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Primitive element theorem
In field theory, the primitive element theorem is a result characterizing the finite degree field extensions that can be generated by a single element. Such a generating element is called a primitive element of the field extension, and the extension is called a simple extension in this case. The theorem states that a finite extension is simple if and only if there are only finitely many intermediate fields. An older result, also often called "primitive element theorem", states that every finite separable extension is simple; it can be seen as a consequence of the former theorem. These theorems imply in particular that all algebraic number fields over the rational numbers, and all extensions in which both fields are finite, are simple.
Terminology
Let $E/F$ be a field extension. An element $\alpha \in E$ is a primitive element for $E/F$ if $E=F(\alpha ),$ i.e. if every element of $E$ can be written as a rational function in $\alpha $ with coefficients in $F$. If there exists such a primitive element, then $E/F$ is referred to as a simple extension.
If the field extension $E/F$ has primitive element $\alpha $ and is of finite degree $n=[E:F]$, then every element x of E can be written uniquely in the form
$x=f_{n-1}{\alpha }^{n-1}+\cdots +f_{1}{\alpha }+f_{0},$
where $f_{i}\in F$ for all i. That is, the set
$\{1,\alpha ,\ldots ,{\alpha }^{n-1}\}$
is a basis for E as a vector space over F.
Example
If one adjoins to the rational numbers $F=\mathbb {Q} $ the two irrational numbers ${\sqrt {2}}$ and ${\sqrt {3}}$ to get the extension field $E=\mathbb {Q} ({\sqrt {2}},{\sqrt {3}})$ of degree 4, one can show this extension is simple, meaning $E=\mathbb {Q} (\alpha )$ for a single $\alpha \in E$. Taking $\alpha ={\sqrt {2}}+{\sqrt {3}}$, the powers 1, α, α2, α3 can be expanded as linear combinations of 1, ${\sqrt {2}}$, ${\sqrt {3}}$, ${\sqrt {6}}$ with integer coefficients. One can solve this system of linear equations for ${\sqrt {2}}$ and ${\sqrt {3}}$ over $\mathbb {Q} (\alpha )$, to obtain ${\sqrt {2}}={\tfrac {1}{2}}(\alpha ^{3}-9\alpha )$ and ${\sqrt {3}}=-{\tfrac {1}{2}}(\alpha ^{3}-11\alpha )$. This shows that α is indeed a primitive element:
$\mathbb {Q} ({\sqrt {2}},{\sqrt {3}})=\mathbb {Q} ({\sqrt {2}}+{\sqrt {3}}).$
The theorems
The classical primitive element theorem states:
Every separable field extension of finite degree is simple.
This theorem applies to algebraic number fields, i.e. finite extensions of the rational numbers Q, since Q has characteristic 0 and therefore every finite extension over Q is separable.
The following primitive element theorem (Ernst Steinitz[1]) is more general:
A finite field extension $E/F$ is simple if and only if there exist only finitely many intermediate fields K with $E\supseteq K\supseteq F$.
Using the fundamental theorem of Galois theory, the former theorem immediately follows from the latter.
Characteristic p
For a non-separable extension $E/F$ of characteristic p, there is nevertheless a primitive element provided the degree [E : F] is p: indeed, there can be no non-trivial intermediate subfields since their degrees would be factors of the prime p.
When [E : F] = p2, there may not be a primitive element (in which case there are infinitely many intermediate fields). The simplest example is $E=\mathbb {F} _{p}(T,U)$, the field of rational functions in two indeterminates T and U over the finite field with p elements, and $F=\mathbb {F} _{p}(T^{p},U^{p})$. In fact, for any α = g(T,U) in $E\setminus F$, the Frobenius endomorphism shows that the element αp lies in F , so α is a root of $f(X)=X^{p}-\alpha ^{p}\in F[X]$, and α cannot be a primitive element (of degree p2 over F), but instead F(α) is a non-trivial intermediate field.
Proof
Starting with a simple finite extension E = F(α), let f be the minimal polynomial of α over F. If K is an intermediate subfield, then let g be the minimal polynomial of α over K, and let L be the field generated over F by the coefficients of g. Then since L ⊆ K, the minimal polynomial of α over L must be a multiple of g, so it is g; this implies that the degree of E over L is the same as that over K, but since L ⊆ K, this means that L = K. Since g is a factor of f, this means that there can be no more intermediate fields than factors of f, so there are only finitely many.
Going in the other direction, if F is finite, then any finite extension E of F is automatically simple, so assume that F is infinite. Then E is generated over F by a finite number of elements, so it's enough to prove that F(α, β) is simple for any two elements α and β in E. But, considering all fields F(α + x β), where x is an element of F, there are only finitely many, so there must be distinct x0 and x1 in F for which F(α + x0 β) = F(α + x1 β). Then simple algebra shows that F(α + x0 β) = F(α, β). For an alternative proof, observe that each of the finite number of intermediate fields is a proper linear subspace of E over F, and that a finite union of proper linear subspaces of a vector space over an infinite field cannot equal the entire space. Then, taking any element in E that is not in any intermediate field, it must generate the whole of E over F.[2][3]
Constructive results
Generally, the set of all primitive elements for a finite separable extension E / F is the complement of a finite collection of proper F-subspaces of E, namely the intermediate fields. This statement says nothing in the case of finite fields, for which there is a computational theory dedicated to finding a generator of the multiplicative group of the field (a cyclic group), which is a fortiori a primitive element (see primitive element (finite field)). Where F is infinite, a pigeonhole principle proof technique considers the linear subspace generated by two elements and proves that there are only finitely many linear combinations
$\gamma =\alpha +c\beta \ $
with c in F, that fail to generate the subfield containing both elements:
as $F(\alpha ,\beta )/F(\alpha +c\beta )$ is a separable extension, if $F(\alpha +c\beta )\subsetneq F(\alpha ,\beta )$ there exists a non-trivial embedding $\sigma :F(\alpha ,\beta )\to {\overline {F}}$ whose restriction to $F(\alpha +c\beta )$ is the identity which means $\sigma (\alpha )+c\sigma (\beta )=\alpha +c\beta $ and $\sigma (\beta )\neq \beta $ so that $c={\frac {\sigma (\alpha )-\alpha }{\beta -\sigma (\beta )}}$. This expression for c can take only $[F(\alpha ):F][F(\beta ):F]$ different values. For all other value of $c\in F$ then $F(\alpha ,\beta )=F(\alpha +c\beta )$.
This is almost immediate as a way of showing how Steinitz' result implies the classical result, and a bound for the number of exceptional c in terms of the number of intermediate fields results (this number being something that can be bounded itself by Galois theory and a priori). Therefore, in this case trial-and-error is a possible practical method to find primitive elements.
History
In his First Memoir of 1831,[4] Évariste Galois sketched a proof of the classical primitive element theorem in the case of a splitting field of a polynomial over the rational numbers. The gaps in his sketch could easily be filled[5] (as remarked by the referee Siméon Denis Poisson; Galois' Memoir was not published until 1846) by exploiting a theorem[6][7] of Joseph-Louis Lagrange from 1771, which Galois certainly knew. It is likely that Lagrange had already been aware of the primitive element theorem for splitting fields.[7] Galois then used this theorem heavily in his development of the Galois group. Since then it has been used in the development of Galois theory and the fundamental theorem of Galois theory. The two primitive element theorems were proved in their modern form by Ernst Steinitz, in an influential article on field theory in 1910;[1] Steinitz called the "classical" one Theorem of the primitive elements and the other one Theorem of the intermediate fields. Emil Artin reformulated Galois theory in the 1930s without the use of the primitive element theorems.[8][9]
References
1. Steinitz, Ernst (1910). "Algebraische Theorie der Körper". Journal für die reine und angewandte Mathematik (in German). 1910 (137): 167–309. doi:10.1515/crll.1910.137.167. ISSN 1435-5345. S2CID 120807300.
2. Theorem 26, Galois Theory, Emil Artin and Arthur N. Milgram, University of Notre Dame Press, 2nd ed., 1944.
3. Lemma 9.19.1 (Primitive element), The Stacks project. Accessed on line July 19, 2023.
4. Neumann, Peter M. (2011). The mathematical writings of Évariste Galois. Zürich: European Mathematical Society. ISBN 978-3-03719-104-0. OCLC 757486602.
5. Tignol, Jean-Pierre (February 2016). Galois' Theory of Algebraic Equations (2 ed.). WORLD SCIENTIFIC. p. 231. doi:10.1142/9719. ISBN 978-981-4704-69-4. OCLC 1020698655.
6. Tignol, Jean-Pierre (February 2016). Galois' Theory of Algebraic Equations (2 ed.). WORLD SCIENTIFIC. p. 135. doi:10.1142/9719. ISBN 978-981-4704-69-4. OCLC 1020698655.
7. Cox, David A. (2012). Galois theory (2nd ed.). Hoboken, NJ: John Wiley & Sons. p. 322. ISBN 978-1-118-21845-7. OCLC 784952441.
8. Kleiner, Israel (2007). "§4.1 Galois theory". A History of Abstract Algebra. Springer. p. 64. ISBN 978-0-8176-4685-1.
9. Artin, Emil (1998). Galois theory. Arthur N. Milgram (Republication of the 1944 revised edition of the 1942 first publication by The University Notre Dame Press ed.). Mineola, N.Y.: Dover Publications. ISBN 0-486-62342-4. OCLC 38144376.
External links
• J. Milne's course notes on fields and Galois theory
• The primitive element theorem at mathreference.com
• The primitive element theorem at planetmath.org
• The primitive element theorem on Ken Brown's website (pdf file)
| Wikipedia |
Arithmetices principia, nova methodo exposita
The 1889 treatise Arithmetices principia, nova methodo exposita (The principles of arithmetic, presented by a new method) by Giuseppe Peano is widely considered to be a seminal document in mathematical logic and set theory,[1][2] introducing what is now the standard axiomatization of the natural numbers, and known as the Peano axioms, as well as some pervasive notations, such as the symbols for the basic set operations ∈, ⊂, ∩, ∪, and A−B.
The treatise is written in Latin, which was already somewhat unusual at the time of publication, Latin having fallen out of favour as the lingua franca of scholarly communications by the end of the 19th century. The use of Latin in spite of this reflected Peano's belief in the universal importance of the work – which is now generally regarded as his most important contribution to arithmetic – and in that of universal communication. Peano would publish later works both in Latin and in his own artificial language, Latino sine flexione, which is a grammatically simplified version of Latin.
Peano also continued to publish mathematical notations in a series from 1895 to 1908 collectively known as Formulario mathematico.
References
1. Segre, Michael (1 January 1994). "Peano's Axioms in their Historical Context". Archive for History of Exact Sciences. 48 (3/4): 201–342. doi:10.1007/bf00375085. S2CID 122070745.
2. Clegg, Brian (7 February 2013). A Brief History of Infinity: The Quest to Think the Unthinkable. Little, Brown Book Group. ISBN 9781472107640.
External links
• English translation (with original Latin): https://github.com/mdnahas/Peano_Book/blob/master/Peano.pdf
• Original treatise (in Latin, scanned) at Internet Archive: https://archive.org/details/arithmeticespri00peangoog
| Wikipedia |
Pseudo-arc
In general topology, the pseudo-arc is the simplest nondegenerate hereditarily indecomposable continuum. The pseudo-arc is an arc-like homogeneous continuum, and played a central role in the classification of homogeneous planar continua. R. H. Bing proved that, in a certain well-defined sense, most continua in Rn, n ≥ 2, are homeomorphic to the pseudo-arc.
History
In 1920, Bronisław Knaster and Kazimierz Kuratowski asked whether a nondegenerate homogeneous continuum in the Euclidean plane R2 must be a Jordan curve. In 1921, Stefan Mazurkiewicz asked whether a nondegenerate continuum in R2 that is homeomorphic to each of its nondegenerate subcontinua must be an arc. In 1922, Knaster discovered the first example of a hereditarily indecomposable continuum K, later named the pseudo-arc, giving a negative answer to a Mazurkiewicz question. In 1948, R. H. Bing proved that Knaster's continuum is homogeneous, i.e. for any two of its points there is a homeomorphism taking one to the other. Yet also in 1948, Edwin Moise showed that Knaster's continuum is homeomorphic to each of its non-degenerate subcontinua. Due to its resemblance to the fundamental property of the arc, namely, being homeomorphic to all its nondegenerate subcontinua, Moise called his example M a pseudo-arc.[lower-alpha 1] Bing's construction is a modification of Moise's construction of M, which he had first heard described in a lecture. In 1951, Bing proved that all hereditarily indecomposable arc-like continua are homeomorphic — this implies that Knaster's K, Moise's M, and Bing's B are all homeomorphic. Bing also proved that the pseudo-arc is typical among the continua in a Euclidean space of dimension at least 2 or an infinite-dimensional separable Hilbert space.[lower-alpha 2] Bing and F. Burton Jones constructed a decomposable planar continuum that admits an open map onto the circle, with each point preimage homeomorphic to the pseudo-arc, called the circle of pseudo-arcs. Bing and Jones also showed that it is homogeneous. In 2016 Logan Hoehn and Lex Oversteegen classified all planar homogeneous continua, up to a homeomorphism, as the circle, pseudo-arc and circle of pseudo-arcs. In 2019 Hoehn and Oversteegen showed that the pseudo-arc is topologically the only, other than the arc, hereditarily equivalent planar continuum, thus providing a complete solution to the planar case of Mazurkiewicz's problem from 1921.
Construction
The following construction of the pseudo-arc follows (Wayne Lewis 1999) harv error: no target: CITEREFWayne_Lewis1999 (help).
Chains
At the heart of the definition of the pseudo-arc is the concept of a chain, which is defined as follows:
A chain is a finite collection of open sets ${\mathcal {C}}=\{C_{1},C_{2},\ldots ,C_{n}\}$ in a metric space such that $C_{i}\cap C_{j}\neq \emptyset $ if and only if $|i-j|\leq 1.$ The elements of a chain are called its links, and a chain is called an ε-chain if each of its links has diameter less than ε.
While being the simplest of the type of spaces listed above, the pseudo-arc is actually very complex. The concept of a chain being crooked (defined below) is what endows the pseudo-arc with its complexity. Informally, it requires a chain to follow a certain recursive zig-zag pattern in another chain. To 'move' from the mth link of the larger chain to the nth, the smaller chain must first move in a crooked manner from the mth link to the (n-1)th link, then in a crooked manner to the (m+1)th link, and then finally to the nth link.
More formally:
Let ${\mathcal {C}}$ and ${\mathcal {D}}$ be chains such that
1. each link of ${\mathcal {D}}$ is a subset of a link of ${\mathcal {C}}$, and
2. for any indices i, j, m, and n with $D_{i}\cap C_{m}\neq \emptyset $, $D_{j}\cap C_{n}\neq \emptyset $, and $m<n-2$, there exist indices $k$ and $\ell $ with $i<k<\ell <j$ (or $i>k>\ell >j$) and $D_{k}\subseteq C_{n-1}$ and $D_{\ell }\subseteq C_{m+1}.$
Then ${\mathcal {D}}$ is crooked in ${\mathcal {C}}.$
Pseudo-arc
For any collection C of sets, let $C^{*}$ denote the union of all of the elements of C. That is, let
$C^{*}=\bigcup _{S\in C}S.$
The pseudo-arc is defined as follows:
Let p and q be distinct points in the plane and $\left\{{\mathcal {C}}^{i}\right\}_{i\in \mathbb {N} }$ be a sequence of chains in the plane such that for each i,
1. the first link of ${\mathcal {C}}^{i}$ contains p and the last link contains q,
2. the chain ${\mathcal {C}}^{i}$ is a $1/2^{i}$-chain,
3. the closure of each link of ${\mathcal {C}}^{i+1}$ is a subset of some link of ${\mathcal {C}}^{i}$, and
4. the chain ${\mathcal {C}}^{i+1}$ is crooked in ${\mathcal {C}}^{i}$.
Let
$P=\bigcap _{i\in \mathbb {N} }\left({\mathcal {C}}^{i}\right)^{*}.$
Then P is a pseudo-arc.
Notes
1. Henderson (1960) later showed that a decomposable continuum homeomorphic to all its nondegenerate subcontinua must be an arc.
2. The history of the discovery of the pseudo-arc is described in Nadler (1992), pp. 228–229.
References
• Bing, R.H. (1948), "A homogeneous indecomposable plane continuum", Duke Mathematical Journal, 15 (3): 729–742, doi:10.1215/S0012-7094-48-01563-4
• Bing, R.H. (1951), "Concerning hereditarily indecomposable continua", Pacific Journal of Mathematics, 1: 43–51, doi:10.2140/pjm.1951.1.43
• Bing, R.H.; Jones, F. Burton (1959), "Another homogeneous plane continuum", Transactions of the American Mathematical Society, 90 (1): 171–192, doi:10.1090/S0002-9947-1959-0100823-3
• Henderson, George W. (1960), "Proof that every compact decomposable continuum which is topologically equivalent to each of its nondegenerate subcontinua is an arc", Annals of Mathematics, 2nd series, 72 (3): 421–428, doi:10.2307/1970224
• Hoehn, Logan C.; Oversteegen, Lex G. (2016), "A complete classification of homogeneous plane continua", Acta Mathematica, 216 (2): 177–216, doi:10.1007/s11511-016-0138-0
• Hoehn, Logan C.; Oversteegen, Lex G. (2020), "A complete classification of hereditarily equivalent plane continua", Advances in Mathematics, 368: 107131, arXiv:1812.08846, doi:10.1016/j.aim.2020.107131
• Irwin, Trevor; Solecki, Sławomir (2006), "Projective Fraïssé limits and the pseudo-arc", Transactions of the American Mathematical Society, 358 (7): 3077–3096, doi:10.1090/S0002-9947-06-03928-6
• Kawamura, Kazuhiro (2005), "On a conjecture of Wood", Glasgow Mathematical Journal, 47 (1): 1–5, doi:10.1017/S0017089504002186
• Knaster, Bronisław (1922), "Un continu dont tout sous-continu est indécomposable", Fundamenta Mathematicae, 3: 247–286, doi:10.4064/fm-3-1-247-286
• Lewis, Wayne (1999), "The Pseudo-Arc", Boletín de la Sociedad Matemática Mexicana, 5 (1): 25–77
• Lewis, Wayne; Minc, Piotr (2010), "Drawing the pseudo-arc" (PDF), Houston Journal of Mathematics, 36: 905–934
• Moise, Edwin (1948), "An indecomposable plane continuum which is homeomorphic to each of its nondegenerate subcontinua", Transactions of the American Mathematical Society, 63 (3): 581–594, doi:10.1090/S0002-9947-1948-0025733-4
• Nadler, Sam B., Jr. (1992), Continuum theory. An introduction, Monographs and Textbooks in Pure and Applied Mathematics, vol. 158, Marcel Dekker, Inc., New York, ISBN 0-8247-8659-9{{citation}}: CS1 maint: multiple names: authors list (link)
• Rambla, Fernando (2006), "A counterexample to Wood's conjecture", Journal of Mathematical Analysis and Applications, 317 (2): 659–667, doi:10.1016/j.jmaa.2005.07.064
• Rempe-Gillen, Lasse (2016), Arc-like continua, Julia sets of entire functions, and Eremenko's Conjecture, arXiv:1610.06278
| Wikipedia |
The spider and the fly problem
The spider and the fly problem is a recreational geodesics problem with an unintuitive solution.
Problem
In the typical version of the puzzle, an otherwise empty cuboid room 30 feet long, 12 feet wide and 12 feet high contains a spider and a fly. The spider is 1 foot below the ceiling and horizontally centred on one 12′×12′ wall. The fly is 1 foot above the floor and horizontally centred on the opposite wall. The problem is to find the minimum distance the spider must crawl along the walls, ceiling and/or floor to reach the fly, which remains stationary.
Solutions
A naive solution is for the spider to remain horizontally centred, and crawl up to the ceiling, across it and down to the fly, giving a distance of 42 feet. The shortest distance strictly abiding by the rules, 40 feet, is obtained by constructing an appropriate net of the room and connecting the spider and fly with a straight line, but counter-intuitively this optimal path used five of the six faces of the cuboid and can easily be missed.[1]
A lateral thinking solution involves the spider attaching dragline silk to the wall to lower itself to the floor, and crawling 30 feet across it and 1 foot up the opposite wall, giving a crawl distance of 31 feet. Similarly, it can climb to the ceiling, cross it, then attach the silk to lower itself 11 feet, also a 31-foot crawl.[2]
lwhbanon−o
22551127261
22991131301
28881136342
28971135341
2611101136351
33661139372
33751138371
34871141392
34961140391
3012121142402
3013111141401
38541142411
3414131147452
3415121146451
3815151153503
3816141152502
3615152251501
3715151251501
3715152151501
3817131151501
4017162256551
4020201160582
3821211159581
4021191159581
For a room of length l, width w and height h, the spider a distance b below the ceiling, and the fly a distance a above the floor, the optimal distance o is ${\sqrt {(w+h)^{2}+(b+l+a)^{2}}}$ while the naive distance n is $l+h-|b-a|$.
This table gives integer solutions for l, w ≤ 40, h ≤ w and o < n, sorted by ascending o then n−o, with the original values in bold.
History
The problem was originally posed by Henry Dudeney in the English newspaper Weekly Dispatch on 14 June 1903, presented in The Canterbury Puzzles (1907) and described by Martin Gardner.[3]
References
1. Distances on the surface of a cuboid, Henry Bottomley
2. Weisstein, Eric W. "Spider and Fly Problem". Mathworld.wolfram.com. Retrieved 1 March 2019.
3. Darling, David. "spider-and-fly problem". Daviddarling.info. Retrieved 1 March 2019.
| Wikipedia |
Stacks Project
The Stacks Project is an open source collaborative mathematics textbook writing project with the aim to cover "algebraic stacks and the algebraic geometry needed to define them".[1][2][3][4] As of July 2022, the book consists of 115 chapters[5] (excluding the license and index chapters) spreading over 7500 pages. The maintainer of the project, who reviews and accepts the changes, is Aise Johan de Jong.[1][2]
See also
• Kerodon a Stacks project inspired online textbook on categorical homotopy theory maintained by Jacob Lurie
References
1. "Stacks Project — About". Stacks.math.columbia.edu. Retrieved 2020-04-01.
2. "Aise Johan de Jong receives 2022 Steele Prize for Mathematical Exposition". ams.org. Retrieved 2021-12-25.
3. "Stacks Project". swmath.org. Retrieved 2021-12-25.
4. Douglas, Michael R. How will we do mathematics in 2030? (Speech). MIT Center for Brains, Minds & Machines. Retrieved 2021-12-25.
5. "Stacks Project — Chapters". Stacks.math.columbia.edu. Retrieved 2020-04-01.
External links
• Project website
• The Stacks Project at the nLab
• Latest from the Stacks Project (as of 2013) (Accessed 2020-04-01)
| Wikipedia |
Zahlbericht
In mathematics, the Zahlbericht (number report) was a report on algebraic number theory by Hilbert (1897, 1998, (English translation)).
History
In 1893 the German Mathematical Society invited Hilbert and Minkowski to write reports on the theory of numbers. They agreed that Minkowski would cover the more elementary parts of number theory while Hilbert would cover algebraic number theory. Minkowski eventually abandoned his report, while Hilbert's report was published in 1897. It was reprinted in volume 1 of his collected works, and republished in an English translation in 1998. Corry (1996) and Schappacher (2005) and the English introduction to (Hilbert 1998) give detailed discussions of the history and influence of Hilbert's Zahlbericht.
Some earlier reports on number theory include the report by H. J. S. Smith in 6 parts between 1859 and 1865, reprinted in Smith (1965), and the report by Brill & Noether (1894). Hasse (1926, 1927, 1930) wrote an update of Hilbert's Zahlbericht that covered class field theory (republished in 1 volume as (Hasse 1970)).
Contents
Part 1 covers the theory of general number fields, including ideals, discriminants, differents, units, and ideal classes.
Part 2 covers Galois number fields, including in particular Hilbert's theorem 90.
Part 3 covers quadratic number fields, including the theory of genera, and class numbers of quadratic fields.
Part 4 covers cyclotomic fields, including the Kronecker–Weber theorem (theorem 131), the Hilbert–Speiser theorem (theorem 132), and the Eisenstein reciprocity law for lth power residues (theorem 140) .
Part 5 covers Kummer number fields, and ends with Kummer's proof of Fermat's last theorem for regular primes.
References
• Brill, A.; Noether, M. (1894), "Die Entwickelung der Theorie der algebraischen Functionen in älterer und neuerer Zeit", Jahresbericht der Deutschen Mathematiker-Vereinigung (in German), 3: 107–566, ISSN 0012-0456
• Corry, Leo (1996), Modern algebra and the rise of mathematical structures, Science Networks. Historical Studies, vol. 17, Birkhäuser Verlag, ISBN 978-3-7643-5311-7, MR 1391720
• Hasse, H. (1926), "Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper. I: Klassenkörpertheorie.", Jahresbericht der Deutschen Mathematiker-Vereinigung (in German), 35: 1–55
• Hasse, H. (1927), "Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper. Teil Ia: Beweise zu I.", Jahresbericht der Deutschen Mathematiker-Vereinigung (in German), 36: 233–311
• Hasse, H. (1930), "Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper. Teil II: Reziprozitätsgesetz", Jahresbericht der Deutschen Mathematiker-Vereinigung (in German), Ergänzungsband 6
• Hasse, Helmut (1970) [1930], Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper. Teil I: Klassenkörpertheorie. Teil Ia: Beweise zu Teil I. Teil II: Reziprozitätsgesetz (3rd ed.), Physica-Verlag, ISBN 978-3-7908-0010-4, MR 0266893
• Hilbert, David (1897), "Die Theorie der algebraischen Zahlkörper", Jahresbericht der Deutschen Mathematiker-Vereinigung (in German), 4: 175–546, ISSN 0012-0456
• Hilbert, David (1998), The theory of algebraic number fields, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-662-03545-0, ISBN 978-3-540-62779-1, MR 1646901
• Schappacher, N. (2005), "Chapter 54. David Hilbert, Report on algebraic number fields", in Grattan-Guinness, Ivor (ed.), Landmark writings in western mathematics 1640–1940, Elsevier B. V., Amsterdam, ISBN 978-0-444-50871-3, MR 2169816
• Smith, Henry John Stephen (1965) [1894], Glaisher, J. W. L. (ed.), The Collected Mathematical Papers of Henry John Stephen Smith, vol. I, New York: AMS Chelsea Publishing, ISBN 978-0-8284-0187-6, volume 1volume 2
External links
Wikisource has original text related to this article:
Die Theorie der algebraischen Zahlkörper.
• Introduction to the English Edition of Hilbert's Zahlbericht
| Wikipedia |
Theano (philosopher)
Theano (/θiˈænoʊ/; Greek: Θεανώ) was a 6th-century BC Pythagorean philosopher. She has been called the wife or student of Pythagoras, although others see her as the wife of Brontinus. Her place of birth and the identity of her father is uncertain as well. Many Pythagorean writings were attributed to her in antiquity, including some letters and a few fragments from philosophical treatises, although these are all regarded as spurious by modern scholars.
Theano
Θεανώ
Bornc. 6th century BC
Croton or Metapontum
SpousePythagoras or Brontinus
EraAncient Greek philosophy
SchoolPythagoreanism
Life
Little is known about the life of Theano, and the few details on her life from ancient testimony are contradictory.[1] From the current historical evidence, it is not currently possible to conclude whether or not she even existed, or was the invention of the later Pythagoreans who attached her name to their writings. According to Porphyry, she came from Crete and was the daughter of Pythonax.[2][3] In the catalog of Aristoxenus of Tarentum quoted by Iamblichus, she is the wife of Brontinus, and from Metapontum in Magna Graecia, while Diogenes Laertius reports a tradition from Hermesianax where she came from Crotone, married Pythagoras, and was the daughter of Brontinus.[4][5][3]
Writings
Many writings were attributed to Theano in antiquity[6] - The Suda[3] attributes to her works with the titles Pythagorean Apophthegms, Advice to Women, On Pythagoras, On Virtue and Philosophical Commentaries, which have not survived. In addition, a short fragment attributed to her from a work titled On Piety is preserved in the Anthologium of Stobaeus, and several epistles have survived through medieval manuscript traditions that are attributed to her.[7]
These writings are all widely considered by modern scholarship to be pseudepigrapha,[1][8] works that were written long after Theano's death by later Pythagoreans, which attempt to correct doctrinal disputes with later philosophers[9] or apply Pythagorean philosophy to a woman's life.[1] Some sources claim that Theano wrote about either the doctrine of the golden mean in philosophy, or the golden ratio in mathematics, but there is no evidence from the time to justify this claim.[10]
On Piety
The surviving fragment of On Piety preserved in Stobaeus concerns a Pythagorean analogy between numbers and objects;
I have learned that many of the Greeks suppose Pythagoras said that everything came to be from number. This statement, however, poses a difficulty—how something that does not even exist is thought to beget things. But he did not say that things came to be from number, but according to number. For in number is the primary ordering, by virtue of whose presence, in the realm of things that can be counted, too, something takes its place as first, something as second, and the rest follow in order.[9]
Walter Burkert notes that this statement, that "number does not even exist" contradicts the Platonic idealism of the Neopythagoreans and Neoplatonists, and attributes it to the Hellenistic period, before the advent of Neopythagoreanism in the early roman period.[9]
Letters
The various surviving letters deal with domestic concerns: how a woman should bring up children, how she should treat servants, and how she should behave virtuously towards her husband.[1]
The preserved letters are as follows:[6]
• To Eubule: On caring for infants.
• To Euclides: A short letter to a physician who is ill.
• To Eurydice: On behavior when a husband is unfaithful.
• To Callisto: On etiquette towards maids.
• To Nicostrate: On behavior when a husband is unfaithful.
• To Rhodope: On a philosopher named Cleon.
• To Timonides: Addressed to an unfaithful lover
There are also references to a letter addressed To Timareta, which is referenced by Julius Pollux in his Onomasticon for its use of the word wikt:οἰκοδεσπότης.[6]
Notes
1. Plant 2004, p. 69.
2. Porphyry, Life of Pythagoras, 4
3. Suda, Theano.
4. Suda, Pythagoras.
5. Laërtius 1925, 42.
6. Thesleff 1961, p. 22-23.
7. Hercher 1873.
8. Voula Lambropoulou, Some Pythagorean female virtues, in Richard Hawley, Barbara Levick, (1995), Women in antiquity: new assessments, page 133. Routledge
9. Burkert 1972, p. 61.
10. Deakin 2013.
References
Ancient testimony
• Laërtius, Diogenes (1925). "Pythagoreans: Pythagoras" . Lives of the Eminent Philosophers. Vol. 2:8. Translated by Hicks, Robert Drew (Two volume ed.). Loeb Classical Library.
• Porphyry. Life of Pythagoras. Translated by Kenneth Sylvan Guthrie – via Tertullian Project.
• Iamblichus. Life of Pythagoras. Translated by Thomas Taylor. Retrieved 20 June 2023.
• "Puthagoras". Suda On Line. Retrieved 2023-05-04.
• "Theano". Suda On Line. Retrieved 2023-05-04.
Modern scholarship
• Burkert, Walter (1972). Lore and Science in Ancient Pythagoreanism. Harvard University Press. ISBN 978-0-674-53918-1. Retrieved 20 June 2023.
• Deakin, Michael A.B. (15 April 2013). "Theano: the world's first female mathematician?". International Journal of Mathematical Education in Science and Technology. 44 (3): 350–364. doi:10.1080/0020739X.2012.729614.
• Hercher, Rudolf (1873). "Pythagoreans". Epistolographoi hellenikoi. Epistolographi graeci, recensuit, recognovit, adnotatione critica et indicibus instruxit Rudolphus Hercher; accedunt Francisci Boissonadii ad Synesium notae ineditae (in Ancient Greek and Latin). Parisiis A.F. Didot. pp. 603–607. Retrieved 20 June 2023.</ref>
• Plant, Ian Michael (2004). Women writers of ancient Greece and Rome: an anthology. University of Oklahoma Press. p. 68-75. ISBN 978-0-8061-3621-9.
• Thesleff, Holger (1961). An Introduction to the Pythagorean Writtings of the Hellenistic Period.
• Zhmud, Leonid (31 May 2012). Pythagoras and the Early Pythagoreans. OUP Oxford. ISBN 978-0-19-928931-8.
Further reading
Wikiquote has quotations related to Theano (philosopher).
• Dancy, R. M. (1989). "On A History of Women Philosophers, Vol. I". Hypatia. 4 (1): 160–171. ISSN 0887-5367.
• Huizenga, Annette (27 March 2013). Moral Education for Women in the Pastoral and Pythagorean Letters: Philosophers of the Household. BRILL. ISBN 978-90-04-24518-1. Retrieved 20 June 2023.
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Thébault's theorem
Thébault's theorem is the name given variously to one of the geometry problems proposed by the French mathematician Victor Thébault, individually known as Thébault's problem I, II, and III.
Thébault's problem I
Given any parallelogram, construct on its sides four squares external to the parallelogram. The quadrilateral formed by joining the centers of those four squares is a square.[1]
It is a special case of van Aubel's theorem and a square version of the Napoleon's theorem.
Thébault's problem II
Given a square, construct equilateral triangles on two adjacent edges, either both inside or both outside the square. Then the triangle formed by joining the vertex of the square distant from both triangles and the vertices of the triangles distant from the square is equilateral.[2]
Thébault's problem III
Given any triangle ABC, and any point M on BC, construct the incircle and circumcircle of the triangle. Then construct two additional circles, each tangent to AM, BC, and to the circumcircle. Then their centers and the center of the incircle are colinear.[3][4]
Until 2003, academia thought this third problem of Thébault the most difficult to prove. It was published in the American Mathematical Monthly in 1938, and proved by Dutch mathematician H. Streefkerk in 1973. However, in 2003, Jean-Louis Ayme discovered that Y. Sawayama, an instructor at The Central Military School of Tokyo, independently proposed and solved this problem in 1905.[5]
An "external" version of this theorem, where the incircle is replaced by an excircle and the two additional circles are external to the circumcircle, is found in Shay Gueron (2002). [6] A proof based on Casey's theorem is in the paper.
References
1. http://www.cut-the-knot.org/Curriculum/Geometry/Thebault1.shtml (retrieved 2016-01-27)
2. http://www.cut-the-knot.org/Curriculum/Geometry/Thebault2.shtml (retrieved 2016-01-27)
3. http://www.cut-the-knot.org/Curriculum/Geometry/Thebault3.shtml (retrieved 2016-01-27)
4. Alexander Ostermann, Gerhard Wanner: Geometry by Its History. Springer, 2012, pp. 226–230
5. Ayme, Jean-Louis (2003), "Sawayama and Thébault's theorem" (PDF), Forum Geometricorum, 3: 225–229, MR 2055379
6. Gueron, Shay (April 2002). "Two Applications of the Generalized Ptolemy Theorem" (PDF). The American Mathematical Monthly. 109 (4): 362–370. doi:10.2307/2695499. JSTOR 2695499.
External links
Wikimedia Commons has media related to Thébault's theorem.
• Thébault's problems and variations at cut-the.knot.org
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Theodoor Jacobus Boks
Theodoor Jacobus Boks (15 August 1893, in Elst – 18 July 1961, in Hilversum?)[1] was a Dutch mathematician working on analysis.
Boks obtained his PhD cum laude at Utrecht University in 1921 with the dissertation "Sur les rapports entre les méthodes d'intégration de Riemann et de Lebesgue".[2][3]
References
1. Obituary
2. Theodoor Jacobus Boks at the Album Promotorum of Utrecht University
3. Theodoor Jacobus Boks at the Mathematics Genealogy Project
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Theodor Estermann
Theodor Estermann (5 February 1902 – 29 November 1991) was a German-born American mathematician, working in the field of analytic number theory. The Estermann measure, a measure of the central symmetry of a convex set in the Euclidean plane, is named after him.[1]
He was born in Neubrandenburg, Germany, "to keen Zionists who named him in honour of Herzl."[2] His doctorate, completed in 1925, was supervised by Hans Rademacher. He spent most of his career at University College London, eventually as a professor. Heini Halberstam, Klaus Roth and Robert Charles Vaughan were Ph.D. students of his.
Though Estermann left Germany in 1929, before the Nazis seized power in 1933, some historians count him among the early emigrants who fled Nazi Germany.[3][4]
The physicist Immanuel Estermann was the brother of Theodor Estermann.
References
1. Grünbaum, Branko (1963). "Measures of symmetry for convex sets". In Klee, Victor L. (ed.). Convexity. Proceedings of Symposia in Pure Mathematics. Vol. 7. Providence, Rhode Island: American Mathematical Society. pp. 233–270. MR 0156259.
2. William D. Rubinstein, Michael Jolles, Hilary L. Rubinstein, The Palgrave Dictionary of Anglo-Jewish History, Palgrave Macmillan (2011), p. 260
3. Siegmund-Schultze, Reinhard (2009). Mathematicians Fleeing from Nazi Germany: Individual Fates and Global Impact. Princeton University Press. p. 8. ISBN 978-0-691-12593-0.
4. Pinl, Max; Furtmüller, Lux (1973). "Mathematicians under Hitler". The Leo Baeck Institute Year Book. 18 (1). doi:10.1093/leobaeck/20.1.370.
