where the three perpendiculars from the vertices to the sides meet. The orthocentre itself was known previously, but Chapple writes that its existence was "often taken for granted, but no where demonstrated".[8]
Finance
Chapple learned of the problem of valuation of annuities through his correspondence with John Rowe and Thomas Simpson, and carried out this valuation for Courtenay. In this, he became one of the first mathematicians to work on this problem, along with Simpson, Abraham de Moivre, James Dodson, and William Jones.[4]
References
1. Pengelly, W. (1887), "Prince's "Worthies of Devon" and the "Dictionary of National Biography", part III", Report & Transactions of the Devonshire Association, Devonshire Association for the Advancement of Science Literature & the Arts, 19: 217–348. See in particular "Chapple, William", pp. 316–318.
2. "William Chapple", Witheridge Historical Archive, retrieved 18 November 2019
3. Del Centina, Andrea (2016), "Poncelet's porism: a long story of renewed discoveries, I", Archive for History of Exact Sciences, 70 (1): 1–122, doi:10.1007/s00407-015-0163-y, MR 3437893, S2CID 253898210
4. Bellhouse, David R. (2017), Leases for Lives: Life Contingent Contracts and the Emergence of Actuarial Science in Eighteenth-Century England, Cambridge University Press, p. 79, ISBN 9781108509121
5. Lysons, Daniel (1822), Magna Brittanica; being a concise topographical account of the several counties of Great Britain, Vol. VI: Devonshire, Thomas Cadell, p. 215
6. Milne, Antony (2015), "The Euler and Grace-Danielsson inequalities for nested triangles and tetrahedra: a derivation and generalisation using quantum information theory", Journal of Geometry, 106 (3): 455–463, doi:10.1007/s00022-014-0257-8, MR 3420559
7. Chapple, William (1749), "An essay on the properties of triangles inscribed in, and circumscribed about two given circles", Miscellanea Curiosa Mathematica, The Gentleman's Magazine, vol. 4, pp. 117–124
8. Bogomolny, Alexander, "A Possibly First Proof of the Concurrence of Altitudes", Cut The Knot, retrieved 17 November 2019. See also Chapple's letter with the proof.
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William Charles Brenke
William Charles Brenke (April 12, 1874, Berlin – 1964)[1] was an American mathematician who introduced Brenke polynomials and wrote several undergraduate textbooks.
He received his PhD in mathematics at Harvard under Maxime Bôcher. Brenke taught at the University of Nebraska-Lincoln mathematics department from 1908 to 1944 and was chair of the department from 1934 to 1944. He retired in 1943 but his successor, Ralph Hull, was put on official leave to do war work and returned from leave in 1945.[2]
Publications
• Brenke, W. C. (1930). "On polynomial solutions of a class of differential equations of the second order". Bull. Amer. Math. Soc. 36 (2): 77–84. doi:10.1090/s0002-9904-1930-04888-0. MR 1561893.
• Brenke, W. C. (1933). "On the summability and general sum of a series of Legendre polynomials". Bull. Amer. Math. Soc. 39 (10): 821–824. doi:10.1090/s0002-9904-1933-05753-1. MR 1562735.
References
1. According to ancestry.com Brenke married Kate Read in 1898, had three children, and died in 1964.
2. General History - Department of Mathematics - University of Nebraska-Lincoln
External links
• William Charles Brenke at the Mathematics Genealogy Project
• NEGenWeb Project - Lancaster County Who's Who in Nebraska, 1940
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William Christopher Swinbank
William Christopher Swinbank (8 May 1913 – 28 December 1973) was a British-born meteorological physicist who worked at the UK Meteorological Office, the CSIRO Australia and the NCAR Colorado. His main areas of research were fog prediction, upper atmosphere analysis, wind predictions, hail storms and turbulent fluxes.
Early life
William Swinbank was born on 8 May 1913 in the small coal mining village of Easington in County Durham,[1] UK where his father worked above ground at a local colliery as a mechanical engineer. Swinbank was the oldest of three boys. He won a scholarship to Henry Smith Grammar School in Hartlepool and went up to Durham University (Hatfield College) where he took a Double Honours in Mathematics and Physics, graduating 1934.[2]
Career
After leaving university, Swinbank worked briefly as a schoolteacher and also in industrial research. In 1938 he commenced working as a meteorologist at the UK Met Office.
The Met Office 1938–1948
In 1938, Swinbank began work as a Met Office Technical Officer for the Air Ministry and was attached to various RAF bases to carry out weather forecasting duties. In 1940, he was given the task of investigating the issue of fog, since this was critical to the operations of an airbase. He initially worked with C.S. Durst in the areas of cloud physics and turbulence. In 1942 he moved to the forecasting headquarters at RAF Dunstable in Bedfordshire and continued his research on fog.[2] During this time, he worked with C.H.B. Priestley, P. A Sheppard and Sverre Petterssen and came to the realisation that fog needed to be studied in relation to larger weather systems.[2] He joined the Upper Air Unit at Dunstable which was headed by Petterssen and as the war progressed, Swinbank became more involved in the use of upper air analysis forecasting techniques and further developed the use of isobaric analysis for producing the synoptic charts used for weather forecasting.[2] His research into predicting fog was valuable for guiding returning RAF bombers back to base after bombing raids in the early morning over Germany as fog was most prevalent in these hours.[3] In 1944, RAF Dunstable played an important role in forecasting the weather patterns leading up to the D-Day landings[4] and Swinbank, along with Priestly and Andrezej Berson were involved in this forecasting.[5] After the war, Swinbank returned to the study of fog as well as agricultural physics and worked closely with the meteorologist H. L. Penman.[2] In 1947 Swinbank and Priestley wrote a paper titled The Vertical Transfer of Heat by Turbulence in the Atmosphere, a paper that was acknowledged as being a landmark in micrometeorology[6][7] and laid the foundation for the development of eddy covariance.[8]
CSIRO 1948–1969
In 1948, Swinbank moved to Australia to take up a position in the newly formed section of Meteorological Physics (later renamed Atmospheric Physics) at the Commonwealth Scientific and Industrial Research Organisation in Aspendale, a suburb of Melboune. He worked with C. H. B. Priestley who had joined the organisation two years earlier.[9] His early work at the organisation was in micrometeorolgy where he developed the use of hot wire anemometry for the direct measurement of the turbulent fluxes of heat, water vapour and momentum.[6] This research was presented at the International Symposium on Atmospheric Turbulence in the Boundary Layer at MIT in 1951. In the discussion following Swinbank's presentation, one participant commented thus: "We have had to rely too much on hypothesis and conjecture in the processing and analysis of micrometeorological observations due to the fact that most observers did not measure the totality of atmospheric elements. I want to complement the group in Australia for the collection of the complete, detailed and accurate data on eddy fluxes."[10] Swinbank was an advocate for large scale atmospheric field experiments and between 1962 and 1964 established research projects in Kerang (Vic) and later on in Hay (NSW) for the recording of accurate data for later analysis.[2] These experiments recorded wind, temperature and moisture in the atmosphere, initially to a height of 16 metres, and later to 32 metres. In the later experiments at Hay, measurements were taken up to 1000 metres.[5] Swinbank was instrumental in the establishment of ozone monitoring in Australia.[6]
NCAR 1969–1973
In 1969, Swinbank took leave of absence from the CSIRO to work at the NCAR in Colorado. He was free to pursue his work on turbulence and the boundary layer. In 1971, he was appointed Director of the National Hail Research Experiment (NHRE), and consequently resigned from the CSIRO. The NHRE developed in response to the impacts of hailstorms on crops in the Mid-West. By taking detailed atmospheric measurements within the storm cell, it was hoped that a storm could be seeded with silver oxide to break up the moisture into rain or small hail and so prevent the development of large destructive hailstones.[11] The project ran for five years until 1976.
Personal life
In 1939 Swinbank married Ivy Hook[12] and they had a daughter, Susan. Both Ivy and Susan died in 1940 within a day of each other.[13] In 1942 he met Angela Pinney while they were both working at RAF Dunstable in Bedfordshire and they married in December of that year. In 1948, Swinbank, his wife and two children moved to Australia and lived in Mt Eliza near Melbourne. He came to Australia to work in the newly founded CSIRO. They had four more children. In 1969, Swinbank took up a position at NCAR in Colorado and he died in Boulder on 28 December 1973. In May 1974, the Boundary-Layer Meteorology journal published a memorial edition to which colleagues from around the world contributed articles.[14]
Awards
• 1941 Elected Fellow of the Royal Meteorological Society[15]
• 1968 Winner of the Buchan Prize by the Royal Meteorological Society[6] (in conjunction with A. J. Dyer)
• 1970 Elected Fellow of the Australian Academy of Science[2]
• Fellow of the Australian Institute of Physics[16]
Selected publications
• Priestley, C. H. B and Swinbank, W. C. (1947) Vertical transport of heat by turbulence in the atmosphere Proc. R. Soc. Lond. A189: 543–56
• Swinbank, W. C. (1948) Note on the formation of fog over a snow surface. Quarterly Journal of the Royal Meteorological Society. 1948; Vol 74(Issue 321-322):406-407.
• Swinbank, W. C. (1949) Prediction diagrams for radiation fog. Durham theses, Durham University. Available at Durham E-Theses Online: http://etheses.dur.ac.uk/10469/
• Swinbank, W. C. (1951) The measurement of vertical transfer of heat and water vapor by eddies in the lower atmosphere. Journal of Meteorology. 1951; 8(3):135-145
• Swinbank, W. C. (1956) The physics of natural evaporation. Australian Meteorological Magazine. 1956; (14):58-59.
• Swinbank, W. C. (1960) Wind profile in thermally stratified flow. Nature. 1960; 186(4723):463-464.
• Swinbank, W. C. (1963) Long‐wave radiation from clear skies. Quarterly Journal of the Royal Meteorological Society, 89(381):339-348. doi: 10.1002/QJ.49708938105
• Swinbank, W. C and Dyer A.J. (1967) An experimental study in micro-meteorology. Quarterly Journal of the Royal Meteorological Society. Vol 93 (Issue 398):494-500.
• Swinbank, W. C. & Dyer, A. J. (1968). Micrometeorological expeditions 1962-1964. Melbourne: C.S.I.R.O
• Swinbank, W. C. (1970). Structure of wind and the shearing stress in the planetary boundary layer. Archiv Für Meteorologie, Geophysik Und Bioklimatologie, Serie A, 19, 1-12.
Notes
1. "FreeBMD Entry Info". www.freebmd.org.uk. Retrieved 21 December 2022.
2. "William Christopher Swinbank 1913-1973". Australian Academy of Science. Retrieved 20 December 2022.
3. Petterssen, S (2001). Weathering the Storm: Sverre Petterssen, the D-Day Forecast and the Rise of Modern Meteorology. Ed J R Fleming. p. 128.
4. "The Role of the Met Office in the D-Day Landings" (PDF).
5. Garratt, John; Angus, David; Holper, Paul (1998). Winds of Change:Fifty Years of Achievements in the CSIRO Division of Atmospheric Research 1946-1996. CSIRO Publishing.
6. Dyer, A. J. (1974). "Obituary — William Christopher Swinbank". Boundary-Layer Meteorology. 6 (3–4): I. doi:10.1007/BF02137671. ISSN 0006-8314. S2CID 189836328.
7. Ward, Colin (13 January 2015). "Charles Henry Brian (Bill) Priestley (1915-1998)". CSIROpedia. Retrieved 23 December 2022.
8. Berg, Peter; Huettel, Markus; Glud, Ronnie N.; Reimers, Clare E.; Attard, Karl M. (January 2022). "Aquatic Eddy Covariance: The Method and Its Contributions to Defining Oxygen and Carbon Fluxes in Marine Environments". Annual Review of Marine Science. 14: 431–455. doi:10.1146/annurev-marine-042121-012329. PMID 34587456. S2CID 238230170.
9. Centre for Transformative Innovation, Swinburne University of Technology. "CSIRO Division of Meteorological Physics - Corporate Body - Encyclopedia of Australian Science and Innovation". www.eoas.info. Retrieved 23 December 2022.
10. Hewson, Edgar Wendell (1952). "International Symposium of Turbulence in The Boundary Layer".
11. "NHRE Information release 1971".
12. "FreeBMD District Info". www.freebmd.org.uk. Retrieved 21 December 2022.
13. "Well-known sportswoman's death". Biggleswade Chronicle. 20 March 1940. Retrieved 31 December 2022 – via britishnewspaperarchive.co.uk.
14. "Boundary-Layer Meteorology journal volume 6 issue3".
15. "Proceedings of the Royal Meteorological Society". Quarterly Journal of the Royal Meteorological Society. 68 (293): 76–83. December 1941. doi:10.1002/qj.49706829310.{{cite journal}}: CS1 maint: url-status (link)
16. "Australian Institute of Physics" (PDF).
External links
• List of publications by W C Swinbank while he was working at the CSIRO
• National Hail Research Experiment Film (1974) 18 mins Swinbank was the Technical Advisor.
• National Hail Research Experiment Summary reports (1970 -1976)
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William Davidson Niven
Sir William Davidson Niven KCB FRS (24 March 1842 – 29 May 1917) was a Scottish mathematician and electrical engineer.
Sir William Davidson Niven
Born
William Davidson Niven
(1842-03-24)March 24, 1842
Peterhead, Scotland
DiedMay 29, 1917(1917-05-29) (aged 75)
Sidcup; England
Resting placePeterhead Old Churchyard
57.504068°N 1.790279°W / 57.504068; -1.790279
Alma materUniversity of Aberdeen
Trinity College, Cambridge
Known forEditor of James Clerk Maxwell's papers
Scientific career
FieldsMathematics
InstitutionsRoyal Naval College, Greenwich
InfluencesJames Clerk Maxwell
InfluencedAlfred North Whitehead
After an early teaching career at Cambridge, Niven was Director of Studies at the Royal Naval College, Greenwich, for thirty years.
Life
Niven was born at Peterhead in Aberdeenshire, one of five notable mathematician brothers: Charles and James the best known. He graduated first from the University of Aberdeen,[1] then from Trinity College, Cambridge, where he was a Wrangler and was elected a Fellow of his college.
In 1882 Niven became Director of Studies at the Royal Naval College, Greenwich, succeeding Thomas Archer Hirst.[2] He was appointed a Companion of the Order of the Bath (Civil division) in Queen Victoria's Diamond Jubilee Honours of 1897. He retired in 1903, when he was knighted by being appointed a Knight Commander of the Order of the Bath.[2]
Niven was a colleague of James Clerk Maxwell (1831–1879), whose scientific papers he edited after his death. Among Niven's students was Alfred North Whitehead, to whom he taught mathematics, by instructing him in the physics of Maxwell.[3]
In retirement Niven lived at Eastburn, Sidcup, Kent,[2] where he died in 1917.[1]
Major publications
Niven edited works by J. C. Maxwell:
• 1881: A Treatise on Electricity and Magnetism, 2nd edition
• 1890: The Scientific Papers of James Clerk Maxwell from Biodiversity Heritage Library
Notes
1. Ronny Desmet, Michel Weber, Whitehead. The Algebra of Metaphysics (2010), p. 116
2. 'NIVEN, Sir William Davidson', in Who Was Who 1916–1928 (London: A. & C. Black, 1992 reprint, ISBN 0-7136-3143-0)
3. Frank Northen Magill, Alison Aves (1999) Dictionary of World Biography, p. 3,965
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William Duke (mathematician)
William Drexel Duke (born 1958) is an American mathematician specializing in number theory.
Duke studied at the University of New Mexico and then at New York University (Courant Institute), from which he received his Ph.D. in 1986 under the direction of Peter Sarnak. After a postdoctoral stint at the University of California, San Diego he joined the faculty of Rutgers University, where he stayed until becoming a Professor of Mathematics at the University of California, Los Angeles. Since 2015, he has been Chair of the mathematics department at UCLA.[1]
Honors
Duke gave an Invited Address at the 1998 International Congress of Mathematicians in Berlin.[2][3][4] Duke gave an AMS Invited Address at a 2001 Fall sectional meeting of the American Mathematical Society in Irvine, California.[5] He was selected as a fellow of the American Mathematical Society in 2016 "for contributions to analytic number theory and the theory of automorphic forms".[6]
Duke is an Editorial Board Member for the book series "Monographs in Number Theory" published by World Scientific.[7]
Students
• Amanda Folsom
Selected publications
• Duke, W. (1988) Hyperbolic distribution problems and half-integral weight Maass forms, Inventiones Mathematicae, 92, 73–90.
• Duke, W., Schulze-Pillot, R. (1993) Representation of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids, Duke Mathematical Journal, 71, 143–179.
• Duke, W., Friedlander, J., Iwaniec, H. (1993) Bounds for automorphic L-functions, Inventiones Mathematicae, 112, 1–8.
• Duke, W., Friedlander, J., Iwaniec, H. (1994) Bounds for automorphic L-functions II, Inventiones Mathematicae, 115, 219–239.
• Duke, W., Friedlander, J., Iwaniec, H. (1995), Equidistribution of roots of a quadratic congruence to prime moduli, Annals of Mathematics, 141, 423–441.
• Duke, W. (1995) The critical order of vanishing of automorphic L-functions with large level, Inventiones Mathematicae, 119, 165–174.
• Duke, W., Kowalski, E. (2000), A problem of Linnik for elliptic curves and mean-value estimates for automorphic representations. With an appendix by Dinakar Ramakrishnan, Inventiones Mathematicae, 139, 1–39.
• Duke, W., Friedlander, J., Iwaniec, H. (2002), The subconvexity problem for Artin L-functions, Inventiones Mathematicae, 149, 489–577.
References
1. Faculty profile, Department of Mathematics, University of California, Los Angeles. Accessed May 15, 2016.
2. ICM Plenary and Invited Speakers since 1897, International Mathematical Union. Accessed May 15, 2016.
3. Invited speakers for ICM-98, Notices of the American Mathematical Society, 45 (1998), no. 5, p. 621
4. Duke, William (1998). "Bounds for arithmetic multiplicities". Doc. Math. (Bielefeld) Extra Vol. ICM Berlin, 1998, vol. II. pp. 163–172.
5. AMS Sectional Meeting Invited Addresses, American Mathematical Society. Accessed May 15, 2016
6. 2016 Class of the Fellows of the AMS, American Mathematical Society. Accessed May 15, 2016.