External links
• Theodor Estermann at the Mathematics Genealogy Project
• LMS obituary
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Theodor Schneider
Theodor Schneider (7 May 1911, Frankfurt am Main – 31 October 1988, Freiburg im Breisgau) was a German mathematician, best known for providing proof of what is now known as the Gelfond–Schneider theorem.
Schneider studied from 1929 to 1934 in Frankfurt; he solved Hilbert's 7th problem in his PhD thesis, which then came to be known as the Gelfond–Schneider theorem. Later he became an assistant to Carl Ludwig Siegel in Göttingen, where he stayed until 1953. Then he became a professor in Erlangen (1953–59) and finally until his retirement in Freiburg (1959–1976). During his time in Freiburg he also served as the director of the Mathematical Research Institute of Oberwolfach from 1959 to 1963. His doctoral students include H. P. Schlickewei.
Works
• Einführung in die Theorie der transzendenten Zahlen, Springer 1957 (German, French translation 1959)
• Transzendenzuntersuchungen periodischer Funktionen, Teil 1,2, Journal für die Reine und Angewandte Mathematik, volume 172, 1934, pp. 65–69, 70-74, online: part 1, part 2 (dissertation in which he solved Hilbert's 7th problem, German)
See also
• Schneider–Lang theorem
References
• L.-Ch. Kappe, H.P.Schlickewei, Wolfgang Schwarz Theodor Schneider zum Gedächtnis, Jahresbericht DMV, Bd.92, 1990, S.111-129 (German)
• Wolfgang Schwarz, Jürgen Wolfart: Zur Geschichte des Mathematischen Seminars der Universität Frankfurt am Main von 1914 - 1970, pp. 29, 82-82, 92-94, 97 (German, ps)
• O'Connor, John J.; Robertson, Edmund F., "Theodor Schneider", MacTutor History of Mathematics Archive, University of St Andrews
• Wolfgang Karl Schwarz (2007), "Schneider, Theodor Adam", Neue Deutsche Biographie (in German), vol. 23, Berlin: Duncker & Humblot, pp. 308–309; (full text online)
External links
• Theodor Schneider at the Mathematics Genealogy Project
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Ted Harris (mathematician)
Theodore Edward Harris (11 January 1919 – 3 November 2005) was an American mathematician known for his research on stochastic processes, including such areas as general state-space Markov chains (often now called Harris chains), the theory of branching processes and stochastic models of interacting particle systems such as the contact process. The Harris inequality in statistical physics and percolation theory is named after him.
Ted Harris
Born(1919-01-11)January 11, 1919
DiedNovember 3, 2005(2005-11-03) (aged 86)[1]
Alma materPrinceton University
Known forHarris chain
Scientific career
InstitutionsRAND Corporation
University of Southern California
ThesisSome Theorems on the Bernoullian Multiplicative Process[2] (1947)
Doctoral advisorSamuel S. Wilks
He received his Ph.D. at Princeton University in 1947 under advisor Samuel Wilks. From 1947 until 1966 he worked for the RAND Corporation, heading their mathematics department from 1959 to 1965. From 1966 until retirement in 1989 he was Professor of Mathematics and Electrical Engineering at University of Southern California.
He was elected to the United States National Academy of Sciences in 1988.[3]
Selected publications
Books
• Harris, Theodore E. "The theory of branching processes". Springer-Verlag, Berlin. (1963) 230 pp.
Papers
• Harris, Theodore; Arrow, Kenneth J.; Marschak, Jacob (July 1951). "Optimal inventory policy". Econometrica. 19 (3): 250–272. doi:10.2307/1906813. JSTOR 1906813.
• Harris, T.E. (December 1974). "Contact interactions on a lattice". The Annals of Probability. 2 (6): 969–988. doi:10.1214/aop/1176996493. JSTOR 2959099.
References
1. Alexander, K. S. (1996). "A conversation with Ted Harris". Statistical Science. 11 (2): 150–158. doi:10.1214/ss/1038425658.
2. Ted Harris at the Mathematics Genealogy Project
3. "College Magazine Obituaries: Theodore E. Harris". USC College of Letters, Arts and Sciences. Archived from the original on December 4, 2006. Retrieved 19 December 2012.
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Ted Hill (mathematician)
Theodore Preston Hill (born December 28, 1943) is an American mathematician specializing in probability theory. He is a professor emeritus at the Georgia Institute of Technology and a researcher at California Polytechnic State University, San Luis Obispo.
Ted Hill
Born
Theodore Preston Hill
(1943-12-28) December 28, 1943
Flatbush, New York, U.S.
NationalityAmerican
Alma materB.S., United States Military Academy, 1966
M.S., Stanford University, 1968
Ph.D., University of California, Berkeley, 1977
Known forProbability Theory: Benford's Law, Fair division, Optimal Stopping
Scientific career
FieldsMathematics
InstitutionsGeorgia Institute of Technology
Doctoral advisorLester Dubins
Contributions
Hill is known for his research on mathematical probability theory, in particular for his work on Benford's law,[1] and for his work in the theories of optimal stopping (secretary problems) and fair division, in particular the Hill-Beck land division problem.
Hill has attracted attention for a paper on the theory that men exhibit greater variability than women in genetically controlled traits that he wrote with Sergei Tabachnikov.[2] It was accepted but not published by The Mathematical Intelligencer; a later version authored by Hill alone was peer reviewed and accepted by The New York Journal of Mathematics (NYJM) and retracted after publication. A revised version, again authored by Hill alone, was subsequently peer reviewed again and published in the Journal of Interdisciplinary Mathematics.[3][4]
Education and career
Born in Flatbush, New York, he studied at the United States Military Academy at West Point (Distinguished Graduate of the Class of 1966), and Stanford University (M.S. in Operations Research). After graduating from the U.S. Army Ranger School and serving as an Army Captain in the Combat Engineers of the 25th Infantry Division in Vietnam, he returned to study mathematics at the University of Göttingen (Fulbright Scholar), the University of California at Berkeley (M.A., Ph.D. under advisor Lester Dubins), and as NATO/NSF Postdoctoral Fellow at Leiden University.
He spent most of his career as a professor in the School of Mathematics at the Georgia Institute of Technology, with temporary appointments at Washington University in St. Louis, Tel Aviv University, the University of Hawaii, the University of Göttingen (Fulbright Professor), the University of Costa Rica, the Free University of Amsterdam, the Mexican Centre for Mathematical Research (CIMAT), and as Gauss Professor in the Göttingen Academy of Sciences. He is currently professor emeritus of mathematics at the Georgia Institute of Technology and Research Scholar in Residence at California Polytechnic State University, San Luis Obispo.
Selected publications
• Theodore P. Hill (1995). "A Statistical Derivation of the Significant-Digit Law" (PDF). Statistical Science. 10 (4): 354–363. doi:10.1214/ss/1177009869. MR 1421567.
• Theodore P. Hill (July–August 1998). "The First Digit Phenomenon" (PDF). American Scientist. 86 (4): 358+. Bibcode:1998AmSci..86..358H. doi:10.1511/1998.4.358. S2CID 13553246.
• Theodore P. Hill (July–August 2000). "Mathematical Devices for Getting a Fair Share" (PDF). American Scientist. 88 (4): 325+. Bibcode:2000AmSci..88..325H. doi:10.1511/2000.4.325. S2CID 221539202.
• Theodore P. Hill (March–April 2009). "Knowing When to Stop". American Scientist. 97 (2): 126+. doi:10.1511/2009.77.126. S2CID 124798270.
• Arno Berger & Theodore P. Hill (2015). An Introduction to Benford's Law. Princeton University Press. ISBN 978-0-691-16306-2.
• Theodore P. Hill (2017). Pushing Limits: From West Point to Berkeley and Beyond. American Mathematical Society and Mathematical Association of America. ISBN 978-1-4704-3584-4.
• Theodore P. Hill (2018). "Slicing Sandwiches, States, and Solar Systems". American Scientist. 106 (1): 42–49. doi:10.1511/2018.106.1.42.
• Theodore P. Hill (2020). Pushing Limits: Memoir of a Maverick from Soldier to Scholar. Wise Ink Creative Publishing. ISBN 978-1-63489-351-0.
References
1. Brase, Charles Henry; Brase, Corrinne Pellillo (2014-01-01). Understandable Statistics. Cengage Learning. pp. 436–. ISBN 9781305142909. Retrieved 25 February 2014.
2. Azvolinsky, Anna (2018-09-27). "A Retracted Paper on Sex Differences Ignites Debate". The Scientist. Retrieved 2019-02-01.
3. Hill, Theodore P. (2020-07-13). "Modeling the evolution of differences in variability between sexes". Journal of Interdisciplinary Mathematics. 23 (5): 1009–1031. doi:10.1080/09720502.2020.1769827. S2CID 221060074. Retrieved 7 October 2020.
4. "What really happened when two mathematicians tried to publish a paper on gender differences? The tale of the emails". Retraction Watch. September 17, 2018.
External links
• "Ted Hill's webpage".
• Theodore Preston Hill at the Mathematics Genealogy Project
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Theodore Motzkin
Theodore Samuel Motzkin (26 March 1908 – 15 December 1970) was an Israeli-American mathematician.[1]
Theodore Motzkin
Born(1908-03-26)March 26, 1908
Berlin, German Empire
DiedOctober 15, 1970(1970-10-15) (aged 62)
Los Angeles
NationalityAmerican
Alma materUniversity of Basel
Known forMotzkin transposition theorem
Motzkin number
PIDs that are not EDs
Linear programming
Fourier–Motzkin elimination
Scientific career
InstitutionsUCLA
Doctoral advisorAlexander Ostrowski
Doctoral studentsJohn Selfridge
Rafael Artzy
Biography
Motzkin's father Leo Motzkin, a Ukrainian Jew, went to Berlin at the age of thirteen to study mathematics. He pursued university studies in the topic and was accepted as a graduate student by Leopold Kronecker, but left the field to work for the Zionist movement before finishing a dissertation.[2]
Motzkin grew up in Berlin and started studying mathematics at an early age as well, entering university when he was only 15.[2] He received his Ph.D. in 1934 from the University of Basel under the supervision of Alexander Ostrowski[3] for a thesis on the subject of linear programming[2] (Beiträge zur Theorie der linearen Ungleichungen, "Contributions to the Theory of Linear Inequalities", 1936[4]).
In 1935, Motzkin was appointed to the Hebrew University in Jerusalem, contributing to the development of mathematical terminology in Hebrew.[4] In 1936 he was an Invited Speaker at the International Congress of Mathematicians in Oslo.[5] During World War II, he worked as a cryptographer for the British government.[2]
In 1948, Motzkin moved to the United States. After two years at Harvard and Boston College, he was appointed at UCLA in 1950, becoming a professor in 1960.[4] He worked there until his retirement.[2]
Motzkin married Naomi Orenstein in Jerusalem. The couple had three sons:
• Aryeh Leo Motzkin - Orientalist
• Gabriel Motzkin - philosopher
• Elhanan Motzkin - mathematician
Contributions to mathematics
Motzkin's dissertation contained an important contribution to the nascent theory of linear programming (LP), but its importance was only recognized after an English translation appeared in 1951. He would continue to play an important role in the development of LP while at UCLA.[4] Apart from this, Motzkin published about diverse problems in algebra, graph theory, approximation theory, combinatorics, numerical analysis, algebraic geometry and number theory.[4]
The Motzkin transposition theorem, Motzkin numbers and the Fourier–Motzkin elimination are named after Theodore Motzkin. He first developed the "double description" algorithm of polyhedral combinatorics and computational geometry.[6] He was the first to prove the existence of principal ideal domains that are not Euclidean domains, $ \mathbb {Z} \left[{\frac {1+{\sqrt {-19}}}{2}}\right]$ being his first example.
Motzkin found the first explicit example of a nonnegative polynomial which is not sum of squares, known as the Motzkin polynomial $X^{4}Y^{2}+X^{2}Y^{4}-3X^{2}Y^{2}+1$.[7]
The quote "complete disorder is impossible," describing Ramsey theory, is attributed to him.[8]
See also
• Cyclic polytope
• Pentagram map, a related concept
References
Wikiquote has quotations related to Theodore Motzkin.
1. Motzkin, Theodore S. (1983). David Cantor; Basil Gordon; Bruce Rothschild (eds.). Theodore S. Motzkin: Selected papers. Contemporary Mathematicians. Boston, Mass.: Birkhäuser. pp. xxvi+530. ISBN 3-7643-3087-2. MR 0693096.
2. O'Connor, John J.; Robertson, Edmund F. "Theodore Motzkin". MacTutor History of Mathematics Archive. University of St Andrews.
3. Theodore Motzkin at the Mathematics Genealogy Project
4. Joachim Schwermer (1997). "Motzkin, Theodor Samuel". Neue Deutsche Biographie. Vol. 18. pp. 231 ff.
5. Motzkin, Th. (1936). "Sur le produit des spaces métriques". In: Congrès International des Mathématiciens. pp. 137–138.
6. Motzkin, T. S.; Raiffa, H.; Thompson, G. L.; Thrall, R. M. (1953). "The double description method". Contributions to the theory of games. Annals of Mathematics Studies. Vol. 2. Princeton, N. J.: Princeton University Press. pp. 51–73. MR 0060202.
7. Motzkin, T. S. (1967). "The arithmetic-geometric inequality". Inequalities (Proc. Sympos. Wright-Patterson Air Force Base, Ohio, 1965). New York: Academic Press. pp. 205–224. MR 0223521.
8. Hans Jürgen Prömel (2005). "Complete Disorder is Impossible: The Mathematical Work of Walter Deuber". Combinatorics, Probability and Computing. Cambridge University Press. 14: 3–16. doi:10.1017/S0963548304006674. S2CID 37243306.
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Theodore Strong
Theodore Strong (July 26, 1790 – February 1, 1869) was an American mathematician.
Theodore Strong
Born(1790-07-26)July 26, 1790
South Hadley, Massachusetts
DiedFebruary 1, 1869(1869-02-01) (aged 78)
New Brunswick, New Jersey
Alma materYale College
OccupationMathematician
Spouse
Lucy Dix
(m. 1818)
Children7
Parent(s)Joseph Strong
Sophia Woodbridge Strong
RelativesMaltby Strong (brother)
Benjamin Ruggles Woodbridge (uncle)
Early life
Strong was born in South Hadley, Massachusetts on July 26, 1790. He was the second son of Rev. Joseph Strong and Sophia (née Woodbridge) Strong.[1] Through his paternal grandfather, also known as Rev. Joseph Strong, he was a direct descendant of Joseph Strong, who settled in Dorchester, Massachusetts in 1630, and through his maternal grandfather, Rev. John Woodbridge of South Hadley, he was a direct descendant of Governor Thomas Dudley,[2] and Rev. John Woodbridge, who came to Massachusetts in 1634.[3]
After his father's death, he was adopted by his wealthy unmarried uncle, Benjamin Ruggles Woodbridge, who raised him as his own son. Strong graduated from Yale College in 1812.[1]
Career
Upon his graduation he was appointed Tutor in Mathematics in Hamilton College, then just organized, and in 1816 he was made Professor of Mathematics and Natural Philosophy, and so remained until 1827, when he was called to a similar position in Rutgers College, New Brunswick, N. J., where he also served as the college's longtime vice president.[4] Strong was elected an Associate Fellow of the American Academy of Arts and Sciences in 1832.[5] He retired from Rutgers in 1863.[1]
He published various mathematical papers in the first series of Silliman's Journal, and an Algebra of high order in 1859. A Treatise on the Differential and Integral Calculus was in press at the time of his death. He received the degree of Doctor of Laws from Rutgers College in 1835. He was one of fifty charter members of the National Academy of Sciences, to which he was formally named in 1863, shortly after the death of a son.[4] Strong was also an associate of many other scientific bodies.[1]
He was elected as a member of the American Philosophical Society in 1844.[6]
Personal life
On September 3, 1818, Dr. Strong was married Lucy Dix (1798–1875) of Littleton, Massachusetts. Lucy and her twin brother John Dix were children of Captain John Dix and Huldah (née Warren) Dix. Her sister, Mary Hartwell Warren was the wife of Asa Mahan, the 1st President of Oberlin College.[7] Together, Theodore and Lucy were the parents of seven children, two sons and five daughters:[8]
• Mary Dix Strong (1819–1873), who married U.S. Representative John Van Dyke.[9]
• Sophia Woodbridge Strong (b. 1821), who married Richard Hasluck.[10]
• Sarah Bowers Strong (1823–1839), who died young.[10]
• Harriet Strong (1825–1893), who married U.S. Representative John W. Ferdon.[10]
• Benjamin Ruggles Woodbridge Strong (1827–1907), who married Harriet Anna Hartwell (1827–1909).[10]
• Lucy Dix Strong (1832–1834), who died young.[10]
• Theodore Strong Jr. (1836–1863), who died in Belle Plains, Virginia while serving with the Union Army.[10]
Strong resided in New Brunswick until his death, which took place on February 1, 1869.[10] He was survived by his wife, one son and two daughters.[1]
Descendants
Through his son Woodbridge, he was a grandfather of New Jersey State Senator, Theodore Strong,[2] who married Cornelia Livingston Van Rensselaer, parents of prominent lawyer, Stephen Van Rensselaer Strong.[11][12]
References
1. Dexter, Franklin Bowditch (1912). Biographical Sketches of the Graduates of Yale College with Annals of the College History: September 1805-September 1815. H. Holt. pp. 501-503. Retrieved 8 April 2019.
2. Thurtle, Robert Glenn (2009). Lineage Book of Hereditary Order of Descendants of Colonial Governors. Genealogical Publishing Com. p. 267. ISBN 9780806350875. Retrieved 8 April 2019.
3. Johnson, Rossiter; Brown, John Howard (1904). The Twentieth Century Biographical Dictionary of Notable Americans ... Biographical Society. p. 75. Retrieved 8 April 2019.
4. "Rutgers in the Civil War," Journal of the Rutgers University Libraries, Vol. 66 (2014), page 98
5. "Book of Members, 1780–2010: Chapter B" (PDF). American Academy of Arts and Sciences. Retrieved September 13, 2016.
6. "APS Member History". search.amphilsoc.org. Retrieved 2021-04-12.
7. The Congregational Year-book. Congregational Publishing Society. 1890. p. 31. Retrieved 8 April 2019.
8. Bradley, Joseph P. (1879). A Memoir of Theodore Strong, LL.D.: Prepared at the Request of The National Academy of Science, and Read Before that Body, Thursday Evening, April 17, 1879. Joseph L. Pearson. Retrieved 8 April 2019.
9. Honeyman, Abraham Van Doren (1909). Honeyman family (Honeyman, Honyman, Hunneman, etc.) in Scotland and America, 1548-1908. N.J. Honeyman's Pub. Hs. pp. 222-223. Retrieved 8 April 2019.
10. Dwight, Benjamin Woodbridge (1871). The History of the Descendants of Elder John Strong, of Northampton, Mass. J. Munsell. pp. 362-635. Retrieved 8 April 2019.
11. "Stephen Strong, 68, Trial Lawyer, Dies". The New York Times. 15 February 1975. Retrieved 8 April 2019.
12. Reynolds, Cuyler (1914). Genealogical and Family History of Southern New York and the Hudson River Valley: A Record of the Achievements of Her People in the Making of a Commonwealth and the Building of a Nation. Lewis Historical Publishing Company. p. 1158. Retrieved 8 April 2019.
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| Wikipedia |
Theodore William Chaundy
Theodore William Chaundy (19 January 1889 – 14 April 1966) was an English mathematician who introduced Burchnall–Chaundy theory.
Theodore William Chaundy
Born(1889-01-19)19 January 1889
Oxford, England
Died14 April 1966(1966-04-14) (aged 77)
Alma materBalliol College, Oxford
Known forBurchnall–Chaundy theory
SpouseHilda Weston Dott
Scientific career
FieldsDifferential calculus
InstitutionsOxford University
Doctoral studentsKathleen Ollerenshaw
Chaundy was born to widowed businessman John Chaundy and his second wife Sarah Pates in their shop-cum-home at 49 Broad Street in Oxford. John had eight children, one of whom died as a toddler, with his late first wife and died barely a year after Chaundy was born. The Chaundy home along Broad Street has since been demolished.[1]
Chaundy attended Oxford High School for Boys and read mathematics at Balliol College, Oxford on a scholarship. In 1912 he became a lecturer at Oxford and later named a Fellow of Christ Church, Oxford. He married Hilda Weston Dott (1890–1986) in 1920. They had five children and thirteen grandchildren.[1]
Publications
• Chaundy, Theodore (1935). The differential calculus. Oxford: Clarendon Press.
• Chaundy, T. W.; Barrett, P. R.; Batey, Charles (1954). The printing of mathematics. Aids for authors and editors and rules for compositors and readers at the University Press, Oxford. Oxford University Press. ISBN 9780608111261. MR 0062667.
• Chaundy, T. W. (1969). McLeod, J. Bryce (ed.). Elementary differential equations. Oxford: Clarendon Press. ISBN 978-0-19-853142-5. MR 0257444.
References
Sources
• Ferrar, W. L. (1966), "Theodore William Chaundy", Journal of the London Mathematical Society, Second Series, 41: 755–756, doi:10.1112/jlms/s1-41.1.755, ISSN 0024-6107, MR 0197263
• Papers of Theodore William Chaundy at the National Archives
• Archive of Theodore William Chaundy at the Department of Special Collections and Western Manuscripts, Bodleian Library, University of Oxford
Notes
1. "Burials: John & Maria Chaundy". St Sepulchre's Cemetery.
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Theodore von Kármán Prize
The Theodore von Kármán Prize in applied mathematics is awarded every fifth year to an individual in recognition of his or her notable application of mathematics to mechanics and/or the engineering sciences.[1] This award was established and endowed in 1968 in honor of Theodore von Kármán by the Society for Industrial and Applied Mathematics (SIAM).[2]
List of recipients
• 1972 Geoffrey Ingram Taylor
• 1979 George F. Carrier and Joseph B. Keller
• 1984 Julian D. Cole
• 1989 Paul R. Garabedian
• 1994 Herbert B. Keller
• 1999 Stuart S. Antman, John M. Ball and Simone Zuccher
• 2004 Roland Glowinski
• 2009 Mary F. Wheeler
• 2014 Weinan E and Richard D. James
• 2020 Kaushik Bhattacharya[3]
See also
• List of mathematics awards
References
1. "The Theodore von Kármán Prize". The MacTutor History of Mathematics archive. Archived from the original on 2008-10-12. Retrieved 2009-09-21.
2. "Theodore von Kármán Prize". Society for Industrial and Applied Mathematics. Retrieved 2009-09-21.
3. "Kaushik Bhattacharya Awarded 2020 Theodore von Kármán Prize". caltech.net. September 11, 2020. Retrieved November 27, 2020.
Society for Industrial and Applied Mathematics
Awards
• John von Neumann Lecture
• SIAM Fellowship
• Germund Dahlquist Prize
• George David Birkhoff Prize
• Norbert Wiener Prize in Applied Mathematics
• Ralph E. Kleinman Prize
• J. D. Crawford Prize
• J. H. Wilkinson Prize for Numerical Software
• W. T. and Idalia Reid Prize in Mathematics
• Theodore von Kármán Prize
• George Pólya Prize
• Peter Henrici Prize
• SIAM/ACM Prize in Computational Science and Engineering
• SIAM Prize for Distinguished Service to the Profession
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| Wikipedia |
Square root of 3
The square root of 3 is the positive real number that, when multiplied by itself, gives the number 3. It is denoted mathematically as $ {\sqrt {3}}$ or $3^{1/2}$. It is more precisely called the principal square root of 3 to distinguish it from the negative number with the same property. The square root of 3 is an irrational number. It is also known as Theodorus' constant, after Theodorus of Cyrene, who proved its irrationality.
Square root of 3
The height of an equilateral triangle with sides of length 2 equals the square root of 3.
Representations
Decimal1.7320508075688772935...
Continued fraction$1+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{1+\ddots }}}}}}}}}}$
Binary1.10111011011001111010...
Hexadecimal1.BB67AE8584CAA73B...
As of December 2013, its numerical value in decimal notation had been computed to at least ten billion digits.[1] Its decimal expansion, written here to 65 decimal places, is given by OEIS: A002194:
1.732050807568877293527446341505872366942805253810380628055806
The fraction $ {\frac {97}{56}}$ (1.732142857...) can be used as a good approximation. Despite having a denominator of only 56, it differs from the correct value by less than $ {\frac {1}{10,000}}$ (approximately $ 9.2\times 10^{-5}$, with a relative error of $ 5\times 10^{-5}$). The rounded value of 1.732 is correct to within 0.01% of the actual value.
The fraction $ {\frac {716,035}{413,403}}$ (1.73205080756...) is accurate to $ 1\times 10^{-11}$.
Archimedes reported a range for its value: $ ({\frac {1351}{780}})^{2}>3>({\frac {265}{153}})^{2}$.[2]
The lower limit $ {\frac {1351}{780}}$ is an accurate approximation for ${\sqrt {3}}$ to $ {\frac {1}{608,400}}$ (six decimal places, relative error $ 3\times 10^{-7}$) and the upper limit $ {\frac {265}{153}}$ to $ {\frac {2}{23,409}}$ (four decimal places, relative error $ 1\times 10^{-5}$).
Expressions
It can be expressed as the continued fraction [1; 1, 2, 1, 2, 1, 2, 1, …] (sequence A040001 in the OEIS).
So it is true to say:
${\begin{bmatrix}1&2\\1&3\end{bmatrix}}^{n}={\begin{bmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{bmatrix}}$
then when $n\to \infty $ :
${\sqrt {3}}=2\cdot {\frac {a_{22}}{a_{12}}}-1$
It can also be expressed by generalized continued fractions such as
$[2;-4,-4,-4,...]=2-{\cfrac {1}{4-{\cfrac {1}{4-{\cfrac {1}{4-\ddots }}}}}}$
which is [1; 1, 2, 1, 2, 1, 2, 1, …] evaluated at every second term.
Geometry and trigonometry
The height of an equilateral triangle with edge length 2 is √3. Also, the long leg of a 30-60-90 triangle with hypotenuse 2.
And, the height of a regular hexagon with sides of length 1.
The square root of 3 can be found as the leg length of an equilateral triangle that encompasses a circle with a diameter of 1.
If an equilateral triangle with sides of length 1 is cut into two equal halves, by bisecting an internal angle across to make a right angle with one side, the right angle triangle's hypotenuse is length one, and the sides are of length $ {\frac {1}{2}}$ and $ {\frac {\sqrt {3}}{2}}$. From this, $ \tan {60^{\circ }}={\sqrt {3}}$, $ \sin {60^{\circ }}={\frac {\sqrt {3}}{2}}$, and $ \cos {30^{\circ }}={\frac {\sqrt {3}}{2}}$.
The square root of 3 also appears in algebraic expressions for various other trigonometric constants, including[3] the sines of 3°, 12°, 15°, 21°, 24°, 33°, 39°, 48°, 51°, 57°, 66°, 69°, 75°, 78°, 84°, and 87°.
It is the distance between parallel sides of a regular hexagon with sides of length 1.
It is the length of the space diagonal of a unit cube.
The vesica piscis has a major axis to minor axis ratio equal to $1:{\sqrt {3}}$. This can be shown by constructing two equilateral triangles within it.
Other uses and occurrence
Power engineering
In power engineering, the voltage between two phases in a three-phase system equals $ {\sqrt {3}}$ times the line to neutral voltage. This is because any two phases are 120° apart, and two points on a circle 120 degrees apart are separated by $ {\sqrt {3}}$ times the radius (see geometry examples above).
Special functions
It is known that most roots of the nth derivatives of $J_{\nu }^{(n)}(x)$ (where n < 18 and $J_{\nu }(x)$ is the Bessel function of the first kind of order $\nu $) are transcendental. The only exceptions are the numbers $\pm {\sqrt {3}}$, which are the algebraic roots of both $J_{1}^{(3)}(x)$ and $J_{0}^{(4)}(x)$. [4]
See also
• Square root of 2
• Square root of 5
Other references
• [5]
• [6]
• [7]
References
1. Komsta, Łukasz (December 2013). "Computations | Łukasz Komsta". komsta.net. WordPress. Retrieved September 24, 2016.{{cite web}}: CS1 maint: url-status (link)
2. Knorr, Wilbur R. (June 1976). "Archimedes and the measurement of the circle: a new interpretation". Archive for History of Exact Sciences. 15 (2): 115–140. doi:10.1007/bf00348496. JSTOR 41133444. MR 0497462. S2CID 120954547. Retrieved November 15, 2022 – via SpringerLink.
3. Wiseman, Julian D. A. (June 2008). "Sin and Cos in Surds". JDAWiseman.com. Retrieved November 15, 2022.{{cite web}}: CS1 maint: url-status (link)
4. Lorch, Lee; Muldoon, Martin E. (1995). "Transcendentality of zeros of higher dereivatives of functions involving Bessel functions". International Journal of Mathematics and Mathematical Sciences. 18 (3): 551–560. doi:10.1155/S0161171295000706.
5. S., D.; Jones, M. F. (1968). "22900D approximations to the square roots of the primes less than 100". Mathematics of Computation. 22 (101): 234–235. doi:10.2307/2004806. JSTOR 2004806.
6. Uhler, H. S. (1951). "Approximations exceeding 1300 decimals for ${\sqrt {3}}$, ${\frac {1}{\sqrt {3}}}$, $\sin({\frac {\pi }{3}})$ and distribution of digits in them". Proc. Natl. Acad. Sci. U.S.A. 37 (7): 443–447. doi:10.1073/pnas.37.7.443. PMC 1063398. PMID 16578382.
7. Wells, D. (1997). The Penguin Dictionary of Curious and Interesting Numbers (Revised ed.). London: Penguin Group. p. 23.
• Podestá, Ricardo A. (2020). "A geometric proof that sqrt 3, sqrt 5, and sqrt 7 are irrational". arXiv:2003.06627 [math.GM].
External links
Wikimedia Commons has media related to Square root of 3.
• Theodorus' Constant at MathWorld
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| Wikipedia |
Theodosius of Bithynia
Theodosius of Bithynia (Greek: Θεοδόσιος; c. 169 BC – c. 100 BC) was a Greek astronomer and mathematician who wrote the Sphaerics, a book on the geometry of the sphere.
Life
Theodosius was mentioned by Strabo as among the residents of Bithynia distinguished for their learning, and one whose sons were also mathematicians. He was cited by Vitruvius as having invented a sundial suitable for any place on Earth.[1] He was not born in Tripolis, as is often said due to a confusion originating from Suidas.[2]
His chief work, the Sphaerics (Greek: σφαιρικά), provided the mathematics for spherical astronomy, and may have been based on a work by Eudoxus of Cnidus.[3] It is reasonably complete, and remained the main reference on the subject at least until the time of Pappus of Alexandria (4th century AD).[1] The work was translated into Arabic in the 10th century, and then into Latin in the early 16th century, but these versions were faulty. Francesco Maurolico translated the works later in the 16th century.[1]
In addition to the Sphaerics, two other works by Theodosius have survived: On Habitations, describing the appearances of the heavens at different climes and different times of the year, and On Days and Nights, a study of the apparent motion of the Sun. Both were published in Latin in the 16th century.[4]
Notes
1. Heath 1911, p. 771.
2. Neugebauer 1975, p. 750.
3. Ivor Bulmer-Thomas, "Theodosius of Bithynia," in Dictionary of Scientific Biography, Encyclopedia.com, 2008.
4. Heath 1911, pp. 771–772.
References
• Ivor Bulmer-Thomas, "Theodosius of Bithynia," Dictionary of Scientific Biography 13:319–320.
• also on line "Theodosius of Bithynia." Complete Dictionary of Scientific Biography. 2008. Encyclopedia.com. 25 Mar. 2015 .
• Heath, Thomas Little (1911). "Theodosius of Tripolis" . In Chisholm, Hugh (ed.). Encyclopædia Britannica. Vol. 26 (11th ed.). Cambridge University Press. pp. 771–772.
• Neugebauer, Otto (1975). A History of Ancient Mathematical Astronomy. Springer-Verlag.
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Theoni Pappas
Theoni Pappas (born 1944)[1] is an American mathematics teacher known for her books and calendars concerning popular mathematics.
Pappas is a graduate of the University of California, Berkeley, and earned a master's degree at Stanford University. She became a high school mathematics teacher in 1967.[2]
She is the author of books including:
• Mathematics Appreciation (1986)[3]
• The Joy of Mathematics (1986)[4]
• Greek Cooking for Everyone (with Elvira Monroe, 1989)
• Math Talk: Mathematical Ideas in Poems for Two Voices (1991)[5]
• More Joy of Mathematics: Exploring Mathematics All around You (1991)[6]
• Fractals, Googols, and Other Mathematical Tales (1993)
• The Magic of Mathematics: Discovering the Spell of Mathematics (1994)[7]
• The Music of Reason: Experience the Beauty of Mathematics through Quotations (1995)[8]
• Math for Kids & Other People Too! (1997)[9]
• The Adventures of Penrose: The Mathematical Cat (1997)[10]
• Mathematical Scandals (1997)[11]
• Math-a-Day: A Book of Days for Your Mathematical Year (1999)[12]
• Mathematical Footprints: Discovering Mathematical Impressions All around Us (1999)[13]
• Math Stuff (2002)[14]
• Further Adventures of Penrose the Mathematical Cat (2004)[15]
• Mathematical Snippets: Exploring Mathematical Ideas in Small Bites (2008)[16]
• Numbers and Other Math Ideas Come Alive (2012)[17]
• Do the Math! Math Challenges to Exercise Your Mind (2015)[18]
• More Math Adventures with Penrose the Mathematical Cat (2017)
• Mathematical Journeys: Math Ideas and the Secrets They Hold (2021)
Additionally, she has written a series of annual mathematics calendars in various editions.[19]
References
1. Birth year from VIAF authority control record, accessed 2021-10-31
2. "Theoni Pappas Biography", Grants and awards: Supporters, National Council of Teachers of Mathematics, retrieved 2021-10-31
3. Review of Mathematics Appreciation: A. Dean Hendrickson, The Arithmetic Teacher, JSTOR 41193107
4. Reviews of The Joy of Mathematics: Mary L. Maxwell, The Mathematics Teacher, JSTOR 27966686; Randy Roberts, The Mathematics Teacher, JSTOR 27965463
5. Review of Math Talk: Barbara Mooney, The Mathematics Teacher, JSTOR 27967359
6. Review of More Joy of Mathematics: Robert P. Stutts, The Mathematics Teacher, JSTOR 27968135
7. Review of The Magic of Mathematics: Art Johnson, The Mathematics Teacher, JSTOR 27969131
8. Review of The Music of Reason: Cynthia Barb, The Mathematics Teacher, JSTOR 27969814
9. Review of Math for Kids & Other People Too!: Jackie Faillace Getgood, Mathematics Teaching in the Middle School, JSTOR 41181709
10. Review of The Adventures of Penrose: Shirley Roberts, Teaching Children Mathematics, JSTOR 41197157
11. Review of Mathematical Scandals: Frank Swetz, The Mathematics Teacher, JSTOR 27970557
12. Review of Math-a-Day: Stuart Moskowitz, The Mathematics Teacher, JSTOR 20870615
13. Review of Mathematical Footprints: Jerry Lenz, The Mathematics Teacher, JSTOR 27971527
14. Review of Math Stuff: Andrew Janoosky, The Mathematics Teacher, JSTOR 20871444
15. Review of Further Adventures of Penrose the Mathematical Cat: Nancy W. Lewis, Mathematics Teaching in the Middle School, JSTOR 41182865
16. Review of Mathematical Snippets: Exploring Mathematical Ideas in Small Bites: Betty Jo Rodgers, The Mathematics Teacher, JSTOR 20876437
17. Reviews of Numbers and Other Math Ideas Come Alive: Marla Ann L. Baber, Mathematics Teaching in the Middle School, doi:10.5951/mathteacmiddscho.18.6.0387, JSTOR 10.5951/mathteacmiddscho.18.6.0387; Joseph Bettina, The Mathematics Teacher, doi:10.5951/mathteacher.107.8.0636, JSTOR 10.5951/mathteacher.107.8.0636; Shannon Hammond, The Mathematics Teacher, doi:10.5951/mathteacher.107.3.0238, JSTOR 10.5951/mathteacher.107.3.0238
18. Review of Do the Math!: The Mathematics Teacher, doi:10.5951/mathteacher.109.9.0717,
19. Reviews of Pappas's mathematical calendars:
• 1981, Margaret Holland, The Mathematics Teacher, JSTOR 27962309
• 1983, Margaret Holland, The Mathematics Teacher, JSTOR 27963372
• 1984, Margaret Holland, The Mathematics Teacher, JSTOR 27963915, JSTOR 27963981
• 1985, Beverly K. Kimes, The Mathematics Teacher, JSTOR 27964506; Stanley M. Lucas, The Mathematics Teacher, JSTOR 27964507; Carol Novillis Larson, The Arithmetic Teacher, JSTOR 41192692
• 1986, Beverly K. Kimes, The Mathematics Teacher, JSTOR 27964817; Carol Novillis Larson, The Arithmetic Teacher, JSTOR 41194158
• 1987, Irene W. Kann, The Arithmetic Teacher, JSTOR 41193137
• 1988, Paul M. Midkiff, The Arithmetic Teacher, JSTOR 41193398
• 1990, Karen Doyle Walton, The Mathematics Teacher, JSTOR 27966641; Arnold Friedman, Science Scope, JSTOR 45027769
• 1993, Lawrence Tripp, The Mathematics Teacher, JSTOR 27968363; Jane B. Murphy, The Arithmetic Teacher, JSTOR 41195897
• 1995, L. Lee Osburn, The Mathematics Teacher, JSTOR 27969779
• 1996, Carol Fisher, Teaching Children Mathematics, JSTOR 41196600
• 1997, Stephen Currie, Teaching Children Mathematics, JSTOR 41196805
• 1998, Monte J. Zerger, The Mathematics Teacher, JSTOR 27970559
• 2001, Corbin P. Smith, The Mathematics Teacher, JSTOR 20870738
• 2005, Andrew Johnson, The Mathematics Teacher, JSTOR 27971838
• 2007, Scott Anthony Barba, The Mathematics Teacher, JSTOR 27972358
• 2008, James N. Boyd, The Mathematics Teacher, JSTOR 20876255
• 2010, David Ebert, The Mathematics Teacher, JSTOR 20876641
• 2011, Beth Bos, The Mathematics Teacher, JSTOR 20876924; Karol Yeatts, Teaching Children Mathematics, JSTOR 41199706
• 2012, Bob Horton, The Mathematics Teacher, doi:10.5951/mathteacher.105.5.0396, JSTOR 10.5951/mathteacher.105.5.0396
• 2014, Patricia Baggett, The Mathematics Teacher, doi:10.5951/mathteacher.107.5.0394, JSTOR 10.5951/mathteacher.107.5.0394
• 2016, The Mathematics Teacher, doi:10.5951/mathteacher.109.6.0476, JSTOR 10.5951/mathteacher.109.6.0476
• 2018, The Mathematics Teacher, doi:10.5951/mathteacher.111.6.0477, JSTOR 10.5951/mathteacher.111.6.0477
• 2019, The Mathematics Teacher, doi:10.5951/mathteacher.112.3.0237
Further reading
• Pappas is profiled in Perl, Teri (1993), Women and Numbers: Lives of Women Mathematicians Plus Discovery Activities, Wide World Publishing/Tetra House
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| Wikipedia |
Theophil Henry Hildebrandt
Theophil Henry Hildebrandt (24 July 1888 – 9 October 1980) was an American mathematician who did research on functional analysis and integration theory.[2]
Theophil Henry Hildebrandt
Born(1888-07-24)24 July 1888
Dover, Ohio, US
Died9 October 1980(1980-10-09) (aged 92)
Ann Arbor, Michigan, US
SpouseDora E. Ware (married 1921)
Children4
AwardsChauvenet Prize (1929)[1]
Scientific career
Fieldsmathematician
InstitutionsUniversity of Michigan
ThesisA Contribution to the Foundations of Fréchet's Calcul Fonctionnel (1910)
Doctoral advisorEliakim Hastings Moore
Doctoral studentsRalph Saul Phillips
Charles Earl Rickart
John V. Wehausen
Hildebrandt was born in Dover, Ohio, graduated from high school at age 14 and at age 17 in 1905 received his bachelor's degree from the University of Illinois. As a graduate student at the University of Chicago he earned his master's degree in 1906 and his PhD in 1910, with thesis A Contribution to the Foundations of Fréchet's Calcul Fonctionnel written under the direction of E. H. Moore.[3] He became an instructor at the University of Michigan in 1909 and then a full professor in 1923, serving as chair of the mathematics department from 1934 until his retirement in 1957. His doctoral students include Ralph S. Phillips, Charles Earl Rickart, and John V. Wehausen.[3]
In 1929 Hildebrandt received the Chauvenet Prize for his 1926 expository article The Borel theorem and its generalizations.[1] He served two years, 1945 and 1946, as president of the American Mathematical Society. The U. of Michigan established in 1962 in his honor the T. H. Hildebrandt Research Instructorships, which were changed in 1974 to assistant professorships.