7. Monographs in Number Theory, (series info), World Scientific. Accessed May 15, 2016
External links
• William Duke's webpage at UCLA
• William Drexel Duke at the Mathematics Genealogy Project
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William Elwood Byerly
William Elwood Byerly (13 December 1849 – 20 December 1935) was an American mathematician at Harvard University where he was the "Perkins Professor of Mathematics". He was noted for his excellent teaching and textbooks.[1] Byerly was the first to receive a Ph.D. from Harvard, and Harvard's chair "William Elwood Byerly Professor in Mathematics" is named after him. Byerly Hall in Radcliffe Yard, Radcliffe Institute for Advanced Study, Harvard University is also named for him.
William Elwood Byerly
Born1849
Died1935 (aged 85–86)
Textbooks
Among the textbooks he wrote are:
• Elements of the Differential Calculus (1879)
• Harmonic Functions (1906)
• Problems in Differential Calculus
• Introduction to the Calculus of Variations (1917)
• Elements of the Integral Calculus (1881)
• An Elementary Treatise on Fourier's Series (1893)
• An Introduction to the Use of Generalized Coordinates in Mechanics and Physics (1916)
References
• J. L. Coolidge, "William Elwood Byerly—In memoriam", Bull. Amer. Math. Soc. Volume 42, Number 5 (1936), pp. 295–298.
• Edwin H. Hall, "William Elwood Byerly (1849-1935)", Proceedings of the American Academy of Arts and Sciences, Vol. 71, No. 10 (Mar., 1937), pp. 492–494.
Notes
1. National Academy of Sciences (2002). Biographical Memoirs. National Academies Press. p. 255. ISBN 978-0-309-08476-5.
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William Esson
William Esson, FRS (17 May 1838 – 28 August 1916) was a British mathematician.
William Esson
Born1838
Dundee, Scotland
Died28 August 1916(1916-08-28) (aged 78)
Abingdon, England
NationalityBritish
Alma materSt John's College, Oxford
Known forMathematics of the rate of chemical change
Scientific career
InstitutionsUniversity of Oxford
Early life
He was born in Carnoustie, Scotland.[1]
Esson attended St John's College, Oxford.
Career
He then became a Fellow of Merton College.[2] In 1892, he became the Savilian Professor of Geometry at the University of Oxford, based at New College. He worked on problems in chemistry with Augustus George Vernon Harcourt.
In 1869 he was elected a Fellow of the Royal Society and in 1895 delivered, jointly with Harcourt, their Bakerian Lecture on the Laws of Connexion between the Conditions of a Chemical Change and its Amount. III. Further Researches on the Reaction of Hydrogen Dioxide and Hydrogen Iodide.[3]
He was on the governing body of Abingdon School until 1900.[4]
Personal life
In 1874, Esson leased 13 Bradmore Road in North Oxford.[5] He died in Abingdon, England.[6]
References
1. Obituary notice, Fellow: Esson, William, Monthly Notices of the Royal Astronomical Society, Vol. 77, p.299, 1917MNRAS..77..299., The SAO/NASA Astrophysics Data System
2. Obituary, Royal Society of Chemistry
3. "Fellow Details". Royal Society. Retrieved 20 January 2017.
4. "School Notes" (PDF). The Abingdonian.
5. Hinchcliffe, Tanis (1992). North Oxford. New Haven & London: Yale University Press. p. 220. ISBN 0-14-071045-0.
6. GRO Register of Deaths: Deaths SEP 1916 2c 348 ABINGDON — Willian Esson, aged 78
Savilian Professors
Chairs established by Sir Henry Savile
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• John Bainbridge (1620)
• John Greaves (1642)
• Seth Ward (1649)
• Christopher Wren (1661)
• Edward Bernard (1673)
• David Gregory (1691)
• John Caswell (1709)
• John Keill (1712)
• James Bradley (1721)
• Thomas Hornsby (1763)
• Abraham Robertson (1810)
• Stephen Rigaud (1827)
• George Johnson (1839)
• William Donkin (1842)
• Charles Pritchard (1870)
• Herbert Turner (1893)
• Harry Plaskett (1932)
• Donald Blackwell (1960)
• George Efstathiou (1994)
• Joseph Silk (1999)
• Steven Balbus (2012)
Savilian Professors
of Geometry
• Henry Briggs (1619)
• Peter Turner (1631)
• John Wallis (1649)
• Edmond Halley (1704)
• Nathaniel Bliss (1742)
• Joseph Betts (1765)
• John Smith (1766)
• Abraham Robertson (1797)
• Stephen Rigaud (1810)
• Baden Powell (1827)
• Henry John Stephen Smith (1861)
• James Joseph Sylvester (1883)
• William Esson (1897)
• Godfrey Harold Hardy (1919)
• Edward Charles Titchmarsh (1931)
• Michael Atiyah (1963)
• Ioan James (1969)
• Richard Taylor (1995)
• Nigel Hitchin (1997)
• Frances Kirwan (2017)
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William Beckner (mathematician)
William Beckner (born September 15, 1941) is an American mathematician, known for his work in harmonic analysis, especially geometric inequalities. He is the Paul V. Montgomery Centennial Memorial Professor in Mathematics at The University of Texas at Austin.
William Beckner
Born (1941-09-15) September 15, 1941
Kirksville, Missouri, USA
NationalityAmerican
Alma materPrinceton University
University of Missouri (Columbia)
Known forSharp Inequalities
Babenko–Beckner inequality
AwardsSalem Prize (1975)
Scientific career
FieldsMathematics
InstitutionsUniversity of Texas, Austin
Princeton University
University of Chicago
Doctoral advisorElias Stein
Doctoral studentsEmanuel Carneiro
Education
Beckner earned his Bachelor of Science in physics from the University of Missouri in Columbia, Missouri in 1963, where he became a member of the Phi Beta Kappa Society. He later earned his Ph.D. in mathematics at Princeton University in Princeton, New Jersey, where his doctoral adviser was Elias Stein. He also completed some postgraduate work in mathematics under adviser A.P. Calderon at the University of Chicago.
Awards and honors
• Salem Prize
• Sloan Fellow
• Fellow of the American Mathematical Society.[1]
Selected publications
• Beckner, William (1975). "Inequalities in Fourier Analysis". Annals of Mathematics. 102 (1): 159–182. doi:10.2307/1970980. JSTOR 1970980.
• Beckner, William (1993). "Sharp Sobolev inequalities on the sphere and the Moser–Trudinger inequality". Annals of Mathematics. 138 (1): 213–242. doi:10.2307/2946638. JSTOR 2946638.
• Beckner, William (1995). "Geometric inequalities in Fourier analysis". Essays on Fourier Analysis in Honor of Elias M. Stein. Princeton University Press. pp. 36–68. ISBN 9780691603650. JSTOR j.ctt7ztk2g.5.
See also
• Babenko–Beckner inequality
• Hirschman uncertainty
References
1. List of Fellows of the American Mathematical Society, retrieved 2012-11-10.
External links
• Beckner's home page
• William Beckner at the Mathematics Genealogy Project
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William Floyd (mathematician)
William J. Floyd is an American mathematician specializing in topology. He is currently a professor at Virginia Polytechnic Institute and State University. Floyd received a PhD in mathematics from Princeton University 1978 under the direction of William Thurston.[1]
Mathematical contributions
Most of Floyd's research is in the areas of geometric topology and geometric group theory.
Floyd and Allen Hatcher classified all the incompressible surfaces in punctured-torus bundles over the circle.[2]
In a 1980 paper[3] Floyd introduced a way to compactify a finitely generated group by adding to it a boundary which came to be called the Floyd boundary.[4][5] Floyd also wrote a number of joint papers with James W. Cannon and Walter R. Parry exploring a combinatorial approach to the Cannon conjecture[6][7][8] using finite subdivision rules. This represents one of the few plausible lines of attack of the conjecture.[9]
References
1. William J. Floyd. Mathematics Genealogy Project. Accessed February 6, 2010
2. Floyd, W.; Hatcher, A. Incompressible surfaces in punctured-torus bundles. Topology and its Applications, vol. 13 (1982), no. 3, pp. 263–282
3. Floyd, William J., Group completions and limit sets of Kleinian groups. Inventiones Mathematicae, vol. 57 (1980), no. 3, pp. 205–218
4. Karlsson, Anders, Free subgroups of groups with nontrivial Floyd boundary. Communications in Algebra, vol. 31 (2003), no. 11, pp. 5361–5376.
5. Buckley, Stephen M.; Kokkendorff, Simon L., Comparing the Floyd and ideal boundaries of a metric space. Transactions of the American Mathematical Society, vol. 361 (2009), no. 2, pp. 715–734
6. J. W. Cannon, W. J. Floyd, W. R. Parry. Sufficiently rich families of planar rings. Annales Academiæ Scientiarium Fennicæ. Mathematica. vol. 24 (1999), no. 2, pp. 265–304.
7. J. W. Cannon, W. J. Floyd, W. R. Parry. Finite subdivision rules. Conformal Geometry and Dynamics, vol. 5 (2001), pp. 153–196.
8. J. W. Cannon, W. J. Floyd, W. R. Parry. Expansion complexes for finite subdivision rules. I. Conformal Geometry and Dynamics, vol. 10 (2006), pp. 63–99.
9. Ilya Kapovich, and Nadia Benakli, in Boundaries of hyperbolic groups, Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001), pp. 39–93, Contemporary Mathematics, 296, American Mathematical Society, Providence, RI, 2002, ISBN 0-8218-2822-3 MR1921706; pp. 63–64
External links
• William Floyd at the Mathematics Genealogy Project
• William Floyd's webpage, Department of Mathematics, Virginia Polytechnic Institute and State University
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William Francis Pohl
William Francis Pohl (16 September 1937 – 9 December 1988)[1] was an American mathematician, specializing in differential geometry and known for the Clifton–Pohl torus.
Pohl received from the University of Chicago his B.A in 1957 and his M.A.1958. He completed his Ph.D. at Berkeley in 1961 under the direction of Shiing-Shen Chern with dissertation Differential Geometry of Higher Order.[2] His dissertation was published in 1962 in the journal Topology[3] and has received over 120 citations in the mathematical literature. He was a member of the mathematics faculty at the University of Minnesota from September 1964 until his untimely death.
Pohl engaged in a famous controversy arguing against Francis Crick[4] but, in view of additional empirical evidence, conceded about 1979 or 1980 that Crick was correct.[5]
Pohl sang liturgical music in Catholic religious services and wrote an article in 1966 from which the journal Sacred Music published an excerpt in 2011.[6]
In the early 1970s, Dr. William F. Pohl, a professor of mathematics at the University of Minnesota, sang the Gregorian chant, mostly solo, while developing a small schola of Chorale volunteers to assist him. Dr. Pohl guided the chant during the aftermath of the Second Vatican Council when all the liturgical books were being revised ? no small task, but as some may recall, he was no small man.
By 1975, in cooperation with Monsignor Richard J. Schuler, pastor of Saint Agnes, and Harold Hughesdon, its master of ceremonies, Dr. Pohl, joined by a number of dedicated volunteers, had begun the custom of singing Sunday vespers weekly and the full office of Tenebrae during Holy Week.
Organist David Bevan arrived from England in 1976 to accompany the Chorale, and he assumed directorship of the Gregorian chant after Dr. Pohl's retirement in 1977.[7]
William Pohl later married Hildegard Bastian (now Hildegard Pohl), and fathered 5 children, Annetta Pohl, Agatha Pohl, Agnes Pohl, Lawrence Pohl, and John Pohl.
Selected publications
• Pohl, William F. (1966). "Connexions in differential geometry of higher order". Transactions of the American Mathematical Society. 125 (2): 310–325. doi:10.1090/s0002-9947-1966-0203628-1. JSTOR 1994357.
• "The self-linking number of a closed space curve (Gauss integral formula treated for disjoint closed space curves linking number)" (PDF). Journal of Mathematics and Mechanics. 17: 975–985. 1968.
• Pohl, William F. (1968). "Some integral formulas for space curves and their generalization". American Journal of Mathematics. 90 (4): 1321–1345. doi:10.2307/2373302. JSTOR 2373302.
• with T. F. Banchoff: Banchoff, Thomas F.; Pohl, William F. (1971). "A generalization of the isoperimetric inequality". Journal of Differential Geometry. 6 (2): 175–192. doi:10.4310/jdg/1214430403. MR 0305319.
• with John Alvord Little: Little, John A.; Pohl, William F. (1971). "On tight immersions of maximal codimension" (PDF). Inventiones Mathematicae. 13 (3): 179–204. Bibcode:1971InMat..13..179L. doi:10.1007/BF01404629. hdl:2027.42/46589. S2CID 54785966.
• with Nicolaas H. Kuiper: Kuiper, Nicolaas H.; Pohl, William F. (1977). "Tight topological embeddings of the real projective plane in E5 ". Inventiones Mathematicae. 42 (1): 177–199. Bibcode:1977InMat..42..177K. doi:10.1007/BF01389787. S2CID 120800935.
• Pohl, William F. (1981). "The probability of linking of random closed curves". Geometry Symposium Utrecht 1980. Lecture Notes in Mathematics. Vol. 894. Springer Berlin Heidelberg. pp. 113–126. doi:10.1007/BFb0096227. ISBN 978-3-540-11167-2.
References
1. Minnesota Historical Society Death Certificate Search, 1904–2001
2. William Francis Pohl at the Mathematics Genealogy Project
3. Pohl, W. F. (1962). "Differential geometry of higher order". Topology. 1 (3): 169–211. doi:10.1016/0040-9383(62)90103-9. hdl:10338.dmlcz/101530.
4. Pohl, W. F.; Roberts, George W. (October 1978). "Topological consideration in the theory of replication of DNA". Journal of Mathematical Biology. 6 (4): 383–402. doi:10.1007/BF02463003. PMID 750633. S2CID 29082243.
5. Pohl, W. F. (March 1980). "DNA and differential geometry". The Mathematical Intelligencer. 3 (1): 20–27. doi:10.1007/BF03023391. S2CID 119798941.
6. Pohl, W. F. (2011). "Liturgical Music and the Liturgical Movement (1966)". Sacred Music. 136 (3): 37.
7. ""Sacred Choral Music" at Saint Agnes, Minneapolis, 2) The Schola Cantorum at Saint Agnes". catholicforum.com. 19 July 2006.
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William Frederick Eberlein
William Frederick Eberlein (June 25, 1917, Shawano, Wisconsin – 1986, Rochester, New York) was an American mathematician, specializing in mathematical analysis and mathematical physics.
Life
Eberlein studied from 1936 to 1942 at the University of Wisconsin and at Harvard University, where he received in 1942 a PhD for the thesis Closure, Convexity, and Linearity in Banach Spaces under the direction of Marshall Stone.[1]
He was married twice—to Mary Bernarda Barry and Patricia Ramsay James. He had four children with Mary Barry, including Patrick Barry Eberlein, another renowned mathematician. Patricia Ramsay James was a mathematician who moved into computer science as the field opened up; their one child is Kristen James Eberlein, the chair of the OASIS Darwin Information Typing Architecture Technical Committee.
Work
Eberlein had academic positions at the Institute for Advanced Study (1947–1948), at the University of Wisconsin (1948–1955), at Wayne State University (1955–1956), and from 1957 at the University of Rochester, where he remained for the rest of his career.[2] His doctoral students include William F. Donoghue, Jr.[3] and A. Wayne Wymore.
Contributions
He worked on functional analysis, harmonic analysis, ergodic theory, mean value theorems, and numerical integration. Eberlein also worked on spacetime models, internal symmetries in gauge theory, and spinors.[2] His name is attached to the Eberlein–Šmulian theorem in functional analysis[4] and the Eberlein compacta in topology.[5]
References
1. William Frederick Eberlein at the Mathematics Genealogy Project
2. A Guide to the W. F. Eberlein Papers, 1936–1986, Briscoe Center for American History, University of Texas at Austin, retrieved 2014-06-19.
3. Gelbaum, Bernard Russell. "In Memoriam: William F. Donoghue, Jr". University of California.
4. Conway, John B. (1990), A Course in Functional Analysis, Graduate Texts in Mathematics, vol. 96, Springer, p. 163, ISBN 9780387972459.
5. Arhangel'skii, A. V. (2003), "Eberlein compacta", in Hart, K. P.; Nagata, Jun-iti; Vaughan, J. E. (eds.), Encyclopedia of General Topology, Elsevier, pp. 145–146, ISBN 9780080530864.
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William Fulton (mathematician)
William Edgar Fulton (born August 29, 1939) is an American mathematician, specializing in algebraic geometry.
William Fulton
William Fulton at Oberwolfach in 2006
Born (1939-08-29) August 29, 1939
Naugatuck, Connecticut, US
NationalityAmerican
Alma materPrinceton University
AwardsLeroy P. Steele Prize (2010)
Scientific career
FieldsMathematics
InstitutionsUniversity of Michigan
University of Chicago
Brown University
Brandeis University
Doctoral advisorGerard Washnitzer
Other academic advisorsJohn Milnor
John Coleman Moore
Goro Shimura
Doctoral studentsRobert Lazarsfeld
Education and career
He received his undergraduate degree from Brown University in 1961 and his doctorate from Princeton University in 1966. His Ph.D. thesis, written under the supervision of Gerard Washnitzer, was on The fundamental group of an algebraic curve.
Fulton worked at Princeton and Brandeis University from 1965 until 1970, when he began teaching at Brown. In 1987 he moved to the University of Chicago.[1] He is, as of 2011, a professor at the University of Michigan.[2]
Fulton is known as the author or coauthor of a number of popular texts, including Algebraic Curves and Representation Theory.
Awards and honors
In 1996 he received the Steele Prize for mathematical exposition for his text Intersection Theory.[1] Fulton is a member of the United States National Academy of Sciences since 1997; a fellow of the American Academy of Arts and Sciences from 1998, and was elected a foreign member of the Royal Swedish Academy of Sciences in 2000.[3] In 2010, he was awarded the Steele Prize for Lifetime Achievement.[4] In 2012 he became a fellow of the American Mathematical Society.[5]
Selected works
• Algebraic Curves: An Introduction To Algebraic Geometry, with Richard Weiss. New York: Benjamin, 1969. Reprint ed.: Redwood City, CA, USA: Addison-Wesley, Advanced Book Classics, 1989. ISBN 0-201-51010-3. Full text online.
• William Fulton (1998), Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2 (2nd ed.), Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4612-1700-8, ISBN 978-3-540-62046-4, MR 1644323 1st edn. 1984.[6]
• Fulton, William; Harris, Joe (1991). Representation Theory, A First Course. Graduate Texts in Mathematics. Vol. 129. Berlin, New York: Springer-Verlag. ISBN 978-0-387-97495-8. MR 1153249.
See also
• Fulton–Hansen connectedness theorem
References
1. Announcement of the 1996 Steele Prizes at the American Mathematical Society web site, accessed July 15, 2009.
2. University of Michigan mathematics department, alphabetical faculty listing, accessed November 13, 2011.
3. "View from the Chair's Office - Fulton Named Distinguished Mel Hochster University Professor". University of Michigan. Retrieved 3 April 2022.
4. http://www.ams.org/ams/press/steele-lifetime-2010.html AMS announcement of 2010 Steele Prize for Lifetime Achievement
5. List of Fellows of the American Mathematical Society, retrieved 2012-12-29.
6. Kleiman, Steven L. (1985). "Review: Intersection theory, by W. Fulton and Introduction to intersection theory in algebraic geometry, by W. Fulton" (PDF). Bull. Amer. Math. Soc. (N.S.). 12 (1): 137–143. doi:10.1090/s0273-0979-1985-15319-4.
External links
• Fulton's home page at the University of Michigan
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William George Horner
William George Horner (9 June 1786 – 22 September 1837) was a British mathematician. Proficient in classics and mathematics, he was a schoolmaster, headmaster and schoolkeeper who wrote extensively on functional equations, number theory and approximation theory, but also on optics. His contribution to approximation theory is honoured in the designation Horner's method, in particular respect of a paper in Philosophical Transactions of the Royal Society of London for 1819. The modern invention of the zoetrope, under the name Daedaleum in 1834, has been attributed to him.[1][2][3]
Horner died comparatively young, before the establishment of specialist, regular scientific periodicals. So, the way others have written about him has tended to diverge, sometimes markedly, from his own prolific, if dispersed, record of publications and the contemporary reception of them.
Family life
The eldest son of the Rev. William Horner, a Wesleyan minister, Horner was born in Bristol. He was educated at Kingswood School, a Wesleyan foundation near Bristol, and at the age of sixteen became an assistant master there. In four years he rose to be headmaster (1806), but left in 1809, setting up his own school, The Classical Seminary, at Grosvenor Place, Bath, which he kept until he died there 22 September 1837. He and his wife Sarah (1787?–1864) had six daughters and two sons.
Physical sciences, optics
Although Horner's article on the Dædalum (zoetrope) appeared in Philosophical Magazine only in January, 1834, he had published on Camera lucida as early as August, 1815.
Mathematics
Horner's name first appears in the list of solvers of the mathematical problems in The Ladies' Diary: or, Woman's Almanack for 1811, continuing in the successive annual issues until that for 1817. Up until the issue for 1816, he is listed as solving all but a few of the fifteen problems each year; several of his answers were printed, along with two problems he proposed. He also contributed to other departments of the Diary, not without distinction, reflecting the fact that he was known to be an all-rounder, competent in the classics as well as in mathematics. Horner was ever vigilant in his reading, as shown by his characteristic return to the Diary for 1821 in a discussion of the Prize Problem, where he reminds readers of an item in (Thomson's) Annals of Philosophy for 1817; several other problems in the Diary that year were solved by his youngest brother, Joseph.
His record in The Gentleman's Diary: or, Mathematical Repository for this period is similar, including one of two published modes of proof in the volume for 1815 of a problem posed the previous year by Thomas Scurr (d. 1836), now dubbed the Butterfly theorem. Leaving the headmastership of Kingswood School would have given him more time for this work, while the appearance of his name in these publications, which were favoured by a network of mathematics teachers, would have helped publicize his own school.
At this stage, Horner's efforts turned more to The Mathematical Repository, edited by Thomas Leybourn, but to contributing occasional articles, rather than the problem section, as well as to Annals of Philosophy, where Horner begins by responding to other contributors and works up to independent articles of his own; he has a careful style with acknowledgements and, more often than not, cannot resist adding further detail.
Several contributions pave the way for, or are otherwise related to, his most celebrated mathematical paper, in Philosophical Transactions of the Royal Society of London in 1819, which was read by title at the closing meeting for the session on 1 July 1819, with Davies Gilbert in the Chair. The article, with significant editorial notes by Thomas Stephens Davies, was reprinted as a commemorative tribute in The Ladies' Diary for 1838. The issue of The Gentleman's Diary for that year contains a short obituary notice. A careful analysis of this paper has appeared recently in Craig Smoryński's History of Mathematics: A Supplement.[4]
While a sequel was read before the Royal Society, publication was declined for Philosophical Transactions, having to await appearance in a sequence of parts in the first two volumes of The Mathematician in the mid-1840s, again largely at the instigation of T. S. Davies.
However, Horner published on diverse topics in The Philosophical Magazine well into the 1830s. Davies mooted an edition of Horner's collected papers, but this project never came to fruition, partly on account of Davies' own early death.
Contemporary reception
Some idea of Horner's standing with his contemporaries is provided by exchanges in the issues of Annals of Philosophy for July and August, 1817. Thomas Thomson, in commending to an enquirer Euler's work on algebra, is under some impression that the English translation is by Horner.[5]: 86 Horner writes promptly to correct this,[5]: 170 supposing the translation to be the work of Peter Barlow. Thomson, a professor in Glasgow, might not have known that the translation, originally published as far back as 1789, was the work of Francis Horner MP, an Edinburgh native, who had died only that February.
Peter Barlow and continued fractions
When Peter Barlow wrote, in 1845, he remembered Davies, but not Horner, asking to borrow a book by Budan (both Davies and Horner were living in Bath at the time). Barlow also had a vague recollection that the material on approximations Horner sent him related to continued fractions, rather than what appeared in the Philosophical Transactions. Horner clearly held Barlow in high regard and it would have been natural for Horner to approach him to request both books and critical advice as Horner draws attention to Barlow's article in New Series of the Mathematical Repository[6] and in his survey of approximation methods in the following volume of the Repository (bound up in 1819). The anonymous reviewer for The Monthly Review in the issue for December, 1820 writes that he has seen Horner's letter to Barlow and that the letter confirms that Horner already had his method of approximation at that date (1818).
The methods of both Barlow and Horner use a nesting of expressions akin to continued fractions. Horner was aware of Lagrange's use of continued fractions at least through his reading of Bonnycastle's Algebra which is also mentioned in the survey article in the Repository. Horner may have rewritten his paper either under guidance or of his own volition, with an eye to publication in Philosophical Transactions. Horner goes on to write on the use of continued fractions in the summation of series in Annals of Philosophy in 1826 and on their use in improvements they yield in the solution of equations in Quarterly Journal of Science, Literature and the Arts running over into 1827; he explicitly cites work of Lagrange. Barlow's memory of events may have been confused by the appearance of this later work.
Publications
• New and important combinations with the Camera Lucida, dated Bath, 15 August 1815, Annals of Philosophy, 6 (Oct. 1815), 281–283.
• I. On Annuities. - II. Imaginary cube roots. - III. Roots of Binomials, dated Bath, 9 September 1816, Annals of Philosophy, 8 (Oct. 1816), 279–284.
• Corrections of the paper inserted in the last number of the Annals, p. 279, dated Bath, 3 October 1816 Annals of Philosophy, 8 (Nov. 1816), 388–389.
• Formulas for estimating the height of mountains, dated Bath, 13 February 1817 Annals of Philosophy, 9 (March, 1817), 251–252.
• On cubic equations, dated Bath, 17 January 1817, Annals of Philosophy, 9 (May, 1817), 378–381.
• Solution of the equation ψnx=x, Annals of Philosophy, 10 (Nov, 1817), 341–346.
• On reversion of series, especially in connection with the equation ψα−1ψαx=x, dated Bath, 10 November 1817, Annals of Philosophy, 11 (Feb, 1818), 108–112.
• On popular methods of approximation, dated Bath, 1819, Math. Rep. New Series, 4 (1819), Part II, 131–136.
• 'A Tribute of Friendship,’ a poem addressed to his friend Thomas Fussell, appended to a 'Funeral Sermon on Mrs. Fussell,’ Bristol, 1820.
• On algebraic transformation, as deducible from first principles, and connected with continuous approximations, and the theory of finite and fluxional differences, including some new modes of numerical solution, one of ten papers read at the table at the meeting of the Royal on 19 June 1823, immediately before the long vacation adjournment until 20 November 1823; one of the three papers of the set not published in Phil. Trans. that year; published in issues in the first two volumes of The Mathematician bound up in 1845 and 1847.
• Extension of Theorem of Fermat, dated 26 December, Annals of Philosophy New Series, 11 (Feb, 1826), 81–83.
• On the solutions of the Function ψzx and their limitations, Art 1-8, dated Bath, 11 February 1826, Annals of Philosophy New Series, 11 (March, 1826), 168–183; Art 9-17, ibid, 11 (April, 1826), 241–246.
• Reply to Mr. Herapath, dated Bath, 2 April 1826, Annals of Philosophy New Series, 11 (May, 1826), 363
• On the use of continued fractions with unrestricted numerators in summation of series, Art 1-4, dated Bath, 24 April 1826, Annals of Philosophy New Series, 11 (June, 1826), 416–421; Art 5-6, ibid, 12 (July, 1826), 48–51.
• 'Natural Magic,’ a pamphlet on optics dealing with virtual images, London, 1832.
• On the properties of the Dædaleum, a new instrument of optical illusion, Phil. Mag., Ser. 3, 4 (Jan, 1834), 36-41.
• On the autoptic spectrum of certain vessels within the eye, as delineated in shadow on the retina, Phil. Mag., Ser. 3, 4 (April, 1834), 262-271.
• Considerations relative to an interesting case in equations, Phil. Mag., Ser. 3, 5 (Sept, 1834), 188-191.
• On the signs of the trigonometrical lines, Phil. Mag., Ser. 3, 6 (Feb, 1836), 86-90.
• On the theory of congeneric surd equations, Communicated by T. S. Davies, Phil. Mag., Ser. 3, 8 (Jan, 1836), 43-50.
• New demonstration of an original proposition in the theory of numbers, Communicated by T. R. Phillips, Phil. Mag., Ser. 3, 11 (Nov, 1837), 456-459.
• 'Questions for the Examination of Pupils on … General History,’ Bath, 1843, 12mo.
A complete edition of Horner's works was promised by Thomas Stephens Davies, but never appeared.
Other contemporary literature
• P. Barlow, On the resolution of the irreducible case in cubic equations, Math. Rep., NS IV (1814), 46-57 [includes Table for the solution of the irreducible case in cubic equations (6pp.)].
• P. Barlow, A new method of approximating towards the roots of equations of all dimensions, Math. Rep., NS IV (1814), No. 12, 67–71.
• T. Holdred, A New Method of Solving Equations with Ease and Expedition; by which the True Value of the Unknown Quantity is Found Without Previous Reduction. With a Supplement, Containing Two Other Methods of Solving Equations, Derived from the Same Principle(Richard Watts. Sold by Davis and Dickson, mathematical and philosophical booksellers, 17, St. Martin's-le-Grand; and by the author, 2, Denzel Street, Clare-Market, 1820), 56pp..
Notes
1. Zoetrope. EarlyCinema.com. Retrieved on 2011-10-11.
2. Glossary – Z. Wernernekes.de. Retrieved on 2011-10-11.
3. Philosophical magazine. Taylor & Francis. 1834. p. 36.
4. History of Mathematics: A Supplement. New York, NY: Springer. 2008. ISBN 9780387754802. esp. Chap. 7
5. Thomas Thomson, ed. (1817). "Annals of Philosophy". X. London. hdl:2027/mdp.39015066710818. {{cite journal}}: Cite journal requires |journal= (help)
6. Thomas Leybourn, ed. (1814). "New Series of the Mathematical Repository". III. {{cite journal}}: Cite journal requires |journal= (help)
References
• Register of Kingswood School, 1748-1922 (1923), p. 89.
• 1861 Census
External links
• Bath: Births, Marriages and Deaths
• Prof. Neville Fletcher, Research School of Physics and Engineering, ANU
• Australian Academy of Science: Interview with Neville Fletcher
Attribution
This article incorporates text from a publication now in the public domain: "Horner, William George". Dictionary of National Biography. London: Smith, Elder & Co. 1885–1900.
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William Galbraith (mathematician)
Rev William Galbraith (1786 – 27 October 1850) was a Scottish mathematician. He taught mathematics and nautical astronomy in Edinburgh, and took an interest in surveying work, becoming an advocate of the extension of the work of triangulating Great Britain.[1]
Early life
He was born at Greenlaw, Berwickshire.[2] Initially he was a schoolmaster. His pupil William Rutherford walked long distances to attend his school at Eccles. Subsequently, he moved to Edinburgh, and graduated A.M. at the University of Edinburgh in 1821.[3]
Surveyor
During the 1830s Galbraith became interested in the surveying problems of Scotland. In 1831 he pointed out that Arthur's Seat had a strongly magnetic peak.[4] In 1837 he pointed out the impact of anomalies in measurement, work that received recognition;[5] it was topical because of the 1836 geological map of Scotland by John MacCulloch, with which critics had found fault on topographical as well as geological grounds.[6] A paper on the locations of places on the River Clyde was recognised in 1837 by a gold medal, from the Society for the Encouragement of the Useful Arts for Scotland.[7]
Galbraith followed with detailed Remarks on the Geographical Position of some Points on the West Coast of Scotland (1838).[8] Having made some accurate surveys of his own, he lobbied for further attention from the national survey.[1]
Later life
About 1832 Galbraith was licensed a minister by the presbytery of Dunse. He married Eleanor Gale in 1833.[3]
Galbraith was buried with his wife in the north-east section of the Grange Cemetery in Edinburgh.[9]
Works
Galbraith's major works combined textbook material with mathematical tables:
• Mathematical and Astronomical Tables (1827):[10] review.[11]
• Trigonometrical Surveying, Levelling, and Railway Engineering (1842)[12]
He edited John Ainslie's 1812 treatise on land surveying (1849),[13] and with William Rutherford revised John Bonnycastle's Algebra.[14]
Notes
1. "National Museums of Scotland - Rule (Detail)".
2. "Biographical notices of Mr. William Galbraith". Monthly Notices of the Royal Astronomical Society. 11 (4): 86. 1851. Bibcode:1851MNRAS..11...86.. doi:10.1093/mnras/11.4.67a.
3. Royal Astronomical Society (1851). Memoirs. Society. pp. 194–. Retrieved 6 November 2012.
4. "The topographical, statistical, and historical gazetteer of Scotland". 1848.
5. Kaiserl. Akademie der Wissenschaften in Wien (1851). Almanach der Kaiserlichen Akademie der Wissenschaften für das Jahr . Harvard University. Wien : K.K. Hof- und Staatsdruckerei.
6. Cumming, David A. "MacCulloch, John". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/17412. (Subscription or UK public library membership required.)
7. The Edinburgh Philosophical Journal. Constable. 1837. pp. 1–. Retrieved 6 November 2012.
8. The Edinburgh New Philosophical Journal. A. and C. Black. 1838. pp. 300–. Retrieved 4 May 2012.
9. "Monuments and monumental inscriptions in Scotland".
10. William Galbraith (1827). Mathematical and Astronomical Tables. Retrieved 4 May 2012.
11. Robert Jameson; Sir William Jardine; Henry Darwin Rogers (1827). The Edinburgh New Philosophical Journal: exhibiting a view of the progressive discoveries and improvements in the sciences and the arts. A. and C. Black. pp. 404–. Retrieved 4 May 2012.
12. "Trigonometrical surveying, levelling, and railway engineering". 1842.
13. "A treatise on land surveying [ed.] by W. Galbraith". 1849.
14. "Rutherford, William" . Dictionary of National Biography. London: Smith, Elder & Co. 1885–1900.
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William Gardiner (mathematician)
William Gardiner (died 1752) was an English mathematician.[1] His logarithmic tables of sines and tangents (Tables of logarithms, 1742) had various reprints and saw use by scientists and other mathematicians.
Works
• A literal exposition of two prophecies cited by St. Matthew out of the Old Testament (1726), reprinted as A literal exposition of two remarkable prophecies in the Old Testament : relating to Jesus Christ the Messiah (1728)
• Practical surveying improved : or, land-measuring, according to the present most correct methods.: 1737 (erroneously given as 1773 in Worldcat), many reprints
• Tables of logarithms, for all numbers from 1 to 102100, and for the sines and tangents to every ten seconds of each degree in the quadrant; as also, for the sines of the first 72 minutes to every single second; with other useful and necessary tables: 1742, reprinted many times, translated in French and Italian[2]
References
1. "Gardiner, William, -1752". National Library of the Czech Republic.
2. Gardiner, William (1796). Tavole logaritmiche (in Italian). Florence: Pietro Allegrini & Gioacchino Pagani.
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William Gasarch
William Ian Gasarch (/ɡəˈsɑːrʃ/ gə-SARSH;[1] born 1959[2]) is an American computer scientist known for his work in computational complexity theory, computability theory, computational learning theory, and Ramsey theory. He is currently a professor at the University of Maryland Department of Computer Science with an affiliate appointment in Mathematics.
William Ian Gasarch
Professor Bill Gasarch at UMD
Born1959 (age 63–64)
NationalityAmerican
Alma materStony Brook University
Harvard University
Known forComputational complexity theory
Computability theory
Computational learning theory
Ramsey theory
Scientific career
FieldsComputer science
InstitutionsUniversity of Maryland, College Park
Doctoral advisorHarry R. Lewis
Websitewww.cs.umd.edu/~gasarch
http://blog.computationalcomplexity.org/
As of 2015 he has supervised over 40 high school students on research projects, including Jacob Lurie. He has co-blogged on computational complexity with Lance Fortnow since 2007. He was book review editor for ACM SIGACT NEWS from 1997 to 2015.
Education
Gasarch received his doctorate in computer science from Harvard in 1985, advised by Harry R. Lewis. His thesis was titled Recursion-Theoretic Techniques in Complexity Theory and Combinatorics.[3] He was hired into a tenure track professorial job at the University of Maryland in the Fall of 1985. He was promoted to Associate Professor with Tenure in 1991, and to Full Professor in 1998.
Work
Gasarch co-founded (with Richard Beigel) the field of Bounded Queries in Recursion Theory[4] and has written many papers in the area capped off by a book on the topic co-authored with Georgia Martin, titled Bounded Queries in Recursion Theory.[5] He has published books such as Problems with a Point,[6] a book with a broad view on mathematics and theoretical computer science which he co-authored with Clyde Kruskal and includes works by other professors such as David Eppstein.[7] He also co-founded the subfield of recursion-theoretic inductive inference named Learning via Queries[8] with Carl Smith. More recently he has been more involved with combinatorics, notably Ramsey Theory.[9][10][11] He has written three surveys of what theorists think of the P vs NP problem: in 2002, 2012, and 2019.[12][13][14] In 2020 he wrote Mathematical Muffin Morsels: Nobody Wants a Small Piece with Erik Metz, Jacob Prinz, and Daniel Smolyak. [15]
Blog
Lance Fortnow began writing a blog on theoretical computer science with an emphasis on complexity theory in 2002.[16] Gasarch was a frequent guest blogger until 2007 when he became an official co-blogger.