Hildebrandt, as an instructor at the U. of Michigan, enrolled in the School of Music and earned a degree in music with a major in organ. He played the organ in his local church. He married Dora E. Ware in 1921, and they had four children.[4]
He died, aged 92, in Ann Arbor, Michigan.
Selected publications
• T. H. Hildebrandt (1923). "On uniform limitedness of sets of functional operations". Bulletin of the American Mathematical Society. 29 (7): 309–315. doi:10.1090/s0002-9904-1923-03730-0. MR 1560736.
• T. H. Hildebrandt (1928). "Über vollstetige linear Transformationen". Acta Mathematica. 51 (1): 311–318. doi:10.1007/bf02545664. MR 1555265.
References
1. Hildebrandt, T. H. (1926). "The Borel Theorem and Its Generalizations". Bulletin of the American Mathematical Society. 32 (5): 423–474. doi:10.1090/S0002-9904-1926-04238-5.
2. AMS Presidents: A Timeline, Theophil Henry Hildebrandt
3. Theophil Henry Hildebrandt at the Mathematics Genealogy Project
4. Theophil Henry Hildebrandt | Faculty History Project
External links
• Theophil Henry Hildebrandt at the Mathematics Genealogy Project
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
Presidents of the American Mathematical Society
1888–1900
• John Howard Van Amringe (1888–1890)
• Emory McClintock (1891–1894)
• George William Hill (1895–1896)
• Simon Newcomb (1897–1898)
• Robert Simpson Woodward (1899–1900)
1901–1924
• E. H. Moore (1901–1902)
• Thomas Fiske (1903–1904)
• William Fogg Osgood (1905–1906)
• Henry Seely White (1907–1908)
• Maxime Bôcher (1909–1910)
• Henry Burchard Fine (1911–1912)
• Edward Burr Van Vleck (1913–1914)
• Ernest William Brown (1915–1916)
• Leonard Eugene Dickson (1917–1918)
• Frank Morley (1919–1920)
• Gilbert Ames Bliss (1921–1922)
• Oswald Veblen (1923–1924)
1925–1950
• George David Birkhoff (1925–1926)
• Virgil Snyder (1927–1928)
• Earle Raymond Hedrick (1929–1930)
• Luther P. Eisenhart (1931–1932)
• Arthur Byron Coble (1933–1934)
• Solomon Lefschetz (1935–1936)
• Robert Lee Moore (1937–1938)
• Griffith C. Evans (1939–1940)
• Marston Morse (1941–1942)
• Marshall H. Stone (1943–1944)
• Theophil Henry Hildebrandt (1945–1946)
• Einar Hille (1947–1948)
• Joseph L. Walsh (1949–1950)
1951–1974
• John von Neumann (1951–1952)
• Gordon Thomas Whyburn (1953–1954)
• Raymond Louis Wilder (1955–1956)
• Richard Brauer (1957–1958)
• Edward J. McShane (1959–1960)
• Deane Montgomery (1961–1962)
• Joseph L. Doob (1963–1964)
• Abraham Adrian Albert (1965–1966)
• Charles B. Morrey Jr. (1967–1968)
• Oscar Zariski (1969–1970)
• Nathan Jacobson (1971–1972)
• Saunders Mac Lane (1973–1974)
1975–2000
• Lipman Bers (1975–1976)
• R. H. Bing (1977–1978)
• Peter Lax (1979–1980)
• Andrew M. Gleason (1981–1982)
• Julia Robinson (1983–1984)
• Irving Kaplansky (1985–1986)
• George Mostow (1987–1988)
• William Browder (1989–1990)
• Michael Artin (1991–1992)
• Ronald Graham (1993–1994)
• Cathleen Synge Morawetz (1995–1996)
• Arthur Jaffe (1997–1998)
• Felix Browder (1999–2000)
2001–2024
• Hyman Bass (2001–2002)
• David Eisenbud (2003–2004)
• James Arthur (2005–2006)
• James Glimm (2007–2008)
• George Andrews (2009–2010)
• Eric Friedlander (2011–2012)
• David Vogan (2013–2014)
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| Wikipedia |
Theorem of Bertini
In mathematics, the theorem of Bertini is an existence and genericity theorem for smooth connected hyperplane sections for smooth projective varieties over algebraically closed fields, introduced by Eugenio Bertini. This is the simplest and broadest of the "Bertini theorems" applying to a linear system of divisors; simplest because there is no restriction on the characteristic of the underlying field, while the extensions require characteristic 0.[1][2]
Statement for hyperplane sections of smooth varieties
Let X be a smooth quasi-projective variety over an algebraically closed field, embedded in a projective space $\mathbf {P} ^{n}$. Let $|H|$ denote the complete system of hyperplane divisors in $\mathbf {P} ^{n}$. Recall that it is the dual space $(\mathbf {P} ^{n})^{\star }$ of $\mathbf {P} ^{n}$ and is isomorphic to $\mathbf {P} ^{n}$.
The theorem of Bertini states that the set of hyperplanes not containing X and with smooth intersection with X contains an open dense subset of the total system of divisors $|H|$. The set itself is open if X is projective. If $\dim(X)\geq 2$, then these intersections (called hyperplane sections of X) are connected, hence irreducible.
The theorem hence asserts that a general hyperplane section not equal to X is smooth, that is: the property of smoothness is generic.
Over an arbitrary field k, there is a dense open subset of the dual space $(\mathbf {P} ^{n})^{\star }$ whose rational points define hyperplanes smooth hyperplane sections of X. When k is infinite, this open subset then has infinitely many rational points and there are infinitely many smooth hyperplane sections in X.
Over a finite field, the above open subset may not contain rational points and in general there is no hyperplanes with smooth intersection with X. However, if we take hypersurfaces of sufficiently big degrees, then the theorem of Bertini holds.[3]
Outline of a proof
We consider the subfibration of the product variety $X\times |H|$ with fiber above $x\in X$ the linear system of hyperplanes that intersect X non-transversally at x.
The rank of the fibration in the product is one less than the codimension of $X\subset \mathbf {P} ^{n}$, so that the total space has lesser dimension than $n$ and so its projection is contained in a divisor of the complete system $|H|$.
General statement
Over any infinite field $k$ of characteristic 0, if X is a smooth quasi-projective $k$-variety, a general member of a linear system of divisors on X is smooth away from the base locus of the system. For clarification, this means that given a linear system $f:X\rightarrow \mathbf {P} ^{n}$, the preimage $f^{-1}(H)$ of a hyperplane H is smooth -- outside the base locus of f -- for all hyperplanes H in some dense open subset of the dual projective space $(\mathbf {P} ^{n})^{\star }$. This theorem also holds in characteristic p>0 when the linear system f is unramified. [4]
Generalizations
The theorem of Bertini has been generalized in various ways. For example, a result due to Steven Kleiman asserts the following (cf. Kleiman's theorem): for a connected algebraic group G, and any homogeneous G-variety X, and two varieties Y and Z mapping to X, let Yσ be the variety obtained by letting σ ∈ G act on Y. Then, there is an open dense subscheme H of G such that for σ ∈ H, $Y^{\sigma }\times _{X}Z$ is either empty or purely of the (expected) dimension dim Y + dim Z − dim X. If, in addition, Y and Z are smooth and the base field has characteristic zero, then H may be taken such that $Y^{\sigma }\times _{X}Z$ is smooth for all $\sigma \in H$, as well. The above theorem of Bertini is the special case where $X=\mathbb {P} ^{n}$ is expressed as the quotient of SLn by the parabolic subgroup of upper triangular matrices, Z is a subvariety and Y is a hyperplane.[5]
Theorem of Bertini has also been generalized to discrete valuation domains or finite fields, or for étale coverings of X.
The theorem is often used for induction steps.
See also
• Grothendieck's connectedness theorem
Notes
1. "Bertini theorems", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
2. Hartshorne, Ch. III.10.
3. Poonen, Bjorn (2004). "Bertini theorems over finite fields". Annals of Mathematics. 160 (3): 1099–1127. doi:10.4007/annals.2004.160.1099.
4. Jouanolou, Jean-Pierre (1983). Théorèmes de Bertini et applications. Boston, MA: Birkhäuser Boston, Inc. p. 89. ISBN 0-8176-3164-X.
5. Kleiman, Steven L. (1974), "The transversality of a general translate", Compositio Mathematica, 28: 287–297, ISSN 0010-437X
References
• Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157
• Bertini and his two fundamental theorems by Steven L. Kleiman, on the life and works of Eugenio Bertini
| Wikipedia |
Gerbaldi's theorem
In linear algebra and projective geometry, Gerbaldi's theorem, proved by Gerbaldi (1882), states that one can find six pairwise apolar linearly independent nondegenerate ternary quadratic forms. These are permuted by the Valentiner group.
References
• Gerbaldi, Francesco (1882), "Sui gruppi di sei coniche in involuzione", Torino Atti (in Italian), XVII: 566–580, JFM 14.0537.02
| Wikipedia |
Heine–Borel theorem
In real analysis the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states:
For a subset S of Euclidean space Rn, the following two statements are equivalent:
• S is closed and bounded
• S is compact, that is, every open cover of S has a finite subcover.
History and motivation
The history of what today is called the Heine–Borel theorem starts in the 19th century, with the search for solid foundations of real analysis. Central to the theory was the concept of uniform continuity and the theorem stating that every continuous function on a closed and bounded interval is uniformly continuous. Peter Gustav Lejeune Dirichlet was the first to prove this and implicitly he used the existence of a finite subcover of a given open cover of a closed interval in his proof.[1] He used this proof in his 1852 lectures, which were published only in 1904.[1] Later Eduard Heine, Karl Weierstrass and Salvatore Pincherle used similar techniques. Émile Borel in 1895 was the first to state and prove a form of what is now called the Heine–Borel theorem. His formulation was restricted to countable covers. Pierre Cousin (1895), Lebesgue (1898) and Schoenflies (1900) generalized it to arbitrary covers.[2]
Proof
If a set is compact, then it must be closed.
Let S be a subset of Rn. Observe first the following: if a is a limit point of S, then any finite collection C of open sets, such that each open set U ∈ C is disjoint from some neighborhood VU of a, fails to be a cover of S. Indeed, the intersection of the finite family of sets VU is a neighborhood W of a in Rn. Since a is a limit point of S, W must contain a point x in S. This x ∈ S is not covered by the family C, because every U in C is disjoint from VU and hence disjoint from W, which contains x.
If S is compact but not closed, then it has a limit point a not in S. Consider a collection C ′ consisting of an open neighborhood N(x) for each x ∈ S, chosen small enough to not intersect some neighborhood Vx of a. Then C ′ is an open cover of S, but any finite subcollection of C ′ has the form of C discussed previously, and thus cannot be an open subcover of S. This contradicts the compactness of S. Hence, every limit point of S is in S, so S is closed.
The proof above applies with almost no change to showing that any compact subset S of a Hausdorff topological space X is closed in X.
If a set is compact, then it is bounded.
Let $S$ be a compact set in $\mathbf {R} ^{n}$, and $U_{x}$ a ball of radius 1 centered at $x\in \mathbf {R} ^{n}$. Then the set of all such balls centered at $x\in S$ is clearly an open cover of $S$, since $\cup _{x\in S}U_{x}$ contains all of $S$. Since $S$ is compact, take a finite subcover of this cover. This subcover is the finite union of balls of radius 1. Consider all pairs of centers of these (finitely many) balls (of radius 1) and let $M$ be the maximum of the distances between them. Then if $C_{p}$ and $C_{q}$ are the centers (respectively) of unit balls containing arbitrary $p,q\in S$, the triangle inequality says:
$d(p,q)\leq d(p,C_{p})+d(C_{p},C_{q})+d(C_{q},q)\leq 1+M+1=M+2.$
So the diameter of $S$ is bounded by $M+2$.
Lemma: A closed subset of a compact set is compact.
Let K be a closed subset of a compact set T in Rn and let CK be an open cover of K. Then U = Rn \ K is an open set and
$C_{T}=C_{K}\cup \{U\}$
is an open cover of T. Since T is compact, then CT has a finite subcover $C_{T}',$ that also covers the smaller set K. Since U does not contain any point of K, the set K is already covered by $C_{K}'=C_{T}'\setminus \{U\},$ that is a finite subcollection of the original collection CK. It is thus possible to extract from any open cover CK of K a finite subcover.
If a set is closed and bounded, then it is compact.
If a set S in Rn is bounded, then it can be enclosed within an n-box
$T_{0}=[-a,a]^{n}$
where a > 0. By the lemma above, it is enough to show that T0 is compact.
Assume, by way of contradiction, that T0 is not compact. Then there exists an infinite open cover C of T0 that does not admit any finite subcover. Through bisection of each of the sides of T0, the box T0 can be broken up into 2n sub n-boxes, each of which has diameter equal to half the diameter of T0. Then at least one of the 2n sections of T0 must require an infinite subcover of C, otherwise C itself would have a finite subcover, by uniting together the finite covers of the sections. Call this section T1.
Likewise, the sides of T1 can be bisected, yielding 2n sections of T1, at least one of which must require an infinite subcover of C. Continuing in like manner yields a decreasing sequence of nested n-boxes:
$T_{0}\supset T_{1}\supset T_{2}\supset \ldots \supset T_{k}\supset \ldots $
where the side length of Tk is (2 a) / 2k, which tends to 0 as k tends to infinity. Let us define a sequence (xk) such that each xk is in Tk. This sequence is Cauchy, so it must converge to some limit L. Since each Tk is closed, and for each k the sequence (xk) is eventually always inside Tk, we see that L ∈ Tk for each k.
Since C covers T0, then it has some member U ∈ C such that L ∈ U. Since U is open, there is an n-ball B(L) ⊆ U. For large enough k, one has Tk ⊆ B(L) ⊆ U, but then the infinite number of members of C needed to cover Tk can be replaced by just one: U, a contradiction.
Thus, T0 is compact. Since S is closed and a subset of the compact set T0, then S is also compact (see the lemma above).
Heine–Borel property
The Heine–Borel theorem does not hold as stated for general metric and topological vector spaces, and this gives rise to the necessity to consider special classes of spaces where this proposition is true. They are called the spaces with the Heine–Borel property.
In the theory of metric spaces
A metric space $(X,d)$ is said to have the Heine–Borel property if each closed bounded[3] set in $X$ is compact.
Many metric spaces fail to have the Heine–Borel property, such as the metric space of rational numbers (or indeed any incomplete metric space). Complete metric spaces may also fail to have the property; for instance, no infinite-dimensional Banach spaces have the Heine–Borel property (as metric spaces). Even more trivially, if the real line is not endowed with the usual metric, it may fail to have the Heine–Borel property.
A metric space $(X,d)$ has a Heine–Borel metric which is Cauchy locally identical to $d$ if and only if it is complete, $\sigma $-compact, and locally compact.[4]
In the theory of topological vector spaces
A topological vector space $X$ is said to have the Heine–Borel property[5] (R.E. Edwards uses the term boundedly compact space[6]) if each closed bounded[7] set in $X$ is compact.[8] No infinite-dimensional Banach spaces have the Heine–Borel property (as topological vector spaces). But some infinite-dimensional Fréchet spaces do have, for instance, the space $C^{\infty }(\Omega )$ of smooth functions on an open set $\Omega \subset \mathbb {R} ^{n}$[6] and the space $H(\Omega )$ of holomorphic functions on an open set $\Omega \subset \mathbb {C} ^{n}$.[6] More generally, any quasi-complete nuclear space has the Heine–Borel property. All Montel spaces have the Heine–Borel property as well.
See also
• Bolzano–Weierstrass theorem
Notes
1. Raman-Sundström, Manya (August–September 2015). "A Pedagogical History of Compactness". American Mathematical Monthly. 122 (7): 619–635. arXiv:1006.4131. doi:10.4169/amer.math.monthly.122.7.619. JSTOR 10.4169/amer.math.monthly.122.7.619. S2CID 119936587.
2. Sundström, Manya Raman (2010). "A pedagogical history of compactness". arXiv:1006.4131v1 [math.HO].
3. A set $B$ in a metric space $(X,d)$ is said to be bounded if it is contained in a ball of a finite radius, i.e. there exists $a\in X$ and $r>0$ such that $B\subseteq \{x\in X:\ d(x,a)\leq r\}$.
4. Williamson & Janos 1987.
5. Kirillov & Gvishiani 1982, Theorem 28.
6. Edwards 1965, 8.4.7.
7. A set $B$ in a topological vector space $X$ is said to be bounded if for each neighborhood of zero $U$ in $X$ there exists a scalar $\lambda $ such that $B\subseteq \lambda \cdot U$.
8. In the case when the topology of a topological vector space $X$ is generated by some metric $d$ this definition is not equivalent to the definition of the Heine–Borel property of $X$ as a metric space, since the notion of bounded set in $X$ as a metric space is different from the notion of bounded set in $X$ as a topological vector space. For instance, the space ${\mathcal {C}}^{\infty }[0,1]$ of smooth functions on the interval $[0,1]$ with the metric $d(x,y)=\sum _{k=0}^{\infty }{\frac {1}{2^{k}}}\cdot {\frac {\max _{t\in [0,1]}|x^{(k)}(t)-y^{(k)}(t)|}{1+\max _{t\in [0,1]}|x^{(k)}(t)-y^{(k)}(t)|}}$ (here $x^{(k)}$ is the $k$-th derivative of the function $x\in {\mathcal {C}}^{\infty }[0,1]$) has the Heine–Borel property as a topological vector space but not as a metric space.
References
• P. Dugac (1989). "Sur la correspondance de Borel et le théorème de Dirichlet–Heine–Weierstrass–Borel–Schoenflies–Lebesgue". Arch. Int. Hist. Sci. 39: 69–110.
• BookOfProofs: Heine-Borel Property
• Jeffreys, H.; Jeffreys, B.S. (1988). Methods of Mathematical Physics. Cambridge University Press. ISBN 978-0521097239.
• Williamson, R.; Janos, L. (1987). "Construction metrics with the Heine-Borel property". Proc. AMS. 100 (3): 567–573. doi:10.1090/S0002-9939-1987-0891165-X.
• Kirillov, A.A.; Gvishiani, A.D. (1982). Theorems and Problems in Functional Analysis. Springer-Verlag New York. ISBN 978-1-4613-8155-6.
• Edwards, R.E. (1965). Functional analysis. Holt, Rinehart and Winston. ISBN 0030505356.
External links
• Ivan Kenig, Dr. Prof. Hans-Christian Graf v. Botthmer, Dmitrij Tiessen, Andreas Timm, Viktor Wittman (2004). The Heine–Borel Theorem. Hannover: Leibniz Universität. Archived from the original (avi • mp4 • mov • swf • streamed video) on 2011-07-19.
• "Borel-Lebesgue covering theorem", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Mathworld "Heine-Borel Theorem"
• "An Analysis of the First Proofs of the Heine-Borel Theorem - Lebesgue's Proof"
| Wikipedia |
Pappus's centroid theorem
In mathematics, Pappus's centroid theorem (also known as the Guldinus theorem, Pappus–Guldinus theorem or Pappus's theorem) is either of two related theorems dealing with the surface areas and volumes of surfaces and solids of revolution.
The theorems are attributed to Pappus of Alexandria[lower-alpha 1] and Paul Guldin.[lower-alpha 2] Pappus's statement of this theorem appears in print for the first time in 1659, but it was known before, by Kepler in 1615 and by Guldin in 1640.[4]
The first theorem
The first theorem states that the surface area A of a surface of revolution generated by rotating a plane curve C about an axis external to C and on the same plane is equal to the product of the arc length s of C and the distance d traveled by the geometric centroid of C:
$A=sd.$
For example, the surface area of the torus with minor radius r and major radius R is
$A=(2\pi r)(2\pi R)=4\pi ^{2}Rr.$
Proof
A curve given by the positive function $f(x)$ is bounded by two points given by:
$a\geq 0$ and $b\geq a$
If $dL$ is an infinitesimal line element tangent to the curve, the length of the curve is given by:
$L=\int _{a}^{b}dL=\int _{a}^{b}{\sqrt {dx^{2}+dy^{2}}}=\int _{a}^{b}{\sqrt {1+\left({\frac {dy}{dx}}\right)^{2}}}\,dx$
The $y$ component of the centroid of this curve is:
${\bar {y}}={\frac {1}{L}}\int _{a}^{b}y\,dL={\frac {1}{L}}\int _{a}^{b}y{\sqrt {1+\left({\frac {dy}{dx}}\right)^{2}}}\,dx$
The area of the surface generated by rotating the curve around the x-axis is given by:
$A=2\pi \int _{a}^{b}y\,dL=2\pi \int _{a}^{b}y{\sqrt {1+\left({\frac {dy}{dx}}\right)^{2}}}\,dx$
Using the last two equations to eliminate the integral we have:
$A=2\pi {\bar {y}}L$
The second theorem
The second theorem states that the volume V of a solid of revolution generated by rotating a plane figure F about an external axis is equal to the product of the area A of F and the distance d traveled by the geometric centroid of F. (The centroid of F is usually different from the centroid of its boundary curve C.) That is:
$V=Ad.$
For example, the volume of the torus with minor radius r and major radius R is
$V=(\pi r^{2})(2\pi R)=2\pi ^{2}Rr^{2}.$
This special case was derived by Johannes Kepler using infinitesimals.[lower-alpha 3]
Proof 1
The area bounded by the two functions:
$y=f(x),\,\qquad y\geq 0$
$y=g(x),\,\qquad f(x)\geq g(x)$
and bounded by the two lines:
$x=a\geq 0$ and $x=b\geq a$
is given by:
$A=\int _{a}^{b}dA=\int _{a}^{b}[f(x)-g(x)]\,dx$
The $x$ component of the centroid of this area is given by:
${\bar {x}}={\frac {1}{A}}\,\int _{a}^{b}x\,[f(x)-g(x)]\,dx$
If this area is rotated about the y-axis, the volume generated can be calculated using the shell method. It is given by:
$V=2\pi \int _{a}^{b}x\,[f(x)-g(x)]\,dx$
Using the last two equations to eliminate the integral we have:
$V=2\pi {\bar {x}}A$
Proof 2
Let $A$ be the area of $F$, $W$ the solid of revolution of $F$, and $V$ the volume of $W$. Suppose $F$ starts in the $xz$-plane and rotates around the $z$-axis. The distance of the centroid of $F$ from the $z$-axis is its $x$-coordinate
$R={\frac {\int _{F}x\,dA}{A}},$
and the theorem states that
$V=Ad=A\cdot 2\pi R=2\pi \int _{F}x\,dA.$
To show this, let $F$ be in the xz-plane, parametrized by $\mathbf {\Phi } (u,v)=(x(u,v),0,z(u,v))$ for $(u,v)\in F^{*}$, a parameter region. Since ${\boldsymbol {\Phi }}$ is essentially a mapping from $\mathbb {R} ^{2}$ to $\mathbb {R} ^{2}$, the area of $F$ is given by the change of variables formula:
$A=\int _{F}dA=\iint _{F^{*}}\left|{\frac {\partial (x,z)}{\partial (u,v)}}\right|\,du\,dv=\iint _{F^{*}}\left|{\frac {\partial x}{\partial u}}{\frac {\partial z}{\partial v}}-{\frac {\partial x}{\partial v}}{\frac {\partial z}{\partial u}}\right|\,du\,dv,$
where $\left|{\tfrac {\partial (x,z)}{\partial (u,v)}}\right|$ is the determinant of the Jacobian matrix of the change of variables.
The solid $W$ has the toroidal parametrization ${\boldsymbol {\Phi }}(u,v,\theta )=(x(u,v)\cos \theta ,x(u,v)\sin \theta ,z(u,v))$ for $(u,v,\theta )$ in the parameter region $W^{*}=F^{*}\times [0,2\pi ]$; and its volume is
$V=\int _{W}dV=\iiint _{W^{*}}\left|{\frac {\partial (x,y,z)}{\partial (u,v,\theta )}}\right|\,du\,dv\,d\theta .$
Expanding,
${\begin{aligned}\left|{\frac {\partial (x,y,z)}{\partial (u,v,\theta )}}\right|&=\left|\det {\begin{bmatrix}{\frac {\partial x}{\partial u}}\cos \theta &{\frac {\partial x}{\partial v}}\cos \theta &-x\sin \theta \\[6pt]{\frac {\partial x}{\partial u}}\sin \theta &{\frac {\partial x}{\partial v}}\sin \theta &x\cos \theta \\[6pt]{\frac {\partial z}{\partial u}}&{\frac {\partial z}{\partial v}}&0\end{bmatrix}}\right|\\[5pt]&=\left|-{\frac {\partial z}{\partial v}}{\frac {\partial x}{\partial u}}\,x+{\frac {\partial z}{\partial u}}{\frac {\partial x}{\partial v}}\,x\right|=\ \left|-x\,{\frac {\partial (x,z)}{\partial (u,v)}}\right|=x\left|{\frac {\partial (x,z)}{\partial (u,v)}}\right|.\end{aligned}}$
The last equality holds because the axis of rotation must be external to $F$, meaning $x\geq 0$. Now,
${\begin{aligned}V&=\iiint _{W^{*}}\left|{\frac {\partial (x,y,z)}{\partial (u,v,\theta )}}\right|\,du\,dv\,d\theta \\[1ex]&=\int _{0}^{2\pi }\!\!\!\!\iint _{F^{*}}x(u,v)\left|{\frac {\partial (x,z)}{\partial (u,v)}}\right|du\,dv\,d\theta \\[6pt]&=2\pi \iint _{F^{*}}x(u,v)\left|{\frac {\partial (x,z)}{\partial (u,v)}}\right|\,du\,dv\\[1ex]&=2\pi \int _{F}x\,dA\end{aligned}}$
by change of variables.
Generalizations
The theorems can be generalized for arbitrary curves and shapes, under appropriate conditions.
Goodman & Goodman[6] generalize the second theorem as follows. If the figure F moves through space so that it remains perpendicular to the curve L traced by the centroid of F, then it sweeps out a solid of volume V = Ad, where A is the area of F and d is the length of L. (This assumes the solid does not intersect itself.) In particular, F may rotate about its centroid during the motion.
However, the corresponding generalization of the first theorem is only true if the curve L traced by the centroid lies in a plane perpendicular to the plane of C.
In n-dimensions
In general, one can generate an $n$ dimensional solid by rotating an $n-p$ dimensional solid $F$ around a $p$ dimensional sphere. This is called an $n$-solid of revolution of species $p$. Let the $p$-th centroid of $F$ be defined by
$R={\frac {\iint _{F}x^{p}\,dA}{A}},$
Then Pappus' theorems generalize to:[7]
Volume of $n$-solid of revolution of species $p$
= (Volume of generating $(n{-}p)$-solid) $\times $ (Surface area of $p$-sphere traced by the $p$-th centroid of the generating solid)
and
Surface area of $n$-solid of revolution of species $p$
= (Surface area of generating $(n{-}p)$-solid) $\times $ (Surface area of $p$-sphere traced by the $p$-th centroid of the generating solid)
The original theorems are the case with $n=3,\,p=1$.
Footnotes
1. See:[1]
They who look at these things are hardly exalted, as were the ancients and all who wrote the finer things. When I see everyone occupied with the rudiments of mathematics and of the material for inquiries that nature sets before us, I am ashamed; I for one have proved things that are much more valuable and offer much application. In order not to end my discourse declaiming this with empty hands, I will give this for the benefit of the readers:
The ratio of solids of complete revolution is compounded of (that) of the revolved figures and (that) of the straight lines similarly drawn to the axes from the centers of gravity in them; that of (solids of) incomplete (revolution) from (that) of the revolved figures and (that) of the arcs that the centers of gravity in them describe, where the (ratio) of these arcs is, of course, (compounded) of (that) of the (lines) drawn and (that) of the angles of revolution that their extremities contain, if these (lines) are also at (right angles) to the axes. These propositions, which are practically a single one, contain many theorems of all kinds, for curves and surfaces and solids, all at once and by one proof, things not yet and things already demonstrated, such as those in the twelfth book of the First Elements.
— Pappus, Collection, Book VII, ¶41‒42
2. "Quantitas rotanda in viam rotationis ducta, producit Potestatem Rotundam uno gradu altiorem, Potestate sive Quantitate rotata."[2] That is: "A quantity in rotation, multiplied by its circular trajectory, creates a circular power of higher degree, power, or quantity in rotation."[3]
3. Theorem XVIII of Kepler's Nova Stereometria Doliorum Vinariorum (1615):[5] "Omnis annulus sectionis circularis vel ellipticae est aequalis cylindro, cujus altitudo aequat longitudinem circumferentiae, quam centrum figurae circumductae descripsit, basis vero eadem est cum sectione annuli." Translation:[3] "Any ring whose cross-section is circular or elliptic is equal to a cylinder whose height equals the length of the circumference covered by the center of the figure during its circular movement, and whose base is equal to the section of the ring."
References
1. Pappus of Alexandria (1986) [c. 320]. Jones, Alexander (ed.). Book 7 of the Collection. Sources in the History of Mathematics and Physical Sciences. Vol. 8. New York: Springer-Verlag. doi:10.1007/978-1-4612-4908-5. ISBN 978-1-4612-4908-5.
2. Guldin, Paul (1640). De centro gravitatis trium specierum quanitatis continuae. Vol. 2. Vienna: Gelbhaar, Cosmerovius. p. 147. Retrieved 2016-08-04.
3. Radelet-de Grave, Patricia (2015-05-19). "Kepler, Cavalieri, Guldin. Polemics with the departed". In Jullien, Vincent (ed.). Seventeenth-Century Indivisibles Revisited. Science Networks. Historical Studies. Vol. 49. Basel: Birkhäuser. p. 68. doi:10.1007/978-3-319-00131-9. hdl:2117/28047. ISBN 978-3-3190-0131-9. ISSN 1421-6329. Retrieved 2016-08-04.
4. Bulmer-Thomas, Ivor (1984). "Guldin's Theorem--Or Pappus's?". Isis. 75 (2): 348–352. ISSN 0021-1753.
5. Kepler, Johannes (1870) [1615]. "Nova Stereometria Doliorum Vinariorum". In Frisch, Christian (ed.). Joannis Kepleri astronomi opera omnia. Vol. 4. Frankfurt: Heyder and Zimmer. p. 582. Retrieved 2016-08-04.
6. Goodman, A. W.; Goodman, G. (1969). "Generalizations of the Theorems of Pappus". The American Mathematical Monthly. 76 (4): 355–366. doi:10.1080/00029890.1969.12000217. JSTOR 2316426.
7. McLaren-Young-Sommerville, Duncan (1958). "8.17 Extensions of Pappus' Theorem". An introduction to the geometry of n dimensions. New York, NY: Dover.
External links
Wikimedia Commons has media related to Pappus-Guldinus theorem.
• Weisstein, Eric W. "Pappus's Centroid Theorem". MathWorld.
| Wikipedia |
Theorem of absolute purity
In algebraic geometry, the theorem of absolute (cohomological) purity is an important theorem in the theory of étale cohomology. It states:[1] given
• a regular scheme X over some base scheme,
• $i:Z\to X$ a closed immersion of a regular scheme of pure codimension r,
• an integer n that is invertible on the base scheme,
• ${\mathcal {F}}$ a locally constant étale sheaf with finite stalks and values in $\mathbb {Z} /n\mathbb {Z} $,
for each integer $m\geq 0$, the map
$\operatorname {H} ^{m}(Z_{\text{ét}};{\mathcal {F}})\to \operatorname {H} _{Z}^{m+2r}(X_{\text{ét}};{\mathcal {F}}(r))$
is bijective, where the map is induced by cup product with $c_{r}(Z)$.
The theorem was introduced in SGA 5 Exposé I, § 3.1.4. as an open problem. Later, Thomason proved it for large n and Gabber in general.
See also
• purity (algebraic geometry)
References
1. A version of the theorem is stated at Déglise, Frédéric; Fasel, Jean; Jin, Fangzhou; Khan, Adeel (2019-02-06). "Borel isomorphism and absolute purity". arXiv:1902.02055 [math.AG].
• Fujiwara, K.: A proof of the absolute purity conjecture (after Gabber). Algebraic geometry 2000, Azumino (Hotaka), pp. 153–183, Adv. Stud. Pure Math. 36, Math. Soc. Japan, Tokyo, 2002
• R. W. Thomason, Absolute cohomological purity, Bull. Soc. Math. France 112 (1984), no. 3, 397–406. MR 794741
| Wikipedia |
Theorem of the cube
In mathematics, the theorem of the cube is a condition for a line bundle over a product of three complete varieties to be trivial. It was a principle discovered, in the context of linear equivalence, by the Italian school of algebraic geometry. The final version of the theorem of the cube was first published by Lang (1959), who credited it to André Weil. A discussion of the history has been given by Kleiman (2005). A treatment by means of sheaf cohomology, and description in terms of the Picard functor, was given by Mumford (2008).
Statement
The theorem states that for any complete varieties U, V and W over an algebraically closed field, and given points u, v and w on them, any invertible sheaf L which has a trivial restriction to each of U× V × {w}, U× {v} × W, and {u} × V × W, is itself trivial. (Mumford p. 55; the result there is slightly stronger, in that one of the varieties need not be complete and can be replaced by a connected scheme.)
Special cases
On a ringed space X, an invertible sheaf L is trivial if isomorphic to OX, as an OX-module. If the base X is a complex manifold, then an invertible sheaf is (the sheaf of sections of) a holomorphic line bundle, and trivial means holomorphically equivalent to a trivial bundle, not just topologically equivalent.