References
1. "Rectangle Free Colorings – William Gasarch". YouTube. May 8, 2017. Retrieved 12 October 2022.
2. "Still Typecasting from Dagstuhl". Computational Complexity Weblog. Lance Fortnow and William Gasarch. Retrieved 27 September 2018.
3. William Gasarch at the Mathematics Genealogy Project
4. http://www.cs.umd.edu/~gasarch/papers/gems.pdf Gems in the Field of Bounded Queries by William Gasarch, 2003
5. https://www.springer.com/us/book/9780817639662 Bounded Queries in Recursion Theory (with Georgia Martin), Birkhauser, 1999
6. https://www.worldscientific.com/worldscibooks/10.1142/11261 Problems with a Point Exploring Math and Computer Science, 2019
7. https://www.worldscientific.com/doi/abs/10.1142/9789813279735_0014 Chapter 14: Is This Problem Too Hard for a High School Math Competition?, 2019
8. http://www.cs.umd.edu/~gasarch/papers/lvqsur.pdf A Survey of Inductive Inference with an Emphasis on Queries, Gasarch and Smith, 1997
9. Gasarch, William; Haeupler, Bernhard (2011). "Lower Bounds on the van der Waerden Numbers: Randomized- and Deterministic-Constructive". Electronic Journal of Combinatorics. 18 (64). arXiv:1005.3749. doi:10.37236/551. S2CID 534179.
10. Gasarch, William; Haeupler, Bernhard (2010). "Rectangle Free Coloring of Grids". arXiv:1005.3750 [math.CO].
11. Gasarch, William; Haeupler, Bernhard (2011). "Proving programs terminate using well orderings, Ramsey Theory, and Matrices". arXiv:1108.3347 [math.CO].
12. Hemaspaandra, Lane A. (2002-06-01). "SIGACT news complexity theory column 36". ACM SIGACT News. 33 (2): 34–47. doi:10.1145/564585.564599. ISSN 0163-5700. S2CID 36828694.
13. Hemaspaandra, Lane A. (2012-06-11). "SIGACT news complexity theory column 74". ACM SIGACT News. 43 (2): 51–52. doi:10.1145/2261417.2261433. ISSN 0163-5700. S2CID 52847337.
14. Gasarch, William I. (2019-03-13). "Guest Column: The Third P=?NP Poll". ACM SIGACT News. 50 (1): 38–59. doi:10.1145/3319627.3319636. ISSN 0163-5700. S2CID 83458626.
15. Gasarch, William; Metz, Erik; Prinz, Jacob; Smolyak, Daniel (28 May 2020). Mathematical Muffin Morsels: Nobody Wants A Small Piece. World Scientific. ISBN 978-981-12-1519-3.
16. http://blog.computationalcomplexity.org/ Computational Complexity Weblog
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William Gentle (headmaster)
William Gentle FRSE (19 January 1877 – 31 March 1964) was a Scottish mathematician and physicist, and headmaster of George Heriot's School from 1926 to 1942. He was President of the Science Teachers Association in Scotland.[1]
Life
He was born in Dundee on 19 January 1877 the third child of Marion Drummond Brand from Torryburn in Fife and William Gentle (1838–1890), a printer. He was raised at 12 Garland Place in Dundee.[2] He attended Rosebank School then Morgan Academy. In 1889 his family moved to 2 Blackwood Crescent in Edinburgh, a Victorian flat in the south side of the city. He then attended George Heriot's School going on to study mathematics and other sciences at the University of Edinburgh, graduating in 1903 with a BSc. He studied geology under Professor James Geikie and mathematics under Professor George Chrystal. In autumn 1903 he returned to George Heriot's School as a teacher of mathematics and physics.[3][4]
He was elected a Fellow of the Royal Society of Edinburgh in 1908. His proposers were James Gordon MacGregor, David Fowler Lowe, Thomas Burns and John Brown Clark.[5] In the First World War he served as an officer in the Royal Field Artillery, and was wounded by shell-fire at Messines Ridge in 1917. Thereafter he concerned himself with supply of food to the troops, at one point finding himself responsible for feeding 7,000 men.[3]
He returned to George Heriot's School after the war. In 1926 he succeeded John Brown Clark as headmaster at a salary of £1,000 per annum. He retired in 1942 and was succeeded by William Carnon.
He died in Edinburgh on 31 March 1964.
Family
In 1924 he married Jessie Currie Ainslie.
References
1. "Gentle biography". www-history.mcs.st-andrews.ac.uk. Retrieved 19 January 2018.
2. Dundee Post Office Directory 1877-78
3. "Gentle biography". www-groups.dcs.st-and.ac.uk. Retrieved 19 January 2018.
4. "Slater's Royal National Commercial Directory of Scotland, 1903, Part 2". National Library of Scotland. Retrieved 19 January 2018.
5. Biographical Index of Former Fellows of the Royal Society of Edinburgh 1783–2002 (PDF). The Royal Society of Edinburgh. July 2006. ISBN 0-902-198-84-X.
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William Goldman (mathematician)
William Mark Goldman (born 1955 in Kansas City, Missouri) is a professor of mathematics at the University of Maryland, College Park (since 1986). He received a B.A. in mathematics from Princeton University in 1977, and a Ph.D. in mathematics from the University of California, Berkeley in 1980.
William Goldman
William Goldman at Bar-Ilan University in 2008
Born (1955-11-17) November 17, 1955
Kansas City, United States
NationalityAmerican
Alma materPrinceton University
University of California, Berkeley
Scientific career
FieldsMathematics
InstitutionsUniversity of Maryland-College Park
Doctoral advisorsMorris Hirsch
William Thurston
Research contributions
Goldman has investigated geometric structures, in various incarnations, on manifolds since his undergraduate thesis, "Affine manifolds and projective geometry on manifolds", supervised by William Thurston and Dennis Sullivan. This work led to work with Morris Hirsch and David Fried on affine structures on manifolds, and work in real projective structures on compact surfaces. In particular he proved that the space of convex real projective structures on a closed orientable surface of genus $g>1$ is homeomorphic to an open cell of dimension $16g-16$. With Suhyoung Choi, he proved that this space is a connected component (the "Hitchin component") of the space of equivalence classes of representations of the fundamental group in ${\rm {SL}}(3,\mathbb {R} )$. Combining this result with Suhyoung Choi's convex decomposition theorem, this led to a complete classification of convex real projective structures on compact surfaces.
His doctoral dissertation, "Discontinuous groups and the Euler class" (supervised by Morris W. Hirsch), characterizes discrete embeddings of surface groups in ${\rm {PSL}}(3,\mathbb {R} )$ in terms of maximal Euler class, proving a converse to the Milnor–Wood inequality for flat bundles. Shortly thereafter he showed that the space of representations of the fundamental group of a closed orientable surface of genus $g>1$ in ${\rm {PSL}}(3,\mathbb {R} )$ has $4g-3$ connected components, distinguished by the Euler class.
With David Fried, he classified compact quotients of Euclidean 3-space by discrete groups of affine transformations, showing that all such manifolds are finite quotients of torus bundles over the circle. The noncompact case is much more interesting, as Grigory Margulis found complete affine manifolds with nonabelian free fundamental group. In his 1990 doctoral thesis, Todd Drumm found examples which are solid handlebodies using polyhedra which have since been called "crooked planes."
Goldman found examples (non-Euclidean nilmanifolds and solvmanifolds) of closed 3-manifolds which fail to admit flat conformal structures.
Generalizing Scott Wolpert's work on the Weil–Petersson symplectic structure on the space of hyperbolic structures on surfaces, he found an algebraic-topological description of a symplectic structure on spaces of representations of a surface group in a reductive Lie group. Traces of representations of the corresponding curves on the surfaces generate a Poisson algebra, whose Lie bracket has a topological description in terms of the intersections of curves. Furthermore, the Hamiltonian vector fields of these trace functions define flows generalizing the Fenchel–Nielsen flows on Teichmüller space. This symplectic structure is invariant under the natural action of the mapping class group, and using the relationship between Dehn twists and the generalized Fenchel–Nielsen flows, he proved the ergodicity of the action of the mapping class group on the SU(2)-character variety with respect to symplectic Lebesgue measure.
Following suggestions of Pierre Deligne, he and John Millson proved that the variety of representations of the fundamental group of a compact Kähler manifold has singularities defined by systems of homogeneous quadratic equations. This leads to various local rigidity results for actions on Hermitian symmetric spaces.
With John Parker, he examined the complex hyperbolic ideal triangle group representations. These are representations of hyperbolic ideal triangle groups to the group of holomorphic isometries of the complex hyperbolic plane such that each standard generator of the triangle group maps to a complex reflection and the products of pairs of generators to parabolics. The space of representations for a given triangle group (modulo conjugacy) is parametrized by a half-open interval. They showed that the representations in a particular range were discrete and conjectured that a representation would be discrete if and only if it was in a specified larger range. This has become known as the Goldman–Parker conjecture and was eventually proven by Richard Schwartz.
Professional service
Goldman also heads a research group at the University of Maryland called the Experimental Geometry Lab, a team developing software (primarily in Mathematica) to explore geometric structures and dynamics in low dimensions. He served on the Board of Governors for The Geometry Center at the University of Minnesota from 1994 to 1996.
He served as Editor-In-Chief of Geometriae Dedicata from 2003 until 2013.
Awards and honors
In 2012 he became a fellow of the American Mathematical Society.[1]
Publications
• Goldman, William M. (1999). Complex hyperbolic geometry. Oxford Mathematical Monographs. Oxford Science Publications. New York: The Clarendon Press, Oxford University Press. xx+316 pp. ISBN 0-19-853793-X. MR 1695450.
• Goldman, William M.; Xia, Eugene Z. (2008). "Rank one Higgs bundles and representations of fundamental groups of Riemann surfaces". Memoirs of the American Mathematical Society. 193 (904): viii+69 pp. arXiv:math/0402429. doi:10.1090/memo/0904. ISSN 0065-9266. MR 2400111. S2CID 2865489.
References
1. List of Fellows of the American Mathematical Society, retrieved 2013-01-19.
External links
• Faculty page at the University of Maryland, College Park
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William B. Gragg
William B. Gragg (1936–2016) ended his career as an Emeritus Professor in the Department of Applied Mathematics at the Naval Postgraduate School. He has made fundamental contributions in numerical analysis, particularly the areas of numerical linear algebra and numerical methods for ordinary differential equations.
William B. Gragg
Born(1936-11-02)2 November 1936
Bakersfield, California
Died25 December 2016(2016-12-25) (aged 80)
Monterey, California
NationalityAmerican
Alma materUCLA
Known forGragg Extrapolation
Scientific career
FieldsMathematics
InstitutionsNaval Postgraduate School
ThesisRepeated extrapolation to the limit in the numerical solution of ordinary differential equations (1964)
Doctoral advisorPeter Henrici
He received his PhD at UCLA in 1964 under the direction of Peter Henrici. His dissertation work resulted in the Gragg Extrapolation method[1] for the numerical solution of ordinary differential equations (sometimes also called the Bulirsch–Stoer algorithm).
Gragg is also well known for his work on the QR algorithm for unitary Hessenberg matrices, on updating the QR factorization,[2] superfast solution of Toeplitz systems,[3] parallel algorithms for solving eigenvalue problems,[4][5] as well as his exposition on the Pade table and its relation to a large number of algorithms in numerical analysis.[6]
References
1. http://epubs.siam.org/doi/pdf/10.1137/0702030 On extrapolation algorithms for ordinary initial value problems, WB Gragg SINUM, vol. 2, no. 3, 1965.
2. Daniel, J. W.; Gragg, W. B.; Kaufman, L.; Stewart, G. W. (1976). "Reorthogonalization and stable algorithms for updating the Gram-Schmidt factorization". Math. Comp. 30 (136): 772–795. doi:10.1090/S0025-5718-1976-0431641-8.
3. Ammar, Gregory S.; Gragg, William B. (1988). "Superfast Solution of Real Positive Definite Toeplitz Systems". SIAM Journal on Matrix Analysis and Applications. 9: 61–76. CiteSeerX 10.1.1.64.8032. doi:10.1137/0609005. hdl:10945/30445.
4. Article title A Parallel Divide and Conquer Algorithm for the Generalized Real Symmetric Definite Tridiagonal Eigenproblem, C.F. Borges and W.B.Gragg, 1992
5. Gragg, W. B.; Reichel, L. (1990). "A divide and conquer method for unitary and orthogonal eigenproblems". Numerische Mathematik. 57: 695–718. doi:10.1007/BF01386438. hdl:10945/29823. S2CID 53684596.
6. Gragg, W. B. (1972). "The Padé Table and Its Relation to Certain Algorithms of Numerical Analysis". SIAM Review. 14: 1–62. doi:10.1137/1014001.
External links
• William B. Gragg at the Mathematics Genealogy Project
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William H. Bossert
William H. Bossert (born 1937) is an American mathematician. He is the David B. Arnold, Jr. Professor of Science, Emeritus at Harvard University. He was the housemaster of Lowell House for 23 years.[1][2] He received his PhD from Harvard in 1963.[3]
Publications
With Edward O. Wilson A primer of population biology (1971)[4]
References
1. Flaherty, Julie (April 15, 1998). "Harvard Breaks a Housemaster Mold" – via NYTimes.com.
2. "William H. Bossert | Harvard John A. Paulson School of Engineering and Applied Sciences". www.seas.harvard.edu.
3. "William Bossert - The Mathematics Genealogy Project". www.genealogy.math.ndsu.nodak.edu. Retrieved 2021-12-31.
4. "A Primer of Population Biology". Oxford University Press. June 1, 1971 – via Oxford University Press.
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William Jack (mathematician)
William Jack FRSE (29 May 1834 – 20 March 1924) was a Scottish mathematician and journalist. He was Editor of the Glasgow Herald newspaper from 1870 to 1876, and Professor of Mathematics at the University of Glasgow from 1879 until 1909.[1]
Life
He was born on 29 May 1834 in Stewarton in Ayrshire the son of Robert Jack of Irvine. He was educated at Irvine Academy, going on to study mathematics at the University of Glasgow, graduating with an MA in 1853. He then continued his studies at the University of Cambridge, graduating with a second MA in 1859.
From 1860 to 1866 he was HM Inspector of Schools for Scotland. In 1866 he accepted the post of Professor of Natural Philosophy (Physics) at Owens College in Manchester and held this position until 1870 when he moved to Glasgow as Editor of the Glasgow Herald newspaper. He left in 1876 to run Macmillan & Co, a London publisher,[2] and in 1879 joined the staff of Glasgow University as Professor of Mathematics. In 1875, he was elected a Fellow of the Royal Society of Edinburgh. His proposers were, William Thomson, Lord Kelvin, James Thomson Bottomley, Allen Thomson and Peter Guthrie Tait.[3]
In 1875 the University of Glasgow awarded him an honorary LLD and in 1902 the University of Manchester awarded him an honorary doctorate (DSc).
In his final working years he lived on the campus of the University of Glasgow, at 10 The College.[4]
He died on 20 March 1924.[5]
Family
He was married to Agnes Jane Nichol (1837–1901), daughter of John Pringle Nichol and sister of Professor John Nichol. Their children included sons William Tullis Jack (b.1862), William Robert Jack (b.1866), Adolphus Alfred Jack (b.1868), and a daughter Agnes Elizabeth Jack (b.1871).[6]
References
1. "University of Glasgow :: Story :: Biography of William Jack". Universitystory.gla.ac.uk. 23 March 1924. Retrieved 12 March 2017.
2. "William Jack". Glasgowwestaddress.co.uk. Retrieved 12 March 2017.
3. Biographical Index of Former Fellows of the Royal Society of Edinburgh 1783–2002 (PDF). The Royal Society of Edinburgh. July 2006. ISBN 0-902-198-84-X.
4. Glasgow Post Office directory 1905–6
5. "Professor Jack. Death of distinguished mathematician. Thirty years in Glasgow University". The Glasgow Herald. 21 March 1924. p. 5. Retrieved 6 January 2016.
6. "William Jack (1834 - 1924) - Genealogy". Geni.com. 20 March 1924. Retrieved 12 March 2017.
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William James Macdonald
William James Macdonald FRSE was born on 14 December 1851 in Scotland.[1] He is known for being a pioneer of the introduction of modern geometry to the mathematical curriculum in schools and for being one of the founding members of the Edinburgh Mathematical Society.[1][2][3]
William James Macdonald
Born(1851-12-14)December 14, 1851
DiedDecember 29, 1941(1941-12-29) (aged 90)
Alma materUniversity of St Andrews
Scientific career
InstitutionsMadras College, Merchiston Castle School, Daniel Stewart’s College
Biography
Macdonald was born in Huntly, Aberdeenshire, but moved to the coastal town of St Andrews when he was young.[1] There he got an education in Madras College,[1][4][5] and became dux of the college in 1868.[1]
After completing his school education he entered the University of St Andrews, where he studied a variety of subjects including mathematics, English literature, Latin, Greek, chemistry, and philosophy.[1][6] While there Macdonald won many prizes, including the Miller prize given to the student who did the best work in 1870, 1871, and 1872, the Gray prize in 1872 for an essay on spectrum analysis, and the Arnott prize, also in 1872.[1][6]
After graduating, Macdonald was appointed assistant to the Mathematics Department in Madras College, but only taught there for a short time before accepting a role as Mathematics Master at Merchiston Castle school in Edinburgh.[1][6] He soon after accepted a role at Daniel Stewart’s College where he spent the rest of his career.[1][6] Between 1898 and 1899, he was the president of the Scottish Secondary Teachers' Association.[1]
He died in Edinburgh on 29 December 1941.[1][6]
Accomplishments
Macdonald was a pioneer in the introduction of modern geometry to the mathematical curriculum. He wrote Higher Geometry: Containing an Introduction to Modern Geometry and Elementary Geometrical Conics,[7] a text which was widely used in schools and colleges to teach geometry.[1]
He was a founding member of the Edinburgh Mathematical Society,[1][3] and was honoured by the society when he was elected as president for 1887-88 session.[8]
On the 1st of February 1886 he accepted a fellowship to the Royal Society of Edinburgh[9] after being proposed by William Swan, John Sturgeon Mackay, George Chrystal, and Sir Thomas Muir.[1]
In June 1914, he was offered the degree of LL.D by the Senatus Academicus of the University of St Andrews, but he respectfully declined the honour.[1][6]
References
1. "William James Macdonald - Biography". Maths History. Retrieved 2022-06-23.
2. Macdonald, W. J. (1888). "Office-Bearers". Proceedings of the Edinburgh Mathematical Society. 7: 1. doi:10.1017/S0013091500030200. ISSN 0013-0915. S2CID 251062914.
3. "EMS Founder Members". Maths History. Retrieved 2022-06-24.
4. "Biographies". www.madrascollegearchive.org.uk. Retrieved 2022-06-24.
5. "MCM Christmas 1912". www.madrascollegearchive.org.uk. Retrieved 2022-06-24.
6. "Biographical Register 1747-1897". arts.st-andrews.ac.uk. Retrieved 2022-06-24.
7. Macdonald, William James (1897). Higher Geometry: Containing an Introduction to Modern Geometry and Elementary Geometrical Conics (2nd ed.). Edinburgh: James Thin.
8. Rankin, R. A. (1983). "The first hundred years (1883–1983)". Proceedings of the Edinburgh Mathematical Society. 26 (2): 147. doi:10.1017/S0013091500016849. ISSN 0013-0915. S2CID 178104090.
9. Royal Society of Edinburgh (1886). Proceedings of the Royal Society of Edinburgh (Volume 13 ed.). Edinburgh. p. 596.{{cite book}}: CS1 maint: location missing publisher (link)
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William John Greenstreet
William John Greenstreet (1861–1930) was an English mathematician who was editor of The Mathematical Gazette for more than thirty years.[1]
William John Greenstreet
Born(1861-08-18)18 August 1861
Shorncliffe Army Camp, Folkestone, Kent
Died28 June 1930(1930-06-28) (aged 68)
Burghfield Common, West Berkshire
Alma materSt John's College, Cambridge
SpouseEthel de Medina Spender
Parent(s)Thomas Greenstreet and Catherine S Greenstreet
Scientific career
FieldsMathematics
InstitutionsMarling School
Life and work
Greenstreet was son of a Royal Artillery's Sergeant. He was educated at Southwark and he entered St John's College, Cambridge in 1879, graduating there in 1883. Then he was mathematics professor in different schools in Framlingham, East Riding and Cardiff before he became Head Master at Marling School in 1891. In 1910 he retired to Burghfield Common with the intention of devoting to literary work.[2]
Greenstreet was founding member of the Mathematical Association and he started the Association's Library given a large collection of books.[3]
References
1. Macaulay et al. 1930, p. 181.
2. Macaulay et al. 1930, p. 182.
3. Goodstein 1974, p. 100.
Bibliography
• Goodstein, R.L. (1974). "The Mathematical Association Library at the University of Leicester". The British Journal for the History of Science. 7 (1): 100–103. doi:10.1017/S0007087400013066. ISSN 0007-0874. S2CID 144469175.
• Macaulay, I.F.S.; Neville, E.H.; Pendlebury, C.; Spender, J.A.; Anderson, W.C.F. (1930). "Obituary: W. J. Greenstreet". The Mathematical Gazette. 15 (209): 181–186. doi:10.1017/S0025557200137258. ISSN 0025-5572. JSTOR 3605787. S2CID 185174470.
External links
• O'Connor, John J.; Robertson, Edmund F., "William John Greenstreet", MacTutor History of Mathematics Archive, University of St Andrews
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William J. LeVeque
William Judson LeVeque (August 9, 1923 – December 1, 2007) was an American mathematician and administrator who worked primarily in number theory. He was executive director of the American Mathematical Society during the 1970s and 1980s when that organization was growing rapidly and greatly increasing its use of computers in academic publishing.
William J. LeVeque
Born(1923-08-09)August 9, 1923
Boulder, Colorado
DiedDecember 1, 2007(2007-12-01) (aged 84)
Alma materCornell University
Known forNumber theory
AwardsSLA PMA Division Award
Scientific career
FieldsMathematics
InstitutionsHarvard University
University of Michigan
Claremont Graduate School
American Mathematical Society
Doctoral advisorBurton W. Jones,
Mark Kac
Doctoral studentsUnderwood Dudley
Life and education
LeVeque was born August 9, 1923, in Boulder, Colorado. He received his Bachelor of Arts degree from the University of Colorado in 1944, and a master's degree in 1945 and a Ph.D. in 1947 from Cornell University.[1]
He was an instructor at Harvard University from 1947 to 1949, then started at University of Michigan as an instructor and rose to professor. In 1970 he moved to the Claremont Graduate School. In 1977 he became executive director of the American Mathematical Society and remained there until his retirement in 1988.[1]
After retirement LeVeque and his wife, Ann, took up sailing and lived on their sailboat for three years while they traveled from Narragansett Bay to Grenada. They then moved to Bainbridge Island, Washington, where he kept active in volunteer activities for the rest of his life.[2] He died December 1, 2007.[3] His son Randall J. LeVeque is a well known applied mathematician.
Work
LeVeque's research interest was number theory, specifically transcendental numbers, uniform distribution, and Diophantine approximation.[3]
He wrote a number of number theory textbooks and reference books, which influenced the development of number theory in the United States. A long-term project was to update Leonard Eugene Dickson's History of the Theory of Numbers. This project eventually produced a six-volume collection titled Reviews in Number Theory.[3] The Special Libraries Association's Physics-Astronomy-Mathematics Division awarded LeVeque its Division Award in 1978 for his contributions to the bibliography of mathematics.[4]
The American Mathematical Society grew rapidly during LeVeque's time as executive director (1977–1988). Revenues tripled from $5 million in 1977 to $14.9 million in 1988. The Society began computerizing at a rapid rate during this period, with Mathematical Reviews first becoming available electronically through existing academic dial-up services; this system later evolved into MathSciNet. Most of the headquarters staff received computer terminals for use in the new operations.[3]
Selected publications
• LeVeque, William J. (2002) [1956]. Topics in Number Theory, Volumes I and II. New York: Dover Publications. ISBN 978-0-486-42539-9.
• Leveque, William J. (1990) [1962]. Elementary Theory of Numbers. New York: Dover Publications. ISBN 978-0-486-66348-7.
• LeVeque, William J., ed. (1969). Studies in number theory. Mathematical Association of America. OCLC 17369.
• LeVeque, William J., ed. (1974). Reviews in number theory, as printed in Mathematical reviews, 1940 through 1972, volumes 1-44 inclusive. American Mathematical Society. OCLC 948152. (6 volumes)
• Leveque, William J. (1996) [1977]. Fundamentals of Number Theory. New York: Dover Publications. ISBN 978-0-486-68906-7.
Further reading
• LeVeque, William J. (July–August 1988). "The AMS—Then, Now, and Soon" (PDF). Notices of the American Mathematical Society. 35 (6): 785–789. Retrieved 2009-01-01. A retrospective by LeVeque of his work at the American Mathematical Society.
Notes
1. "William J. LeVeque Retires as Executive Director" (PDF). Notices of the American Mathematical Society. 35 (6): 783–784. July–August 1988. Retrieved 2009-01-01.
2. "Current Obituaries: William J. LeVeque". Cook Family Funeral Home. Retrieved 2 January 2009.
3. Maxwell, James W. (November 2008). "William J. LeVeque (1923–2007)" (PDF). Notices of the American Mathematical Society. 55 (10): 1261–1262. Retrieved 2009-01-01.
4. "PAM Division Award Winners List". Special Libraries Association. 2007-12-18. Retrieved 2 January 2009.
External links
• William J. LeVeque at the Mathematics Genealogy Project
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William Lloyd Garrison Williams
William Lloyd Garrison Williams (3 October 1888 - 31 January 1976) was an American-Canadian Quaker and mathematician, known for the founding of the Canadian Mathematical Society and overseeing Elbert Frank Cox's doctorate in mathematics.
William Lloyd Garrison Williams
Born(1888-10-03)October 3, 1888
Friendship, Kansas
DiedJanuary 31, 1976(1976-01-31) (aged 87)
NationalityAmerican-Canadian
Other namesLloyd Williams
Education
• Haverford College B.A.
• University of Oxford (1913, M.A.)
• University of Chicago (1925, Ph.D)
OccupationMathematician
Known for
• Canadian Mathematical Congress
• Elbert Frank Cox Ph.D
PartnerAnne Skyes (1917)
Children2
Personal life
Williams was born in Friendship, Kansas to Amanda Dunreath Truex and Nathan Williams.[1] After the death of his mother, he was taken in by his father's first wife's family, the Tominsons, in Indiana.
Williams married Anne Skyes, a pianist from Cincinnati, in 1917.[2] Together, they had two daughters, Hester and mathematician Christine Ayoub.[3]
Academic career
After attending the Quaker Academy, Williams taught in North Dakota.[2] He then studied Classics at Haverford.[1] Subsequently, he was awarded a Rhodes Scholarship, where he studied mathematics at Oxford from 1910 to 1913.[4] Once obtaining an M.A. he took on a faculty position at the Miami University.[2] During the summers, he did Ph.D. work at the University of Chicago.[4] He wrote his Ph.D. on Fundamental Systems of Formal Modular Seminvariants of the Binary Cubic, published in 1920. He then taught briefly at Gettysburg College and William and Mary, before relocating to Cornell.[1]
In 1924, Williams moved again, this time to teach at McGill, to develop their graduate program. He remained there until his retirement in 1954.[5] He was awarded honorary degrees by the University of Montreal, University of Manitoba, Dalhousie University, and Mount Allison University.[1]
Elbert Frank Cox
Williams supervised the Ph.D. of Elbert Frank Cox, the first African American to get a mathematics doctorate, during his time at Cornell. When Williams took up his position at McGill, Cox followed him. Cox was awarded his degree in 1925, utilizing the Erastus Brooks Fellowship.[1]
Unsatisfied with the limited amount of recognition Cox received, Williams petitioned international universities to recognise his student, eventually convincing the Sendai University, Japan.[6]
Canadian Mathematical Congress
Williams founded the Canadian Mathematical Society, formerly the Canadian Mathematical Congress, in 1945. Williams dreamed of a forum to bring Canadian mathematicians together, regardless of race or creed, especially after attending the Toronto Congress in 1924 where all mathematicians of the Central Powers had been excluded.[3] He worked as the treasurer from its founding in 1945 to 1963.[1] He was particularly successful in achieving support from insurance companies, although he was noted by many to be quite zealous and friendly and found support from many.[7]
Currently the Jeffrey-Williams Prize is named in his honour, and awarded to mathematicians who have made outstanding contributions to mathematical research by the Canadian Mathematical Society.[5]
Religious beliefs
A devoted Quaker, Williams was instrumental in the Montreal Quaker community during the 1930s and beyond.[8] These beliefs influenced his views of racial equality and hard work. Williams pioneered Quaker causes. He was a member of the Board of the Canadian Friends Service Committee and chairman of the committee from 1959 until 1963.[3] He founded the Montreal “Save the Children” Fund in 1944.[9] He additionally helped with the purchasing of a building for Montreal friends to use as a meeting house, although it is no longer in use.[2] He is noted for being an official of the Quakers of Montreal.[3]
References
1. O'Connor, J J; Robertson, E F (November 2006). "MacTutor: William Lloyd Garrison Williams". MacTutor. School of Mathematics and Statistics, University of St Andrews, Scotland.{{cite web}}: CS1 maint: url-status (link)
2. Ayoub, Christine (2014). Memories of the Quaker past : stories of thirty-seven senior Quakers. Xlibris Corporation. [State College, PA]. pp. 5–20. ISBN 978-1-4691-6254-6. OCLC 870842742.{{cite book}}: CS1 maint: location missing publisher (link)
3. "My father, Lloyd Williams". Quakers In Canada. Retrieved 2022-06-23.
4. Donaldson, James A.; Fleming, Richard J. (2000-02-01). "Elbert F. Cox: An Early Pioneer". The American Mathematical Monthly. 107 (2): 105–128. doi:10.1080/00029890.2000.12005169. ISSN 0002-9890. S2CID 833917.
5. "Jeffery-Williams Prize". CMS-SMC. Retrieved 2022-06-23.
6. "Elbert Frank Cox, first Black to earn a Ph.D. in mathematics". www.math.buffalo.edu. Retrieved 2022-06-23.
7. MATHEMATICAL CULTURES : the london meetings 2012 2014. [Place of publication not identified]: BIRKHAUSER. 2018. ISBN 978-3-319-80379-1. OCLC 1040201215.
8. Lawson, David (15 December 1977). "A View of Montreal Meeting" (PDF). Friend's Journal. Philadelphia, PA: Friends Publishing Company. 23 (21): 660.
9. Zavitz-Bond, Jane (1997). "Book Review: My Father, Llyod Williams" (PDF). Canadian Quaker History Journal. Toronto, Ontario: Canadian Friends Historical Association (62): 31. ISSN 0319-3934.
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William Lowell Putnam Mathematical Competition
The William Lowell Putnam Mathematical Competition, often abbreviated to Putnam Competition, is an annual mathematics competition for undergraduate college students enrolled at institutions of higher learning in the United States and Canada (regardless of the students' nationalities). It awards a scholarship and cash prizes ranging from $250 to $2,500 for the top students and $5,000 to $25,000 for the top schools, plus one of the top five individual scorers (designated as Putnam Fellows) is awarded a scholarship of up to $12,000 plus tuition at Harvard University (Putnam Fellow Prize Fellowship),[1] the top 100 individual scorers have their names mentioned in the American Mathematical Monthly (alphabetically ordered within rank), and the names and addresses of the top 500 contestants are mailed to all participating institutions. It is widely considered to be the most prestigious university-level mathematics competition in the world, and its difficulty is such that the median score is often zero (out of 120) despite being attempted by students specializing in mathematics.[2][3]
The competition was founded in 1927 by Elizabeth Lowell Putnam in memory of her husband William Lowell Putnam, who was an advocate of intercollegiate intellectual competition. The competition has been offered annually since 1938 and is administered by the Mathematical Association of America.[4]
Competition layout
The Putnam competition takes place on the first Saturday in December, and consists of two three-hour sittings separated by a lunch break. The competition is supervised by faculty members at the participating schools. Each one consists of twelve challenging problems. The problems cover a range of advanced material in undergraduate mathematics, including concepts from group theory, set theory, graph theory, lattice theory, and number theory. [5]
Each of the twelve questions is worth 10 points, and the most frequent scores above zero are 10 points for a complete solution, 9 points for a nearly complete solution, and 1 point for the beginnings of a solution. In earlier years, the twelve questions were worth one point each, with no partial credit given. The competition is considered to be very difficult: it is typically attempted by students specializing in mathematics, but the median score is usually zero or one point out of 120 possible, and there have been only five perfect scores as of 2021.[6][7] In 2003, of the 3,615 students competing, 1,024 (28%) scored 10 or more points, and 42 points was sufficient to make the top percentile.
At a participating college, any student who wishes to take part in the competition may (limited by the number of spots a school receives); but until 2019 the school's official team consisted of three individuals whom it designated in advance. Until 2019, a team's score was the sum of the ranks of its three team members, with the lowest cumulative rank winning. It was entirely possible, even commonplace at some institutions, for the eventual results to show that the "wrong" team was picked—i.e. that some students not on the official team outscored an official team member. For example, in 2010, MIT had two of the top five scorers in the competition and seven of the top 24, while Caltech had just one student in the top five and only four in the top 24; yet Caltech took first place among teams while MIT took second.[8] In 2019 the rules of the competition changed, with a school's team consisting of its top three scorers, and team ranks determined by comparing the sums of the scores of the team members. [5]
Awards
The top five teams win $25,000, $20,000, $15,000, $10,000, and $5,000, in that order, with team members receiving $1,000, $800, $600, $400, and $200, respectively.
The top five individual scorers are named Putnam Fellows and awarded $2,500. The school with the first-place team receives an award of $25,000. Each first-place team member, as well as the winner of the Elizabeth Lowell Putnam Prize, receives $1,000. Sixth through 15th place individuals receive $1,000 and the next ten receive $250. The names of the top 100 students are published in the American Mathematical Monthly, and competition results are published in early April of the year following the competition.[9]
Many Putnam Fellows have gone on to become distinguished researchers in mathematics and other fields, including three Fields Medalists—John Milnor (also an Abel Prize laureate), David Mumford, and Daniel Quillen—and two Nobel laureates in physics—Richard Feynman and Kenneth Wilson.