Restatement using biextensions
Weil's result has been restated in terms of biextensions, a concept now generally used in the duality theory of abelian varieties.[1]
Theorem of the square
The theorem of the square (Lang 1959) (Mumford 2008, p.59) is a corollary (also due to Weil) applying to an abelian variety A. One version of it states that the function φL taking x∈A to T*
x
L⊗L−1 is a group homomorphism from A to Pic(A) (where T*
x
is translation by x on line bundles).
References
• Kleiman, Steven L. (2005), "The Picard scheme", Fundamental algebraic geometry, Math. Surveys Monogr., vol. 123, Providence, R.I.: American Mathematical Society, pp. 235–321, arXiv:math/0504020, Bibcode:2005math......4020K, MR 2223410
• Lang, Serge (1959), Abelian varieties, Interscience Tracts in Pure and Applied Mathematics, vol. 7, New York: Interscience Publishers, Inc., MR 0106225
• Mumford, David (2008) [1970], Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Providence, R.I.: American Mathematical Society, ISBN 978-81-85931-86-9, MR 0282985, OCLC 138290
Notes
1. Alexander Polishchuk, Abelian Varieties, Theta Functions and the Fourier Transform (2003), p. 122.
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Theorem of transition
In algebra, the theorem of transition is said to hold between commutative rings $A\subset B$ if[1][2]
1. $B$ dominates $A$; i.e., for each proper ideal I of A, $IB$ is proper and for each maximal ideal ${\mathfrak {n}}$ of B, ${\mathfrak {n}}\cap A$ is maximal
2. for each maximal ideal ${\mathfrak {m}}$ and ${\mathfrak {m}}$-primary ideal $Q$ of $A$, $\operatorname {length} _{B}(B/QB)$ is finite and moreover
$\operatorname {length} _{B}(B/QB)=\operatorname {length} _{B}(B/{\mathfrak {m}}B)\operatorname {length} _{A}(A/Q).$
Given commutative rings $A\subset B$ such that $B$ dominates $A$ and for each maximal ideal ${\mathfrak {m}}$ of $A$ such that $\operatorname {length} _{B}(B/{\mathfrak {m}}B)$ is finite, the natural inclusion $A\to B$ is a faithfully flat ring homomorphism if and only if the theorem of transition holds between $A\subset B$.[2]
References
1. Nagata 1975, Ch. II, § 19.
2. Matsumura 1986, Ch. 8, Exercise 22.1.
• Nagata, M. (1975). Local Rings. Interscience tracts in pure and applied mathematics. Krieger. ISBN 978-0-88275-228-0.
• Matsumura, Hideyuki (1986). Commutative ring theory. Cambridge Studies in Advanced Mathematics. Vol. 8. Cambridge University Press. ISBN 0-521-36764-6. MR 0879273. Zbl 0603.13001.
| Wikipedia |
Theorem on friends and strangers
The theorem on friends and strangers is a mathematical theorem in an area of mathematics called Ramsey theory.
For the friendship theorem of Paul Erdős, Alfréd Rényi, and Vera T. Sós characterizing graphs in which each two vertices have exactly one neighbor, see friendship graph.
Statement
Suppose a party has six people. Consider any two of them. They might be meeting for the first time—in which case we will call them mutual strangers; or they might have met before—in which case we will call them mutual acquaintances. The theorem says:
In any party of six people, at least three of them are (pairwise) mutual strangers or mutual acquaintances.
Conversion to a graph-theoretic setting
A proof of the theorem requires nothing but a three-step logic. It is convenient to phrase the problem in graph-theoretic language.
Suppose a graph has 6 vertices and every pair of (distinct) vertices is joined by an edge. Such a graph is called a complete graph (because there cannot be any more edges). A complete graph on $n$ vertices is denoted by the symbol $K_{n}$.
Now take a $K_{6}$. It has 15 edges in all. Let the 6 vertices stand for the 6 people in our party. Let the edges be coloured red or blue depending on whether the two people represented by the vertices connected by the edge are mutual strangers or mutual acquaintances, respectively. The theorem now asserts:
No matter how you colour the 15 edges of a $K_{6}$ with red and blue, you cannot avoid having either a red triangle—that is, a triangle all of whose three sides are red, representing three pairs of mutual strangers—or a blue triangle, representing three pairs of mutual acquaintances. In other words, whatever colours you use, there will always be at least one monochromatic triangle ( that is, a triangle all of whose edges have the same color ).
Proof
Choose any one vertex; call it P. There are five edges leaving P. They are each coloured red or blue. The pigeonhole principle says that at least three of them must be of the same colour; for if there are less than three of one colour, say red, then there are at least three that are blue.
Let A, B, C be the other ends of these three edges, all of the same colour, say blue. If any one of AB, BC, CA is blue, then that edge together with the two edges from P to the edge's endpoints forms a blue triangle. If none of AB, BC, CA is blue, then all three edges are red and we have a red triangle, namely, ABC.
Ramsey's paper
The utter simplicity of this argument, which so powerfully produces a very interesting conclusion, is what makes the theorem appealing. In 1930, in a paper entitled 'On a Problem of Formal Logic,' Frank P. Ramsey proved a very general theorem (now known as Ramsey's theorem) of which this theorem is a simple case. This theorem of Ramsey forms the foundation of the area known as Ramsey theory in combinatorics.
Boundaries to the theorem
The conclusion to the theorem does not hold if we replace the party of six people by a party of less than six. To show this, we give a coloring of K5 with red and blue that does not contain a triangle with all edges the same color. We draw K5 as a pentagon surrounding a star (a pentagram). We color the edges of the pentagon red and the edges of the star blue. Thus, 6 is the smallest number for which we can claim the conclusion of the theorem. In Ramsey theory, we write this fact as:
$R(3,3:2)=6.$
References
• V. Krishnamurthy. Culture, Excitement and Relevance of Mathematics, Wiley Eastern, 1990. ISBN 81-224-0272-0.
External links
• Party Acquaintances at cut-the-knot (requires Java)
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Theorem on formal functions
In algebraic geometry, the theorem on formal functions states the following:[1]
Let $f:X\to S$ be a proper morphism of noetherian schemes with a coherent sheaf ${\mathcal {F}}$ on X. Let $S_{0}$ be a closed subscheme of S defined by ${\mathcal {I}}$ and ${\widehat {X}},{\widehat {S}}$ formal completions with respect to $X_{0}=f^{-1}(S_{0})$ and $S_{0}$. Then for each $p\geq 0$ the canonical (continuous) map:
$(R^{p}f_{*}{\mathcal {F}})^{\wedge }\to \varprojlim _{k}R^{p}f_{*}{\mathcal {F}}_{k}$
is an isomorphism of (topological) ${\mathcal {O}}_{\widehat {S}}$-modules, where
• The left term is $\varprojlim R^{p}f_{*}{\mathcal {F}}\otimes _{{\mathcal {O}}_{S}}{\mathcal {O}}_{S}/{{\mathcal {I}}^{k+1}}$.
• ${\mathcal {F}}_{k}={\mathcal {F}}\otimes _{{\mathcal {O}}_{S}}({\mathcal {O}}_{S}/{\mathcal {I}}^{k+1})$
• The canonical map is one obtained by passage to limit.
The theorem is used to deduce some other important theorems: Stein factorization and a version of Zariski's main theorem that says that a proper birational morphism into a normal variety is an isomorphism. Some other corollaries (with the notations as above) are:
Corollary:[2] For any $s\in S$, topologically,
$((R^{p}f_{*}{\mathcal {F}})_{s})^{\wedge }\simeq \varprojlim H^{p}(f^{-1}(s),{\mathcal {F}}\otimes _{{\mathcal {O}}_{S}}({\mathcal {O}}_{s}/{\mathfrak {m}}_{s}^{k}))$
where the completion on the left is with respect to ${\mathfrak {m}}_{s}$.
Corollary:[3] Let r be such that $\operatorname {dim} f^{-1}(s)\leq r$ for all $s\in S$. Then
$R^{i}f_{*}{\mathcal {F}}=0,\quad i>r.$
Corollay:[4] For each $s\in S$, there exists an open neighborhood U of s such that
$R^{i}f_{*}{\mathcal {F}}|_{U}=0,\quad i>\operatorname {dim} f^{-1}(s).$
Corollary:[5] If $f_{*}{\mathcal {O}}_{X}={\mathcal {O}}_{S}$, then $f^{-1}(s)$ is connected for all $s\in S$.
The theorem also leads to the Grothendieck existence theorem, which gives an equivalence between the category of coherent sheaves on a scheme and the category of coherent sheaves on its formal completion (in particular, it yields algebralizability.)
Finally, it is possible to weaken the hypothesis in the theorem; cf. Illusie. According to Illusie (pg. 204), the proof given in EGA III is due to Serre. The original proof (due to Grothendieck) was never published.
The construction of the canonical map
Let the setting be as in the lede. In the proof one uses the following alternative definition of the canonical map.
Let $i':{\widehat {X}}\to X,i:{\widehat {S}}\to S$ be the canonical maps. Then we have the base change map of ${\mathcal {O}}_{\widehat {S}}$-modules
$i^{*}R^{q}f_{*}{\mathcal {F}}\to R^{p}{\widehat {f}}_{*}(i'^{*}{\mathcal {F}})$.
where ${\widehat {f}}:{\widehat {X}}\to {\widehat {S}}$ is induced by $f:X\to S$. Since ${\mathcal {F}}$ is coherent, we can identify $i'^{*}{\mathcal {F}}$ with ${\widehat {\mathcal {F}}}$. Since $R^{q}f_{*}{\mathcal {F}}$ is also coherent (as f is proper), doing the same identification, the above reads:
$(R^{q}f_{*}{\mathcal {F}})^{\wedge }\to R^{p}{\widehat {f}}_{*}{\widehat {\mathcal {F}}}$.
Using $f:X_{n}\to S_{n}$ where $X_{n}=(X_{0},{\mathcal {O}}_{X}/{\mathcal {J}}^{n+1})$ and $S_{n}=(S_{0},{\mathcal {O}}_{S}/{\mathcal {I}}^{n+1})$, one also obtains (after passing to limit):
$R^{q}{\widehat {f}}_{*}{\widehat {\mathcal {F}}}\to \varprojlim R^{p}f_{*}{\mathcal {F}}_{n}$
where ${\mathcal {F}}_{n}$ are as before. One can verify that the composition of the two maps is the same map in the lede. (cf. EGA III-1, section 4)
Notes
1. Grothendieck & Dieudonné 1961, 4.1.5
2. Grothendieck & Dieudonné 1961, 4.2.1
3. Hartshorne 1977, Ch. III. Corollary 11.2
4. The same argument as in the preceding corollary
5. Hartshorne 1977, Ch. III. Corollary 11.3
References
• Grothendieck, Alexandre; Dieudonné, Jean (1961). "Eléments de géométrie algébrique: III. Étude cohomologique des faisceaux cohérents, Première partie". Publications Mathématiques de l'IHÉS. 11. doi:10.1007/bf02684274. MR 0217085.
• Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157
Further reading
• Illusie, Luc. "Topics in Algebraic Geometry" (PDF).
| Wikipedia |
Theorem of the highest weight
In representation theory, a branch of mathematics, the theorem of the highest weight classifies the irreducible representations of a complex semisimple Lie algebra ${\mathfrak {g}}$.[1][2] There is a closely related theorem classifying the irreducible representations of a connected compact Lie group $K$.[3] The theorem states that there is a bijection
$\lambda \mapsto [V^{\lambda }]$
from the set of "dominant integral elements" to the set of equivalence classes of irreducible representations of ${\mathfrak {g}}$ or $K$. The difference between the two results is in the precise notion of "integral" in the definition of a dominant integral element. If $K$ is simply connected, this distinction disappears.
The theorem was originally proved by Élie Cartan in his 1913 paper.[4] The version of the theorem for a compact Lie group is due to Hermann Weyl. The theorem is one of the key pieces of representation theory of semisimple Lie algebras.
Statement
Lie algebra case
See also: Weight (representation theory) § Weights in the representation theory of semisimple Lie algebras
Let ${\mathfrak {g}}$ be a finite-dimensional semisimple complex Lie algebra with Cartan subalgebra ${\mathfrak {h}}$. Let $R$ be the associated root system. We then say that an element $\lambda \in {\mathfrak {h}}^{*}$ is integral[5] if
$2{\frac {\langle \lambda ,\alpha \rangle }{\langle \alpha ,\alpha \rangle }}$
is an integer for each root $\alpha $. Next, we choose a set $R^{+}$ of positive roots and we say that an element $\lambda \in {\mathfrak {h}}^{*}$ is dominant if $\langle \lambda ,\alpha \rangle \geq 0$ for all $\alpha \in R^{+}$. An element $\lambda \in {\mathfrak {h}}^{*}$ dominant integral if it is both dominant and integral. Finally, if $\lambda $ and $\mu $ are in ${\mathfrak {h}}^{*}$, we say that $\lambda $ is higher[6] than $\mu $ if $\lambda -\mu $ is expressible as a linear combination of positive roots with non-negative real coefficients.
A weight $\lambda $ of a representation $V$ of ${\mathfrak {g}}$ is then called a highest weight if $\lambda $ is higher than every other weight $\mu $ of $V$.
The theorem of the highest weight then states:[2]
• If $V$ is a finite-dimensional irreducible representation of ${\mathfrak {g}}$, then $V$ has a unique highest weight, and this highest weight is dominant integral.
• If two finite-dimensional irreducible representations have the same highest weight, they are isomorphic.
• For each dominant integral element $\lambda $, there exists a finite-dimensional irreducible representation with highest weight $\lambda $.
The most difficult part is the last one; the construction of a finite-dimensional irreducible representation with a prescribed highest weight.
The compact group case
See also: Compact group § Representation theory of a connected compact Lie group
Let $K$ be a connected compact Lie group with Lie algebra ${\mathfrak {k}}$ and let ${\mathfrak {g}}:={\mathfrak {k}}+i{\mathfrak {k}}$ be the complexification of ${\mathfrak {g}}$. Let $T$ be a maximal torus in $K$ with Lie algebra ${\mathfrak {t}}$. Then ${\mathfrak {h}}:={\mathfrak {t}}+i{\mathfrak {t}}$ is a Cartan subalgebra of ${\mathfrak {g}}$, and we may form the associated root system $R$. The theory then proceeds in much the same way as in the Lie algebra case, with one crucial difference: the notion of integrality is different. Specifically, we say that an element $\lambda \in {\mathfrak {h}}$ is analytically integral[7] if
$\langle \lambda ,H\rangle $
is an integer whenever
$e^{2\pi H}=I$
where $I$ is the identity element of $K$. Every analytically integral element is integral in the Lie algebra sense,[8] but there may be integral elements in the Lie algebra sense that are not analytically integral. This distinction reflects the fact that if $K$ is not simply connected, there may be representations of ${\mathfrak {g}}$ that do not come from representations of $K$. On the other hand, if $K$ is simply connected, the notions of "integral" and "analytically integral" coincide.[3]
The theorem of the highest weight for representations of $K$[9] is then the same as in the Lie algebra case, except that "integral" is replaced by "analytically integral."
Proofs
There are at least four proofs:
• Hermann Weyl's original proof from the compact group point of view,[10] based on the Weyl character formula and the Peter–Weyl theorem.
• The theory of Verma modules contains the highest weight theorem. This is the approach taken in many standard textbooks (e.g., Humphreys and Part II of Hall).
• The Borel–Weil–Bott theorem constructs an irreducible representation as the space of global sections of an ample line bundle; the highest weight theorem results as a consequence. (The approach uses a fair bit of algebraic geometry but yields a very quick proof.)
• The invariant theoretic approach: one constructs irreducible representations as subrepresentations of a tensor power of the standard representations. This approach is essentially due to H. Weyl and works quite well for classical groups.
See also
• Classifying finite-dimensional representations of Lie algebras
• Representation theory of a connected compact Lie group
• Weights in the representation theory of semisimple Lie algebras
Notes
1. Dixmier 1996, Theorem 7.2.6.
2. Hall 2015 Theorems 9.4 and 9.5
3. Hall 2015 Theorem 12.6
4. Knapp, A. W. (2003). "Reviewed work: Matrix Groups: An Introduction to Lie Group Theory, Andrew Baker; Lie Groups: An Introduction through Linear Groups, Wulf Rossmann". The American Mathematical Monthly. 110 (5): 446–455. doi:10.2307/3647845. JSTOR 3647845.
5. Hall 2015 Section 8.7
6. Hall 2015 Section 8.8
7. Hall 2015 Definition 12.4
8. Hall 2015 Proposition 12.7
9. Hall 2015 Corollary 13.20
10. Hall 2015 Chapter 12
References
• Dixmier, Jacques (1996) [1974], Enveloping algebras, Graduate Studies in Mathematics, vol. 11, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0560-2, MR 0498740
• Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103.
• Hall, Brian C. (2015), Lie groups, Lie algebras, and representations: An elementary introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, ISBN 978-3319134666
• Humphreys, James E. (1972a), Introduction to Lie Algebras and Representation Theory, Birkhäuser, ISBN 978-0-387-90053-7.
| Wikipedia |
Pons asinorum
In geometry, the statement that the angles opposite the equal sides of an isosceles triangle are themselves equal is known as the pons asinorum (Latin: [ˈpõːs asɪˈnoːrũː], English: /ˈpɒnz ˌæsɪˈnɔːrəm/ PONZ ass-i-NOR-əm), typically translated as "bridge of asses". This statement is Proposition 5 of Book 1 in Euclid's Elements, and is also known as the isosceles triangle theorem. Its converse is also true: if two angles of a triangle are equal, then the sides opposite them are also equal. The term is also applied to the Pythagorean theorem.[1]
Pons asinorum is also used metaphorically for a problem or challenge which acts as a test of critical thinking, referring to the "ass' bridge's" ability to separate capable and incapable reasoners. Its first known usage in this context was in 1645.[2]
A persistent piece of mathematical folklore claims that an artificial intelligence program discovered an original and more elegant proof of this theorem.[3][4] In fact, Marvin Minsky recounts that he had rediscovered the Pappus proof (which he was not aware of) by simulating what a mechanical theorem prover might do.[5][6]
Proofs
Euclid and Proclus
Euclid's statement of the pons asinorum includes a second conclusion that if the equal sides of the triangle are extended below the base, then the angles between the extensions and the base are also equal. Euclid's proof involves drawing auxiliary lines to these extensions. But, as Euclid's commentator Proclus points out, Euclid never uses the second conclusion and his proof can be simplified somewhat by drawing the auxiliary lines to the sides of the triangle instead, the rest of the proof proceeding in more or less the same way.
There has been much speculation and debate as to why Euclid added the second conclusion to the theorem, given that it makes the proof more complicated. One plausible explanation, given by Proclus, is that the second conclusion can be used in possible objections to the proofs of later propositions where Euclid does not cover every case.[7] The proof relies heavily on what is today called side-angle-side, the previous proposition in the Elements.
Proclus' variation of Euclid's proof proceeds as follows:[8]
Let ABC be an isosceles triangle with AB and AC being the equal sides. Pick an arbitrary point D on side AB and construct E on AC so that AD = AE. Draw the lines BE, DC and DE.
Consider the triangles BAE and CAD; BA = CA, AE = AD, and $\angle A$ is equal to itself, so by side-angle-side, the triangles are congruent and corresponding sides and angles are equal.
Therefore $\angle ABE=\angle ACD$ and $\angle ADC=\angle AEB$, and BE = CD.
Since AB = AC and AD = AE, BD = CE by subtraction of equal parts.
Now consider the triangles DBE and ECD; BD = CE, BE = CD, and $\angle DBE=\angle ECD$ have just been shown, so applying side-angle-side again, the triangles are congruent.
Therefore $\angle BDE=\angle CED$ and $\angle BED=\angle CDE$.
Since $\angle BDE=\angle CED$ and $\angle CDE=\angle BED$, $\angle BDC=\angle CEB$ by subtraction of equal parts.
Consider a third pair of triangles, BDC and CEB; DB = EC, DC = EB, and $\angle BDC=\angle CEB$, so applying side-angle-side a third time, the triangles are congruent.
In particular, angle CBD = BCE, which was to be proved.
Pappus
Proclus gives a much shorter proof attributed to Pappus of Alexandria. This is not only simpler but it requires no additional construction at all. The method of proof is to apply side-angle-side to the triangle and its mirror image. More modern authors, in imitation of the method of proof given for the previous proposition have described this as picking up the triangle, turning it over and laying it down upon itself.[9][6] This method is lampooned by Charles Lutwidge Dodgson in Euclid and his Modern Rivals, calling it an "Irish bull" because it apparently requires the triangle to be in two places at once.[10]
The proof is as follows:[11]
Let ABC be an isosceles triangle with AB and AC being the equal sides.
Consider the triangles ABC and ACB, where ACB is considered a second triangle with vertices A, C and B corresponding respectively to A, B and C in the original triangle.
$\angle A$ is equal to itself, AB = AC and AC = AB, so by side-angle-side, triangles ABC and ACB are congruent.
In particular, $\angle B=\angle C$.[12]
Others
A standard textbook method is to construct the bisector of the angle at A.[13] This is simpler than Euclid's proof, but Euclid does not present the construction of an angle bisector until proposition 9. So the order of presentation of the Euclid's propositions would have to be changed to avoid the possibility of circular reasoning.
The proof proceeds as follows:[14]
As before, let the triangle be ABC with AB = AC.
Construct the angle bisector of $\angle BAC$ and extend it to meet BC at X.
AB = AC and AX is equal to itself.
Furthermore, $\angle BAX=\angle CAX$, so, applying side-angle-side, triangle BAX and triangle CAX are congruent.
It follows that the angles at B and C are equal.
Legendre uses a similar construction in Éléments de géométrie, but taking X to be the midpoint of BC.[15] The proof is similar but side-side-side must be used instead of side-angle-side, and side-side-side is not given by Euclid until later in the Elements.
In 1876, while a member of the United States Congress, future President James A. Garfield developed a proof using the trapezoid, which was published in the New England Journal of Education.[16] Mathematics historian William Dunham wrote that Garfield's trapezoid work was "really a very clever proof."[17] According to the Journal, Garfield arrived at the proof "in mathematical amusements and discussions with other members of congress."[18]
In inner product spaces
The isosceles triangle theorem holds in inner product spaces over the real or complex numbers. In such spaces, it takes a form that says of vectors x, y, and z that if[19]
$x+y+z=0{\text{ and }}\|x\|=\|y\|,$
then
$\|x-z\|=\|y-z\|.$
Since
$\|x-z\|^{2}=\|x\|^{2}-2x\cdot z+\|z\|^{2},$
and
$x\cdot z=\|x\|\|z\|\cos \theta $
where θ is the angle between the two vectors, the conclusion of this inner product space form of the theorem is equivalent to the statement about equality of angles.
Etymology and related terms
Another medieval term for the pons asinorum was Elefuga which, according to Roger Bacon, comes from Greek elegia "misery", and Latin fuga "flight", that is "flight of the wretches". Though this etymology is dubious, it is echoed in Chaucer's use of the term "flemyng of wreches" for the theorem.[20]
There are two possible explanations for the name pons asinorum, the simplest being that the diagram used resembles an actual bridge. But the more popular explanation is that it is the first real test in the Elements of the intelligence of the reader and functions as a "bridge" to the harder propositions that follow.[21] Gauss supposedly once espoused a similar belief in the necessity of immediately understanding Euler's identity as a benchmark pursuant to becoming a first-class mathematician.[22]
Similarly, the name Dulcarnon was given to the 47th proposition of Book I of Euclid, better known as the Pythagorean theorem, after the Arabic Dhū 'l qarnain ذُو ٱلْقَرْنَيْن, meaning "the owner of the two horns", because diagrams of the theorem showed two smaller squares like horns at the top of the figure. The term is also used as a metaphor for a dilemma.[20] The theorem was also sometimes called "the Windmill" for similar reasons.[23]
Metaphorical usage
Uses of the pons asinorum as a metaphor for a test of critical thinking include:
• Richard Aungerville's 14th century Philobiblon contains the passage "Quot Euclidis discipulos retrojecit Elefuga quasi scopulos eminens et abruptus, qui nullo scalarum suffragio scandi posset! Durus, inquiunt, est his sermo; quis potest eum audire?", which compares the theorem to a steep cliff that no ladder may help scale and asks how many would-be geometers have been turned away.[20]
• The term pons asinorum, in both its meanings as a bridge and as a test, is used as a metaphor for finding the middle term of a syllogism.[20]
• The 18th-century poet Thomas Campbell wrote a humorous poem called "Pons asinorum" where a geometry class assails the theorem as a company of soldiers might charge a fortress; the battle was not without casualties.[24]
• Economist John Stuart Mill called Ricardo's Law of Rent the pons asinorum of economics.[25]
• Pons Asinorum is the name given to a particular configuration[26] of a Rubik's Cube.
• Eric Raymond referred to the issue of syntactically-significant whitespace in the Python programming language as its pons asinorum.[27]
• The Finnish aasinsilta and Swedish åsnebrygga is a literary technique where a tenuous, even contrived connection between two arguments or topics, which is almost but not quite a non sequitur, is used as an awkward transition between them.[28] In serious text, it is considered a stylistic error, since it belongs properly to the stream of consciousness- or causerie-style writing. Typical examples are ending a section by telling what the next section is about, without bothering to explain why the topics are related, expanding a casual mention into a detailed treatment, or finding a contrived connection between the topics (e.g. "We bought some red wine; speaking of red liquids, tomorrow is the World Blood Donor Day").
• In Dutch, ezelsbruggetje ('little bridge of asses') is the word for a mnemonic. The same is true for the German Eselsbrücke.
• In Czech, oslí můstek has two meanings – it can describe either a contrived connection between two topics or a mnemonic.
References
1. Smith, David Eugene (1925). History Of Mathematics. Vol. II. Ginn And Company. pp. 284. It formed at bridge across which fools could not hope to pass, and was therefore known as the pons asinorum, or bridge of fools.¹
1. The term is sometimes applied to the Pythagorean Theorem.
2. Pons asinorum — Definition and More from the Free Merriam
3. Jaakko Hintikka, "On Creativity in Reasoning", in Ake E. Andersson, N.E. Sahlin, eds., The Complexity of Creativity, 2013, ISBN 9401587884, p. 72
4. A. Battersby, Mathematics in Management, 1966, quoted in Deakin
5. Jeremy Bernstein, "Profiles: A.I." (interview with Marvin Minsky), The New Yorker December 14, 1981, p. 50-126
6. Michael A.B. Deakin, "From Pappus to Today: The History of a Proof", The Mathematical Gazette 74:467:6-11 (March 1990) JSTOR 3618841
7. Heath pp. 251–255
8. Following Proclus p. 53
9. For example F. Cuthbertson Primer of geometry (1876 Oxford) p. 7
10. Charles Lutwidge Dodgson, Euclid and his Modern Rivals Act I Scene II §6
11. Following Proclus p. 54
12. Heath p. 254 for section
13. For example J.M. Wilson Elementary geometry (1878 Oxford) p. 20
14. Following Wilson
15. A. M. Legendre Éléments de géométrie (1876 Libr. de Firmin-Didot et Cie) p. 14
16. G., J. A. (1876). "PONS ASINORUM". New England Journal of Education. 3 (14): 161. ISSN 2578-4145. JSTOR 44764657.
17. Dunham, William (1994). The Mathematical Universe: An Alphabetical Journey Through the Great Proofs, Problems, and Personalities. Wiley & Sons. p. 99. Bibcode:1994muaa.book.....D. ISBN 9780471536567.
18. Kolpas, Sid J. "Mathematical Treasure: Garfield's Proof of the Pythagorean Theorem". Mathematical Assoc. of America. Archived from the original on December 6, 2021. Retrieved December 22, 2021.
19. J. R. Retherford, Hilbert Space, Cambridge University Press, 1993, page 27.
20. A. F. West & H. D. Thompson "On Dulcarnon, Elefuga And Pons Asinorum as Fanciful Names For Geometrical Propositions" The Princeton University bulletin Vol. 3 No. 4 (1891) p. 84
21. D.E. Smith History of Mathematics (1958 Dover) p. 284
22. Derbyshire, John (2003). Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. 500 Fifth Street, NW, Washington D.C. 20001: Joseph Henry Press. p. 202. ISBN 0-309-08549-7. first-class mathematician.{{cite book}}: CS1 maint: location (link)
23. Charles Lutwidge Dodgson, Euclid and his Modern Rivals Act I Scene II §1
24. W.E. Aytoun (Ed.) The poetical works of Thomas Campbell (1864, Little, Brown) p. 385 Google Books
25. John Stuart Mill Principles of Political Economy (1866: Longmans, Green, Reader, and Dyer) Book 2, Chapter 16, p. 261
26. Reid, Michael (28 October 2006). "Rubik's Cube patterns". www.cflmath.com. Archived from the original on 12 December 2012. Retrieved 22 September 2019.
27. Eric S. Raymond, "Why Python?", Linux Journal, April 30, 2000
28. Aasinsilta on laiskurin apuneuvo | Yle Uutiset | yle.fi
External links
Look up pons asinorum in Wiktionary, the free dictionary.
Wikisource has original text related to this article:
Proposition 5 of Euclid's Elements
• Pons asinorum at PlanetMath.
• D. E. Joyce's presentation of Euclid's Elements
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| Wikipedia |
Theorema Egregium
Gauss's Theorema Egregium (Latin for "Remarkable Theorem") is a major result of differential geometry, proved by Carl Friedrich Gauss in 1827, that concerns the curvature of surfaces. The theorem says that Gaussian curvature can be determined entirely by measuring angles, distances and their rates on a surface, without reference to the particular manner in which the surface is embedded in the ambient 3-dimensional Euclidean space. In other words, the Gaussian curvature of a surface does not change if one bends the surface without stretching it. Thus the Gaussian curvature is an intrinsic invariant of a surface.
Gauss presented the theorem in this manner (translated from Latin):
Thus the formula of the preceding article leads itself to the remarkable Theorem. If a curved surface is developed upon any other surface whatever, the measure of curvature in each point remains unchanged.
The theorem is "remarkable" because the starting definition of Gaussian curvature makes direct use of position of the surface in space. So it is quite surprising that the result does not depend on its embedding in spite of all bending and twisting deformations undergone.
In modern mathematical terminology, the theorem may be stated as follows:
The Gaussian curvature of a surface is invariant under local isometry.
Elementary applications
A sphere of radius R has constant Gaussian curvature which is equal to 1/R2. At the same time, a plane has zero Gaussian curvature. As a corollary of Theorema Egregium, a piece of paper cannot be bent onto a sphere without crumpling. Conversely, the surface of a sphere cannot be unfolded onto a flat plane without distorting the distances. If one were to step on an empty egg shell, its edges have to split in expansion before being flattened. Mathematically, a sphere and a plane are not isometric, even locally. This fact is significant for cartography: it implies that no planar (flat) map of Earth can be perfect, even for a portion of the Earth's surface. Thus every cartographic projection necessarily distorts at least some distances.[1]
The catenoid and the helicoid are two very different-looking surfaces. Nevertheless, each of them can be continuously bent into the other: they are locally isometric. It follows from Theorema Egregium that under this bending the Gaussian curvature at any two corresponding points of the catenoid and helicoid is always the same. Thus isometry is simply bending and twisting of a surface without internal crumpling or tearing, in other words without extra tension, compression, or shear.
An application of the theorem is seen when a flat object is somewhat folded or bent along a line, creating rigidity in the perpendicular direction. This is of practical use in construction, as well as in a common pizza-eating strategy: A flat slice of pizza can be seen as a surface with constant Gaussian curvature 0. Gently bending a slice must then roughly maintain this curvature (assuming the bend is roughly a local isometry). If one bends a slice horizontally along a radius, non-zero principal curvatures are created along the bend, dictating that the other principal curvature at these points must be zero. This creates rigidity in the direction perpendicular to the fold, an attribute desirable for eating pizza, as it holds its shape long enough to be consumed without a mess. This same principle is used for strengthening in corrugated materials, most familiarly corrugated fiberboard and corrugated galvanised iron,[2] and in some forms of potato chips.
See also
• Second fundamental form
• Gaussian curvature
• Differential geometry of surfaces
• Carl Friedrich Gauss#Theorema Egregium
Notes
1. Geodetical applications were one of the primary motivations for Gauss's "investigations of the curved surfaces".
2. wired.com
References
• Gauss, C. F. (2005). Pesic, Peter (ed.). General Investigations of Curved Surfaces (Paperback ed.). Dover Publications. ISBN 0-486-44645-X.
• O'Neill, Barrett (1966). Elementary Differential Geometry. New York: Academic Press. pp. 271–275.
• Stoker, J. J. (1969). "The Partial Differential Equations of Surface Theory". Differential Geometry. New York: Wiley. pp. 133–150. ISBN 0-471-82825-4.
External links
• Theorema Egregium on Mathworld
• Dominic Vella : Some wrinkles in Gauss' Theoreme : Mathematics of everday objects from Pizza to Umbrellas and Parachutes (G. I. Taylor Lecture) on YouTube, 30 January 2023
| Wikipedia |
Cartan's theorems A and B
In mathematics, Cartan's theorems A and B are two results proved by Henri Cartan around 1951, concerning a coherent sheaf F on a Stein manifold X. They are significant both as applied to several complex variables, and in the general development of sheaf cohomology.
Theorem A — F is spanned by its global sections.
Theorem B is stated in cohomological terms (a formulation that Cartan (1953, p. 51) attributes to J.-P. Serre):
Theorem B — Hp(X, F) = 0 for all p > 0.
Analogous properties were established by Serre (1957) for coherent sheaves in algebraic geometry, when X is an affine scheme. The analogue of Theorem B in this context is as follows (Hartshorne 1977, Theorem III.3.7):
Theorem B (Scheme theoretic analogue) — Let X be an affine scheme, F a quasi-coherent sheaf of OX-modules for the Zariski topology on X. Then Hp(X, F) = 0 for all p > 0.
These theorems have many important applications. For instance, they imply that a holomorphic function on a closed complex submanifold, Z, of a Stein manifold X can be extended to a holomorphic function on all of X. At a deeper level, these theorems were used by Jean-Pierre Serre to prove the GAGA theorem.
Theorem B is sharp in the sense that if H1(X, F) = 0 for all coherent sheaves F on a complex manifold X (resp. quasi-coherent sheaves F on a noetherian scheme X), then X is Stein (resp. affine); see (Serre 1956) (resp. (Serre 1957) and (Hartshorne 1977, Theorem III.3.7)).
See also
• Cousin problems
References
• Cartan, H. (1953), "Variétés analytiques complexes et cohomologie", Colloque tenu à Bruxelles: 41–55, Zbl 0053.05301.
• Gunning, Robert C.; Rossi, Hugo (1965), Analytic Functions of Several Complex Variables, Prentice Hall, doi:10.1090/chel/368, ISBN 9780821821657.
• Hartshorne, Robin (1977). Algebraic Geometry. Graduate Texts in Mathematics. Vol. 52. Berlin, New York: Springer-Verlag. doi:10.1007/978-1-4757-3849-0. ISBN 978-0-387-90244-9. MR 0463157. Zbl 0367.14001..
• Serre, Jean-Pierre (1956), "Géométrie algébrique et géométrie analytique", Annales de l'Institut Fourier, 6: 1–42, doi:10.5802/aif.59, ISSN 0373-0956, MR 0082175
• Serre, Jean-Pierre (1957), "Sur la cohomologie des variétés algébriques", Journal de Mathématiques Pures et Appliquées, 36: 1–16, Zbl 0078.34604
• Serre, Jean-Pierre (2 December 2013). "35. Sur la cohomologie des variétés algébriques". Oeuvres - Collected Papers I: 1949 - 1959. pp. 469–484. ISBN 978-3-642-39815-5.
| Wikipedia |
Theoretical computer science
Theoretical computer science (TCS) is a subset of general computer science and mathematics that focuses on mathematical aspects of computer science such as the theory of computation, formal language theory, the lambda calculus and type theory.
It is difficult to circumscribe the theoretical areas precisely. The ACM's Special Interest Group on Algorithms and Computation Theory (SIGACT) provides the following description:[1]
TCS covers a wide variety of topics including algorithms, data structures, computational complexity, parallel and distributed computation, probabilistic computation, quantum computation, automata theory, information theory, cryptography, program semantics and verification, algorithmic game theory, machine learning, computational biology, computational economics, computational geometry, and computational number theory and algebra. Work in this field is often distinguished by its emphasis on mathematical technique and rigor.
History
While logical inference and mathematical proof had existed previously, in 1931 Kurt Gödel proved with his incompleteness theorem that there are fundamental limitations on what statements could be proved or disproved.
Information theory was added to the field with a 1948 mathematical theory of communication by Claude Shannon. In the same decade, Donald Hebb introduced a mathematical model of learning in the brain. With mounting biological data supporting this hypothesis with some modification, the fields of neural networks and parallel distributed processing were established. In 1971, Stephen Cook and, working independently, Leonid Levin, proved that there exist practically relevant problems that are NP-complete – a landmark result in computational complexity theory.
With the development of quantum mechanics in the beginning of the 20th century came the concept that mathematical operations could be performed on an entire particle wavefunction. In other words, one could compute functions on multiple states simultaneously. This led to the concept of a quantum computer in the latter half of the 20th century that took off in the 1990s when Peter Shor showed that such methods could be used to factor large numbers in polynomial time, which, if implemented, would render some modern public key cryptography algorithms like RSA insecure.
Modern theoretical computer science research is based on these basic developments, but includes many other mathematical and interdisciplinary problems that have been posed, as shown below:
$P\rightarrow Q\,$ P = NP ?