Winners
Top-scoring teams
Year First Second Third Fourth Fifth
1938 Toronto UC Berkeley Columbia
1939 Brooklyn College MIT Mississippi Woman's
1940 Toronto Yale Columbia
1941 Brooklyn College UPenn MIT
1942 Toronto Yale MIT City College of NY
1946 Toronto MIT Brooklyn College Carnegie Tech
1947 Harvard Yale Columbia UPenn
1948 Brooklyn College Toronto Harvard City College of NY and McGill
1949 Harvard Toronto Carnegie Tech City College of NY
1950 Caltech Harvard NYU Toronto
1951 Cornell Harvard Cooper Union City College of NY
1952 Queen's Brooklyn Polytech Harvard MIT
1953 Harvard City College of NY Cornell UC Berkeley
1954 Cornell Harvard MIT Toronto
1955 Harvard Toronto Yale Kenyon
1956 Harvard Columbia Queen's MIT
1957 Harvard Columbia Cornell Caltech
1958 (Spring) Brooklyn Polytech Harvard Toronto Manitoba
1958 (Fall) Harvard Toronto Caltech Cornell
1959 Brooklyn Polytech Caltech Toronto Harvard Case Tech
1960 UC Berkeley Harvard MIT Michigan State Cornell
1961 Michigan State MIT Caltech Harvard Dartmouth
1962 Caltech Dartmouth Harvard Queen's UCLA
1963 Michigan State Brooklyn College UPenn Caltech MIT
1964 Caltech MIT Harvard Case Tech UC Berkeley
1965 Harvard MIT Toronto Princeton Caltech
1966 Harvard MIT Chicago Michigan Princeton
1967 Michigan State Caltech Harvard MIT Michigan
1968 MIT Waterloo UCLA Michigan State Kansas
1969 MIT Rice Chicago Harvard Yale
1970 Chicago MIT Toronto Illinois Tech Caltech
1971 Caltech Chicago Harvard UC Davis MIT
1972 Caltech Oberlin Harvard Swarthmore MIT
1973 Caltech British Columbia Chicago Harvard Princeton
1974 Waterloo Chicago Caltech MIT British Columbia
1975 Caltech Chicago MIT Princeton Harvard
1976 Caltech Washington U in StL Princeton Case Western Reserve and MIT
1977 Washington U in StL UC Davis Caltech Princeton MIT
1978 Case Western Reserve Washington U in StL Waterloo Harvard Caltech
1979 MIT Caltech Princeton Stanford Waterloo
1980 Washington U in StL Harvard Maryland Chicago UC Berkeley
1981 Washington U in StL Princeton Harvard Stanford Maryland
1982 Harvard Waterloo Caltech Yale Princeton
1983 Caltech Washington U in StL Waterloo Princeton Chicago
1984 UC Davis and Washington U in StL Harvard Princeton Yale
1985 Harvard Princeton UC Berkeley Rice Waterloo
1986 Harvard Washington U in StL UC Berkeley Yale MIT
1987 Harvard Princeton Carnegie Mellon UC Berkeley MIT
1988 Harvard Princeton Rice Waterloo Caltech
1989 Harvard Princeton Waterloo Yale Rice
1990 Harvard Duke Waterloo Yale Washington U in StL
1991 Harvard Waterloo Harvey Mudd Stanford Yale
1992 Harvard Toronto Waterloo Princeton Cornell
1993 Duke Harvard Miami University MIT Michigan
1994 Harvard Cornell MIT Princeton Waterloo
1995 Harvard Cornell MIT Toronto Princeton
1996 Duke Princeton Harvard Washington U in StL Caltech
1997 Harvard Duke Princeton MIT Washington U in StL
1998 Harvard MIT Princeton Caltech Waterloo
1999 Waterloo Harvard Duke Michigan Chicago
2000 Duke MIT Harvard Caltech Toronto
2001 Harvard MIT Duke UC Berkeley Stanford
2002 Harvard Princeton Duke UC Berkeley Stanford
2003 MIT Harvard Duke Caltech Harvey Mudd
2004 MIT Princeton Duke Waterloo Caltech
2005 Harvard Princeton Duke MIT Waterloo
2006 Princeton Harvard MIT Toronto Chicago
2007 Harvard Princeton MIT Stanford Duke
2008 Harvard Princeton MIT Stanford Caltech
2009 MIT Harvard Caltech Stanford Princeton
2010 Caltech MIT Harvard UC Berkeley Waterloo
2011 Harvard Carnegie Mellon Caltech Stanford MIT
2012 Harvard MIT UCLA Stony Brook Carnegie Mellon
2013 MIT Carnegie Mellon Stanford Harvard Caltech
2014 MIT Harvard RPI Waterloo Carnegie Mellon
2015 MIT Carnegie Mellon Princeton Stanford Harvard
2016 Carnegie Mellon Princeton Harvard MIT Stanford
2017 MIT Harvard Princeton Toronto UCLA
2018 Harvard MIT UCLA Columbia Stanford
2019 MIT Harvard Stanford UCLA Waterloo
2021 MIT Princeton Harvard Stanford UCLA
2022 MIT Harvard Stanford Maryland Yale
Teams ranked by historical performance
Below is a table of teams by the number of appearances in the top five and number of titles.
The following table lists Teams finishing in Top Five (as of 2021 competition):
Top Five Team (s)
67 Harvard
52 MIT
33 Caltech
32 Princeton
20 Waterloo
19 Toronto
16 Stanford
13 Yale
12 Duke
11 Chicago, Washington University in St. Louis
10 UC Berkeley, Cornell
9 Carnegie Mellon (including former Carnegie Tech)
7 UCLA
6 Columbia
5 Brooklyn College, City College of New York, Michigan State
4 Case Western Reserve (including former Case Tech), Michigan, Rice
3 Brooklyn Polytech, UC Davis, Queen's, Penn, Maryland
2 British Columbia, Dartmouth, Harvey Mudd
1 Cooper Union, Illinois Tech, Kansas, Kenyon, Manitoba, McGill, Miami University, RPI
NYU, Oberlin, Stony Brook, Swarthmore, William Carey (under former name of Mississippi Woman's)
For a recent analysis, the following table lists teams that finished in the top five since 2000 (as of 2021 competition):
Top Five Team (s)
21 Harvard, MIT
13 Stanford
11 Princeton
8 Caltech
7 Duke
6 Carnegie Mellon
5 UCLA, Waterloo
3 UC Berkeley, Toronto
1 Chicago, Harvey Mudd, Stony Brook, Yale, RPI, Columbia, Maryland
The following table lists Teams with First place finishes (as of 2021 competition):
First Place Team (s)
30 Harvard
12 MIT
10 Caltech
4 Toronto, Washington University in St. Louis
3 Brooklyn College, Duke, Michigan State
2 Brooklyn Polytech, Cornell, Waterloo
1 UC Berkeley, UC Davis, Carnegie Mellon, Case Western Reserve, Chicago, Princeton, Queen's
Putnam Fellows
Since the first competition, the top five (or six, in case of a tie) scorers in the competition have been named Putnam Fellows. Within the top five, Putnam Fellows are not ranked. Students are not allowed to participate in the Putnam Competition more than four times. For example, if a high school senior chooses to officially participate, he/she effectively chooses to forfeit one of his/her years of eligibility in college (see Gabriel Carroll). This makes it even more of a remarkable feat to become a Putnam Fellow four times. In the history of the Competition, only eight students have been Putnam Fellows four times, with twenty-three others winning the award three times. The following table lists these students:
Name School Years
Don Coppersmith MIT 1968 1969 1970 1971
Arthur Rubin Purdue, Caltech 1970 1971 1972 1973
Bjorn M. Poonen Harvard 1985 1986 1987 1988
Ravi D. Vakil Toronto 1988 1989 1990 1991
Gabriel D. Carroll UC Berkeley, Harvard 2000 2001 2002 2003
Reid W. Barton MIT 2001 2002 2003 2004
Daniel Kane MIT 2003 2004 2005 2006
Brian R. Lawrence Caltech 2007 2008 2010 2011
Edward L. Kaplan Carnegie Tech 1939 1940 1941
Andrew M. Gleason Yale 1940 1941 1942
Donald J. Newman City College of NY 1948 1949 1950
James B. Herreshoff IV UC Berkeley 1951 1952 1953
Samuel Jacob Klein City College of NY 1953 1959 1960
Randall L. Dougherty UC Berkeley 1978 1979 1980
Eric D. Carlson Michigan State 1980 1982 1983
David W. Ash Waterloo 1981 1982 1983
Noam D. Elkies Columbia 1982 1983 1984
David J. Grabiner Princeton 1986 1987 1988
David J. Moews Harvard 1986 1987 1988
J. P. Grossman Toronto 1993 1994 1995
Kiran S. Kedlaya Harvard 1993 1994 1995
Lenhard L. Ng Harvard 1993 1994 1995
Ciprian Manolescu Harvard 1997 1998 2000
Aaron C. Pixton Princeton 2004 2005 2007
Yufei Zhao MIT 2006 2008 2009
Arnav Tripathy Harvard 2007 2008 2009
Seok Hyeong Lee Stanford 2008 2010 2011
Evan M. O'Dorney UC Berkeley, Harvard 2011 2012 2013
Zipei Nie MIT 2012 2013 2014
David H. Yang MIT 2013 2014 2015
Yunkun Zhou MIT 2015 2016 2017
Shengtong Zhang MIT 2018 2019 2021
The following table lists all Putnam fellows from 1938 to present, with the years they placed in the top five.[10] Ioana Dumitriu was the first woman to become a Putnam Fellow, in 1996.
Name (School) Year (s)
George W. Mackey (Rice) 1938
Irving Kaplansky (Toronto) 1938
Michael J. Norris (College of St. Thomas) 1938
Robert W. Gibson (Fort Hays Kansas State) 1938
Bernard Sherman (Brooklyn College) 1938, 1939
Abraham Hillman (Brooklyn College) 1939
Richard P. Feynman (MIT) 1939
William Nierenberg (City College of NY) 1939
Edward L. Kaplan (Carnegie Tech) 1939, 1940, 1941
John Cotton Maynard (Toronto) 1940
Robert Maughan Snow (George Washington) 1940
W. J. R. Crosby (Toronto) 1940
Andrew M. Gleason (Yale) 1940, 1941, 1942
Paul C. Rosenbloom (UPenn) 1941
Richard F. Arens (UCLA) 1941
Samuel I. Askovitz (UPenn) 1941
Harold Victor Lyons (Toronto) 1942
Harvey Cohn (City College of NY) 1942
Melvin A. Preston (Toronto) 1942
Warren S. Loud (MIT) 1942
Donald A. Fraser (Toronto) 1946
Eugenio Calabi (MIT) 1946
Felix Browder (MIT) 1946
J. Arthur Greenwood (Harvard) 1946
Maxwell A. Rosenlicht (Columbia) 1946, 1947
Clarence Wilson Hewlett, Jr. (Harvard) 1947
William Turanski (UPenn) 1947
Eoin L. Whitney (Alberta) 1947, 1948
W. Forrest Stinespring (Harvard) 1947, 1949
George F. D. Duff (Toronto) 1948
Harry Gonshor (McGill) 1948
Leonard Geller (Brooklyn College) 1948
Robert L. Mills (Columbia) 1948
Donald J. Newman (City College of NY) 1948, 1949, 1950
Ariel Zemach (Harvard) 1949
David L. Yarmush (Harvard) 1949
John W. Milnor (Princeton) 1949, 1950
John P. Mayberry (Toronto) 1950
Richard J. Semple (Toronto) 1950
Z. Alexander Melzak (British Columbia) 1950
Arthur P. Dempster (Toronto) 1951
Harold Widom (City College of NY) 1951
Herbert C. Kranzer (NYU) 1951
Peter John Redmond (Cooper Union) 1951
James B. Herreshoff IV (UC Berkeley) 1951, 1952, 1953
Eugene R. Rodemich (Washington U in StL) 1952
Gerhard Rayna (Harvard) 1952
Richard G. Swan (Princeton) 1952
Walter Lewis Baily, Jr. (MIT) 1952
Marshall L. Freimer (Harvard) 1953
Norman Bauman (Harvard) 1953
Tai Tsun Wu (Minnesota) 1953
Samuel Jacob Klein (City College of NY) 1953, 1959, 1960
Benjamin Muckenhoupt (Harvard) 1954
James Daniel Bjorken (MIT) 1954
Leonard Evens (Cornell) 1954
William P. Hanf (UC Berkeley) 1954
Kenneth G. Wilson (Harvard) 1954, 1956
Howard C. Rumsey, Jr. (Caltech) 1955
Jack Towber (Brooklyn College) 1955
David B. Mumford (Harvard) 1955, 1956
Trevor Barker (Kenyon) 1955, 1956
Everett C. Dade (Harvard) 1955, 1957
Richard Michael Friedberg (Harvard) 1956
David M. Bloom (Columbia) 1956, 1957
J. Ian Richards (Minnesota) 1957
Richard T. Bumby (MIT) 1957
Rohit J. Parikh (Harvard) 1957
David R. Brillinger (Toronto) Spring 1958
Donald J. C. Bures (Queen's) Spring 1958
Lawrence A. Shepp (Brooklyn Polytech) Spring 1958
Richard M. Dudley (Harvard) Spring 1958
Joseph Lipman (Toronto) Spring 1958, Fall 1958
Alan Gaisford Waterman (San Diego State) Fall 1958
John Rex Forrester Hewett (Toronto) Fall 1958
Robert C. Hartshorne (Harvard) Fall 1958
Alfred W. Hales (Caltech) Fall 1958, 1959
Daniel G. Quillen (Harvard) 1959
Donald Passman (Brooklyn Polytech) 1959
Donald S. Gorman (Harvard) 1959
Martin Isaacs (Brooklyn Polytech) 1959
Stephen L. Adler (Harvard) 1959
Stephen Lichtenbaum (Harvard) 1959
Jon H. Folkman (UC Berkeley) 1960
Louis Jaeckel (UCLA) 1960
Melvin Hochster (Harvard) 1960
William R. Emerson (Caltech) 1960
Barry Wolk (Manitoba) 1961
Elwyn R. Berlekamp (MIT) 1961
Edward Anton Bender (Caltech) 1961, 1962
John Hathaway Lindsey (Caltech) 1961, 1962
William C. Waterhouse (Harvard) 1961, 1962
John William Wood (Harvard) 1962
Robert S. Strichartz (Dartmouth) 1962
Joel H. Spencer (MIT) 1963
Lawrence A. Zalcman (Dartmouth) 1963
Lawrence J. Corwin (Harvard) 1963
Robert E. Greene (Michigan State) 1963
Stephen E. Crick, Jr. (Michigan State) 1963
Barry B. MacKichan (Harvard) 1964
Fred William Roush (UNC Chapel Hill) 1964
Roger E. Howe (Harvard) 1964
Rufus (Robert) Bowen (UC Berkeley) 1964, 1965
Vern Sheridan Poythress (Caltech) 1964
Andreas R. Blass (Detroit) 1965
Barry Simon (Harvard) 1965
Daniel Fendel (Harvard) 1965
Lon M. Rosen (Toronto) 1965
Marshall W. Buck (Harvard) 1966
Robert E. Maas (Santa Clara) 1966
Robert S. Winternitz (MIT) 1966
Theodore C. Chang (MIT) 1966
Richard C. Schroeppel (MIT) 1966, 1967
David R. Haynor (Harvard) 1967
Dennis A. Hejhal (Chicago) 1967
Don B. Zagier (MIT) 1967
Peter L. Montgomery (UC Berkeley) 1967
Dean G. Huffman (Yale) 1968
Gerald S. Gras (MIT) 1968
Neal Koblitz (Harvard) 1968
Gerald A. Edgar (UC Santa Barbara) 1968, 1969
Don Coppersmith (MIT) 1968, 1969, 1970, 1971
Alan R. Beale (Rice) 1969
Steven K. Winkler (MIT) 1969, 1970
Robert A. Oliver (Chicago) 1969, 1970
Jeffrey Lagarias (MIT) 1970
Jockum Aniansson (Yale) 1970
Arthur Rubin (Purdue, Caltech) 1970, 1971, 1972, 1973
Dale Peterson (Yale) 1971
David Shucker (Swarthmore) 1971
Robert Israel (Chicago) 1971
Michael Yoder (Caltech) 1971, 1972
Arthur Rothstein (Reed) 1972
David Vogan (Chicago) 1972
Dean Hickerson (UC Davis) 1972
Ira Gessel (Harvard) 1972
Angelos J. Tsirimokos (Princeton) 1973
Matthew L. Ginsberg (Wesleyan) 1973
Peter G. De Buda (Toronto) 1973
David J. Anick (MIT) 1973, 1975
Grant M. Roberts (Waterloo) 1974
James B. Saxe (Union) 1974
Karl C. Rubin (Princeton) 1974
Philip N. Strenski (Armstrong State) 1974
Thomas G. Goodwillie (Harvard) 1974, 1975
Ernest S. Davis (MIT) 1975
Franklin T. Adams (Chicago) 1975
Christopher L. Henley (Caltech) 1975, 1976
David J. Wright (Cornell) 1976
Nathaniel S. Kuhn (Harvard) 1976
Paul M. Herdig (Case Western Reserve) 1976
Philip I. Harrington (Washington U in StL) 1976
Steven T. Tschantz (UC Berkeley) 1976, 1978
Adam L. Stephanides (Chicago) 1977, 1981
Michael Roberts (MIT) 1977
Paul A. Vojta (Minnesota) 1977
Stephen W. Modzelewski (Harvard) 1977
Russell D. Lyons (Case Western Reserve) 1977, 1978
Mark R. Kleiman (Princeton) 1978
Peter W. Shor (Caltech) 1978
Randall L. Dougherty (UC Berkeley) 1978, 1979, 1980
Charles H. Walter (Princeton) 1979
Mark G. Pleszkoch (Virginia) 1979
Miller Puckette (MIT) 1979
Richard Mifflin (Rice) 1979
Daniel J. Goldstein (Chicago) 1980
Laurence E. Penn (Harvard) 1980
Michael Raship (Harvard) 1980
Eric D. Carlson (Michigan State) 1980, 1982, 1983
Robin A. Pemantle (UC Berkeley) 1981
Scott R. Fluhrer (Case Western Reserve) 1981
David W. Ash (Waterloo) 1981, 1982, 1983
Michael J. Larsen (Harvard) 1981, 1983
Brian R. Hunt (Maryland) 1982
Edward A. Shpiz (Washington U in StL) 1982
Noam D. Elkies (Columbia) 1982, 1983, 1984
Gregg N. Patruno (Princeton) 1983
Benji N. Fisher (Harvard) 1984
Daniel W. Johnson (Rose-Hulman Tech) 1984
Richard A. Stong (Washington U in StL) 1984
Michael Reid (Harvard) 1984, 1987
Everett W. Howe (Caltech) 1985
Keith A. Ramsay (Chicago) 1985
Martin V. Hildebrand (Williams) 1985
Douglas S. Jungreis (Harvard) 1985, 1986
Bjorn M. Poonen (Harvard) 1985, 1986, 1987, 1988
David I. Zuckerman (Harvard) 1986
Waldemar P. Horwat (MIT) 1986
David J. Grabiner (Princeton) 1986, 1987, 1988
David J. Moews (Harvard) 1986, 1987, 1988
Constantin S. Teleman (Harvard) 1987
John S. Tillinghast (UC Davis) 1987
Jeremy A. Kahn (Harvard) 1988
Ravi D. Vakil (Toronto) 1988, 1989, 1990, 1991
Andrew H. Kresch (Yale) 1989
Christos A. Athanasiadis (MIT) 1989
Colin M. Springer (Waterloo) 1989
Sihao Wu (Yale) 1989
William P. Cross (Caltech) 1989
Jordan Lampe (UC Berkeley) 1990
Raymond M. Sidney (Harvard) 1990
Eric K. Wepsic (Harvard) 1990, 1991
Jordan S. Ellenberg (Harvard) 1990, 1992
Joshua B. Fischman (Princeton) 1991
Xi Chen (Missouri–Rolla) 1991
Samuel A. Kutin (Harvard) 1991, 1992
Jeffrey M. Vanderkam (Duke) 1992
Serban M. Nacu (Harvard) 1992
Adam M. Logan (Princeton) 1992, 1993
Craig B. Gentry (Duke) 1993
Wei-Hwa Huang (Caltech) 1993
J. P. Grossman (Toronto) 1993, 1994, 1995
Kiran S. Kedlaya (Harvard) 1993, 1994, 1995
Lenhard L. Ng (Harvard) 1993, 1994, 1995
William R. Mann (Princeton) 1994
Jeremy L. Bem (Cornell) 1994, 1996
Sergey V. Levin (Harvard) 1995
Yevgeniy Dodis (NYU) 1995
Dragos N. Oprea (Harvard) 1996
Ioana Dumitriu (NYU) 1996
Robert D. Kleinberg (Cornell) 1996
Stephen S. Wang (Harvard) 1996
Daniel K. Schepler (Washington U in StL) 1996, 1997
Ovidiu Savin (Pittsburgh) 1997
Patrick K. Corn (Harvard) 1997
Samuel Grushevsky (Harvard) 1997
Mike L. Develin (Harvard) 1997, 1998
Ciprian Manolescu (Harvard) 1997, 1998, 2000
Ari M. Turner (Princeton) 1998
Nathan G. Curtis (Duke) 1998
Kevin D. Lacker (Duke) 1998, 2001
Christopher C. Mihelich (Harvard) 1999
Colin A. Percival (Simon Fraser) 1999
Davesh Maulik (Harvard) 1999
Derek I.E. Kisman (Waterloo) 1999
Sabin Cautis (Waterloo) 1999
Abhinav Kumar (MIT) 1999, 2000
Pavlo Pylyavskyy (MIT) 2000
Alexander B. Schwartz (Harvard) 2000, 2002
Gabriel D. Carroll (UC Berkeley, Harvard) 2000, 2001, 2002, 2003
George Lee, Jr. (Harvard) 2001
Jan K. Siwanowicz (City College of NY) 2001
Reid W. Barton (MIT) 2001, 2002, 2003, 2004
Deniss Cebikins (MIT) 2002
Melanie E. Wood (Duke) 2002
Ralph C. Furmaniak (Waterloo) 2003
Ana Caraiani (Princeton) 2003, 2004
Daniel M. Kane (MIT) 2003, 2004, 2005, 2006
Vladimir V. Barzov (MIT) 2004
Aaron Pixton (Princeton) 2004, 2005, 2007
Oleg I. Golberg (MIT) 2005
Matthew M. Ince (MIT) 2005
Ricky I. Liu (Harvard) 2005
Tiankai Liu (Harvard) 2005, 2006
Hansheng Diao (MIT) 2006
Po-Ru Loh (Caltech) 2006
Yufei Zhao (MIT) 2006, 2008, 2009
Jason C. Bland (Caltech) 2007
Brian R. Lawrence (Caltech) 2007, 2008, 2010, 2011
Qingchun Ren (MIT) 2007, 2009
Xuancheng Shao (MIT) 2007
Arnav Tripathy (Harvard) 2007, 2008, 2009
Seok Hyeong Lee (Stanford) 2008, 2010, 2011
Bohua Zhan (MIT) 2008
William A. Johnson (U of Washington) 2009
Xiaosheng Mu (Yale) 2009, 2011
Yu Deng (MIT) 2010
Colin P. Sandon (MIT) 2010
Alex (Lin) Zhai (Harvard) 2010
Samuel S. Elder (Caltech) 2011
Evan M. O'Dorney (Harvard) 2011, 2012, 2013
Benjamin P. Gunby (MIT) 2012
Eric K. Larson (Harvard) 2012
Mitchell M. Lee (MIT) 2012, 2013
Zipei Nie (MIT) 2012, 2013, 2014
Bobby C. Shen (MIT) 2013, 2014
David H. Yang (MIT) 2013, 2014, 2015
Ravi Jagadeesan (Harvard) 2014
Mark A. Sellke (MIT) 2014
Lingfu Zhang (MIT) 2014
Pakawut Jiradilok (Harvard) 2015
Bumsoo Kim (Princeton) 2015
Gyujin Oh (Stanford) 2015
Daniel Spivak (Waterloo) 2015
Yunkun Zhou (MIT) 2015, 2016, 2017
Joshua D. Brakensiek (Carnegie Mellon) 2016
Dong Ryul Kim (Harvard) 2016, 2018
Thomas E. Swayze (Carnegie Mellon) 2016
Samuel Zbarsky (Carnegie Mellon) 2016
David Stoner (Harvard) 2017, 2018
Ömer Cerrahoğlu (MIT) 2017
Jiyang Gao (MIT) 2017
Junyao Peng (MIT) 2017
Ashwin Sah (MIT) 2017, 2019
Yuan Yao (MIT) 2018, 2019
Shengtong Zhang (MIT) 2018, 2019, 2021
Shyam Narayanan (Harvard) 2018
Kevin Sun (MIT) 2019
Daniel Zhu (MIT) 2019, 2021, 2022
Andrew Gu (MIT) 2021
Michael Ren (MIT) 2021
Edward Wan (MIT) 2021
Mingyang Deng (MIT) 2022
Papon Lapate (MIT) 2022
Brian Liu (MIT) 2022
Luke Robitaille (MIT) 2022
Elizabeth Lowell Putnam Award winners
Since 1992, the Elizabeth Lowell Putnam Award has been available to be awarded to a female participant with a high score, with three awards being made for the first time in 2019.[11] The year(s) in which they were Fellows are in bold. Ioana Dumitriu was the first woman to become a Putnam Fellow, in 1996.
NameSchoolYear (s)
Dana PascoviciDartmouth1992
Ruth A. Britto-PacumioMIT1994
Ioana DumitriuNYU1995, 1996, 1997
Wai Ling YeeWaterloo1999
Melanie E. WoodDuke2001, 2002
Ana CaraianiPrinceton2003, 2004
Alison B. MillerHarvard2005, 2006, 2007
Viktoriya KrakovnaToronto2008
Yinghui WangMIT2011
Fei SongVirginia2011
Xiao WuYale2013
Simona DiaconuPrinceton2016
Ni YanUCLA2017
Danielle WangMIT2015, 2018
Laura PiersonHarvard2019
Qi QiMIT2019
Hanzhi ZhengStanford2019
Dain KimMIT2021
Binwei YanMIT2022
See also
• List of mathematics awards
References
1. "William Lowell Putnam Competition Prizes". math.scu.edu. Santa Clara University.
2. "Putnam Competition | Mathematical Association of America". www.maa.org. Retrieved 18 April 2018.
3. David Arney; George Rosenstein. "The Harvard-United States Military Academy Mathematics Competition of 1933: Genesis of the William Lowell Putnam Mathematical Competition". www.westpoint.edu. Retrieved 18 April 2018.
4. "History of the Putnam Prize". Santa Clara University – Mathematics & Computer Science. Retrieved 7 November 2021.
5. "Putnam Competition | Mathematical Association of America". www.maa.org. Retrieved Mar 27, 2020.
6. "82nd Putnam Competition Announces Top Students in Undergraduate Mathematics". newsroom.maa.org. Retrieved 2022-12-01.
7. "MIT students dominate annual Putnam Mathematical Competition". MIT News | Massachusetts Institute of Technology. Retrieved 2022-12-01.
8. Fujimori, Jessica (3 May 2016). "At Putnam, students rise to the challenge". news.mit.edu. MIT News. Retrieved 18 April 2018.
9. Weisstein, Eric (17 April 2018). "Putnam Mathematical Competition". mathworld.wolfram.com. Wolfram Web Resources. Retrieved 18 April 2018.
10. "List of Previous Putnam Winners" (PDF). Mathematical Association of America. Retrieved December 30, 2020.
11. "Top Students and Teams Announced in Putnam Competition". www.maa.org. Mathematical Association of America.
External links
Wikimedia Commons has media related to Putnam Fellows.
• William Lowell Putnam Mathematical Competition results
• William Lowell Putnam Competition problems, solutions, and results archive
• Archive of Problems 1938–2003
• Searchable data base for information about careers of Putnam Fellows
• A comprehensive history of the Putnam competition An electronic update of Gallian's 2004 paper (PDF)
American mathematics
Organizations
• AMS
• MAA
• SIAM
• AMATYC
• AWM
Institutions
• AIM
• CIMS
• IAS
• ICERM
• IMA
• IPAM
• MBI
• SLMath
• SAMSI
• Geometry Center
Competitions
• MATHCOUNTS
• AMC
• AIME
• USAMO
• MOP
• Putnam Competition
• Integration Bee
| Wikipedia |
William Kantor
William M. Kantor (born September 19, 1944) is an American mathematician who works in finite group theory and finite geometries, particularly in computational aspects of these subjects.
William Kantor
William Kantor in Oberwolfach 2011
Born
William M. Kantor
(1944-09-19) September 19, 1944
Other namesBill Kantor
Scientific career
FieldsMathematics
InstitutionsUniversity of Oregon, University of Illinois at Chicago
Thesis2-transitive symmetric designs (1968)
Doctoral advisorsPeter Dembowski, Richard Hubert Bruck
Notable studentsMark Ronan
Websitehttps://pages.uoregon.edu/kantor/
Education and career
Kantor graduated with a bachelor's degree from Brooklyn College in 1964.[1] He went on to graduate studies at the University of Wisconsin, receiving his PhD in 1968 under the supervision of Peter Dembowski and R. H. Bruck.[2] He then worked at the University of Illinois at Chicago from 1968 to 1971 before moving in 1971 to the University of Oregon, where he remained for the rest of his career.
Kantor's research mostly involves finite groups, often in relation to finite geometries and computation. Algorithms developed by him have found use, for example, in the GAP computer algebra system.[3]
Kantor has written over 170 papers,[4] and has advised 7 PhD students.[2]
Significant publications
Books and monographs
• Kantor, W. M. (1979). Classical groups from a nonclassical viewpoint. Oxford University, Mathematical Institute, Oxford. MR 0578539.
• Kantor, William M.; Seress, Ákos (2001). "Black box classical groups". Memoirs of the American Mathematical Society. 149 (708): 0. CiteSeerX 10.1.1.294.1011. doi:10.1090/memo/0708. ISSN 0065-9266. MR 1804385.
Journal articles
• Calderbank, R.; Kantor, W. M. (1986). "The Geometry of Two-Weight Codes". Bulletin of the London Mathematical Society. 18 (2): 97–122. doi:10.1112/blms/18.2.97. ISSN 0024-6093. MR 0818812.
• Kantor, William M.; Lubotzky, Alexander (1990). "The probability of generating a finite classical group". Geometriae Dedicata. 36 (1). doi:10.1007/BF00181465. ISSN 0046-5755. MR 1065213. S2CID 6771.
• Kantor, William M. (1985). "Homogeneous designs and geometric lattices". Journal of Combinatorial Theory, Series A. 38 (1): 66–74. doi:10.1016/0097-3165(85)90022-6. ISSN 0097-3165. MR 0773556.
Awards and honors
• In 2013, Kantor was named a fellow of the American Mathematical Society as a member of the inaugural class of fellows.[5]
• In 2004, a conference "Finite geometries, groups, and computation" was held in honor of Kantor's 60th birthday.[6]
• In 1998, Kantor gave an invited talk at the International Congress of Mathematicians in Berlin.[7][8]
References
1. William Kantor on LinkedIn
2. William M. Kantor at the Mathematics Genealogy Project
3. "GAP source code". Official GAP system website. Retrieved September 27, 2019.
4. "William Kantor author profile". MathSciNet. American Mathematical Society.
5. "Fellows of the AMS: Inaugural Class" (PDF). Notices of the American Mathematical Society. 60 (5): 631–637. May 2013. Retrieved September 24, 2019.
6. Hulpke, Alexander; Liebler, Robert; Penttila, Tim; Seress, Ákos, eds. (2006). Finite geometries, groups, and computation. Berlin New York: Walter de Gruyter. ISBN 978-3-11-019974-1. MR 2256928. OCLC 181078514.
7. "Invited Lectures in Algebra at the Berlin ICM". Retrieved September 24, 2019.
8. Kantor, William M. (1998). "Simple groups in computational group theory". Doc. Math. (Bielefeld) Extra Vol. ICM Berlin, 1998, vol. II. pp. 77–86.
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William Makeham
William Matthew Makeham (11 September 1826 – 17 November 1891) was an English actuary and mathematician.
Makeham was responsible for proposing the age-independent Makeham term in the Gompertz–Makeham law of mortality that, together with the exponentially age-dependent Gompertz term, was one of the most effective theories to describe human mortality.[1]
Makeham was responsible for two important studies on human mortality:[2]
• Makeham (1860). "On the Law of Mortality and the Construction of Annuity Tables". The Assurance Magazine, and Journal of the Institute of Actuaries. 8 (6): 301–310. doi:10.1017/S204616580000126X.
• Makeham (1874). "On an Application of the Theory of the Composition of Decremental Forces". Journal of the Institute of Actuaries. 18: 317–322.
He had one wife, Hepzibah Reed, and seven children, William, Amy, Elizabeth, Thomas, Frederick, Emily, and George.
References
1. Dale, Andrew I. (1999). A History of Inverse Probability: From Thomas Bayes to Karl Pearson (2nd ed.). Springer. pp. 489 ff. ISBN 9780387988078.
2. Wolfram Mathworld, "Makeham Curve"
| Wikipedia |
William Marrat
William Marrat (1772–1852) was an English printer, publisher and educator, known as a mathematician and antiquarian.
Life
Born at Sibsey, Lincolnshire, on 6 April 1772, Marrat was self-taught through wide reading and study of modern language. While at Boston, Lincolnshire, he for some years worked as a printer and publisher. In 1811–12 he, in conjunction with Pishey Thompson, ran The Enquirer, or Literary, Mathematical, and Philosophical Repository, Boston.[1]
At other times Marrat was a teacher of mathematics, in Lincolnshire and elsewhere. He lived in New York City from 1817 to 1820, and edited there The Scientific Journal (imprint "Perth Amboy, N. J. and New York", 1818, nine numbers). He returned to England, and at Liverpool, where he settled in 1821.[1] George Boole taught at his school, in 1833.[2]
From 1833 to 1836 Marrat was mathematical tutor in a school at Exeter, but on the death of his wife he returned to Liverpool. He died suddenly there on 26 March 1852, and was buried at the necropolis near that city.[1]
Works
Marrat was for fifty years a contributor to mathematical serials, including The Ladies' Diary and The Gentlemen's Diary, The Receptacle, The Student, and the Leeds Correspondent. His first book was An Introduction to the Theory and Practice of Mechanics, Boston, 1810, pp. 468. During 1814–16 he wrote The History of Lincolnshire, which came out in parts, and after three volumes had been published, it was stopped: Marrat alleged this was a consequence of Sir Joseph Banks's refusal to allow access to his papers. In 1816 his Historical Description of Stamford was published at Lincoln.[1]
An anonymous Geometrical System of Conic Sections, Cambridge, 1822, was ascribed to Marrat in the catalogue of the Liverpool Free Library. He compiled Lunar Tables, Liverpool, 1823, and wrote The Elements of Mechanical Philosophy, 1825. At this period he compiled the Liverpool Tide Table, and was a contributor to Blackwood's Magazine.[1]
Family
His son Frederick Price Marrat was known as a conchologist.[1]
Notes
1. Lee, Sidney, ed. (1893). "Marrat, William" . Dictionary of National Biography. Vol. 36. London: Smith, Elder & Co.
2. Des MacHale (1985). George Boole: his life and work. Boole Press. p. 19. ISBN 978-0-906783-05-4.
Attribution
• This article incorporates text from a publication now in the public domain: Lee, Sidney, ed. (1893). "Marrat, William". Dictionary of National Biography. Vol. 36. London: Smith, Elder & Co.
| Wikipedia |
W. T. Martin
William Ted Martin (June 4, 1911 – May 30, 2004) was an American mathematician, who worked on mathematical analysis, several complex variables, and probability theory. He is known for the Cameron–Martin theorem and for his 1948 book Several complex variables, co-authored with Salomon Bochner.
W. T. Martin
Born(1911-06-04)June 4, 1911
Arkansas
DiedMay 30, 2004(2004-05-30) (aged 92)
NationalityAmerican
Alma materUniversity of Illinois
Scientific career
FieldsMathematics
Doctoral advisorRobert Carmichael
Biography
He was born on June 4, 1911, in Arkansas.
W. T. Martin received his B.A. in mathematics from the University of Arkansas in 1930. He did graduate work at the University of Illinois at Urbana–Champaign, where he received his M.A. in 1931 and his Ph.D. in 1934 under the direction of Robert Carmichael.[1] He studied under a National Research Council postdoctoral fellowship at the Institute for Advanced Study in Princeton from 1934 to 1936.[2] In 1936 Martin became an instructor at MIT and in 1938 a faculty member there. He collaborated with several fellow MIT faculty members, notably Norbert Wiener, R. H. Cameron, Stefan Bergman, and Salomon Bochner. During the 1940s Martin and R. H. Cameron wrote a series of papers extending Norbert Wiener's early work on mathematical models of Brownian motion.[3] During the 1950s W. T. Martin wrote with Salomon Bochner a series of papers that proved basic results in the theory of several complex variables.
Martin was the department head for the MIT mathematics department from 1947 to 1968. During this time he oversaw the hiring of 24 faculty members in the mathematics department. He initiated MIT's C. L. E. Moore Instructorship Program in 1949.[4] He spent his entire career at MIT, except for the years from 1943 to 1946, when he left MIT to become the head of the mathematics department of Syracuse University[5] and, in the academic year 1951–1952, when he was on sabbatical at the Institute for Advanced Study.[2] Martin did important editorial work and co-authored three influential books: Several complex variables (1948), Elementary differential equations (1956), and Differential space, quantum space, and prediction (1966).[5] Beginning in 1961, Martin involved himself in developing math curricula for English-speaking African nations, serving as chair of the Steering Committee of the Education Development Center's African Mathematics Program and visited Africa regularly from 1961 to 1975.[6]
He retired to Block Island and died on May 30, 2004.[5]
Selected publications
• with Norbert Wiener: Wiener, N.; Martin, W. T. (1937). "Taylor's series of entire functions of smooth growth". Duke Math. J. 3 (2): 213–223. doi:10.1215/s0012-7094-37-00314-4. MR 1545980.
• with Norbert Wiener: "Taylor's series of smooth growth in the unit circle". Duke Math. J. 4 (2): 384–392. 1938. doi:10.1215/s0012-7094-38-00430-2. MR 1546059.
• with Stefan Bergman: Bergman, Stefan; Martin, W. T. (1940). "A modified moment problem in two variables". Duke Math. J. 6 (2): 389–407. doi:10.1215/s0012-7094-40-00630-5. MR 0001993.