Mathematical logic Automata theory Number theory Graph theory Computability theory Computational complexity theory
GNITIRW-TERCES $\Gamma \vdash x:{\text{Int}}$
Cryptography Type theory Category theory Computational geometry Combinatorial optimization Quantum computing theory
Topics
Algorithms
Main article: Algorithm
An algorithm is a step-by-step procedure for calculations. Algorithms are used for calculation, data processing, and automated reasoning.
An algorithm is an effective method expressed as a finite list[2] of well-defined instructions[3] for calculating a function.[4] Starting from an initial state and initial input (perhaps empty),[5] the instructions describe a computation that, when executed, proceeds through a finite[6] number of well-defined successive states, eventually producing "output"[7] and terminating at a final ending state. The transition from one state to the next is not necessarily deterministic; some algorithms, known as randomized algorithms, incorporate random input.[8]
Automata theory
Automata theory is the study of abstract machines and automata, as well as the computational problems that can be solved using them. It is a theory in theoretical computer science, under discrete mathematics (a section of mathematics and also of computer science). Automata comes from the Greek word αὐτόματα meaning "self-acting".
Automata Theory is the study of self-operating virtual machines to help in the logical understanding of input and output process, without or with intermediate stage(s) of computation (or any function/process).
Coding theory
Main article: Coding theory
Coding theory is the study of the properties of codes and their fitness for a specific application. Codes are used for data compression, cryptography, error-correction and more recently also for network coding. Codes are studied by various scientific disciplines—such as information theory, electrical engineering, mathematics, and computer science—for the purpose of designing efficient and reliable data transmission methods. This typically involves the removal of redundancy and the correction (or detection) of errors in the transmitted data.
Computational biology
Computational biology involves the development and application of data-analytical and theoretical methods, mathematical modeling and computational simulation techniques to the study of biological, behavioral, and social systems.[9] The field is broadly defined and includes foundations in computer science, applied mathematics, animation, statistics, biochemistry, chemistry, biophysics, molecular biology, genetics, genomics, ecology, evolution, anatomy, neuroscience, and visualization.[10]
Computational biology is different from biological computation, which is a subfield of computer science and computer engineering using bioengineering and biology to build computers, but is similar to bioinformatics, which is an interdisciplinary science using computers to store and process biological data.
Computational complexity theory
Main article: Computational complexity theory
Computational complexity theory is a branch of the theory of computation that focuses on classifying computational problems according to their inherent difficulty, and relating those classes to each other. A computational problem is understood to be a task that is in principle amenable to being solved by a computer, which is equivalent to stating that the problem may be solved by mechanical application of mathematical steps, such as an algorithm.
A problem is regarded as inherently difficult if its solution requires significant resources, whatever the algorithm used. The theory formalizes this intuition, by introducing mathematical models of computation to study these problems and quantifying the amount of resources needed to solve them, such as time and storage. Other complexity measures are also used, such as the amount of communication (used in communication complexity), the number of gates in a circuit (used in circuit complexity) and the number of processors (used in parallel computing). One of the roles of computational complexity theory is to determine the practical limits on what computers can and cannot do.
Computational geometry
Computational geometry is a branch of computer science devoted to the study of algorithms that can be stated in terms of geometry. Some purely geometrical problems arise out of the study of computational geometric algorithms, and such problems are also considered to be part of computational geometry.
The main impetus for the development of computational geometry as a discipline was progress in computer graphics and computer-aided design and manufacturing (CAD/CAM), but many problems in computational geometry are classical in nature, and may come from mathematical visualization.
Other important applications of computational geometry include robotics (motion planning and visibility problems), geographic information systems (GIS) (geometrical location and search, route planning), integrated circuit design (IC geometry design and verification), computer-aided engineering (CAE) (mesh generation), computer vision (3D reconstruction).
Computational learning theory
Theoretical results in machine learning mainly deal with a type of inductive learning called supervised learning. In supervised learning, an algorithm is given samples that are labeled in some useful way. For example, the samples might be descriptions of mushrooms, and the labels could be whether or not the mushrooms are edible. The algorithm takes these previously labeled samples and uses them to induce a classifier. This classifier is a function that assigns labels to samples including the samples that have never been previously seen by the algorithm. The goal of the supervised learning algorithm is to optimize some measure of performance such as minimizing the number of mistakes made on new samples.
Computational number theory
Main article: Computational number theory
Computational number theory, also known as algorithmic number theory, is the study of algorithms for performing number theoretic computations. The best known problem in the field is integer factorization.
Cryptography
Main article: Cryptography
Cryptography is the practice and study of techniques for secure communication in the presence of third parties (called adversaries).[11] More generally, it is about constructing and analyzing protocols that overcome the influence of adversaries[12] and that are related to various aspects in information security such as data confidentiality, data integrity, authentication, and non-repudiation.[13] Modern cryptography intersects the disciplines of mathematics, computer science, and electrical engineering. Applications of cryptography include ATM cards, computer passwords, and electronic commerce.
Modern cryptography is heavily based on mathematical theory and computer science practice; cryptographic algorithms are designed around computational hardness assumptions, making such algorithms hard to break in practice by any adversary. It is theoretically possible to break such a system, but it is infeasible to do so by any known practical means. These schemes are therefore termed computationally secure; theoretical advances, e.g., improvements in integer factorization algorithms, and faster computing technology require these solutions to be continually adapted. There exist information-theoretically secure schemes that provably cannot be broken even with unlimited computing power—an example is the one-time pad—but these schemes are more difficult to implement than the best theoretically breakable but computationally secure mechanisms.
Data structures
Main article: Data structure
A data structure is a particular way of organizing data in a computer so that it can be used efficiently.[14][15]
Different kinds of data structures are suited to different kinds of applications, and some are highly specialized to specific tasks. For example, databases use B-tree indexes for small percentages of data retrieval and compilers and databases use dynamic hash tables as look up tables.
Data structures provide a means to manage large amounts of data efficiently for uses such as large databases and internet indexing services. Usually, efficient data structures are key to designing efficient algorithms. Some formal design methods and programming languages emphasize data structures, rather than algorithms, as the key organizing factor in software design. Storing and retrieving can be carried out on data stored in both main memory and in secondary memory.
Distributed computation
Distributed computing studies distributed systems. A distributed system is a software system in which components located on networked computers communicate and coordinate their actions by passing messages.[16] The components interact with each other in order to achieve a common goal. Three significant characteristics of distributed systems are: concurrency of components, lack of a global clock, and independent failure of components.[16] Examples of distributed systems vary from SOA-based systems to massively multiplayer online games to peer-to-peer applications, and blockchain networks like Bitcoin.
A computer program that runs in a distributed system is called a distributed program, and distributed programming is the process of writing such programs.[17] There are many alternatives for the message passing mechanism, including RPC-like connectors and message queues. An important goal and challenge of distributed systems is location transparency.
Information-based complexity
Main article: Information-based complexity
Information-based complexity (IBC) studies optimal algorithms and computational complexity for continuous problems. IBC has studied continuous problems as path integration, partial differential equations, systems of ordinary differential equations, nonlinear equations, integral equations, fixed points, and very-high-dimensional integration.
Formal methods
Main article: Formal methods
Formal methods are a particular kind of mathematics based techniques for the specification, development and verification of software and hardware systems.[18] The use of formal methods for software and hardware design is motivated by the expectation that, as in other engineering disciplines, performing appropriate mathematical analysis can contribute to the reliability and robustness of a design.[19]
Formal methods are best described as the application of a fairly broad variety of theoretical computer science fundamentals, in particular logic calculi, formal languages, automata theory, and program semantics, but also type systems and algebraic data types to problems in software and hardware specification and verification.[20]
Information theory
Main article: Information theory
Information theory is a branch of applied mathematics, electrical engineering, and computer science involving the quantification of information. Information theory was developed by Claude E. Shannon to find fundamental limits on signal processing operations such as compressing data and on reliably storing and communicating data. Since its inception it has broadened to find applications in many other areas, including statistical inference, natural language processing, cryptography, neurobiology,[21] the evolution[22] and function[23] of molecular codes, model selection in statistics,[24] thermal physics,[25] quantum computing, linguistics, plagiarism detection,[26] pattern recognition, anomaly detection and other forms of data analysis.[27]
Applications of fundamental topics of information theory include lossless data compression (e.g. ZIP files), lossy data compression (e.g. MP3s and JPEGs), and channel coding (e.g. for Digital Subscriber Line (DSL)). The field is at the intersection of mathematics, statistics, computer science, physics, neurobiology, and electrical engineering. Its impact has been crucial to the success of the Voyager missions to deep space, the invention of the compact disc, the feasibility of mobile phones, the development of the Internet, the study of linguistics and of human perception, the understanding of black holes, and numerous other fields. Important sub-fields of information theory are source coding, channel coding, algorithmic complexity theory, algorithmic information theory, information-theoretic security, and measures of information.
Machine learning
Main article: Machine learning
Machine learning is a scientific discipline that deals with the construction and study of algorithms that can learn from data.[28] Such algorithms operate by building a model based on inputs[29]: 2 and using that to make predictions or decisions, rather than following only explicitly programmed instructions.
Machine learning can be considered a subfield of computer science and statistics. It has strong ties to artificial intelligence and optimization, which deliver methods, theory and application domains to the field. Machine learning is employed in a range of computing tasks where designing and programming explicit, rule-based algorithms is infeasible. Example applications include spam filtering, optical character recognition (OCR),[30] search engines and computer vision. Machine learning is sometimes conflated with data mining,[31] although that focuses more on exploratory data analysis.[32] Machine learning and pattern recognition "can be viewed as two facets of the same field."[29]: vii
Parallel computation
Parallel computing is a form of computation in which many calculations are carried out simultaneously,[33] operating on the principle that large problems can often be divided into smaller ones, which are then solved "in parallel". There are several different forms of parallel computing: bit-level, instruction level, data, and task parallelism. Parallelism has been employed for many years, mainly in high-performance computing, but interest in it has grown lately due to the physical constraints preventing frequency scaling.[34] As power consumption (and consequently heat generation) by computers has become a concern in recent years,[35] parallel computing has become the dominant paradigm in computer architecture, mainly in the form of multi-core processors.[36]
Parallel computer programs are more difficult to write than sequential ones,[37] because concurrency introduces several new classes of potential software bugs, of which race conditions are the most common. Communication and synchronization between the different subtasks are typically some of the greatest obstacles to getting good parallel program performance.
The maximum possible speed-up of a single program as a result of parallelization is known as Amdahl's law.
Programming language theory and program semantics
Programming language theory is a branch of computer science that deals with the design, implementation, analysis, characterization, and classification of programming languages and their individual features. It falls within the discipline of theoretical computer science, both depending on and affecting mathematics, software engineering, and linguistics. It is an active research area, with numerous dedicated academic journals.
In programming language theory, semantics is the field concerned with the rigorous mathematical study of the meaning of programming languages. It does so by evaluating the meaning of syntactically legal strings defined by a specific programming language, showing the computation involved. In such a case that the evaluation would be of syntactically illegal strings, the result would be non-computation. Semantics describes the processes a computer follows when executing a program in that specific language. This can be shown by describing the relationship between the input and output of a program, or an explanation of how the program will execute on a certain platform, hence creating a model of computation.
Quantum computation
Main article: Quantum computation
A quantum computer is a computation system that makes direct use of quantum-mechanical phenomena, such as superposition and entanglement, to perform operations on data.[38] Quantum computers are different from digital computers based on transistors. Whereas digital computers require data to be encoded into binary digits (bits), each of which is always in one of two definite states (0 or 1), quantum computation uses qubits (quantum bits), which can be in superpositions of states. A theoretical model is the quantum Turing machine, also known as the universal quantum computer. Quantum computers share theoretical similarities with non-deterministic and probabilistic computers; one example is the ability to be in more than one state simultaneously. The field of quantum computing was first introduced by Yuri Manin in 1980[39] and Richard Feynman in 1982.[40][41] A quantum computer with spins as quantum bits was also formulated for use as a quantum space–time in 1968.[42]
As of 2014, quantum computing is still in its infancy but experiments have been carried out in which quantum computational operations were executed on a very small number of qubits.[43] Both practical and theoretical research continues, and many national governments and military funding agencies support quantum computing research to develop quantum computers for both civilian and national security purposes, such as cryptanalysis.[44]
Symbolic computation
Main article: Symbolic computation
Computer algebra, also called symbolic computation or algebraic computation is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressions and other mathematical objects. Although, properly speaking, computer algebra should be a subfield of scientific computing, they are generally considered as distinct fields because scientific computing is usually based on numerical computation with approximate floating point numbers, while symbolic computation emphasizes exact computation with expressions containing variables that have not any given value and are thus manipulated as symbols (therefore the name of symbolic computation).
Software applications that perform symbolic calculations are called computer algebra systems, with the term system alluding to the complexity of the main applications that include, at least, a method to represent mathematical data in a computer, a user programming language (usually different from the language used for the implementation), a dedicated memory manager, a user interface for the input/output of mathematical expressions, a large set of routines to perform usual operations, like simplification of expressions, differentiation using chain rule, polynomial factorization, indefinite integration, etc.
Very-large-scale integration
Very-large-scale integration (VLSI) is the process of creating an integrated circuit (IC) by combining thousands of transistors into a single chip. VLSI began in the 1970s when complex semiconductor and communication technologies were being developed. The microprocessor is a VLSI device. Before the introduction of VLSI technology most ICs had a limited set of functions they could perform. An electronic circuit might consist of a CPU, ROM, RAM and other glue logic. VLSI allows IC makers to add all of these circuits into one chip.
Organizations
• European Association for Theoretical Computer Science
• SIGACT
• Simons Institute for the Theory of Computing
Journals and newsletters
• Discrete Mathematics and Theoretical Computer Science
• Information and Computation
• Theory of Computing (open access journal)
• Formal Aspects of Computing
• Journal of the ACM
• SIAM Journal on Computing (SICOMP)
• SIGACT News
• Theoretical Computer Science
• Theory of Computing Systems
• TheoretiCS (open access journal)
• International Journal of Foundations of Computer Science
• Chicago Journal of Theoretical Computer Science (open access journal)
• Foundations and Trends in Theoretical Computer Science
• Journal of Automata, Languages and Combinatorics
• Acta Informatica
• Fundamenta Informaticae
• ACM Transactions on Computation Theory
• Computational Complexity
• Journal of Complexity
• ACM Transactions on Algorithms
• Information Processing Letters
• Open Computer Science (open access journal)
Conferences
• Annual ACM Symposium on Theory of Computing (STOC)[45]
• Annual IEEE Symposium on Foundations of Computer Science (FOCS)[45]
• Innovations in Theoretical Computer Science (ITCS)
• Mathematical Foundations of Computer Science (MFCS)[46]
• International Computer Science Symposium in Russia (CSR)[47]
• ACM–SIAM Symposium on Discrete Algorithms (SODA)[45]
• IEEE Symposium on Logic in Computer Science (LICS)[45]
• Computational Complexity Conference (CCC)[48]
• International Colloquium on Automata, Languages and Programming (ICALP)[48]
• Annual Symposium on Computational Geometry (SoCG)[48]
• ACM Symposium on Principles of Distributed Computing (PODC)[45]
• ACM Symposium on Parallelism in Algorithms and Architectures (SPAA)[48]
• Annual Conference on Learning Theory (COLT)[48]
• Symposium on Theoretical Aspects of Computer Science (STACS)[48]
• European Symposium on Algorithms (ESA)[48]
• Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX)[48]
• Workshop on Randomization and Computation (RANDOM)[48]
• International Symposium on Algorithms and Computation (ISAAC)[48]
• International Symposium on Fundamentals of Computation Theory (FCT)[49]
• International Workshop on Graph-Theoretic Concepts in Computer Science (WG)
See also
• Formal science
• Unsolved problems in computer science
• Sun–Ni law
Notes
1. "SIGACT". Retrieved 2017-01-19.
2. "Any classical mathematical algorithm, for example, can be described in a finite number of English words". Rogers, Hartley Jr. (1967). Theory of Recursive Functions and Effective Computability. McGraw-Hill. Page 2.
3. Well defined with respect to the agent that executes the algorithm: "There is a computing agent, usually human, which can react to the instructions and carry out the computations" (Rogers 1967, p. 2).
4. "an algorithm is a procedure for computing a function (with respect to some chosen notation for integers) ... this limitation (to numerical functions) results in no loss of generality", (Rogers 1967, p. 1).
5. "An algorithm has zero or more inputs, i.e., quantities which are given to it initially before the algorithm begins" (Knuth 1973:5).
6. "A procedure which has all the characteristics of an algorithm except that it possibly lacks finiteness may be called a 'computational method'" (Knuth 1973:5).
7. "An algorithm has one or more outputs, i.e. quantities which have a specified relation to the inputs" (Knuth 1973:5).
8. Whether or not a process with random interior processes (not including the input) is an algorithm is debatable. Rogers opines that: "a computation is carried out in a discrete stepwise fashion, without the use of continuous methods or analog devices . . . carried forward deterministically, without resort to random methods or devices, e.g., dice" (Rogers 1967, p. 2).
9. "NIH working definition of bioinformatics and computational biology" (PDF). Biomedical Information Science and Technology Initiative. 17 July 2000. Archived from the original (PDF) on 5 September 2012. Retrieved 18 August 2012.
10. "About the CCMB". Center for Computational Molecular Biology. Retrieved 18 August 2012.
11. Rivest, Ronald L. (1990). "Cryptology". In J. Van Leeuwen (ed.). Handbook of Theoretical Computer Science. Vol. 1. Elsevier.
12. Bellare, Mihir; Rogaway, Phillip (21 September 2005). "Introduction". Introduction to Modern Cryptography. p. 10.
13. Menezes, A. J.; van Oorschot, P. C.; Vanstone, S. A. (1997). Handbook of Applied Cryptography. ISBN 978-0-8493-8523-0.
14. Paul E. Black (ed.), entry for data structure in Dictionary of Algorithms and Data Structures. U.S. National Institute of Standards and Technology. 15 December 2004. Online version Accessed May 21, 2009.
15. Entry data structure in the Encyclopædia Britannica (2009) Online entry accessed on May 21, 2009.
16. Coulouris, George; Jean Dollimore; Tim Kindberg; Gordon Blair (2011). Distributed Systems: Concepts and Design (5th ed.). Boston: Addison-Wesley. ISBN 978-0-132-14301-1.
17. Andrews (2000) harvtxt error: no target: CITEREFAndrews2000 (help). Dolev (2000) harvtxt error: no target: CITEREFDolev2000 (help). Ghosh (2007) harvtxt error: no target: CITEREFGhosh2007 (help), p. 10.
18. R. W. Butler (2001-08-06). "What is Formal Methods?". Retrieved 2006-11-16.
19. C. Michael Holloway. "Why Engineers Should Consider Formal Methods" (PDF). 16th Digital Avionics Systems Conference (27–30 October 1997). Archived from the original (PDF) on 16 November 2006. Retrieved 2006-11-16.
20. Monin, pp.3–4
21. F. Rieke; D. Warland; R Ruyter van Steveninck; W Bialek (1997). Spikes: Exploring the Neural Code. The MIT press. ISBN 978-0262681087.
22. Huelsenbeck, J. P.; Ronquist, F.; Nielsen, R.; Bollback, J. P. (2001-12-14). "Bayesian Inference of Phylogeny and Its Impact on Evolutionary Biology". Science. American Association for the Advancement of Science (AAAS). 294 (5550): 2310–2314. Bibcode:2001Sci...294.2310H. doi:10.1126/science.1065889. ISSN 0036-8075. PMID 11743192. S2CID 2138288.
23. Rando Allikmets, Wyeth W. Wasserman, Amy Hutchinson, Philip Smallwood, Jeremy Nathans, Peter K. Rogan, Thomas D. Schneider, Michael Dean (1998) Organization of the ABCR gene: analysis of promoter and splice junction sequences, Gene 215:1, 111–122
24. Burnham, K. P. and Anderson D. R. (2002) Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach, Second Edition (Springer Science, New York) ISBN 978-0-387-95364-9.
25. Jaynes, E. T. (1957-05-15). "Information Theory and Statistical Mechanics". Physical Review. American Physical Society (APS). 106 (4): 620–630. Bibcode:1957PhRv..106..620J. doi:10.1103/physrev.106.620. ISSN 0031-899X.
26. Charles H. Bennett, Ming Li, and Bin Ma (2003) Chain Letters and Evolutionary Histories, Scientific American 288:6, 76–81
27. David R. Anderson (November 1, 2003). "Some background on why people in the empirical sciences may want to better understand the information-theoretic methods" (PDF). Archived from the original (PDF) on July 23, 2011. Retrieved 2010-06-23.
28. Ron Kovahi; Foster Provost (1998). "Glossary of terms". Machine Learning. 30: 271–274. doi:10.1023/A:1007411609915.
29. C. M. Bishop (2006). Pattern Recognition and Machine Learning. Springer. ISBN 978-0-387-31073-2.
30. Wernick, Yang, Brankov, Yourganov and Strother, Machine Learning in Medical Imaging, IEEE Signal Processing Magazine, vol. 27, no. 4, July 2010, pp. 25–38
31. Mannila, Heikki (1996). Data mining: machine learning, statistics, and databases. Int'l Conf. Scientific and Statistical Database Management. IEEE Computer Society.
32. Friedman, Jerome H. (1998). "Data Mining and Statistics: What's the connection?". Computing Science and Statistics. 29 (1): 3–9.
33. Gottlieb, Allan; Almasi, George S. (1989). Highly parallel computing. Redwood City, Calif.: Benjamin/Cummings. ISBN 978-0-8053-0177-9.
34. S.V. Adve et al. (November 2008). "Parallel Computing Research at Illinois: The UPCRC Agenda" Archived 2008-12-09 at the Wayback Machine (PDF). Parallel@Illinois, University of Illinois at Urbana-Champaign. "The main techniques for these performance benefits – increased clock frequency and smarter but increasingly complex architectures – are now hitting the so-called power wall. The computer industry has accepted that future performance increases must largely come from increasing the number of processors (or cores) on a die, rather than making a single core go faster."
35. Asanovic et al. Old [conventional wisdom]: Power is free, but transistors are expensive. New [conventional wisdom] is [that] power is expensive, but transistors are "free".
36. Asanovic, Krste et al. (December 18, 2006). "The Landscape of Parallel Computing Research: A View from Berkeley" (PDF). University of California, Berkeley. Technical Report No. UCB/EECS-2006-183. "Old [conventional wisdom]: Increasing clock frequency is the primary method of improving processor performance. New [conventional wisdom]: Increasing parallelism is the primary method of improving processor performance ... Even representatives from Intel, a company generally associated with the 'higher clock-speed is better' position, warned that traditional approaches to maximizing performance through maximizing clock speed have been pushed to their limit."
37. Hennessy, John L.; Patterson, David A.; Larus, James R. (1999). Computer organization and design : the hardware/software interface (2. ed., 3rd print. ed.). San Francisco: Kaufmann. ISBN 978-1-55860-428-5.
38. "Quantum Computing with Molecules" article in Scientific American by Neil Gershenfeld and Isaac L. Chuang
39. Manin, Yu. I. (1980). Vychislimoe i nevychislimoe [Computable and Noncomputable] (in Russian). Sov.Radio. pp. 13–15. Archived from the original on 10 May 2013. Retrieved 4 March 2013.
40. Feynman, R. P. (1982). "Simulating physics with computers". International Journal of Theoretical Physics. 21 (6): 467–488. Bibcode:1982IJTP...21..467F. CiteSeerX 10.1.1.45.9310. doi:10.1007/BF02650179. S2CID 124545445.
41. Deutsch, David (1992-01-06). "Quantum computation". Physics World. 5 (6): 57–61. doi:10.1088/2058-7058/5/6/38.
42. Finkelstein, David (1968). "Space-Time Structure in High Energy Interactions". In Gudehus, T.; Kaiser, G. (eds.). Fundamental Interactions at High Energy. New York: Gordon & Breach.
43. "New qubit control bodes well for future of quantum computing". Retrieved 26 October 2014.
44. Quantum Information Science and Technology Roadmap for a sense of where the research is heading.
45. The 2007 Australian Ranking of ICT Conferences Archived 2009-10-02 at the Wayback Machine: tier A+.
46. MFCS 2017
47. CSR 2018
48. The 2007 Australian Ranking of ICT Conferences Archived 2009-10-02 at the Wayback Machine: tier A.
49. FCT 2011 (retrieved 2013-06-03)
Further reading
• Martin Davis, Ron Sigal, Elaine J. Weyuker, Computability, complexity, and languages: fundamentals of theoretical computer science, 2nd ed., Academic Press, 1994, ISBN 0-12-206382-1. Covers theory of computation, but also program semantics and quantification theory. Aimed at graduate students.
External links
• SIGACT directory of additional theory links (archived 15 July 2017)
• Theory Matters Wiki Theoretical Computer Science (TCS) Advocacy Wiki
• List of academic conferences in the area of theoretical computer science at confsearch
• Theoretical Computer Science – StackExchange, a Question and Answer site for researchers in theoretical computer science
• Computer Science Animated
• Theory of computation at the Massachusetts Institute of Technology
Computer science
Note: This template roughly follows the 2012 ACM Computing Classification System.
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• Printed circuit board
• Peripheral
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• Energy consumption (Green computing)
• Electronic design automation
• Hardware acceleration
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| Wikipedia |
Mathematical statistics
Mathematical statistics is the application of probability theory, a branch of mathematics, to statistics, as opposed to techniques for collecting statistical data. Specific mathematical techniques which are used for this include mathematical analysis, linear algebra, stochastic analysis, differential equations, and measure theory.[1][2]
Statistics
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Introduction
Statistical data collection is concerned with the planning of studies, especially with the design of randomized experiments and with the planning of surveys using random sampling. The initial analysis of the data often follows the study protocol specified prior to the study being conducted. The data from a study can also be analyzed to consider secondary hypotheses inspired by the initial results, or to suggest new studies. A secondary analysis of the data from a planned study uses tools from data analysis, and the process of doing this is mathematical statistics.
Data analysis is divided into:
• descriptive statistics – the part of statistics that describes data, i.e. summarises the data and their typical properties.
• inferential statistics – the part of statistics that draws conclusions from data (using some model for the data): For example, inferential statistics involves selecting a model for the data, checking whether the data fulfill the conditions of a particular model, and with quantifying the involved uncertainty (e.g. using confidence intervals).
While the tools of data analysis work best on data from randomized studies, they are also applied to other kinds of data. For example, from natural experiments and observational studies, in which case the inference is dependent on the model chosen by the statistician, and so subjective.[3][4]
Topics
The following are some of the important topics in mathematical statistics:[5][6]
Probability distributions
Main article: Probability distribution
A probability distribution is a function that assigns a probability to each measurable subset of the possible outcomes of a random experiment, survey, or procedure of statistical inference. Examples are found in experiments whose sample space is non-numerical, where the distribution would be a categorical distribution; experiments whose sample space is encoded by discrete random variables, where the distribution can be specified by a probability mass function; and experiments with sample spaces encoded by continuous random variables, where the distribution can be specified by a probability density function. More complex experiments, such as those involving stochastic processes defined in continuous time, may demand the use of more general probability measures.
A probability distribution can either be univariate or multivariate. A univariate distribution gives the probabilities of a single random variable taking on various alternative values; a multivariate distribution (a joint probability distribution) gives the probabilities of a random vector—a set of two or more random variables—taking on various combinations of values. Important and commonly encountered univariate probability distributions include the binomial distribution, the hypergeometric distribution, and the normal distribution. The multivariate normal distribution is a commonly encountered multivariate distribution.
Special distributions
• Normal distribution, the most common continuous distribution
• Bernoulli distribution, for the outcome of a single Bernoulli trial (e.g. success/failure, yes/no)
• Binomial distribution, for the number of "positive occurrences" (e.g. successes, yes votes, etc.) given a fixed total number of independent occurrences
• Negative binomial distribution, for binomial-type observations but where the quantity of interest is the number of failures before a given number of successes occurs
• Geometric distribution, for binomial-type observations but where the quantity of interest is the number of failures before the first success; a special case of the negative binomial distribution, where the number of successes is one.
• Discrete uniform distribution, for a finite set of values (e.g. the outcome of a fair die)
• Continuous uniform distribution, for continuously distributed values
• Poisson distribution, for the number of occurrences of a Poisson-type event in a given period of time
• Exponential distribution, for the time before the next Poisson-type event occurs
• Gamma distribution, for the time before the next k Poisson-type events occur
• Chi-squared distribution, the distribution of a sum of squared standard normal variables; useful e.g. for inference regarding the sample variance of normally distributed samples (see chi-squared test)
• Student's t distribution, the distribution of the ratio of a standard normal variable and the square root of a scaled chi squared variable; useful for inference regarding the mean of normally distributed samples with unknown variance (see Student's t-test)
• Beta distribution, for a single probability (real number between 0 and 1); conjugate to the Bernoulli distribution and binomial distribution
Statistical inference
Main article: Statistical inference
Statistical inference is the process of drawing conclusions from data that are subject to random variation, for example, observational errors or sampling variation.[7] Initial requirements of such a system of procedures for inference and induction are that the system should produce reasonable answers when applied to well-defined situations and that it should be general enough to be applied across a range of situations. Inferential statistics are used to test hypotheses and make estimations using sample data. Whereas descriptive statistics describe a sample, inferential statistics infer predictions about a larger population that the sample represents.
The outcome of statistical inference may be an answer to the question "what should be done next?", where this might be a decision about making further experiments or surveys, or about drawing a conclusion before implementing some organizational or governmental policy. For the most part, statistical inference makes propositions about populations, using data drawn from the population of interest via some form of random sampling. More generally, data about a random process is obtained from its observed behavior during a finite period of time. Given a parameter or hypothesis about which one wishes to make inference, statistical inference most often uses:
• a statistical model of the random process that is supposed to generate the data, which is known when randomization has been used, and
• a particular realization of the random process; i.e., a set of data.
Regression
Main article: Regression analysis
In statistics, regression analysis is a statistical process for estimating the relationships among variables. It includes many ways for modeling and analyzing several variables, when the focus is on the relationship between a dependent variable and one or more independent variables. More specifically, regression analysis helps one understand how the typical value of the dependent variable (or 'criterion variable') changes when any one of the independent variables is varied, while the other independent variables are held fixed. Most commonly, regression analysis estimates the conditional expectation of the dependent variable given the independent variables – that is, the average value of the dependent variable when the independent variables are fixed. Less commonly, the focus is on a quantile, or other location parameter of the conditional distribution of the dependent variable given the independent variables. In all cases, the estimation target is a function of the independent variables called the regression function. In regression analysis, it is also of interest to characterize the variation of the dependent variable around the regression function which can be described by a probability distribution.
Many techniques for carrying out regression analysis have been developed. Familiar methods, such as linear regression, are parametric, in that the regression function is defined in terms of a finite number of unknown parameters that are estimated from the data (e.g. using ordinary least squares). Nonparametric regression refers to techniques that allow the regression function to lie in a specified set of functions, which may be infinite-dimensional.
Nonparametric statistics
Main article: Nonparametric statistics
Nonparametric statistics are values calculated from data in a way that is not based on parameterized families of probability distributions. They include both descriptive and inferential statistics. The typical parameters are the mean, variance, etc. Unlike parametric statistics, nonparametric statistics make no assumptions about the probability distributions of the variables being assessed.[8]
Non-parametric methods are widely used for studying populations that take on a ranked order (such as movie reviews receiving one to four stars). The use of non-parametric methods may be necessary when data have a ranking but no clear numerical interpretation, such as when assessing preferences. In terms of levels of measurement, non-parametric methods result in "ordinal" data.
As non-parametric methods make fewer assumptions, their applicability is much wider than the corresponding parametric methods. In particular, they may be applied in situations where less is known about the application in question. Also, due to the reliance on fewer assumptions, non-parametric methods are more robust.
One drawback of non-parametric methods is that since they do not rely on assumptions, they are generally less powerful than their parametric counterparts.[9] Low power non-parametric tests are problematic because a common use of these methods is for when a sample has a low sample size.[9] Many parametric methods are proven to be the most powerful tests through methods such as the Neyman–Pearson lemma and the Likelihood-ratio test.
Another justification for the use of non-parametric methods is simplicity. In certain cases, even when the use of parametric methods is justified, non-parametric methods may be easier to use. Due both to this simplicity and to their greater robustness, non-parametric methods are seen by some statisticians as leaving less room for improper use and misunderstanding.
Statistics, mathematics, and mathematical statistics
Mathematical statistics is a key subset of the discipline of statistics. Statistical theorists study and improve statistical procedures with mathematics, and statistical research often raises mathematical questions.
Mathematicians and statisticians like Gauss, Laplace, and C. S. Peirce used decision theory with probability distributions and loss functions (or utility functions). The decision-theoretic approach to statistical inference was reinvigorated by Abraham Wald and his successors,[10][11][12][13][14][15][16] and makes extensive use of scientific computing, analysis, and optimization; for the design of experiments, statisticians use algebra and combinatorics. But while statistical practice often relies on probability and decision theory, their application can be controversial [4]
See also
• Asymptotic theory (statistics)
References
1. Kannan, D.; Lakshmikantham, V., eds. (2002). Handbook of stochastic analysis and applications. New York: M. Dekker. ISBN 0824706609.
2. Schervish, Mark J. (1995). Theory of statistics (Corr. 2nd print. ed.). New York: Springer. ISBN 0387945466.
3. Freedman, D.A. (2005) Statistical Models: Theory and Practice, Cambridge University Press. ISBN 978-0-521-67105-7
4. Freedman, David A. (2010). Collier, David; Sekhon, Jasjeet S.; Stark, Philp B. (eds.). Statistical Models and Causal Inference: A Dialogue with the Social Sciences. Cambridge University Press. ISBN 978-0-521-12390-7.
5. Hogg, R. V., A. Craig, and J. W. McKean. "Intro to Mathematical Statistics." (2005).
6. Larsen, Richard J. and Marx, Morris L. "An Introduction to Mathematical Statistics and Its Applications" (2012). Prentice Hall.
7. Upton, G., Cook, I. (2008) Oxford Dictionary of Statistics, OUP. ISBN 978-0-19-954145-4
8. "Research Nonparametric Methods". Carnegie Mellon University. Retrieved August 30, 2022.{{cite web}}: CS1 maint: url-status (link)
9. "Nonparametric Tests". sphweb.bumc.bu.edu. Retrieved 2022-08-31.
10. Wald, Abraham (1947). Sequential analysis. New York: John Wiley and Sons. ISBN 0-471-91806-7. See Dover reprint, 2004: ISBN 0-486-43912-7
11. Wald, Abraham (1950). Statistical Decision Functions. John Wiley and Sons, New York.
12. Lehmann, Erich (1997). Testing Statistical Hypotheses (2nd ed.). ISBN 0-387-94919-4.
13. Lehmann, Erich; Cassella, George (1998). Theory of Point Estimation (2nd ed.). ISBN 0-387-98502-6.
14. Bickel, Peter J.; Doksum, Kjell A. (2001). Mathematical Statistics: Basic and Selected Topics. Vol. 1 (Second (updated printing 2007) ed.). Pearson Prentice-Hall.
15. Le Cam, Lucien (1986). Asymptotic Methods in Statistical Decision Theory. Springer-Verlag. ISBN 0-387-96307-3.
16. Liese, Friedrich & Miescke, Klaus-J. (2008). Statistical Decision Theory: Estimation, Testing, and Selection. Springer.
Further reading
• Borovkov, A. A. (1999). Mathematical Statistics. CRC Press. ISBN 90-5699-018-7
• Virtual Laboratories in Probability and Statistics (Univ. of Ala.-Huntsville)
• StatiBot, interactive online expert system on statistical tests.
• Ray, Manohar; Sharma, Har Swarup (1966). Mathematical Statistics. Ram Prasad & Sons. ISBN 978-9383385188
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| Wikipedia |
Theory of Lie groups
In mathematics, Theory of Lie groups is a series of books on Lie groups by Claude Chevalley (1946, 1951, 1955). The first in the series was one of the earliest books on Lie groups to treat them from the global point of view, and for many years was the standard text on Lie groups. The second and third volumes, on algebraic groups and Lie algebras, were written in French, and later reprinted bound together as one volume. Apparently further volumes were planned but not published, though his lectures (Chevalley 2005) on the classification of semisimple algebraic groups could be considered as a continuation of the series.
References
• Chevalley, Claude (1946), Theory of Lie Groups. I, Princeton Mathematical Series, vol. 8, Princeton University Press, ISBN 978-0-691-04990-8, MR 0015396
• Chevalley, Claude (1951), Théorie des groupes de Lie. Tome II. Groupes algébriques, Actualités Sci. Ind., vol. 1152, Hermann & Cie., Paris, MR 0051242
• Chevalley, Claude (1955), Théorie des groupes de Lie. Tome III. Théorèmes généraux sur les algèbres de Lie, Actualités Sci. Ind., vol. 1226, Hermann & Cie, Paris, MR 0068552
• Chevalley, Claude (1968), Théorie des groupes de Lie : Groupes algébriques, théorèmes généraux sur les algèbres de Lie (in French), vol. 8, Paris: Hermann, Reprint of volumes II and III bound as one volume
• Chevalley, Claude (2005) [1958], Cartier, P. (ed.), Classification des groupes algébriques semi-simples, Collected works., vol. 3, Berlin, New York: Springer-Verlag, ISBN 978-3-540-23031-1, MR 0106966
• Smith, P. A. (1947), "Review: Claude Chevalley, The theory of Lie groups, I", Bull. Amer. Math. Soc., 53 (9): 884–887, doi:10.1090/s0002-9904-1947-08876-5
| Wikipedia |
Theories of iterated inductive definitions
In set theory and logic, Buchholz's ID hierarchy is a hierarchy of subsystems of first-order arithmetic. The systems/theories $ID_{\nu }$ are referred to as "the formal theories of ν-times iterated inductive definitions". IDν extends PA by ν iterated least fixed points of monotone operators.