• Martin, W. T. (1944). "Mappings by means of systems of analytic functions of several complex variables". Bull. Amer. Math. Soc. 50 (1): 5–19. doi:10.1090/s0002-9904-1944-08043-9. MR 0009641.
• with R. H. Cameron: Cameron, R. H.; Martin, W. T. (1944). "Transformations of Wiener integrals under translations". The Annals of Mathematics. 45 (2): 386–396. doi:10.2307/1969276. JSTOR 1969276. (2nd most cited of all Cameron and Martin's papers)
• with R. H. Cameron: Cameron, R. H.; Martin, W. T. (1947). "The orthogonal development of non-linear functionals in series of Fourier–Hermite functionals". The Annals of Mathematics. 48 (2): 385–392. doi:10.2307/1969178. JSTOR 1969178. (most cited of all Cameron and Martin's papers)
• with Salomon Bochner: Several complex variables. Princeton, N. J.: Princeton University Press. 1948. (216 pages)
• with Eric Reissner: Elementary differential equations. Cambridge, Massachusetts: Addison-Wesley. 1956, 260 pages{{cite book}}: CS1 maint: postscript (link); 2nd edn. Reading, Mass.: Addison-Wesley. 1961, 331 pages{{cite book}}: CS1 maint: postscript (link); Reprinting of 2nd edn. NY: Dover. 1986. ISBN 0486650243.
• as co-editor with editors Norbert Wiener, Armand Siegel, and Bayard Rankin: Differential space, quantum systems, and prediction. Cambridge, Massachusetts: M.I.T. Press. 1966. (176 pages, essays)
References
1. W. T. Martin at the Mathematics Genealogy Project
2. Martin, William T., Community of Scholars Profile, IAS
3. Kac, Mark (1985). Enigmas of Chance. New York: Harper & Row. p. 113. ISBN 0520059867.
4. Jackson, Allyn (Sep 2004). "William Ted Martin (1911 – 2004)" (PDF). Notices of the AMS. 51 (8): 919.
5. "Longtime math department head Ted Martin dies at age 92". MITnews. 4 June 2004.
6. Jackson, Allyn (Sep 2004). "William Ted Martin (1911 – 2004)" (PDF). Notices of the AMS. 51 (8): 919.
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William G. McCallum
William G. McCallum (born 1956 in Sydney, Australia) is a University Distinguished Professor of Mathematics and was Head of the Department of Mathematics at the University of Arizona from 2009 to 2013.
Education and professional work
He was educated at North Sydney Boys High School.[1] He received his Ph.D. in Mathematics from Harvard University in 1984, under the supervision of Barry Mazur. After spending two years at the University of California, Berkeley, and one at the Mathematical Sciences Research Institute in Berkeley, he joined the faculty at the University of Arizona in 1987. In 1989 he joined the Harvard calculus consortium, and is the lead author of the consortium's multivariable calculus and college algebra texts. In 1993–94 he spent a year at the Institut des Hautes Études Scientifiques, and in 1995–96 he spent a year at the Institute for Advanced Study on a Centennial Fellowship from the American Mathematical Society.
In 2006 he founded the Institute for Mathematics & Education at the University of Arizona. He was Director of the Institute until 2009 and again starting in 2013. In 2009–2010 he was one of the lead writers for the Common Core State Standards in Mathematics. His professional interests include arithmetical algebraic geometry and mathematics education.
Selected honors and awards
• 2012: Fellow of the American Mathematical Society.[2]
• 2012: The Mary P. Dolciani Award, administered by the Mathematical Association of America
• 2012: The American Mathematical Society Distinguished Public Service Award
• 2006: University of Arizona College of Science Galileo Circle Fellow.
• 2005: National Science Foundation’s Director's Award for Distinguished Teaching Scholars
• 1996: The University of Arizona College of Science Innovation in Teaching Award
• 1995: The American Mathematical Society Centennial Research Fellowship.
Current projects
• Institute for Mathematics and Education
• Common Core State Standards in Mathematics
• Illustrative Mathematics Project
• Standards Progressions for the Common Core
• Tools for the Common Core Blog
• The Klein Project
• Mathematical Models at the University of Arizona
References
1. NSBHS Higher School Certificate 1973
2. List of Fellows of the American Mathematical Society, retrieved 2015-01-12
External links
• The U.S. Common Core State Standards, paper presented at ICME 12, Seoul, Korea (slides for this talk)
• Restoring and Balancing, in Usiskin, Anderson, and Zotto (eds), Future Curricular Trends in School Algebra and Geometry, Information Age Publishing (2010)
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William Menasco
William W. Menasco is a topologist and a professor at the University at Buffalo. He is best known for his work in knot theory.
Biography
Menasco received his B.A. from the University of California, Los Angeles in 1975, and his Ph.D. from the University of California, Berkeley in 1981, where his advisor was Robion Kirby. He served as assistant professor at Rutgers University from 1981 to 1984. He then taught as a visiting professor at the University at Buffalo where he became an assistant professor in 1985, an associate professor in 1991. In 1994 he became a professor at the University at Buffalo where he currently serves.[1]
Work
Menasco proved that a link with an alternating diagram, such as an alternating link, will be non-split if and only if the diagram is connected.
Menasco, along with Morwen Thistlethwaite proved the Tait flyping conjecture, which states that, given any two reduced alternating diagrams $D1,D2$ of an oriented, prime alternating link, $D1$ may be transformed to $D2$ by means of a sequence of certain simple moves called flypes.[2]
References
1. Homepage of William W. Menasco (continued)
2. Weisstein, Eric W. "Tait's Knot Conjectures". MathWorld.
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William Mollison (mathematician)
William Loudon Mollison (19 September 1851 – 10 March 1929)[1] was a Scottish mathematician and academic.[2] From 1915 to 1929, he was Master of Clare College, Cambridge.[3]
William Mollison
38th Master of Clare College, Cambridge
In office
March 1915 – March 1929
Preceded byEdward Atkinson
Succeeded byGodfrey Wilson
Personal details
Born(1851-09-19)19 September 1851
Aberdeen, Scotland
Died10 March 1929(1929-03-10) (aged 77)
London, England
Resting placeAscension Parish Burial Ground, Cambridge
SpouseEllen Mayhew
ChildrenThree
EducationAberdeen Grammar School
Alma materUniversity of Aberdeen
Clare College, Cambridge
Early life and education
Mollison was born on 19 September 1851 in Aberdeen, Scotland.[4][5] He was educated at Aberdeen Grammar School, then an all-boys grammar school.[6] He studied mathematics and natural philosophy at the University of Aberdeen, graduating in 1872 with a first class degree.[7] That year, he was awarded the Ferguson Scholarship by Aberdeen and matriculated into Clare College, Cambridge to continue his mathematical studies.[4][6] He became a Foundation Scholar in 1873.[4] His private tutor while at Cambridge was Edward Routh.[8] He graduated from the University of Cambridge in 1876 as the Second Wrangler.[4][8][9]
Career
On 29 April 1876, Mollison was elected a Fellow of Clare College, Cambridge.[10] He was an examiner for the University of St Andrews between 1876 and 1880.[5] He was a mathematics lecturer at Jesus College, Cambridge from 1877 to 1882, and at Clare College from 1882.[6] In addition to his college teaching, he was a private tutor or "coach" in mathematics.[4]
Due to ill health, he moved from teaching a large number of students, privately and through his college, into administration.[4] He was appointed junior tutor of Clare College in 1880,[6] and was made its senior tutor in May 1894.[11] He was elected a member of the Council of the Senate of the University Of Cambridge in 1892,[4] and appointed Secretary of the General Board of Studies of the University in 1904: he stepped down from both these posts in 1920.[6] He served as locum tenens for the then Master (Edward Atkinson) from 1913 to 1915.[6] Mollison was unanimously elected as Atkinson successor as the 38th Master of Clare College, Cambridge in March 1915.[12]
Personal life
Mollison was married to Ellen Mayhew.[6] They had one son and two daughters,[13][1] one of whom pre-deceased him.[1][6] His wife died in 1917, and he provided the endowment for the Mayhew Prize, a mathematics prize awarded by the University of Cambridge, in her honour.[6] His son, William Mayhew Mollison, was a distinguished ear, nose and throat surgeon,[14] and his son Patrick Mollison, a noted haematologist.[15]
Mollison died on 10 March 1929 in London, England; he was aged 77.[7][6] His funeral was held at the chapel of Clare College, Cambridge,[4] and he was buried in the Ascension Parish Burial Ground alongside his wife.[4]
References
1. { The Times (London, England), Monday, 11 March 1929; pg. 21; Issue 45148.}
2. "Mollison, William Loudon". Who's Who & Who Was Who. Vol. 1920–2016 (April 2014 online ed.). A & C Black. Retrieved 18 November 2018. (Subscription or UK public library membership required.)
3. Alumni Cantabrigienses: A Biographical List of All Known Students, Graduates and Holders of Office at the University of Cambridge, from the Earliest Times to 1900, John Venn/John Archibald Venn Cambridge University Press > (10 volumes 1922 to 1953) Part II. 1752–1900 Vol. iv. Kahlenberg – Oyler, (1947) p434
4. Knewstubb, Peter (2012). "William Loudon Mollison (1851–1929)" (PDF). Clare Association Annual: 65–67. Retrieved 18 November 2018.
5. The New International Year Book. New York City, NY: Dodd, Mead and Company. 1930. p. 581.
6. "Mollison, William Loudon". Who Was Who. Oxford University Press. 1 December 2007. doi:10.1093/ww/9780199540884.013.U214230. Retrieved 18 November 2018.
7. "Births, Deaths, Marriages and Obituaries – Master of Clare". Aberdeen Press and Journal. No. 23148. 12 March 1929. p. 11.
8. "CAMBRIDGE UNIVERSITY – Mathematical Tripos". The Leeds Mercury. No. 11795. 29 January 1876.
9. Tanner, Joseph Robson (1917). The historical register of the University of Cambridge, being a supplement to the Calendar with a record of University offices, honours and distinctions to the year 1910 (PDF). Cambridge University Press. Retrieved 18 November 2018.
10. "Scotchmen at Cambridge University". The Dundee Courier & Argus. No. 7103. 1 May 1876.
11. "The Tutorship of Clare". Cambridge Independent Press. No. 4065. 4 May 1894. p. 5.
12. "New Master of Clare". Cambridge Independent Press. No. 5155. 2 April 1915. p. 5.
13. "1891 Census Swanage, Dorset, England RG12/1641 page 2". Retrieved 16 September 2020.
14. "Mollison, William Mayhew (1878–1967)". Plarr's Lives of the Fellows. The Royal College of Surgeons of England. 18 September 2014. Retrieved 18 November 2018.
15. "Professor Patrick Mollison". The Daily Telegraph. 18 January 2012. Retrieved 18 November 2018.
External links
• Portraits of William Loudon Mollison at the National Portrait Gallery, London
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William Neile
William Neile (7 December 1637 – 24 August 1670) was an English mathematician and founder member of the Royal Society. His major mathematical work, the rectification of the semicubical parabola, was carried out when he was aged nineteen, and was published by John Wallis. By carrying out the determination of arc lengths on a curve given algebraically, in other words by extending to algebraic curves generally with Cartesian geometry a basic concept from differential geometry, it represented a major advance in what would become infinitesimal calculus. His name also appears as Neil.
Life
Neile was born at Bishopsthorpe, the eldest son of Sir Paul Neile MP for Ripon and Newark. His grandfather was Richard Neile, the Archbishop of York.[1] He entered Wadham College, Oxford as a gentleman-commoner in 1652, matriculating in 1655. He was taught by John Wilkins and Seth Ward.[1]
In 1657, he became a student at the Middle Temple. In the same year he gave his exact rectification of the semicubical parabola and communicated his discovery to William Brouncker, Christopher Wren and others connected with Gresham College. His demonstration was published by Wallis in De Cycloide (1659). The general formula for rectification by definite integral was in effect discovered by Hendrik van Heuraet in 1659. In 1673 Wallis asserted that Christiaan Huyghens, who was advancing his own claim to have influenced Heuraet, was also slighting the priority of Neile. [1][2]
Neile was elected a fellow of the Royal Society on 7 January 1663 and a member of the council on 11 April 1666. He entered the debate on the theory of motion, as a critic of the empiricist stance of other members. His own theory of motion was held up from publication by unfavourable peer review by Wallis, in 1667; a revision was communicated to the society on 29 April 1669. Neile objected to Wren's 1668 work on collision as lacking discussion of causality: he asked for discussion of the nature of momentum. His own work was much influenced by ideas drawn from the De Corpore of Thomas Hobbes.[1][3][4]
He made astronomical observations with instruments erected on the roof of his father's residence, the “Hill House” (later called Waltham Place) at White Waltham in Berkshire, where he died at the age of 32. A white marble monument in the parish church of White Waltham commemorates him and an inscribed slab in the floor marks his burial-place. He belonged to the privy council of King Charles II.[1]
Notes
1. "Neile, William" . Dictionary of National Biography. London: Smith, Elder & Co. 1885–1900.
2. "Hendrik van Heuraet - Biography".
3. Andrew Pyle (editor), Dictionary of Seventeenth Century British Philosophers (2000), article on Neile, pp. 602-3.
4. Jon Parkin, Science, Religion and Politics in Restoration England: Richard Cumberland's De Legibus Naturae (1999), pp. 136-7.
References
• This article incorporates text from a publication now in the public domain: "Neile, William". Dictionary of National Biography. London: Smith, Elder & Co. 1885–1900.
External links
• O'Connor, John J.; Robertson, Edmund F., "William Neile", MacTutor History of Mathematics Archive, University of St Andrews
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William P. Byers
William Paul Byers (born 1943) is a Canadian mathematician and philosopher; professor emeritus in mathematics and statistics at Concordia University in Montreal, Quebec, Canada.
He completed a BSc ('64), and an MSc ('65) from McGill University, and obtained his PhD ('69) from the University of California, Berkeley. His dissertation, Anosov Flows, was supervised by Stephen Smale.[1]
His area of interests include dynamical systems and the philosophy of mathematics.
Books
Byers is the author of three books on mathematics:
• How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics (Princeton University Press, 2007)[2]
• The Blind Spot: Science and the Crisis of Uncertainty (Princeton University Press, 2011)[3]
• Deep Thinking: What Mathematics Can Teach Us About the Mind (World Scientific, 2015)[4]
See also
• List of people from Montreal
References
1. William P. Byers at the Mathematics Genealogy Project
2. Reviews of How Mathematicians Think:
• Grattan-Guinness, Ivor, zbMATH, Zbl 1123.00003{{citation}}: CS1 maint: untitled periodical (link)
• Rogovchenko, Svitlana P., zbMATH, Zbl 1191.00009{{citation}}: CS1 maint: untitled periodical (link)
• Chaitin, Gregory (July 25, 2007), "Review", New Scientist
• Hersh, Reuben (December 2007), "Review" (PDF), Notices of the American Mathematical Society, 54 (11): 1496–1499
• Kennard, Lee (April 2008), Math Horizons, 15 (4): 24–25, doi:10.1080/10724117.2008.11974771, JSTOR 25678755, S2CID 125265631{{citation}}: CS1 maint: untitled periodical (link)
• Wolff, Ken (May 2008), The Mathematics Teacher, 101 (9): 696, JSTOR 20876252{{citation}}: CS1 maint: untitled periodical (link)
• Baylis, John (March 2009), The Mathematical Gazette, 93 (526): 170–173, doi:10.1017/S0025557200184463, JSTOR 40378699, S2CID 185022987{{citation}}: CS1 maint: untitled periodical (link)
• Keyfitz, Barbara Lee (Winter 2009), University of Toronto Quarterly, 78 (1): 141–143, doi:10.1353/utq.0.0372, S2CID 120506445{{citation}}: CS1 maint: untitled periodical (link)
3. Reviews of The Blind Spot:
• "Review", Quill & Quire, Ontario Arts Council, 26 July 2011
• "Nonfiction book review", Publishers Weekly
• Jubin, Brenda (April 2011), "Review", Seeking Alpha
• "Review", Kirkus Reviews, May 2011
• Taylor, Paul (April 2012), "Review", Mathematics Today, Institute of Mathematics & its Applications
• Sears, Ruthmae (October 2012), The Mathematics Teacher, 106 (3): 238, doi:10.5951/mathteacher.106.3.0238, JSTOR 10.5951/mathteacher.106.3.0238{{citation}}: CS1 maint: untitled periodical (link)
• Edwards, Matthew (February 2013), "Review", The Actuary
4. Review of Deep Thinking:
• Bultheel, Adhemar (February 2015), "Review", EMS Reviews, European Mathematical Society
External links
Wikimedia Commons has media related to William P. Byers.
• Roberts, Russ (May 16, 2011). "Byers on The Blind Spot, Science, and Uncertainty". EconTalk. Library of Economics and Liberty.
Authority control
International
• ISNI
• VIAF
National
• France
• BnF data
• Germany
• Israel
• United States
• Czech Republic
• Korea
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Academics
• MathSciNet
• Mathematics Genealogy Project
• zbMATH
Other
• IdRef
| Wikipedia |
William Payne (mathematician)
William Payne (unknown – c. 1779) was an English mathematician[1] and the author of books about mathematics, draughts, and whist. Payne was the brother of prominent London bookseller Thomas Payne, who sold his works and published some of them.[1][2][3]
Payne's first book, An Introduction to the Game of Draughts, was published in 1756.[1][2] The dedication and preface were written by Samuel Johnson.[2][3]
Payne's second book, An Introduction to Geometry: Containing the Most Useful Propositions in Euclid, & Other Authors, was published in 1767.[4]
The book Maxims for Playing the Game of Whist; With All Necessary Calculations, and Laws of the Game was published anonymously in 1773; published by his brother Thomas, it is believed to have been written by William Payne.[5]
References
1. Courtney, A bibliography, p. 74.
2. Hanley, p. 181.
3. Boswell, p. 192.
4. Payne, William (1767). An Introduction to Geometry: Containing the Most Useful Propositions in Euclid, & Other Authors. London: H. Hart.
5. Courtney, English whist, p. 360.
Sources
• Courtney, William Prideaux (1894). English whist and English whist players. Richard Bentley and Son.
• Courtney, William Prideaux; Smith, David Nichol (ed.) (1915). A bibliography of Samuel Johnson. Clarendon Press.
• Boswell, James (1888). The Life of Samuel Johnson. Swan Schonnenheim, Lowrey & Co.
• Hanley, Brian J (2001). Samuel Johnson as Book Reviewer: A Duty to Examine the Labors of the Learned. University of Delaware Press.
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