Definition
Original definition
The formal theory IDω (and IDν in general) is an extension of Peano Arithmetic, formulated in the language LID, by the following axioms:
• $\forall y\forall x({\mathfrak {M}}_{y}(P_{y}^{\mathfrak {M}},x)\rightarrow x\in P_{y}^{\mathfrak {M}})$
• $\forall y(\forall x({\mathfrak {M}}_{y}(F,x)\rightarrow F(x))\rightarrow \forall x(x\in P_{y}^{\mathfrak {M}}\rightarrow F(x)))$ for every LID-formula F(x)
• $\forall y\forall x_{0}\forall x_{1}(P_{<y}^{\mathfrak {M}}x_{0}x_{1}\leftrightarrow x_{0}<y\land x_{1}\in P_{x_{0}}^{\mathfrak {M}})$
The theory IDν with ν ≠ ω is defined as:
• $\forall y\forall x(Z_{y}(P_{y}^{\mathfrak {M}},x)\rightarrow x\in P_{y}^{\mathfrak {M}})$
• $\forall x({\mathfrak {M}}_{u}(F,x)\rightarrow F(x))\rightarrow \forall x(P_{u}^{\mathfrak {M}}x\rightarrow F(x))$ for every LID-formula F(x) and each u < ν
• $\forall y\forall x_{0}\forall x_{1}(P_{<y}^{\mathfrak {M}}x_{0}x_{1}\leftrightarrow x_{0}<y\land x_{1}\in P_{x_{0}}^{\mathfrak {M}})$
ID1
A set $I\subseteq \mathbb {N} $ is called inductively defined if for some monotonic operator $\Gamma :P(N)\rightarrow P(N)$, $LFP(\Gamma )=I$, where $LFP(f)$ denotes the least fixed point of $f$. The language of ID1, $L_{ID_{1}}$, is obtained from that of first-order number theory, $L_{\mathbb {N} }$, by the addition of a set (or predicate) constant IA for every X-positive formula A(X, x) in LN[X] that only contains X (a new set variable) and x (a number variable) as free variables. The term X-positive means that X only occurs positively in A (X is never on the left of an implication). We allow ourselves a bit of set-theoretic notation:
• $F(x)=\{x\in N\mid F(x)\}$
• $s\in F$ means $F(s)$
• For two formulas $F$ and $G$, $F\subseteq G$ means $\forall xF(x)\rightarrow G(x)$.
Then ID1 contains the axioms of first-order number theory (PA) with the induction scheme extended to the new language as well as these axioms:
• $(ID_{1})^{1}:A(I_{A})\subseteq I_{A}$
• $(ID_{1})^{2}:A(F)\subseteq F\rightarrow I_{A}\subseteq F$
Where $F(x)$ ranges over all $L_{ID_{1}}$ formulas.
Note that $(ID_{1})^{1}$ expresses that $I_{A}$ is closed under the arithmetically definable set operator $\Gamma _{A}(S)=\{x\in \mathbb {N} \mid \mathbb {N} \models A(S,x)\}$, while $(ID_{1})^{2}$ expresses that $I_{A}$ is the least such (at least among sets definable in $L_{ID_{1}}$).
Thus, $I_{A}$ is meant to be the least pre-fixed-point, and hence the least fixed point of the operator $\Gamma _{A}$.
IDν
To define the system of ν-times iterated inductive definitions, where ν is an ordinal, let $\prec $ be a primitive recursive well-ordering of order type ν. We use Greek letters to denote elements of the field of $\prec $. The language of IDν, $L_{ID_{\nu }}$ is obtained from $L_{\mathbb {N} }$ by the addition of a binary predicate constant JA for every X-positive $L_{\mathbb {N} }[X,Y]$ formula $A(X,Y,\mu ,x)$ that contains at most the shown free variables, where X is again a unary (set) variable, and Y is a fresh binary predicate variable. We write $x\in J_{A}^{\mu }$ instead of $J_{A}(\mu ,x)$, thinking of x as a distinguished variable in the latter formula.
The system IDν is now obtained from the system of first-order number theory (PA) by expanding the induction scheme to the new language and adding the scheme $(TI_{\nu }):TI(\prec ,F)$ expressing transfinite induction along $\prec $ for an arbitrary $L_{ID_{\nu }}$ formula $F$ as well as the axioms:
• $(ID_{\nu })^{1}:\forall \mu \prec \nu ;A^{\mu }(J_{A}^{\mu })\subseteq J_{A}^{\mu }$
• $(ID_{\nu })^{2}:\forall \mu \prec \nu ;A^{\mu }(F)\subseteq F\rightarrow J_{A}^{\mu }\subseteq F$
where $F(x)$ is an arbitrary $L_{ID_{\nu }}$ formula. In $(ID_{\nu })^{1}$ and $(ID_{\nu })^{2}$ we used the abbreviation $A^{\mu }(F)$ for the formula $A(F,(\lambda \gamma y;\gamma \prec \mu \land y\in J_{A}^{\gamma }),\mu ,x)$, where $x$ is the distinguished variable. We see that these express that each $J_{A}^{\mu }$, for $\mu \prec \nu $, is the least fixed point (among definable sets) for the operator $\Gamma _{A}^{\mu }(S)=\{n\in \mathbb {N} |(\mathbb {N} ,(J_{A}^{\gamma })_{\gamma \prec \mu }\}$. Note how all the previous sets $J_{A}^{\gamma }$, for $\gamma \prec \mu $, are used as parameters.
We then define $ ID_{\prec \nu }=\bigcup _{\xi \prec \nu }ID_{\xi }$.
Variants
${\widehat {\mathsf {ID}}}_{\nu }$ - ${\widehat {\mathsf {ID}}}_{\nu }$ is a weakened version of ${\mathsf {ID}}_{\nu }$. In the system of ${\widehat {\mathsf {ID}}}_{\nu }$, a set $I\subseteq \mathbb {N} $ is instead called inductively defined if for some monotonic operator $\Gamma :P(N)\rightarrow P(N)$, $I$ is a fixed point of $\Gamma $, rather than the least fixed point. This subtle difference makes the system significantly weaker: $PTO({\widehat {\mathsf {ID}}}_{1})=\psi (\Omega ^{\varepsilon _{0}})$, while $PTO({\mathsf {ID}}_{1})=\psi (\varepsilon _{\Omega +1})$.
${\mathsf {ID}}_{\nu }\#$ is ${\widehat {\mathsf {ID}}}_{\nu }$ weakened even further. In ${\mathsf {ID}}_{\nu }\#$, not only does it use fixed points rather than least fixed points, and has induction only for positive formulas. This once again subtle difference makes the system even weaker: $PTO({\mathsf {ID}}_{1}\#)=\psi (\Omega ^{\omega })$, while $PTO({\widehat {\mathsf {ID}}}_{1})=\psi (\Omega ^{\varepsilon _{0}})$.
${\mathsf {W-ID}}_{\nu }$ is the weakest of all variants of ${\mathsf {ID}}_{\nu }$, based on W-types. The amount of weakening compared to regular iterated inductive definitions is identical to removing bar induction given a certain subsystem of second-order arithmetic. $PTO({\mathsf {W-ID}}_{1})=\psi _{0}(\Omega \times \omega )$, while $PTO({\mathsf {ID}}_{1})=\psi (\varepsilon _{\Omega +1})$.
${\mathsf {U(ID}}_{\nu }{\mathsf {)}}$ is an "unfolding" strengthening of ${\mathsf {ID}}_{\nu }$. It is not exactly a first-order arithmetic system, but captures what one can get by predicative reasoning based on ν-times iterated generalized inductive definitions. The amount of increase in strength is identical to the increase from $\varepsilon _{0}$ to $\Gamma _{0}$: $PTO({\mathsf {ID}}_{1})=\psi (\varepsilon _{\Omega +1})$, while $PTO({\mathsf {U(ID}}_{1}{\mathsf {)}})=\psi (\Gamma _{\Omega +1})$.
Results
• Let ν > 0. If a ∈ T0 contains no symbol Dμ with ν < μ, then "a ∈ W0" is provable in IDν.
• IDω is contained in $\Pi _{1}^{1}-CA+BI$.
• If a $\Pi _{2}^{0}$-sentence $\forall x\exists y\varphi (x,y)(\varphi \in \Sigma _{1}^{0})$ is provable in IDν, then there exists $p\in N$ such that $\forall n\geq p\exists k<H_{D_{0}D_{\nu }^{n}0}(1)\varphi (n,k)$.
• If the sentence A is provable in IDν for all ν < ω, then there exists k ∈ N such that $\vdash _{k}^{D_{\nu }^{k}0}A^{N}$.
Proof-theoretic ordinals
• The proof-theoretic ordinal of ID<ν is equal to $\psi _{0}(\Omega _{\nu })$.
• The proof-theoretic ordinal of IDν is equal to $\psi _{0}(\varepsilon _{\Omega _{\nu }+1})=\psi _{0}(\Omega _{\nu +1})$ .
• The proof-theoretic ordinal of ${\widehat {ID}}_{<\omega }$ is equal to $\Gamma _{0}$.
• The proof-theoretic ordinal of ${\widehat {ID}}_{\nu }$ for $\nu <\omega $ is equal to $\varphi (\varphi (\nu ,0),0)$.
• The proof-theoretic ordinal of ${\widehat {ID}}_{\varphi (\alpha ,\beta )}$ is equal to $\varphi (1,0,\varphi (\alpha +1,\beta -1))$.
• The proof-theoretic ordinal of ${\widehat {ID}}_{<\varphi (0,\alpha )}$ for $\alpha >1$ is equal to $\varphi (1,\alpha ,0)$.
• The proof-theoretic ordinal of ${\widehat {ID}}_{<\nu }$ for $\nu \geq \varepsilon _{0}$ is equal to $\varphi (1,\nu ,0)$.
• The proof-theoretic ordinal of $ID_{\nu }\#$ is equal to $\varphi (\omega ^{\nu },0)$.
• The proof-theoretic ordinal of $ID_{<\nu }\#$ is equal to $\varphi (0,\omega ^{\nu +1})$.
• The proof-theoretic ordinal of $W{\textrm {-}}ID_{\varphi (\alpha ,\beta )}$ is equal to $\psi _{0}(\Omega _{\varphi (\alpha ,\beta )}\times \varphi (\alpha +1,\beta -1))$.
• The proof-theoretic ordinal of $W{\textrm {-}}ID_{<\varphi (\alpha ,\beta )}$ is equal to $\psi _{0}(\varphi (\alpha +1,\beta -1)^{\Omega _{\varphi (\alpha ,\beta -1)}+1})$.
• The proof-theoretic ordinal of $U(ID_{\nu })$ is equal to $\psi _{0}(\varphi (\nu ,0,\Omega +1))$.
• The proof-theoretic ordinal of $U(ID_{<\nu })$ is equal to $\psi _{0}(\Omega ^{\Omega +\varphi (\nu ,0,\Omega )})$.
• The proof-theoretic ordinal of ID1 (the Bachmann-Howard ordinal) is also the proof-theoretic ordinal of ${\mathsf {KP}}$, ${\mathsf {KP\omega }}$, ${\mathsf {CZF}}$ and ${\mathsf {ML_{1}V}}$.
• The proof-theoretic ordinal of W-IDω ($\psi _{0}(\Omega _{\omega }\varepsilon _{0})$) is also the proof-theoretic ordinal of ${\mathsf {W-KPI}}$.
• The proof-theoretic ordinal of IDω (the Takeuti-Feferman-Buchholz ordinal) is also the proof-theoretic ordinal of ${\mathsf {KPI}}$, $\Pi _{1}^{1}-{\mathsf {CA}}+{\mathsf {BI}}$ and $\Delta _{2}^{1}-{\mathsf {CA}}+{\mathsf {BI}}$.
• The proof-theoretic ordinal of ID<ω^ω ($\psi _{0}(\Omega _{\omega ^{\omega }})$) is also the proof-theoretic ordinal of $\Delta _{2}^{1}-{\mathsf {CR}}$.
• The proof-theoretic ordinal of ID<ε0 ($\psi _{0}(\Omega _{\varepsilon _{0}})$) is also the proof-theoretic ordinal of $\Delta _{2}^{1}-{\mathsf {CA}}$ and $\Sigma _{2}^{1}-{\mathsf {AC}}$.
References
• An independence result for $\Pi _{1}^{1}-CA+BI$
• Iterated inductive definitions and subsystems of analysis: recent proof-theoretical studies
• Iterated inductive definitions in nLab
• Lemma for the intuitionistic theory of iterated inductive definitions
• Iterated Inductive Definitions and $\Sigma _{2}^{1}-AC$
• Large countable ordinals and numbers in Agda
• Ordinal analysis in nLab
Large countable ordinals
• First infinite ordinal ω
• Epsilon numbers ε0
• Feferman–Schütte ordinal Γ0
• Ackermann ordinal θ(Ω2)
• small Veblen ordinal θ(Ωω)
• large Veblen ordinal θ(ΩΩ)
• Bachmann–Howard ordinal ψ(εΩ+1)
• Buchholz's ordinal ψ0(Ωω)
• Takeuti–Feferman–Buchholz ordinal ψ(εΩω+1)
• Proof-theoretic ordinals of the theories of iterated inductive definitions
• Nonrecursive ordinal ≥ ωCK
1
| Wikipedia |
Theoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experimental tools to probe these phenomena.
The advancement of science generally depends on the interplay between experimental studies and theory. In some cases, theoretical physics adheres to standards of mathematical rigour while giving little weight to experiments and observations.[lower-alpha 1] For example, while developing special relativity, Albert Einstein was concerned with the Lorentz transformation which left Maxwell's equations invariant, but was apparently uninterested in the Michelson–Morley experiment on Earth's drift through a luminiferous aether.[1] Conversely, Einstein was awarded the Nobel Prize for explaining the photoelectric effect, previously an experimental result lacking a theoretical formulation.[2]
Overview
A physical theory is a model of physical events. It is judged by the extent to which its predictions agree with empirical observations. The quality of a physical theory is also judged on its ability to make new predictions which can be verified by new observations. A physical theory differs from a mathematical theorem in that while both are based on some form of axioms, judgment of mathematical applicability is not based on agreement with any experimental results.[3][4] A physical theory similarly differs from a mathematical theory, in the sense that the word "theory" has a different meaning in mathematical terms.[lower-alpha 2]
$\mathrm {Ric} =kg$ The equations for an Einstein manifold, used in general relativity to describe the curvature of spacetime
A physical theory involves one or more relationships between various measurable quantities. Archimedes realized that a ship floats by displacing its mass of water, Pythagoras understood the relation between the length of a vibrating string and the musical tone it produces.[5][6] Other examples include entropy as a measure of the uncertainty regarding the positions and motions of unseen particles and the quantum mechanical idea that (action and) energy are not continuously variable.
Theoretical physics consists of several different approaches. In this regard, theoretical particle physics forms a good example. For instance: "phenomenologists" might employ (semi-) empirical formulas and heuristics to agree with experimental results, often without deep physical understanding.[lower-alpha 3] "Modelers" (also called "model-builders") often appear much like phenomenologists, but try to model speculative theories that have certain desirable features (rather than on experimental data), or apply the techniques of mathematical modeling to physics problems.[lower-alpha 4] Some attempt to create approximate theories, called effective theories, because fully developed theories may be regarded as unsolvable or too complicated. Other theorists may try to unify, formalise, reinterpret or generalise extant theories, or create completely new ones altogether.[lower-alpha 5] Sometimes the vision provided by pure mathematical systems can provide clues to how a physical system might be modeled;[lower-alpha 6] e.g., the notion, due to Riemann and others, that space itself might be curved. Theoretical problems that need computational investigation are often the concern of computational physics.
Theoretical advances may consist in setting aside old, incorrect paradigms (e.g., aether theory of light propagation, caloric theory of heat, burning consisting of evolving phlogiston, or astronomical bodies revolving around the Earth) or may be an alternative model that provides answers that are more accurate or that can be more widely applied. In the latter case, a correspondence principle will be required to recover the previously known result.[7][8] Sometimes though, advances may proceed along different paths. For example, an essentially correct theory may need some conceptual or factual revisions; atomic theory, first postulated millennia ago (by several thinkers in Greece and India) and the two-fluid theory of electricity[9] are two cases in this point. However, an exception to all the above is the wave–particle duality, a theory combining aspects of different, opposing models via the Bohr complementarity principle.
Physical theories become accepted if they are able to make correct predictions and no (or few) incorrect ones. The theory should have, at least as a secondary objective, a certain economy and elegance (compare to mathematical beauty), a notion sometimes called "Occam's razor" after the 13th-century English philosopher William of Occam (or Ockham), in which the simpler of two theories that describe the same matter just as adequately is preferred (but conceptual simplicity may mean mathematical complexity).[10] They are also more likely to be accepted if they connect a wide range of phenomena. Testing the consequences of a theory is part of the scientific method.
Physical theories can be grouped into three categories: mainstream theories, proposed theories and fringe theories.
History
Theoretical physics began at least 2,300 years ago, under the Pre-socratic philosophy, and continued by Plato and Aristotle, whose views held sway for a millennium. During the rise of medieval universities, the only acknowledged intellectual disciplines were the seven liberal arts of the Trivium like grammar, logic, and rhetoric and of the Quadrivium like arithmetic, geometry, music and astronomy. During the Middle Ages and Renaissance, the concept of experimental science, the counterpoint to theory, began with scholars such as Ibn al-Haytham and Francis Bacon. As the Scientific Revolution gathered pace, the concepts of matter, energy, space, time and causality slowly began to acquire the form we know today, and other sciences spun off from the rubric of natural philosophy. Thus began the modern era of theory with the Copernican paradigm shift in astronomy, soon followed by Johannes Kepler's expressions for planetary orbits, which summarized the meticulous observations of Tycho Brahe; the works of these men (alongside Galileo's) can perhaps be considered to constitute the Scientific Revolution.
The great push toward the modern concept of explanation started with Galileo, one of the few physicists who was both a consummate theoretician and a great experimentalist. The analytic geometry and mechanics of Descartes were incorporated into the calculus and mechanics of Isaac Newton, another theoretician/experimentalist of the highest order, writing Principia Mathematica.[11] In it contained a grand synthesis of the work of Copernicus, Galileo and Kepler; as well as Newton's theories of mechanics and gravitation, which held sway as worldviews until the early 20th century. Simultaneously, progress was also made in optics (in particular colour theory and the ancient science of geometrical optics), courtesy of Newton, Descartes and the Dutchmen Snell and Huygens. In the 18th and 19th centuries Joseph-Louis Lagrange, Leonhard Euler and William Rowan Hamilton would extend the theory of classical mechanics considerably.[12] They picked up the interactive intertwining of mathematics and physics begun two millennia earlier by Pythagoras.
Among the great conceptual achievements of the 19th and 20th centuries were the consolidation of the idea of energy (as well as its global conservation) by the inclusion of heat, electricity and magnetism, and then light. The laws of thermodynamics, and most importantly the introduction of the singular concept of entropy began to provide a macroscopic explanation for the properties of matter. Statistical mechanics (followed by statistical physics and Quantum statistical mechanics) emerged as an offshoot of thermodynamics late in the 19th century. Another important event in the 19th century was the discovery of electromagnetic theory, unifying the previously separate phenomena of electricity, magnetism and light.
The pillars of modern physics, and perhaps the most revolutionary theories in the history of physics, have been relativity theory and quantum mechanics. Newtonian mechanics was subsumed under special relativity and Newton's gravity was given a kinematic explanation by general relativity. Quantum mechanics led to an understanding of blackbody radiation (which indeed, was an original motivation for the theory) and of anomalies in the specific heats of solids — and finally to an understanding of the internal structures of atoms and molecules. Quantum mechanics soon gave way to the formulation of quantum field theory (QFT), begun in the late 1920s. In the aftermath of World War 2, more progress brought much renewed interest in QFT, which had since the early efforts, stagnated. The same period also saw fresh attacks on the problems of superconductivity and phase transitions, as well as the first applications of QFT in the area of theoretical condensed matter. The 1960s and 70s saw the formulation of the Standard model of particle physics using QFT and progress in condensed matter physics (theoretical foundations of superconductivity and critical phenomena, among others), in parallel to the applications of relativity to problems in astronomy and cosmology respectively.
All of these achievements depended on the theoretical physics as a moving force both to suggest experiments and to consolidate results — often by ingenious application of existing mathematics, or, as in the case of Descartes and Newton (with Leibniz), by inventing new mathematics. Fourier's studies of heat conduction led to a new branch of mathematics: infinite, orthogonal series.[13]
Modern theoretical physics attempts to unify theories and explain phenomena in further attempts to understand the Universe, from the cosmological to the elementary particle scale. Where experimentation cannot be done, theoretical physics still tries to advance through the use of mathematical models.
Mainstream theories
Mainstream theories (sometimes referred to as central theories) are the body of knowledge of both factual and scientific views and possess a usual scientific quality of the tests of repeatability, consistency with existing well-established science and experimentation. There do exist mainstream theories that are generally accepted theories based solely upon their effects explaining a wide variety of data, although the detection, explanation, and possible composition are subjects of debate.
Examples
• Analog models of gravity
• Big Bang
• Causality
• Chaos theory
• Classical field theory
• Classical mechanics
• Condensed matter physics (including solid state physics and the electronic structure of materials)
• Conservation law
• Conservation of angular momentum
• Conservation of energy
• Conservation of mass
• Conservation of momentum
• Continuum mechanics
• Cosmic censorship hypothesis
• Cosmological Constant
• CPT symmetry
• Dark matter
• Dynamics
• Dynamo theory
• Electromagnetism
• Electroweak interaction
• Field theory
• Fluctuation theorem
• Fluid dynamics
• Fluid mechanics
• Fundamental interaction
• General relativity
• Gravitational constant
• Heisenberg's uncertainty principle
• Kinetic theory of gases
• Laws of thermodynamics
• Maxwell's equations
• Newton's laws of motion
• Pauli exclusion principle
• Perturbation theory (quantum mechanics)
• Physical cosmology
• Planck constant
• Poincaré recurrence theorem
• Quantum biology
• Quantum chaos
• Quantum chromodynamics
• Quantum complexity theory
• Quantum computing
• Quantum dynamics
• Quantum electrochemistry
• Quantum electrodynamics
• Quantum field theory
• Quantum field theory in curved spacetime
• Quantum geometry
• Quantum information theory
• Quantum logic
• Quantum mechanics
• Quantum optics
• Quantum physics
• Quantum thermodynamics
• Relativistic quantum mechanics
• Scattering theory
• Solid mechanics
• Special relativity
• Spin–statistics theorem
• Spontaneous symmetry breaking
• Standard Model
• Statistical mechanics
• Statistical physics
• Theory of relativity
• Thermodynamics
• Wave–particle duality
• Weak interaction
Proposed theories
The proposed theories of physics are usually relatively new theories which deal with the study of physics which include scientific approaches, means for determining the validity of models and new types of reasoning used to arrive at the theory. However, some proposed theories include theories that have been around for decades and have eluded methods of discovery and testing. Proposed theories can include fringe theories in the process of becoming established (and, sometimes, gaining wider acceptance). Proposed theories usually have not been tested. In addition to the theories like those listed below, there are also different interpretations of quantum mechanics, which may or may not be considered different theories since it is debatable whether they yield different predictions for physical experiments, even in principle. For example, AdS/CFT correspondence, Chern–Simons theory, graviton, magnetic monopole, string theory, theory of everything.
Fringe theories
Fringe theories include any new area of scientific endeavor in the process of becoming established and some proposed theories. It can include speculative sciences. This includes physics fields and physical theories presented in accordance with known evidence, and a body of associated predictions have been made according to that theory.
Some fringe theories go on to become a widely accepted part of physics. Other fringe theories end up being disproven. Some fringe theories are a form of protoscience and others are a form of pseudoscience. The falsification of the original theory sometimes leads to reformulation of the theory.
Examples
• Aether (classical element)
• Luminiferous aether
• Digital physics
• Electrogravitics
• Stochastic electrodynamics
• Tesla's dynamic theory of gravity
Thought experiments vs real experiments
"Thought" experiments are situations created in one's mind, asking a question akin to "suppose you are in this situation, assuming such is true, what would follow?". They are usually created to investigate phenomena that are not readily experienced in every-day situations. Famous examples of such thought experiments are Schrödinger's cat, the EPR thought experiment, simple illustrations of time dilation, and so on. These usually lead to real experiments designed to verify that the conclusion (and therefore the assumptions) of the thought experiments are correct. The EPR thought experiment led to the Bell inequalities, which were then tested to various degrees of rigor, leading to the acceptance of the current formulation of quantum mechanics and probabilism as a working hypothesis.
See also
• List of theoretical physicists
• Philosophy of physics
• Symmetry in quantum mechanics
• Timeline of developments in theoretical physics
Notes
1. There is some debate as to whether or not theoretical physics uses mathematics to build intuition and illustrativeness to extract physical insight (especially when normal experience fails), rather than as a tool in formalizing theories. This links to the question of it using mathematics in a less formally rigorous, and more intuitive or heuristic way than, say, mathematical physics.
2. Sometimes the word "theory" can be used ambiguously in this sense, not to describe scientific theories, but research (sub)fields and programmes. Examples: relativity theory, quantum field theory, string theory.
3. The work of Johann Balmer and Johannes Rydberg in spectroscopy, and the semi-empirical mass formula of nuclear physics are good candidates for examples of this approach.
4. The Ptolemaic and Copernican models of the Solar system, the Bohr model of hydrogen atoms and nuclear shell model are good candidates for examples of this approach.
5. Arguably these are the most celebrated theories in physics: Newton's theory of gravitation, Einstein's theory of relativity and Maxwell's theory of electromagnetism share some of these attributes.
6. This approach is often favoured by (pure) mathematicians and mathematical physicists.
References
1. van Dongen, Jeroen (2009). "On the role of the Michelson-Morley experiment: Einstein in Chicago". Archive for History of Exact Sciences. 63 (6): 655–663. arXiv:0908.1545. doi:10.1007/s00407-009-0050-5.
2. "The Nobel Prize in Physics 1921". The Nobel Foundation. Retrieved 2008-10-09.
3. Theorems and Theories Archived 2014-08-19 at the Wayback Machine, Sam Nelson.
4. Mark C. Chu-Carroll, March 13, 2007:Theorems, Lemmas, and Corollaries. Good Math, Bad Math blog.
5. Singiresu S. Rao (2007). Vibration of Continuous Systems (illustrated ed.). John Wiley & Sons. 5,12. ISBN 978-0471771715. ISBN 9780471771715
6. Eli Maor (2007). The Pythagorean Theorem: A 4,000-year History (illustrated ed.). Princeton University Press. pp. 18–20. ISBN 978-0691125268. ISBN 9780691125268
7. Bokulich, Alisa, "Bohr's Correspondence Principle", The Stanford Encyclopedia of Philosophy (Spring 2014 Edition), Edward N. Zalta (ed.)
8. Enc. Britannica (1994), pg 844.
9. Enc. Britannica (1994), pg 834.
10. Simplicity in the Philosophy of Science (retrieved 19 Aug 2014), Internet Encyclopedia of Philosophy.
11. See 'Correspondence of Isaac Newton, vol.2, 1676–1687' ed. H W Turnbull, Cambridge University Press 1960; at page 297, document #235, letter from Hooke to Newton dated 24 November 1679.
12. Penrose, R (2004). The Road to Reality. Jonathan Cape. p. 471.
13. Penrose, R (2004). "9: Fourier decompositions and hyperfunctions". The Road to Reality. Jonathan Cape.
Further reading
• Physical Sciences. 1994. {{cite book}}: |work= ignored (help)
• Duhem, Pierre. La théorie physique - Son objet, sa structure, (in French). 2nd edition - 1914. English translation: The physical theory - its purpose, its structure. Republished by Joseph Vrin philosophical bookstore (1981), ISBN 2711602214.
• Feynman, et al. The Feynman Lectures on Physics (3 vol.). First edition: Addison–Wesley, (1964, 1966).
Bestselling three-volume textbook covering the span of physics. Reference for both (under)graduate student and professional researcher alike.
• Landau et al. Course of Theoretical Physics.
Famous series of books dealing with theoretical concepts in physics covering 10 volumes, translated into many languages and reprinted over many editions. Often known simply as "Landau and Lifschits" or "Landau-Lifschits" in the literature.
• Longair, MS. Theoretical Concepts in Physics: An Alternative View of Theoretical Reasoning in Physics. Cambridge University Press; 2d edition (4 Dec 2003). ISBN 052152878X. ISBN 978-0521528788
• Planck, Max (1909). Eight Lectures on theoretical physics. Library of Alexandria. ISBN 1465521887, ISBN 9781465521880.
A set of lectures given in 1909 at Columbia University.
• Sommerfeld, Arnold. Vorlesungen über theoretische Physik (Lectures on Theoretical Physics); German, 6 volumes.
A series of lessons from a master educator of theoretical physicists.
External links
Wikibooks has a book on the topic of: Introduction to Theoretical Physics
• MIT Center for Theoretical Physics
• How to become a GOOD Theoretical Physicist, a website made by Gerard 't Hooft
Major branches of physics
Divisions
• Pure
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Approaches
• Experimental
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Classical
• Classical mechanics
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• Classical electromagnetism
• Classical optics
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Modern
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| Wikipedia |
Theorem
In mathematics, a theorem is a statement that has been proved, or can be proved.[lower-alpha 1][2][3] The proof of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems.
In mainstream mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice (ZFC), or of a less powerful theory, such as Peano arithmetic.[lower-alpha 2] Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as theorems only the most important results, and use the terms lemma, proposition and corollary for less important theorems.
In mathematical logic, the concepts of theorems and proofs have been formalized in order to allow mathematical reasoning about them. In this context, statements become well-formed formulas of some formal language. A theory consists of some basis statements called axioms, and some deducing rules (sometimes included in the axioms). The theorems of the theory are the statements that can be derived from the axioms by using the deducing rules.[lower-alpha 3] This formalization led to proof theory, which allows proving general theorems about theorems and proofs. In particular, Gödel's incompleteness theorems show that every consistent theory containing the natural numbers has true statements on natural numbers that are not theorems of the theory (that is they cannot be proved inside the theory).
As the axioms are often abstractions of properties of the physical world, theorems may be considered as expressing some truth, but in contrast to the notion of a scientific law, which is experimental, the justification of the truth of a theorem is purely deductive.[6][7]
Theoremhood and truth
Until the end of the 19th century and the foundational crisis of mathematics, all mathematical theories were built from a few basic properties that were considered as self-evident; for example, the facts that every natural number has a successor, and that there is exactly one line that passes through two given distinct points. These basic properties that were considered as absolutely evident were called postulates or axioms; for example Euclid's postulates. All theorems were proved by using implicitly or explicitly these basic properties, and, because of the evidence of these basic properties, a proved theorem was considered as a definitive truth, unless there was an error in the proof. For example, the sum of the interior angles of a triangle equals 180°, and this was considered as an undoubtable fact.
One aspect of the foundational crisis of mathematics was the discovery of non-Euclidean geometries that do not lead to any contradiction, although, in such geometries, the sum of the angles of a triangle is different from 180°. So, the property "the sum of the angles of a triangle equals 180°" is either true or false, depending whether Euclid's fifth postulate is assumed or denied. Similarly, the use of "evident" basic properties of sets leads to the contradiction of Russel's paradox. This has been resolved by elaborating the rules that are allowed for manipulating sets.
This crisis has been resolved by revisiting the foundations of mathematics to make them more rigorous. In these new foundations, a theorem is a well-formed formula of a mathematical theory that can be proved from the axioms and inference rules of the theory. So, the above theorem on the sum of the angles of a triangle becomes: Under the axioms and inference rules of Euclidean geometry, the sum of the interior angles of a triangle equals 180°. Similarly, Russel's paradox disappears because, in an axiomatized set theory, the set of all sets cannot be expressed with a well-formed formula. More precisely, if the set of all sets can be expressed with a well-formed formula, this implies that the theory is inconsistent, and every well-formed assertion, as well as its negation, is a theorem.
In this context, the validity of a theorem depends only on the correctness of its proof. It is independent from the truth, or even the significance of the axioms. This does not mean that the significance of the axioms is uninteresting, but only that the validity of a theorem is independent from the significance of the axioms. This independence may be useful by allowing the use of results of some area of mathematics in apparently unrelated areas.
An important consequence of this way of thinking about mathematics is that it allows defining mathematical theories and theorems as mathematical objects, and to prove theorems about them. Examples are Gödel's incompleteness theorems. In particular, there are well-formed assertions than can be proved to not be a theorem of the ambient theory, although they can be proved in a wider theory. An example is Goodstein's theorem, which can be stated in Peano arithmetic, but is proved to be not provable in Peano arithmetic. However, it is provable in some more general theories, such as Zermelo–Fraenkel set theory.
Epistemological considerations
Many mathematical theorems are conditional statements, whose proofs deduce conclusions from conditions known as hypotheses or premises. In light of the interpretation of proof as justification of truth, the conclusion is often viewed as a necessary consequence of the hypotheses. Namely, that the conclusion is true in case the hypotheses are true—without any further assumptions. However, the conditional could also be interpreted differently in certain deductive systems, depending on the meanings assigned to the derivation rules and the conditional symbol (e.g., non-classical logic).
Although theorems can be written in a completely symbolic form (e.g., as propositions in propositional calculus), they are often expressed informally in a natural language such as English for better readability. The same is true of proofs, which are often expressed as logically organized and clearly worded informal arguments, intended to convince readers of the truth of the statement of the theorem beyond any doubt, and from which a formal symbolic proof can in principle be constructed.
In addition to the better readability, informal arguments are typically easier to check than purely symbolic ones—indeed, many mathematicians would express a preference for a proof that not only demonstrates the validity of a theorem, but also explains in some way why it is obviously true. In some cases, one might even be able to substantiate a theorem by using a picture as its proof.
Because theorems lie at the core of mathematics, they are also central to its aesthetics. Theorems are often described as being "trivial", or "difficult", or "deep", or even "beautiful". These subjective judgments vary not only from person to person, but also with time and culture: for example, as a proof is obtained, simplified or better understood, a theorem that was once difficult may become trivial.[8] On the other hand, a deep theorem may be stated simply, but its proof may involve surprising and subtle connections between disparate areas of mathematics. Fermat's Last Theorem is a particularly well-known example of such a theorem.[9]
Informal account of theorems
Logically, many theorems are of the form of an indicative conditional: If A, then B. Such a theorem does not assert B — only that B is a necessary consequence of A. In this case, A is called the hypothesis of the theorem ("hypothesis" here means something very different from a conjecture), and B the conclusion of the theorem. The two together (without the proof) are called the proposition or statement of the theorem (e.g. "If A, then B" is the proposition). Alternatively, A and B can be also termed the antecedent and the consequent, respectively.[10] The theorem "If n is an even natural number, then n/2 is a natural number" is a typical example in which the hypothesis is "n is an even natural number", and the conclusion is "n/2 is also a natural number".
In order for a theorem be proved, it must be in principle expressible as a precise, formal statement. However, theorems are usually expressed in natural language rather than in a completely symbolic form—with the presumption that a formal statement can be derived from the informal one.
It is common in mathematics to choose a number of hypotheses within a given language and declare that the theory consists of all statements provable from these hypotheses. These hypotheses form the foundational basis of the theory and are called axioms or postulates. The field of mathematics known as proof theory studies formal languages, axioms and the structure of proofs.
Some theorems are "trivial", in the sense that they follow from definitions, axioms, and other theorems in obvious ways and do not contain any surprising insights. Some, on the other hand, may be called "deep", because their proofs may be long and difficult, involve areas of mathematics superficially distinct from the statement of the theorem itself, or show surprising connections between disparate areas of mathematics.[11] A theorem might be simple to state and yet be deep. An excellent example is Fermat's Last Theorem,[9] and there are many other examples of simple yet deep theorems in number theory and combinatorics, among other areas.
Other theorems have a known proof that cannot easily be written down. The most prominent examples are the four color theorem and the Kepler conjecture. Both of these theorems are only known to be true by reducing them to a computational search that is then verified by a computer program. Initially, many mathematicians did not accept this form of proof, but it has become more widely accepted. The mathematician Doron Zeilberger has even gone so far as to claim that these are possibly the only nontrivial results that mathematicians have ever proved.[12] Many mathematical theorems can be reduced to more straightforward computation, including polynomial identities, trigonometric identities[13] and hypergeometric identities.[14]
Relation with scientific theories
Theorems in mathematics and theories in science are fundamentally different in their epistemology. A scientific theory cannot be proved; its key attribute is that it is falsifiable, that is, it makes predictions about the natural world that are testable by experiments. Any disagreement between prediction and experiment demonstrates the incorrectness of the scientific theory, or at least limits its accuracy or domain of validity. Mathematical theorems, on the other hand, are purely abstract formal statements: the proof of a theorem cannot involve experiments or other empirical evidence in the same way such evidence is used to support scientific theories.[6]
Nonetheless, there is some degree of empiricism and data collection involved in the discovery of mathematical theorems. By establishing a pattern, sometimes with the use of a powerful computer, mathematicians may have an idea of what to prove, and in some cases even a plan for how to set about doing the proof. It is also possible to find a single counter-example and so establish the impossibility of a proof for the proposition as-stated, and possibly suggest restricted forms of the original proposition that might have feasible proofs.
For example, both the Collatz conjecture and the Riemann hypothesis are well-known unsolved problems; they have been extensively studied through empirical checks, but remain unproven. The Collatz conjecture has been verified for start values up to about 2.88 × 1018. The Riemann hypothesis has been verified to hold for the first 10 trillion non-trivial zeroes of the zeta function. Although most mathematicians can tolerate supposing that the conjecture and the hypothesis are true, neither of these propositions is considered proved.
Such evidence does not constitute proof. For example, the Mertens conjecture is a statement about natural numbers that is now known to be false, but no explicit counterexample (i.e., a natural number n for which the Mertens function M(n) equals or exceeds the square root of n) is known: all numbers less than 1014 have the Mertens property, and the smallest number that does not have this property is only known to be less than the exponential of 1.59 × 1040, which is approximately 10 to the power 4.3 × 1039. Since the number of particles in the universe is generally considered less than 10 to the power 100 (a googol), there is no hope to find an explicit counterexample by exhaustive search.
The word "theory" also exists in mathematics, to denote a body of mathematical axioms, definitions and theorems, as in, for example, group theory (see mathematical theory). There are also "theorems" in science, particularly physics, and in engineering, but they often have statements and proofs in which physical assumptions and intuition play an important role; the physical axioms on which such "theorems" are based are themselves falsifiable.
Terminology
A number of different terms for mathematical statements exist; these terms indicate the role statements play in a particular subject. The distinction between different terms is sometimes rather arbitrary, and the usage of some terms has evolved over time.
• An axiom or postulate is a fundamental assumption regarding the object of study, that is accepted without proof. A related concept is that of a definition, which gives the meaning of a word or a phrase in terms of known concepts. Classical geometry discerns between axioms, which are general statements; and postulates, which are statements about geometrical objects.[15] Historically, axioms were regarded as "self-evident"; today they are merely assumed to be true.
• A conjecture is an unproved statement that is believed to be true. Conjectures are usually made in public, and named after their maker (for example, Goldbach's conjecture and Collatz conjecture). The term hypothesis is also used in this sense (for example, Riemann hypothesis), which should not be confused with "hypothesis" as the premise of a proof. Other terms are also used on occasion, for example problem when people are not sure whether the statement should be believed to be true. Fermat's Last Theorem was historically called a theorem, although, for centuries, it was only a conjecture.
• A theorem is a statement that has been proven to be true based on axioms and other theorems.
• A proposition is a theorem of lesser importance, or one that is considered so elementary or immediately obvious, that it may be stated without proof. This should not be confused with "proposition" as used in propositional logic. In classical geometry the term "proposition" was used differently: in Euclid's Elements (c. 300 BCE), all theorems and geometric constructions were called "propositions" regardless of their importance.
• A lemma is an "accessory proposition" - a proposition with little applicability outside its use in a particular proof. Over time a lemma may gain in importance and be considered a theorem, though the term "lemma" is usually kept as part of its name (e.g. Gauss's lemma, Zorn's lemma, and the fundamental lemma).
• A corollary is a proposition that follows immediately from another theorem or axiom, with little or no required proof.[16] A corollary may also be a restatement of a theorem in a simpler form, or for a special case: for example, the theorem "all internal angles in a rectangle are right angles" has a corollary that "all internal angles in a square are right angles" - a square being a special case of a rectangle.
• A generalization of a theorem is a theorem with a similar statement but a broader scope, from which the original theorem can be deduced as a special case (a corollary). [lower-alpha 4]
Other terms may also be used for historical or customary reasons, for example:
• An identity is a theorem stating an equality between two expressions, that holds for any value within its domain (e.g. Bézout's identity and Vandermonde's identity).
• A rule is a theorem that establishes a useful formula (e.g. Bayes' rule and Cramer's rule).
• A law or principle is a theorem with wide applicability (e.g. the law of large numbers, law of cosines, Kolmogorov's zero–one law, Harnack's principle, the least-upper-bound principle, and the pigeonhole principle).[lower-alpha 5]
A few well-known theorems have even more idiosyncratic names, for example, the division algorithm, Euler's formula, and the Banach–Tarski paradox.
Layout
A theorem and its proof are typically laid out as follows:
Theorem (name of the person who proved it, along with year of discovery or publication of the proof)
Statement of theorem (sometimes called the proposition)
Proof
Description of proof
End
The end of the proof may be signaled by the letters Q.E.D. (quod erat demonstrandum) or by one of the tombstone marks, such as "□" or "∎", meaning "end of proof", introduced by Paul Halmos following their use in magazines to mark the end of an article.[17]
The exact style depends on the author or publication. Many publications provide instructions or macros for typesetting in the house style.
It is common for a theorem to be preceded by definitions describing the exact meaning of the terms used in the theorem. It is also common for a theorem to be preceded by a number of propositions or lemmas which are then used in the proof. However, lemmas are sometimes embedded in the proof of a theorem, either with nested proofs, or with their proofs presented after the proof of the theorem.
Corollaries to a theorem are either presented between the theorem and the proof, or directly after the proof. Sometimes, corollaries have proofs of their own that explain why they follow from the theorem.
Lore
It has been estimated that over a quarter of a million theorems are proved every year.[18]
The well-known aphorism, "A mathematician is a device for turning coffee into theorems", is probably due to Alfréd Rényi, although it is often attributed to Rényi's colleague Paul Erdős (and Rényi may have been thinking of Erdős), who was famous for the many theorems he produced, the number of his collaborations, and his coffee drinking.[19]
The classification of finite simple groups is regarded by some to be the longest proof of a theorem. It comprises tens of thousands of pages in 500 journal articles by some 100 authors. These papers are together believed to give a complete proof, and several ongoing projects hope to shorten and simplify this proof.[20] Another theorem of this type is the four color theorem whose computer generated proof is too long for a human to read. It is among the longest known proofs of a theorem whose statement can be easily understood by a layman.
Theorems in logic
In mathematical logic, a formal theory is a set of sentences within a formal language. A sentence is a well-formed formula with no free variables. A sentence that is a member of a theory is one of its theorems, and the theory is the set of its theorems. Usually a theory is understood to be closed under the relation of logical consequence. Some accounts define a theory to be closed under the semantic consequence relation ($\models $), while others define it to be closed under the syntactic consequence, or derivability relation ($\vdash $).[21][22][23][24][25][26][27][28][29][30]
For a theory to be closed under a derivability relation, it must be associated with a deductive system that specifies how the theorems are derived. The deductive system may be stated explicitly, or it may be clear from the context. The closure of the empty set under the relation of logical consequence yields the set that contains just those sentences that are the theorems of the deductive system.
In the broad sense in which the term is used within logic, a theorem does not have to be true, since the theory that contains it may be unsound relative to a given semantics, or relative to the standard interpretation of the underlying language. A theory that is inconsistent has all sentences as theorems.
The definition of theorems as sentences of a formal language is useful within proof theory, which is a branch of mathematics that studies the structure of formal proofs and the structure of provable formulas. It is also important in model theory, which is concerned with the relationship between formal theories and structures that are able to provide a semantics for them through interpretation.
Although theorems may be uninterpreted sentences, in practice mathematicians are more interested in the meanings of the sentences, i.e. in the propositions they express. What makes formal theorems useful and interesting is that they may be interpreted as true propositions and their derivations may be interpreted as a proof of their truth. A theorem whose interpretation is a true statement about a formal system (as opposed to within a formal system) is called a metatheorem.
Some important theorems in mathematical logic are:
• Compactness of first-order logic
• Completeness of first-order logic
• Gödel's incompleteness theorems of first-order arithmetic
• Consistency of first-order arithmetic
• Tarski's undefinability theorem
• Church-Turing theorem of undecidability
• Löb's theorem
• Löwenheim–Skolem theorem
• Lindström's theorem
• Craig's theorem
• Cut-elimination theorem
Syntax and semantics
The concept of a formal theorem is fundamentally syntactic, in contrast to the notion of a true proposition, which introduces semantics. Different deductive systems can yield other interpretations, depending on the presumptions of the derivation rules (i.e. belief, justification or other modalities). The soundness of a formal system depends on whether or not all of its theorems are also validities. A validity is a formula that is true under any possible interpretation (for example, in classical propositional logic, validities are tautologies). A formal system is considered semantically complete when all of its theorems are also tautologies.
Interpretation of a formal theorem
Main article: Interpretation (logic)
Theorems and theories
Main articles: Theory and Theory (mathematical logic)
See also
• List of theorems
• List of theorems called fundamental
• Formula
• Inference
• Toy theorem
Notes
1. In general, the distinction is weak, as the standard way to prove that a statement is provable consists of proving it. However, in mathematical logic, one considers often the set of all theorems of a theory, although one cannot prove them individually.
2. An exception is the original Wiles's proof of Fermat's Last Theorem, which relies implicitly on Grothendieck universes, whose existence requires the addition of a new axiom to set theory.[4] This reliance on a new axiom of set theory has since been removed.[5] Nevertheless, it is rather astonishing that the first proof of a statement expressed in elementary arithmetic involves the existence of very large infinite sets.
3. A theory is often identified with the set of its theorems. This is avoided here for clarity, and also for not depending on set theory.
4. Often, when the less general or "corollary"-like theorem is proven first, it is because the proof of the more general form requires the simpler, corollary-like form, for use as a what is functionally a lemma, or "helper" theorem.
5. The word law can also refer to an axiom, a rule of inference, or, in probability theory, a probability distribution.
References
1. Elisha Scott Loomis. "The Pythagorean proposition: its demonstrations analyzed and classified, and bibliography of sources for data of the four kinds of proofs" (PDF). Education Resources Information Center. Institute of Education Sciences (IES) of the U.S. Department of Education. Retrieved 2010-09-26. Originally published in 1940 and reprinted in 1968 by National Council of Teachers of Mathematics.
2. "Definition of THEOREM". www.merriam-webster.com. Retrieved 2019-11-02.
3. "Theorem | Definition of Theorem by Lexico". Lexico Dictionaries | English. Archived from the original on November 2, 2019. Retrieved 2019-11-02.
4. McLarty, Colin (2010). "What does it take to prove Fermat's last theorem? Grothendieck and the logic of number theory". The Review of Symbolic Logic. Cambridge University Press. 13 (3): 359--377.
5. McLarty, Colin (2020). "The large structures of Grothendieck founded on finite order arithmetic". Bulletin of Symbolic Logic. Cambridge University Press. 16 (2): 296--325.
6. Markie, Peter (2017), "Rationalism vs. Empiricism", in Zalta, Edward N. (ed.), The Stanford Encyclopedia of Philosophy (Fall 2017 ed.), Metaphysics Research Lab, Stanford University, retrieved 2019-11-02
7. However, both theorems and scientific law are the result of investigations. See Heath 1897 Introduction, The terminology of Archimedes, p. clxxxii:"theorem (θεὼρνμα) from θεωρεἳν to investigate"
8. Weisstein, Eric W. "Theorem". mathworld.wolfram.com. Retrieved 2019-11-02.
9. Darmon, Henri; Diamond, Fred; Taylor, Richard (2007-09-09). "Fermat's Last Theorem" (PDF). McGill University – Department of Mathematics and Statistics. Retrieved 2019-11-01.
10. "Implication". intrologic.stanford.edu. Retrieved 2019-11-02.
11. Weisstein, Eric W. "Deep Theorem". MathWorld.
12. Doron Zeilberger. "Opinion 51".
13. Such as the derivation of the formula for $\tan(\alpha +\beta )$ from the addition formulas of sine and cosine.
14. Petkovsek et al. 1996.
15. Wentworth, G.; Smith, D.E. (1913). Plane Geometry. Ginn & Co. Articles 46, 47.
16. Wentworth & Smith, article 51
17. "Earliest Uses of Symbols of Set Theory and Logic". jeff560.tripod.com. Retrieved 2 November 2019.
18. Hoffman 1998, p. 204.
19. Hoffman 1998, p. 7.
20. An enormous theorem: the classification of finite simple groups, Richard Elwes, Plus Magazine, Issue 41 December 2006.
21. Boolos, et al 2007, p. 191.
22. Chiswell and Hodges, p. 172.
23. Enderton, p. 148
24. Hedman, p. 89.
25. Hinman, p. 139.
26. Hodges, p. 33.
27. Johnstone, p. 21.
28. Monk, p. 208.
29. Rautenberg, p. 81.
30. van Dalen, p. 104.
References
• Boolos, George; Burgess, John; Jeffrey, Richard (2007). Computability and Logic (5th ed.). Cambridge University Press.
• Chiswell, Ian; Hodges, Wilfred (2007). Mathematical Logic. Oxford University Press.
• Enderton, Herbert (2001). A Mathematical Introduction to Logic (2nd ed.). Harcourt Academic Press.
• Heath, Sir Thomas Little (1897). The works of Archimedes. Dover. Retrieved 2009-11-15.
• Hedman, Shawn (2004). A First Course in Logic. Oxford University Press.
• Hinman, Peter (2005). Fundamentals of Mathematical Logic. Wellesley, MA: A K Peters.
• Hoffman, P. (1998). The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth. Hyperion, New York. ISBN 1-85702-829-5.
• Hodges, Wilfrid (1993). Model Theory. Cambridge University Press.
• Hunter, Geoffrey (1996) [1973]. Metalogic: An Introduction to the Metatheory of Standard First Order Logic. University of California Press. ISBN 0-520-02356-0.
• Johnstone, P. T. (1987). Notes on Logic and Set Theory. Cambridge University Press.
• Mates, Benson (1972). Elementary Logic. Oxford University Press. ISBN 0-19-501491-X.
• Monk, J. Donald (1976). Mathematical Logic. Springer-Verlag.
• Petkovsek, Marko; Wilf, Herbert; Zeilberger, Doron (1996). A = B. A.K. Peters, Wellesley, Massachusetts. ISBN 1-56881-063-6.
• Rautenberg, Wolfgang (2010). A Concise Introduction to Mathematical Logic (3rd ed.). Springer.
• van Dalen, Dirk (1994). Logic and Structure (3rd ed.). Springer-Verlag.
External links
Look up theorem in Wiktionary, the free dictionary.
• Media related to Theorems at Wikimedia Commons
• Weisstein, Eric W. "Theorem". MathWorld.
• Theorem of the Day
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| Wikipedia |
Theory (mathematical logic)
In mathematical logic, a theory (also called a formal theory) is a set of sentences in a formal language. In most scenarios a deductive system is first understood from context, after which an element $\phi \in T$ of a deductively closed theory $T$ is then called a theorem of the theory. In many deductive systems there is usually a subset $\Sigma \subseteq T$ that is called "the set of axioms" of the theory $T$, in which case the deductive system is also called an "axiomatic system". By definition, every axiom is automatically a theorem. A first-order theory is a set of first-order sentences (theorems) recursively obtained by the inference rules of the system applied to the set of axioms.
General theories (as expressed in formal language)
When defining theories for foundational purposes, additional care must be taken, as normal set-theoretic language may not be appropriate.
The construction of a theory begins by specifying a definite non-empty conceptual class ${\mathcal {E}}$, the elements of which are called statements. These initial statements are often called the primitive elements or elementary statements of the theory—to distinguish them from other statements that may be derived from them.
A theory ${\mathcal {T}}$ is a conceptual class consisting of certain of these elementary statements. The elementary statements that belong to ${\mathcal {T}}$ are called the elementary theorems of ${\mathcal {T}}$ and are said to be true. In this way, a theory can be seen as a way of designating a subset of ${\mathcal {E}}$ that only contain statements that are true.
This general way of designating a theory stipulates that the truth of any of its elementary statements is not known without reference to ${\mathcal {T}}$. Thus the same elementary statement may be true with respect to one theory but false with respect to another. This is reminiscent of the case in ordinary language where statements such as "He is an honest person" cannot be judged true or false without interpreting who "he" is, and, for that matter, what an "honest person" is under this theory.[1]
Subtheories and extensions
A theory ${\mathcal {S}}$ is a subtheory of a theory ${\mathcal {T}}$ if ${\mathcal {S}}$ is a subset of ${\mathcal {T}}$. If ${\mathcal {T}}$ is a subset of ${\mathcal {S}}$ then ${\mathcal {S}}$ is called an extension or a supertheory of ${\mathcal {T}}$
Deductive theories
A theory is said to be a deductive theory if ${\mathcal {T}}$ is an inductive class, which is to say that its content is based on some formal deductive system and that some of its elementary statements are taken as axioms. In a deductive theory, any sentence that is a logical consequence of one or more of the axioms is also a sentence of that theory.[1] More formally, if $\vdash $ is a Tarski-style consequence relation, then ${\mathcal {T}}$ is closed under $\vdash $ (and so each of its theorems is a logical consequence of its axioms) if and only if, for all sentences $\phi $ in the language of the theory ${\mathcal {T}}$, if ${\mathcal {T}}\vdash \phi $, then $\phi \in {\mathcal {T}}$; or, equivalently, if ${\mathcal {T}}'$ is a finite subset of ${\mathcal {T}}$ (possibly the set of axioms of ${\mathcal {T}}$ in the case of finitely axiomatizable theories) and ${\mathcal {T}}'\vdash \phi $, then $\phi \in {\mathcal {T}}'$, and therefore $\phi \in {\mathcal {T}}$.
Consistency and completeness
Main articles: Consistency and Completeness (logic)
A syntactically consistent theory is a theory from which not every sentence in the underlying language can be proven (with respect to some deductive system, which is usually clear from context). In a deductive system (such as first-order logic) that satisfies the principle of explosion, this is equivalent to requiring that there is no sentence φ such that both φ and its negation can be proven from the theory.
A satisfiable theory is a theory that has a model. This means there is a structure M that satisfies every sentence in the theory. Any satisfiable theory is syntactically consistent, because the structure satisfying the theory will satisfy exactly one of φ and the negation of φ, for each sentence φ.
A consistent theory is sometimes defined to be a syntactically consistent theory, and sometimes defined to be a satisfiable theory. For first-order logic, the most important case, it follows from the completeness theorem that the two meanings coincide.[2] In other logics, such as second-order logic, there are syntactically consistent theories that are not satisfiable, such as ω-inconsistent theories.
A complete consistent theory (or just a complete theory) is a consistent theory ${\mathcal {T}}$ such that for every sentence φ in its language, either φ is provable from ${\mathcal {T}}$ or ${\mathcal {T}}$ $\cup $ {φ} is inconsistent. For theories closed under logical consequence, this means that for every sentence φ, either φ or its negation is contained in the theory.[3] An incomplete theory is a consistent theory that is not complete.
(see also ω-consistent theory for a stronger notion of consistency.)
Interpretation of a theory
Main article: Interpretation (logic)
An interpretation of a theory is the relationship between a theory and some subject matter when there is a many-to-one correspondence between certain elementary statements of the theory, and certain statements related to the subject matter. If every elementary statement in the theory has a correspondent it is called a full interpretation, otherwise it is called a partial interpretation.[4]
Theories associated with a structure
Each structure has several associated theories. The complete theory of a structure A is the set of all first-order sentences over the signature of A that are satisfied by A. It is denoted by Th(A). More generally, the theory of K, a class of σ-structures, is the set of all first-order σ-sentences that are satisfied by all structures in K, and is denoted by Th(K). Clearly Th(A) = Th({A}). These notions can also be defined with respect to other logics.
For each σ-structure A, there are several associated theories in a larger signature σ' that extends σ by adding one new constant symbol for each element of the domain of A. (If the new constant symbols are identified with the elements of A that they represent, σ' can be taken to be σ $\cup $ A.) The cardinality of σ' is thus the larger of the cardinality of σ and the cardinality of A.
The diagram of A consists of all atomic or negated atomic σ'-sentences that are satisfied by A and is denoted by diagA. The positive diagram of A is the set of all atomic σ'-sentences that A satisfies. It is denoted by diag+A. The elementary diagram of A is the set eldiagA of all first-order σ'-sentences that are satisfied by A or, equivalently, the complete (first-order) theory of the natural expansion of A to the signature σ'.
First-order theories
Further information: List of first-order theories
A first-order theory ${\mathcal {QS}}$ is a set of sentences in a first-order formal language ${\mathcal {Q}}$.
Derivation in a first-order theory
Main article: First-order logic § Deductive systems
There are many formal derivation ("proof") systems for first-order logic. These include Hilbert-style deductive systems, natural deduction, the sequent calculus, the tableaux method and resolution.
Syntactic consequence in a first-order theory
Main article: First-order logic § Validity, satisfiability, and logical consequence
A formula A is a syntactic consequence of a first-order theory ${\mathcal {QS}}$ if there is a derivation of A using only formulas in ${\mathcal {QS}}$ as non-logical axioms. Such a formula A is also called a theorem of ${\mathcal {QS}}$. The notation "${\mathcal {QS}}\vdash A$" indicates A is a theorem of ${\mathcal {QS}}$.
Interpretation of a first-order theory
Main article: Structure (mathematical logic)
An interpretation of a first-order theory provides a semantics for the formulas of the theory. An interpretation is said to satisfy a formula if the formula is true according to the interpretation. A model of a first-order theory ${\mathcal {QS}}$ is an interpretation in which every formula of ${\mathcal {QS}}$ is satisfied.
First-order theories with identity
Main article: First-order logic § Equality and its axioms
A first-order theory ${\mathcal {QS}}$ is a first-order theory with identity if ${\mathcal {QS}}$ includes the identity relation symbol "=" and the reflexivity and substitution axiom schemes for this symbol.
Topics related to first-order theories
• Compactness theorem
• Consistent set
• Deduction theorem
• Enumeration theorem
• Lindenbaum's lemma
• Löwenheim–Skolem theorem
Examples
One way to specify a theory is to define a set of axioms in a particular language. The theory can be taken to include just those axioms, or their logical or provable consequences, as desired. Theories obtained this way include ZFC and Peano arithmetic.
A second way to specify a theory is to begin with a structure, and let the theory be the set of sentences that are satisfied by the structure. This is a method for producing complete theories through the semantic route, with examples including the set of true sentences under the structure (N, +, ×, 0, 1, =), where N is the set of natural numbers, and the set of true sentences under the structure (R, +, ×, 0, 1, =), where R is the set of real numbers. The first of these, called the theory of true arithmetic, cannot be written as the set of logical consequences of any enumerable set of axioms. The theory of (R, +, ×, 0, 1, =) was shown by Tarski to be decidable; it is the theory of real closed fields (see Decidability of first-order theories of the real numbers for more).
See also
• Axiomatic system
• Interpretability
• List of first-order theories
• Mathematical theory
References
1. Haskell Curry, Foundations of Mathematical Logic, 2010.
2. Weiss, William; D'Mello, Cherie (2015). "Fundamentals of Model Theory" (PDF). University of Toronto — Department of Mathematics.
3. "Completeness (in logic) - Encyclopedia of Mathematics". www.encyclopediaofmath.org. Retrieved 2019-11-01.
4. Haskell Curry (1963). Foundations of Mathematical Logic. Mcgraw Hill. Here: p.48
Further reading
• Hodges, Wilfrid (1997). A shorter model theory. Cambridge University Press. ISBN 0-521-58713-1.
Mathematical logic
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• Axiom
• list
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• of arithmetic:
• Peano
• second-order
• elementary function
• primitive recursive
• Robinson
• Skolem
• of the real numbers
• Tarski's axiomatization
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• canonical
• minimal axioms
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• Elements
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• Tarski's
• non-Euclidean
• Principia Mathematica
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Distribution (mathematics)
Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative.
This article is about generalized functions in mathematical analysis. For the concept of distributions in probability theory, see Probability distribution. For artificial landscapes, see Test functions for optimization. For other uses, see Distribution § Mathematics.
Distributions are widely used in the theory of partial differential equations, where it may be easier to establish the existence of distributional solutions (weak solutions) than classical solutions, or where appropriate classical solutions may not exist. Distributions are also important in physics and engineering where many problems naturally lead to differential equations whose solutions or initial conditions are singular, such as the Dirac delta function.
A function $f$ is normally thought of as acting on the points in the function domain by "sending" a point $x$ in the domain to the point $f(x).$ Instead of acting on points, distribution theory reinterprets functions such as $f$ as acting on test functions in a certain way. In applications to physics and engineering, test functions are usually infinitely differentiable complex-valued (or real-valued) functions with compact support that are defined on some given non-empty open subset $U\subseteq \mathbb {R} ^{n}$. (Bump functions are examples of test functions.) The set of all such test functions forms a vector space that is denoted by $C_{c}^{\infty }(U)$ or ${\mathcal {D}}(U).$
Most commonly encountered functions, including all continuous maps $f:\mathbb {R} \to \mathbb {R} $ if using $U:=\mathbb {R} ,$ can be canonically reinterpreted as acting via "integration against a test function." Explicitly, this means that such a function $f$ "acts on" a test function $\psi \in {\mathcal {D}}(\mathbb {R} )$ by "sending" it to the number $ \int _{\mathbb {R} }f\,\psi \,dx,$ which is often denoted by $D_{f}(\psi ).$ This new action $ \psi \mapsto D_{f}(\psi )$ of $f$ defines a scalar-valued map $D_{f}:{\mathcal {D}}(\mathbb {R} )\to \mathbb {C} ,$ whose domain is the space of test functions ${\mathcal {D}}(\mathbb {R} ).$ This functional $D_{f}$ turns out to have the two defining properties of what is known as a distribution on $U=\mathbb {R} $: it is linear, and it is also continuous when ${\mathcal {D}}(\mathbb {R} )$ is given a certain topology called the canonical LF topology. The action (the integration $ \psi \mapsto \int _{\mathbb {R} }f\,\psi \,dx$) of this distribution $D_{f}$ on a test function $\psi $ can be interpreted as a weighted average of the distribution on the support of the test function, even if the values of the distribution at a single point are not well-defined. Distributions like $D_{f}$ that arise from functions in this way are prototypical examples of distributions, but there exist many distributions that cannot be defined by integration against any function. Examples of the latter include the Dirac delta function and distributions defined to act by integration of test functions $ \psi \mapsto \int _{U}\psi d\mu $ against certain measures $\mu $ on $U.$ Nonetheless, it is still always possible to reduce any arbitrary distribution down to a simpler family of related distributions that do arise via such actions of integration.
More generally, a distribution on $U$ is by definition a linear functional on $C_{c}^{\infty }(U)$ that is continuous when $C_{c}^{\infty }(U)$ is given a topology called the canonical LF topology. This leads to the space of (all) distributions on $U$, usually denoted by ${\mathcal {D}}'(U)$ (note the prime), which by definition is the space of all distributions on $U$ (that is, it is the continuous dual space of $C_{c}^{\infty }(U)$); it is these distributions that are the main focus of this article.
Definitions of the appropriate topologies on spaces of test functions and distributions are given in the article on spaces of test functions and distributions. This article is primarily concerned with the definition of distributions, together with their properties and some important examples.
History
The practical use of distributions can be traced back to the use of Green functions in the 1830s to solve ordinary differential equations, but was not formalized until much later. According to Kolmogorov & Fomin (1957), generalized functions originated in the work of Sergei Sobolev (1936) on second-order hyperbolic partial differential equations, and the ideas were developed in somewhat extended form by Laurent Schwartz in the late 1940s. According to his autobiography, Schwartz introduced the term "distribution" by analogy with a distribution of electrical charge, possibly including not only point charges but also dipoles and so on. Gårding (1997) comments that although the ideas in the transformative book by Schwartz (1951) were not entirely new, it was Schwartz's broad attack and conviction that distributions would be useful almost everywhere in analysis that made the difference.
Notation
The following notation will be used throughout this article:
• $n$ is a fixed positive integer and $U$ is a fixed non-empty open subset of Euclidean space $\mathbb {R} ^{n}.$
• $\mathbb {N} =\{0,1,2,\ldots \}$ denotes the natural numbers.
• $k$ will denote a non-negative integer or $\infty .$
• If $f$ is a function then $\operatorname {Dom} (f)$ will denote its domain and the support of $f,$ denoted by $\operatorname {supp} (f),$ is defined to be the closure of the set $\{x\in \operatorname {Dom} (f):f(x)\neq 0\}$ in $\operatorname {Dom} (f).$
• For two functions $f,g:U\to \mathbb {C} ,$ the following notation defines a canonical pairing:
$\langle f,g\rangle :=\int _{U}f(x)g(x)\,dx.$ :=\int _{U}f(x)g(x)\,dx.}
• A multi-index of size $n$ is an element in $\mathbb {N} ^{n}$ (given that $n$ is fixed, if the size of multi-indices is omitted then the size should be assumed to be $n$). The length of a multi-index $\alpha =(\alpha _{1},\ldots ,\alpha _{n})\in \mathbb {N} ^{n}$ is defined as $\alpha _{1}+\cdots +\alpha _{n}$ and denoted by $|\alpha |.$ Multi-indices are particularly useful when dealing with functions of several variables, in particular, we introduce the following notations for a given multi-index $\alpha =(\alpha _{1},\ldots ,\alpha _{n})\in \mathbb {N} ^{n}$:
${\begin{aligned}x^{\alpha }&=x_{1}^{\alpha _{1}}\cdots x_{n}^{\alpha _{n}}\\\partial ^{\alpha }&={\frac {\partial ^{|\alpha |}}{\partial x_{1}^{\alpha _{1}}\cdots \partial x_{n}^{\alpha _{n}}}}\end{aligned}}$
We also introduce a partial order of all multi-indices by $\beta \geq \alpha $ if and only if $\beta _{i}\geq \alpha _{i}$ for all $1\leq i\leq n.$ When $\beta \geq \alpha $ we define their multi-index binomial coefficient as:
${\binom {\beta }{\alpha }}:={\binom {\beta _{1}}{\alpha _{1}}}\cdots {\binom {\beta _{n}}{\alpha _{n}}}.$
Definitions of test functions and distributions
In this section, some basic notions and definitions needed to define real-valued distributions on U are introduced. Further discussion of the topologies on the spaces of test functions and distributions is given in the article on spaces of test functions and distributions.
Notation:
1. Let $k\in \{0,1,2,\ldots ,\infty \}.$
2. Let $C^{k}(U)$ denote the vector space of all k-times continuously differentiable real or complex-valued functions on U.
3. For any compact subset $K\subseteq U,$ let $C^{k}(K)$ and $C^{k}(K;U)$ both denote the vector space of all those functions $f\in C^{k}(U)$ such that $\operatorname {supp} (f)\subseteq K.$
• If $f\in C^{k}(K)$ then the domain of $f$ is U and not K. So although $C^{k}(K)$ depends on both K and U, only K is typically indicated. The justification for this common practice is detailed below. The notation $C^{k}(K;U)$ will only be used when the notation $C^{k}(K)$ risks being ambiguous.
• Every $C^{k}(K)$ contains the constant 0 map, even if $K=\varnothing .$
4. Let $C_{c}^{k}(U)$ denote the set of all $f\in C^{k}(U)$ such that $f\in C^{k}(K)$ for some compact subset K of U.
• Equivalently, $C_{c}^{k}(U)$ is the set of all $f\in C^{k}(U)$ such that $f$ has compact support.
• $C_{c}^{k}(U)$ is equal to the union of all $C^{k}(K)$ as $K\subseteq U$ ranges over all compact subsets of $U.$
• If $f$ is a real-valued function on $U$, then $f$ is an element of $C_{c}^{k}(U)$ if and only if $f$ is a $C^{k}$ bump function. Every real-valued test function on $U$ is also a complex-valued test function on $U.$
For all $j,k\in \{0,1,2,\ldots ,\infty \}$ and any compact subsets $K$ and $L$ of $U$, we have:
${\begin{aligned}C^{k}(K)&\subseteq C_{c}^{k}(U)\subseteq C^{k}(U)\\C^{k}(K)&\subseteq C^{k}(L)&&{\text{if }}K\subseteq L\\C^{k}(K)&\subseteq C^{j}(K)&&{\text{if }}j\leq k\\C_{c}^{k}(U)&\subseteq C_{c}^{j}(U)&&{\text{if }}j\leq k\\C^{k}(U)&\subseteq C^{j}(U)&&{\text{if }}j\leq k\\\end{aligned}}$
Definition: Elements of $C_{c}^{\infty }(U)$ are called test functions on U and $C_{c}^{\infty }(U)$ is called the space of test functions on U. We will use both ${\mathcal {D}}(U)$ and $C_{c}^{\infty }(U)$ to denote this space.
Distributions on U are continuous linear functionals on $C_{c}^{\infty }(U)$ when this vector space is endowed with a particular topology called the canonical LF-topology. The following proposition states two necessary and sufficient conditions for the continuity of a linear function on $C_{c}^{\infty }(U)$ that are often straightforward to verify.
Proposition: A linear functional T on $C_{c}^{\infty }(U)$ is continuous, and therefore a distribution, if and only if any of the following equivalent conditions is satisfied:
1. For every compact subset $K\subseteq U$ there exist constants $C>0$ and $N\in \mathbb {N} $ (dependent on $K$) such that for all $f\in C_{c}^{\infty }(U)$ with support contained in $K$,[1][2]
$|T(f)|\leq C\sup\{|\partial ^{\alpha }f(x)|:x\in U,|\alpha |\leq N\}.$
2. For every compact subset $K\subseteq U$ and every sequence $\{f_{i}\}_{i=1}^{\infty }$ in $C_{c}^{\infty }(U)$ whose supports are contained in $K$, if $\{\partial ^{\alpha }f_{i}\}_{i=1}^{\infty }$ converges uniformly to zero on $U$ for every multi-index $\alpha $, then $T(f_{i})\to 0.$
Topology on Ck(U)
We now introduce the seminorms that will define the topology on $C^{k}(U).$ Different authors sometimes use different families of seminorms so we list the most common families below. However, the resulting topology is the same no matter which family is used.
Suppose $k\in \{0,1,2,\ldots ,\infty \}$ and $K$ is an arbitrary compact subset of $U.$ Suppose $i$ an integer such that $0\leq i\leq k$[note 1] and $p$ is a multi-index with length $|p|\leq k.$ For $K\neq \varnothing ,$ define:
${\begin{alignedat}{4}{\text{ (1) }}\ &s_{p,K}(f)&&:=\sup _{x_{0}\in K}\left|\partial ^{p}f(x_{0})\right|\\[4pt]{\text{ (2) }}\ &q_{i,K}(f)&&:=\sup _{|p|\leq i}\left(\sup _{x_{0}\in K}\left|\partial ^{p}f(x_{0})\right|\right)=\sup _{|p|\leq i}\left(s_{p,K}(f)\right)\\[4pt]{\text{ (3) }}\ &r_{i,K}(f)&&:=\sup _{\stackrel {|p|\leq i}{x_{0}\in K}}\left|\partial ^{p}f(x_{0})\right|\\[4pt]{\text{ (4) }}\ &t_{i,K}(f)&&:=\sup _{x_{0}\in K}\left(\sum _{|p|\leq i}\left|\partial ^{p}f(x_{0})\right|\right)\end{alignedat}}$
while for $K=\varnothing ,$ define all the functions above to be the constant 0 map.
All of the functions above are non-negative $\mathbb {R} $-valued[note 2] seminorms on $C^{k}(U).$ As explained in this article, every set of seminorms on a vector space induces a locally convex vector topology.
Each of the following sets of seminorms
${\begin{alignedat}{4}A~:=\quad &\{q_{i,K}&&:\;K{\text{ compact and }}\;&&i\in \mathbb {N} {\text{ satisfies }}\;&&0\leq i\leq k\}\\B~:=\quad &\{r_{i,K}&&:\;K{\text{ compact and }}\;&&i\in \mathbb {N} {\text{ satisfies }}\;&&0\leq i\leq k\}\\C~:=\quad &\{t_{i,K}&&:\;K{\text{ compact and }}\;&&i\in \mathbb {N} {\text{ satisfies }}\;&&0\leq i\leq k\}\\D~:=\quad &\{s_{p,K}&&:\;K{\text{ compact and }}\;&&p\in \mathbb {N} ^{n}{\text{ satisfies }}\;&&|p|\leq k\}\end{alignedat}}$
generate the same locally convex vector topology on $C^{k}(U)$ (so for example, the topology generated by the seminorms in $A$ is equal to the topology generated by those in $C$).
The vector space $C^{k}(U)$ is endowed with the locally convex topology induced by any one of the four families $A,B,C,D$ of seminorms described above. This topology is also equal to the vector topology induced by all of the seminorms in $A\cup B\cup C\cup D.$
With this topology, $C^{k}(U)$ becomes a locally convex Fréchet space that is not normable. Every element of $A\cup B\cup C\cup D$ is a continuous seminorm on $C^{k}(U).$ Under this topology, a net $(f_{i})_{i\in I}$ in $C^{k}(U)$ converges to $f\in C^{k}(U)$ if and only if for every multi-index $p$ with $|p|<k+1$ and every compact $K,$ the net of partial derivatives $\left(\partial ^{p}f_{i}\right)_{i\in I}$ converges uniformly to $\partial ^{p}f$ on $K.$[3] For any $k\in \{0,1,2,\ldots ,\infty \},$ any (von Neumann) bounded subset of $C^{k+1}(U)$ is a relatively compact subset of $C^{k}(U).$[4] In particular, a subset of $C^{\infty }(U)$ is bounded if and only if it is bounded in $C^{i}(U)$ for all $i\in \mathbb {N} .$[4] The space $C^{k}(U)$ is a Montel space if and only if $k=\infty .$[5]
A subset $W$ of $C^{\infty }(U)$ is open in this topology if and only if there exists $i\in \mathbb {N} $ such that $W$ is open when $C^{\infty }(U)$ is endowed with the subspace topology induced on it by $C^{i}(U).$
Topology on Ck(K)
As before, fix $k\in \{0,1,2,\ldots ,\infty \}.$ Recall that if $K$ is any compact subset of $U$ then $C^{k}(K)\subseteq C^{k}(U).$
Assumption: For any compact subset $K\subseteq U,$ we will henceforth assume that $C^{k}(K)$ is endowed with the subspace topology it inherits from the Fréchet space $C^{k}(U).$
If $k$ is finite then $C^{k}(K)$ is a Banach space[6] with a topology that can be defined by the norm
$r_{K}(f):=\sup _{|p|<k}\left(\sup _{x_{0}\in K}\left|\partial ^{p}f(x_{0})\right|\right).$
And when $k=2,$ then $C^{k}(K)$ is even a Hilbert space.[6]
Trivial extensions and independence of Ck(K)'s topology from U
Suppose $U$ is an open subset of $\mathbb {R} ^{n}$ and $K\subseteq U$ is a compact subset. By definition, elements of $C^{k}(K)$ are functions with domain $U$ (in symbols, $C^{k}(K)\subseteq C^{k}(U)$), so the space $C^{k}(K)$ and its topology depend on $U;$ to make this dependence on the open set $U$ clear, temporarily denote $C^{k}(K)$ by $C^{k}(K;U).$ Importantly, changing the set $U$ to a different open subset $U'$ (with $K\subseteq U'$) will change the set $C^{k}(K)$ from $C^{k}(K;U)$ to $C^{k}(K;U'),$[note 3] so that elements of $C^{k}(K)$ will be functions with domain $U'$ instead of $U.$ Despite $C^{k}(K)$ depending on the open set ($U{\text{ or }}U'$), the standard notation for $C^{k}(K)$ makes no mention of it. This is justified because, as this subsection will now explain, the space $C^{k}(K;U)$ is canonically identified as a subspace of $C^{k}(K;U')$ (both algebraically and topologically).
It is enough to explain how to canonically identify $C^{k}(K;U)$ with $C^{k}(K;U')$ when one of $U$ and $U'$ is a subset of the other. The reason is that if $V$ and $W$ are arbitrary open subsets of $\mathbb {R} ^{n}$ containing $K$ then the open set $U:=V\cap W$ also contains $K,$ so that each of $C^{k}(K;V)$ and $C^{k}(K;W)$ is canonically identified with $C^{k}(K;V\cap W)$ and now by transitivity, $C^{k}(K;V)$ is thus identified with $C^{k}(K;W).$ So assume $U\subseteq V$ are open subsets of $\mathbb {R} ^{n}$ containing $K.$
Given $f\in C_{c}^{k}(U),$ its trivial extension to $V$ is the function $F:V\to \mathbb {C} $ defined by:
$F(x)={\begin{cases}f(x)&x\in U,\\0&{\text{otherwise}}.\end{cases}}$
This trivial extension belongs to $C^{k}(V)$ (because $f\in C_{c}^{k}(U)$ has compact support) and it will be denoted by $I(f)$ (that is, $I(f):=F$). The assignment $f\mapsto I(f)$ thus induces a map $I:C_{c}^{k}(U)\to C^{k}(V)$ that sends a function in $C_{c}^{k}(U)$ to its trivial extension on $V.$ This map is a linear injection and for every compact subset $K\subseteq U$ (where $K$ is also a compact subset of $V$ since $K\subseteq U\subseteq V$),
${\begin{alignedat}{4}I\left(C^{k}(K;U)\right)&~=~C^{k}(K;V)\qquad {\text{ and thus }}\\I\left(C_{c}^{k}(U)\right)&~\subseteq ~C_{c}^{k}(V).\end{alignedat}}$
If $I$ is restricted to $C^{k}(K;U)$ then the following induced linear map is a homeomorphism (linear homeomorphisms are called TVS-isomorphisms):
${\begin{alignedat}{4}\,&C^{k}(K;U)&&\to \,&&C^{k}(K;V)\\&f&&\mapsto \,&&I(f)\\\end{alignedat}}$
and thus the next map is a topological embedding:
${\begin{alignedat}{4}\,&C^{k}(K;U)&&\to \,&&C^{k}(V)\\&f&&\mapsto \,&&I(f).\\\end{alignedat}}$
Using the injection
$I:C_{c}^{k}(U)\to C^{k}(V)$
the vector space $C_{c}^{k}(U)$ is canonically identified with its image in $C_{c}^{k}(V)\subseteq C^{k}(V).$ Because $C^{k}(K;U)\subseteq C_{c}^{k}(U),$ through this identification, $C^{k}(K;U)$ can also be considered as a subset of $C^{k}(V).$ Thus the topology on $C^{k}(K;U)$ is independent of the open subset $U$ of $\mathbb {R} ^{n}$ that contains $K,$[7] which justifies the practice of writing $C^{k}(K)$ instead of $C^{k}(K;U).$
Canonical LF topology
Main article: Spaces of test functions and distributions
See also: LF-space and Topology of uniform convergence
Recall that $C_{c}^{k}(U)$ denotes all functions in $C^{k}(U)$ that have compact support in $U,$ where note that $C_{c}^{k}(U)$ is the union of all $C^{k}(K)$ as $K$ ranges over all compact subsets of $U.$ Moreover, for each $k,\,C_{c}^{k}(U)$ is a dense subset of $C^{k}(U).$ The special case when $k=\infty $ gives us the space of test functions.
$C_{c}^{\infty }(U)$ is called the space of test functions on $U$ and it may also be denoted by ${\mathcal {D}}(U).$ Unless indicated otherwise, it is endowed with a topology called the canonical LF topology, whose definition is given in the article: Spaces of test functions and distributions.
The canonical LF-topology is not metrizable and importantly, it is strictly finer than the subspace topology that $C^{\infty }(U)$ induces on $C_{c}^{\infty }(U).$ However, the canonical LF-topology does make $C_{c}^{\infty }(U)$ into a complete reflexive nuclear[8] Montel[9] bornological barrelled Mackey space; the same is true of its strong dual space (that is, the space of all distributions with its usual topology). The canonical LF-topology can be defined in various ways.
Distributions
See also: Continuous linear functional
As discussed earlier, continuous linear functionals on a $C_{c}^{\infty }(U)$ are known as distributions on $U.$ Other equivalent definitions are described below.
By definition, a distribution on $U$ is a continuous linear functional on $C_{c}^{\infty }(U).$ Said differently, a distribution on $U$ is an element of the continuous dual space of $C_{c}^{\infty }(U)$ when $C_{c}^{\infty }(U)$ is endowed with its canonical LF topology.
There is a canonical duality pairing between a distribution $T$ on $U$ and a test function $f\in C_{c}^{\infty }(U),$ which is denoted using angle brackets by
${\begin{cases}{\mathcal {D}}'(U)\times C_{c}^{\infty }(U)\to \mathbb {R} \\(T,f)\mapsto \langle T,f\rangle :=T(f)\end{cases}}$ :=T(f)\end{cases}}}
One interprets this notation as the distribution $T$ acting on the test function $f$ to give a scalar, or symmetrically as the test function $f$ acting on the distribution $T.$
Characterizations of distributions
Proposition. If $T$ is a linear functional on $C_{c}^{\infty }(U)$ then the following are equivalent:
1. T is a distribution;
2. T is continuous;
3. T is continuous at the origin;
4. T is uniformly continuous;
5. T is a bounded operator;
6. T is sequentially continuous;
• explicitly, for every sequence $\left(f_{i}\right)_{i=1}^{\infty }$ in $C_{c}^{\infty }(U)$ that converges in $C_{c}^{\infty }(U)$ to some $f\in C_{c}^{\infty }(U),$ $ \lim _{i\to \infty }T\left(f_{i}\right)=T(f);$[note 4]
7. T is sequentially continuous at the origin; in other words, T maps null sequences[note 5] to null sequences;
• explicitly, for every sequence $\left(f_{i}\right)_{i=1}^{\infty }$ in $C_{c}^{\infty }(U)$ that converges in $C_{c}^{\infty }(U)$ to the origin (such a sequence is called a null sequence), $ \lim _{i\to \infty }T\left(f_{i}\right)=0;$
• a null sequence is by definition any sequence that converges to the origin;
8. T maps null sequences to bounded subsets;
• explicitly, for every sequence $\left(f_{i}\right)_{i=1}^{\infty }$ in $C_{c}^{\infty }(U)$ that converges in $C_{c}^{\infty }(U)$ to the origin, the sequence $\left(T\left(f_{i}\right)\right)_{i=1}^{\infty }$ is bounded;
9. T maps Mackey convergent null sequences to bounded subsets;
• explicitly, for every Mackey convergent null sequence $\left(f_{i}\right)_{i=1}^{\infty }$ in $C_{c}^{\infty }(U),$ the sequence $\left(T\left(f_{i}\right)\right)_{i=1}^{\infty }$ is bounded;
• a sequence $f_{\bullet }=\left(f_{i}\right)_{i=1}^{\infty }$ is said to be Mackey convergent to the origin if there exists a divergent sequence $r_{\bullet }=\left(r_{i}\right)_{i=1}^{\infty }\to \infty $ of positive real numbers such that the sequence $\left(r_{i}f_{i}\right)_{i=1}^{\infty }$ is bounded; every sequence that is Mackey convergent to the origin necessarily converges to the origin (in the usual sense);
10. The kernel of T is a closed subspace of $C_{c}^{\infty }(U);$
11. The graph of T is closed;
12. There exists a continuous seminorm $g$ on $C_{c}^{\infty }(U)$ such that $|T|\leq g;$
13. There exists a constant $C>0$ and a finite subset $\{g_{1},\ldots ,g_{m}\}\subseteq {\mathcal {P}}$ (where ${\mathcal {P}}$ is any collection of continuous seminorms that defines the canonical LF topology on $C_{c}^{\infty }(U)$) such that $|T|\leq C(g_{1}+\cdots +g_{m});$[note 6]
14. For every compact subset $K\subseteq U$ there exist constants $C>0$ and $N\in \mathbb {N} $ such that for all $f\in C^{\infty }(K),$[1]
$|T(f)|\leq C\sup\{|\partial ^{\alpha }f(x)|:x\in U,|\alpha |\leq N\};$
15. For every compact subset $K\subseteq U$ there exist constants $C_{K}>0$ and $N_{K}\in \mathbb {N} $ such that for all $f\in C_{c}^{\infty }(U)$ with support contained in $K,$[10]
$|T(f)|\leq C_{K}\sup\{|\partial ^{\alpha }f(x)|:x\in K,|\alpha |\leq N_{K}\};$
16. For any compact subset $K\subseteq U$ and any sequence $\{f_{i}\}_{i=1}^{\infty }$ in $C^{\infty }(K),$ if $\{\partial ^{p}f_{i}\}_{i=1}^{\infty }$ converges uniformly to zero for all multi-indices $p,$ then $T(f_{i})\to 0;$
Topology on the space of distributions and its relation to the weak-* topology
The set of all distributions on $U$ is the continuous dual space of $C_{c}^{\infty }(U),$ which when endowed with the strong dual topology is denoted by ${\mathcal {D}}'(U).$ Importantly, unless indicated otherwise, the topology on ${\mathcal {D}}'(U)$ is the strong dual topology; if the topology is instead the weak-* topology then this will be indicated. Neither topology is metrizable although unlike the weak-* topology, the strong dual topology makes ${\mathcal {D}}'(U)$ into a complete nuclear space, to name just a few of its desirable properties.
Neither $C_{c}^{\infty }(U)$ nor its strong dual ${\mathcal {D}}'(U)$ is a sequential space and so neither of their topologies can be fully described by sequences (in other words, defining only what sequences converge in these spaces is not enough to fully/correctly define their topologies). However, a sequence in ${\mathcal {D}}'(U)$ converges in the strong dual topology if and only if it converges in the weak-* topology (this leads many authors to use pointwise convergence to define the convergence of a sequence of distributions; this is fine for sequences but this is not guaranteed to extend to the convergence of nets of distributions because a net may converge pointwise but fail to converge in the strong dual topology). More information about the topology that ${\mathcal {D}}'(U)$ is endowed with can be found in the article on spaces of test functions and distributions and the articles on polar topologies and dual systems.
A linear map from ${\mathcal {D}}'(U)$ into another locally convex topological vector space (such as any normed space) is continuous if and only if it is sequentially continuous at the origin. However, this is no longer guaranteed if the map is not linear or for maps valued in more general topological spaces (for example, that are not also locally convex topological vector spaces). The same is true of maps from $C_{c}^{\infty }(U)$ (more generally, this is true of maps from any locally convex bornological space).
Localization of distributions
There is no way to define the value of a distribution in ${\mathcal {D}}'(U)$ at a particular point of U. However, as is the case with functions, distributions on U restrict to give distributions on open subsets of U. Furthermore, distributions are locally determined in the sense that a distribution on all of U can be assembled from a distribution on an open cover of U satisfying some compatibility conditions on the overlaps. Such a structure is known as a sheaf.
Extensions and restrictions to an open subset
Let $V\subseteq U$ be open subsets of $\mathbb {R} ^{n}.$ Every function $f\in {\mathcal {D}}(V)$ can be extended by zero from its domain V to a function on U by setting it equal to $0$ on the complement $U\setminus V.$ This extension is a smooth compactly supported function called the trivial extension of $f$ to $U$ and it will be denoted by $E_{VU}(f).$ This assignment $f\mapsto E_{VU}(f)$ defines the trivial extension operator $E_{VU}:{\mathcal {D}}(V)\to {\mathcal {D}}(U),$ which is a continuous injective linear map. It is used to canonically identify ${\mathcal {D}}(V)$ as a vector subspace of ${\mathcal {D}}(U)$ (although not as a topological subspace). Its transpose (explained here)
$\rho _{VU}:={}^{t}E_{VU}:{\mathcal {D}}'(U)\to {\mathcal {D}}'(V),$
is called the restriction to $V$ of distributions in $U$[11] and as the name suggests, the image $\rho _{VU}(T)$ of a distribution $T\in {\mathcal {D}}'(U)$ under this map is a distribution on $V$ called the restriction of $T$ to $V.$ The defining condition of the restriction $\rho _{VU}(T)$ is:
$\langle \rho _{VU}T,\phi \rangle =\langle T,E_{VU}\phi \rangle \quad {\text{ for all }}\phi \in {\mathcal {D}}(V).$
If $V\neq U$ then the (continuous injective linear) trivial extension map $E_{VU}:{\mathcal {D}}(V)\to {\mathcal {D}}(U)$ is not a topological embedding (in other words, if this linear injection was used to identify ${\mathcal {D}}(V)$ as a subset of ${\mathcal {D}}(U)$ then ${\mathcal {D}}(V)$'s topology would strictly finer than the subspace topology that ${\mathcal {D}}(U)$ induces on it; importantly, it would not be a topological subspace since that requires equality of topologies) and its range is also not dense in its codomain ${\mathcal {D}}(U).$[11] Consequently if $V\neq U$ then the restriction mapping is neither injective nor surjective.[11] A distribution $S\in {\mathcal {D}}'(V)$ is said to be extendible to U if it belongs to the range of the transpose of $E_{VU}$ and it is called extendible if it is extendable to $\mathbb {R} ^{n}.$[11]
Unless $U=V,$ the restriction to V is neither injective nor surjective. Lack of surjectivity follows since distributions can blow up towards the boundary of V. For instance, if $U=\mathbb {R} $ and $V=(0,2),$ then the distribution
$T(x)=\sum _{n=1}^{\infty }n\,\delta \left(x-{\frac {1}{n}}\right)$
is in ${\mathcal {D}}'(V)$ but admits no extension to ${\mathcal {D}}'(U).$
Gluing and distributions that vanish in a set
Theorem[12] — Let $(U_{i})_{i\in I}$ be a collection of open subsets of $\mathbb {R} ^{n}.$ For each $i\in I,$ let $T_{i}\in {\mathcal {D}}'(U_{i})$ and suppose that for all $i,j\in I,$ the restriction of $T_{i}$ to $U_{i}\cap U_{j}$ is equal to the restriction of $T_{j}$ to $U_{i}\cap U_{j}$ (note that both restrictions are elements of ${\mathcal {D}}'(U_{i}\cap U_{j})$). Then there exists a unique $ T\in {\mathcal {D}}'(\bigcup _{i\in I}U_{i})$ such that for all $i\in I,$ the restriction of T to $U_{i}$ is equal to $T_{i}.$
Let V be an open subset of U. $T\in {\mathcal {D}}'(U)$ is said to vanish in V if for all $f\in {\mathcal {D}}(U)$ such that $\operatorname {supp} (f)\subseteq V$ we have $Tf=0.$ T vanishes in V if and only if the restriction of T to V is equal to 0, or equivalently, if and only if T lies in the kernel of the restriction map $\rho _{VU}.$
Corollary[12] — Let $(U_{i})_{i\in I}$ be a collection of open subsets of $\mathbb {R} ^{n}$ and let $ T\in {\mathcal {D}}'(\bigcup _{i\in I}U_{i}).$ $T=0$ if and only if for each $i\in I,$ the restriction of T to $U_{i}$ is equal to 0.
Corollary[12] — The union of all open subsets of U in which a distribution T vanishes is an open subset of U in which T vanishes.
Support of a distribution
This last corollary implies that for every distribution T on U, there exists a unique largest subset V of U such that T vanishes in V (and does not vanish in any open subset of U that is not contained in V); the complement in U of this unique largest open subset is called the support of T.[12] Thus
$\operatorname {supp} (T)=U\setminus \bigcup \{V\mid \rho _{VU}T=0\}.$
If $f$ is a locally integrable function on U and if $D_{f}$ is its associated distribution, then the support of $D_{f}$ is the smallest closed subset of U in the complement of which $f$ is almost everywhere equal to 0.[12] If $f$ is continuous, then the support of $D_{f}$ is equal to the closure of the set of points in U at which $f$ does not vanish.[12] The support of the distribution associated with the Dirac measure at a point $x_{0}$ is the set $\{x_{0}\}.$[12] If the support of a test function $f$ does not intersect the support of a distribution T then $Tf=0.$ A distribution T is 0 if and only if its support is empty. If $f\in C^{\infty }(U)$ is identically 1 on some open set containing the support of a distribution T then $fT=T.$ If the support of a distribution T is compact then it has finite order and there is a constant $C$ and a non-negative integer $N$ such that:[7]
$|T\phi |\leq C\|\phi \|_{N}:=C\sup \left\{\left|\partial ^{\alpha }\phi (x)\right|:x\in U,|\alpha |\leq N\right\}\quad {\text{ for all }}\phi \in {\mathcal {D}}(U).$
If T has compact support, then it has a unique extension to a continuous linear functional ${\widehat {T}}$ on $C^{\infty }(U)$; this function can be defined by ${\widehat {T}}(f):=T(\psi f),$ where $\psi \in {\mathcal {D}}(U)$ is any function that is identically 1 on an open set containing the support of T.[7]
If $S,T\in {\mathcal {D}}'(U)$ and $\lambda \neq 0$ then $\operatorname {supp} (S+T)\subseteq \operatorname {supp} (S)\cup \operatorname {supp} (T)$ and $\operatorname {supp} (\lambda T)=\operatorname {supp} (T).$ Thus, distributions with support in a given subset $A\subseteq U$ form a vector subspace of ${\mathcal {D}}'(U).$[13] Furthermore, if $P$ is a differential operator in U, then for all distributions T on U and all $f\in C^{\infty }(U)$ we have $\operatorname {supp} (P(x,\partial )T)\subseteq \operatorname {supp} (T)$ and $\operatorname {supp} (fT)\subseteq \operatorname {supp} (f)\cap \operatorname {supp} (T).$[13]
Support in a point set and Dirac measures
For any $x\in U,$ let $\delta _{x}\in {\mathcal {D}}'(U)$ denote the distribution induced by the Dirac measure at $x.$ For any $x_{0}\in U$ and distribution $T\in {\mathcal {D}}'(U),$ the support of T is contained in $\{x_{0}\}$ if and only if T is a finite linear combination of derivatives of the Dirac measure at $x_{0}.$[14] If in addition the order of T is $\leq k$ then there exist constants $\alpha _{p}$ such that:[15]
$T=\sum _{|p|\leq k}\alpha _{p}\partial ^{p}\delta _{x_{0}}.$
Said differently, if T has support at a single point $\{P\},$ then T is in fact a finite linear combination of distributional derivatives of the $\delta $ function at P. That is, there exists an integer m and complex constants $a_{\alpha }$ such that
$T=\sum _{|\alpha |\leq m}a_{\alpha }\partial ^{\alpha }(\tau _{P}\delta )$
where $\tau _{P}$ is the translation operator.
Distribution with compact support
Theorem[7] — Suppose T is a distribution on U with compact support K. There exists a continuous function $f$ defined on U and a multi-index p such that
$T=\partial ^{p}f,$
where the derivatives are understood in the sense of distributions. That is, for all test functions $\phi $ on U,
$T\phi =(-1)^{|p|}\int _{U}f(x)(\partial ^{p}\phi )(x)\,dx.$
Distributions of finite order with support in an open subset
Theorem[7] — Suppose T is a distribution on U with compact support K and let V be an open subset of U containing K. Since every distribution with compact support has finite order, take N to be the order of T and define $P:=\{0,1,\ldots ,N+2\}^{n}.$ There exists a family of continuous functions $(f_{p})_{p\in P}$ defined on U with support in V such that
$T=\sum _{p\in P}\partial ^{p}f_{p},$
where the derivatives are understood in the sense of distributions. That is, for all test functions $\phi $ on U,
$T\phi =\sum _{p\in P}(-1)^{|p|}\int _{U}f_{p}(x)(\partial ^{p}\phi )(x)\,dx.$
Global structure of distributions
The formal definition of distributions exhibits them as a subspace of a very large space, namely the topological dual of ${\mathcal {D}}(U)$ (or the Schwartz space ${\mathcal {S}}(\mathbb {R} ^{n})$ for tempered distributions). It is not immediately clear from the definition how exotic a distribution might be. To answer this question, it is instructive to see distributions built up from a smaller space, namely the space of continuous functions. Roughly, any distribution is locally a (multiple) derivative of a continuous function. A precise version of this result, given below, holds for distributions of compact support, tempered distributions, and general distributions. Generally speaking, no proper subset of the space of distributions contains all continuous functions and is closed under differentiation. This says that distributions are not particularly exotic objects; they are only as complicated as necessary.
Distributions as sheaves
Theorem[16] — Let T be a distribution on U. There exists a sequence $(T_{i})_{i=1}^{\infty }$ in ${\mathcal {D}}'(U)$ such that each Ti has compact support and every compact subset $K\subseteq U$ intersects the support of only finitely many $T_{i},$ and the sequence of partial sums $(S_{j})_{j=1}^{\infty },$ defined by $S_{j}:=T_{1}+\cdots +T_{j},$ converges in ${\mathcal {D}}'(U)$ to T; in other words we have:
$T=\sum _{i=1}^{\infty }T_{i}.$
Recall that a sequence converges in ${\mathcal {D}}'(U)$ (with its strong dual topology) if and only if it converges pointwise.
Decomposition of distributions as sums of derivatives of continuous functions
By combining the above results, one may express any distribution on U as the sum of a series of distributions with compact support, where each of these distributions can in turn be written as a finite sum of distributional derivatives of continuous functions on U. In other words, for arbitrary $T\in {\mathcal {D}}'(U)$ we can write:
$T=\sum _{i=1}^{\infty }\sum _{p\in P_{i}}\partial ^{p}f_{ip},$
where $P_{1},P_{2},\ldots $ are finite sets of multi-indices and the functions $f_{ip}$ are continuous.
Theorem[17] — Let T be a distribution on U. For every multi-index p there exists a continuous function $g_{p}$ on U such that
1. any compact subset K of U intersects the support of only finitely many $g_{p},$ and
2. $T=\sum \nolimits _{p}\partial ^{p}g_{p}.$
Moreover, if T has finite order, then one can choose $g_{p}$ in such a way that only finitely many of them are non-zero.
Note that the infinite sum above is well-defined as a distribution. The value of T for a given $f\in {\mathcal {D}}(U)$ can be computed using the finitely many $g_{\alpha }$ that intersect the support of $f.$
Operations on distributions
Many operations which are defined on smooth functions with compact support can also be defined for distributions. In general, if $A:{\mathcal {D}}(U)\to {\mathcal {D}}(U)$ is a linear map that is continuous with respect to the weak topology, then it is not always possible to extend $A$ to a map $A':{\mathcal {D}}'(U)\to {\mathcal {D}}'(U)$ by classic extension theorems of topology or linear functional analysis.[note 7] The “distributional” extension of the above linear continuous operator A is possible if and only if A admits a Schwartz adjoint, that is another linear continuous operator B of the same type such that $<Af,g>=<f,Bg>,$ for every pair of test functions. In that condition, B is unique and the extension A’ is the transpose of the Schwartz adjoint B. [18]
Preliminaries: Transpose of a linear operator
Main article: Transpose of a linear map
Operations on distributions and spaces of distributions are often defined using the transpose of a linear operator. This is because the transpose allows for a unified presentation of the many definitions in the theory of distributions and also because its properties are well-known in functional analysis.[19] For instance, the well-known Hermitian adjoint of a linear operator between Hilbert spaces is just the operator's transpose (but with the Riesz representation theorem used to identify each Hilbert space with its continuous dual space). In general, the transpose of a continuous linear map $A:X\to Y$ is the linear map
${}^{t}A:Y'\to X'\qquad {\text{ defined by }}\qquad {}^{t}A(y'):=y'\circ A,$
or equivalently, it is the unique map satisfying $\langle y',A(x)\rangle =\left\langle {}^{t}A(y'),x\right\rangle $ for all $x\in X$ and all $y'\in Y'$ (the prime symbol in $y'$ does not denote a derivative of any kind; it merely indicates that $y'$ is an element of the continuous dual space $Y'$). Since $A$ is continuous, the transpose ${}^{t}A:Y'\to X'$ is also continuous when both duals are endowed with their respective strong dual topologies; it is also continuous when both duals are endowed with their respective weak* topologies (see the articles polar topology and dual system for more details).
In the context of distributions, the characterization of the transpose can be refined slightly. Let $A:{\mathcal {D}}(U)\to {\mathcal {D}}(U)$ be a continuous linear map. Then by definition, the transpose of $A$ is the unique linear operator $A^{t}:{\mathcal {D}}'(U)\to {\mathcal {D}}'(U)$ that satisfies:
$\langle {}^{t}A(T),\phi \rangle =\langle T,A(\phi )\rangle \quad {\text{ for all }}\phi \in {\mathcal {D}}(U){\text{ and all }}T\in {\mathcal {D}}'(U).$
Since ${\mathcal {D}}(U)$ is dense in ${\mathcal {D}}'(U)$ (here, ${\mathcal {D}}(U)$ actually refers to the set of distributions $\left\{D_{\psi }:\psi \in {\mathcal {D}}(U)\right\}$) it is sufficient that the defining equality hold for all distributions of the form $T=D_{\psi }$ where $\psi \in {\mathcal {D}}(U).$ Explicitly, this means that a continuous linear map $B:{\mathcal {D}}'(U)\to {\mathcal {D}}'(U)$ is equal to ${}^{t}A$ if and only if the condition below holds:
$\langle B(D_{\psi }),\phi \rangle =\langle {}^{t}A(D_{\psi }),\phi \rangle \quad {\text{ for all }}\phi ,\psi \in {\mathcal {D}}(U)$
where the right-hand side equals $\langle {}^{t}A(D_{\psi }),\phi \rangle =\langle D_{\psi },A(\phi )\rangle =\langle \psi ,A(\phi )\rangle =\int _{U}\psi \cdot A(\phi )\,dx.$
Differentiation of distributions
Let $A:{\mathcal {D}}(U)\to {\mathcal {D}}(U)$ be the partial derivative operator ${\tfrac {\partial }{\partial x_{k}}}.$ To extend $A$ we compute its transpose:
${\begin{aligned}\langle {}^{t}A(D_{\psi }),\phi \rangle &=\int _{U}\psi (A\phi )\,dx&&{\text{(See above.)}}\\&=\int _{U}\psi {\frac {\partial \phi }{\partial x_{k}}}\,dx\\[4pt]&=-\int _{U}\phi {\frac {\partial \psi }{\partial x_{k}}}\,dx&&{\text{(integration by parts)}}\\[4pt]&=-\left\langle {\frac {\partial \psi }{\partial x_{k}}},\phi \right\rangle \\[4pt]&=-\langle A\psi ,\phi \rangle =\langle -A\psi ,\phi \rangle \end{aligned}}$
Therefore ${}^{t}A=-A.$ Thus, the partial derivative of $T$ with respect to the coordinate $x_{k}$ is defined by the formula
$\left\langle {\frac {\partial T}{\partial x_{k}}},\phi \right\rangle =-\left\langle T,{\frac {\partial \phi }{\partial x_{k}}}\right\rangle \qquad {\text{ for all }}\phi \in {\mathcal {D}}(U).$
With this definition, every distribution is infinitely differentiable, and the derivative in the direction $x_{k}$ is a linear operator on ${\mathcal {D}}'(U).$
More generally, if $\alpha $ is an arbitrary multi-index, then the partial derivative $\partial ^{\alpha }T$ of the distribution $T\in {\mathcal {D}}'(U)$ is defined by
$\langle \partial ^{\alpha }T,\phi \rangle =(-1)^{|\alpha |}\langle T,\partial ^{\alpha }\phi \rangle \qquad {\text{ for all }}\phi \in {\mathcal {D}}(U).$
Differentiation of distributions is a continuous operator on ${\mathcal {D}}'(U);$ this is an important and desirable property that is not shared by most other notions of differentiation.
If $T$ is a distribution in $\mathbb {R} $ then
$\lim _{x\to 0}{\frac {T-\tau _{x}T}{x}}=T'\in {\mathcal {D}}'(\mathbb {R} ),$
where $T'$ is the derivative of $T$ and $\tau _{x}$ is a translation by $x;$ thus the derivative of $T$ may be viewed as a limit of quotients.[20]
Differential operators acting on smooth functions
A linear differential operator in $U$ with smooth coefficients acts on the space of smooth functions on $U.$ Given such an operator $ P:=\sum _{\alpha }c_{\alpha }\partial ^{\alpha },$ we would like to define a continuous linear map, $D_{P}$ that extends the action of $P$ on $C^{\infty }(U)$ to distributions on $U.$ In other words, we would like to define $D_{P}$ such that the following diagram commutes:
${\begin{matrix}{\mathcal {D}}'(U)&{\stackrel {D_{P}}{\longrightarrow }}&{\mathcal {D}}'(U)\\[2pt]\uparrow &&\uparrow \\[2pt]C^{\infty }(U)&{\stackrel {P}{\longrightarrow }}&C^{\infty }(U)\end{matrix}}$
where the vertical maps are given by assigning $f\in C^{\infty }(U)$ its canonical distribution $D_{f}\in {\mathcal {D}}'(U),$ which is defined by:
$D_{f}(\phi )=\langle f,\phi \rangle :=\int _{U}f(x)\phi (x)\,dx\quad {\text{ for all }}\phi \in {\mathcal {D}}(U).$ :=\int _{U}f(x)\phi (x)\,dx\quad {\text{ for all }}\phi \in {\mathcal {D}}(U).}
With this notation, the diagram commuting is equivalent to:
$D_{P(f)}=D_{P}D_{f}\qquad {\text{ for all }}f\in C^{\infty }(U).$
To find $D_{P},$ the transpose ${}^{t}P:{\mathcal {D}}'(U)\to {\mathcal {D}}'(U)$ of the continuous induced map $P:{\mathcal {D}}(U)\to {\mathcal {D}}(U)$ defined by $\phi \mapsto P(\phi )$ is considered in the lemma below. This leads to the following definition of the differential operator on $U$ called the formal transpose of $P,$ which will be denoted by $P_{*}$ to avoid confusion with the transpose map, that is defined by
$P_{*}:=\sum _{\alpha }b_{\alpha }\partial ^{\alpha }\quad {\text{ where }}\quad b_{\alpha }:=\sum _{\beta \geq \alpha }(-1)^{|\beta |}{\binom {\beta }{\alpha }}\partial ^{\beta -\alpha }c_{\beta }.$
Lemma — Let $P$ be a linear differential operator with smooth coefficients in $U.$ Then for all $\phi \in {\mathcal {D}}(U)$ we have
$\left\langle {}^{t}P(D_{f}),\phi \right\rangle =\left\langle D_{P_{*}(f)},\phi \right\rangle ,$
which is equivalent to:
${}^{t}P(D_{f})=D_{P_{*}(f)}.$
Proof
As discussed above, for any $\phi \in {\mathcal {D}}(U),$ the transpose may be calculated by:
${\begin{aligned}\left\langle {}^{t}P(D_{f}),\phi \right\rangle &=\int _{U}f(x)P(\phi )(x)\,dx\\&=\int _{U}f(x)\left[\sum \nolimits _{\alpha }c_{\alpha }(x)(\partial ^{\alpha }\phi )(x)\right]\,dx\\&=\sum \nolimits _{\alpha }\int _{U}f(x)c_{\alpha }(x)(\partial ^{\alpha }\phi )(x)\,dx\\&=\sum \nolimits _{\alpha }(-1)^{|\alpha |}\int _{U}\phi (x)(\partial ^{\alpha }(c_{\alpha }f))(x)\,dx\end{aligned}}$
For the last line we used integration by parts combined with the fact that $\phi $ and therefore all the functions $f(x)c_{\alpha }(x)\partial ^{\alpha }\phi (x)$ have compact support.[note 8] Continuing the calculation above, for all $\phi \in {\mathcal {D}}(U):$
${\begin{aligned}\left\langle {}^{t}P(D_{f}),\phi \right\rangle &=\sum \nolimits _{\alpha }(-1)^{|\alpha |}\int _{U}\phi (x)(\partial ^{\alpha }(c_{\alpha }f))(x)\,dx&&{\text{As shown above}}\\[4pt]&=\int _{U}\phi (x)\sum \nolimits _{\alpha }(-1)^{|\alpha |}(\partial ^{\alpha }(c_{\alpha }f))(x)\,dx\\[4pt]&=\int _{U}\phi (x)\sum _{\alpha }\left[\sum _{\gamma \leq \alpha }{\binom {\alpha }{\gamma }}(\partial ^{\gamma }c_{\alpha })(x)(\partial ^{\alpha -\gamma }f)(x)\right]\,dx&&{\text{Leibniz rule}}\\&=\int _{U}\phi (x)\left[\sum _{\alpha }\sum _{\gamma \leq \alpha }(-1)^{|\alpha |}{\binom {\alpha }{\gamma }}(\partial ^{\gamma }c_{\alpha })(x)(\partial ^{\alpha -\gamma }f)(x)\right]\,dx\\&=\int _{U}\phi (x)\left[\sum _{\alpha }\left[\sum _{\beta \geq \alpha }(-1)^{|\beta |}{\binom {\beta }{\alpha }}\left(\partial ^{\beta -\alpha }c_{\beta }\right)(x)\right](\partial ^{\alpha }f)(x)\right]\,dx&&{\text{Grouping terms by derivatives of }}f\\&=\int _{U}\phi (x)\left[\sum \nolimits _{\alpha }b_{\alpha }(x)(\partial ^{\alpha }f)(x)\right]\,dx&&b_{\alpha }:=\sum _{\beta \geq \alpha }(-1)^{|\beta |}{\binom {\beta }{\alpha }}\partial ^{\beta -\alpha }c_{\beta }\\&=\left\langle \left(\sum \nolimits _{\alpha }b_{\alpha }\partial ^{\alpha }\right)(f),\phi \right\rangle \end{aligned}}$
The Lemma combined with the fact that the formal transpose of the formal transpose is the original differential operator, that is, $P_{**}=P,$[21] enables us to arrive at the correct definition: the formal transpose induces the (continuous) canonical linear operator $P_{*}:C_{c}^{\infty }(U)\to C_{c}^{\infty }(U)$ defined by $\phi \mapsto P_{*}(\phi ).$ We claim that the transpose of this map, ${}^{t}P_{*}:{\mathcal {D}}'(U)\to {\mathcal {D}}'(U),$ can be taken as $D_{P}.$ To see this, for every $\phi \in {\mathcal {D}}(U),$ compute its action on a distribution of the form $D_{f}$ with $f\in C^{\infty }(U)$:
${\begin{aligned}\left\langle {}^{t}P_{*}\left(D_{f}\right),\phi \right\rangle &=\left\langle D_{P_{**}(f)},\phi \right\rangle &&{\text{Using Lemma above with }}P_{*}{\text{ in place of }}P\\&=\left\langle D_{P(f)},\phi \right\rangle &&P_{**}=P\end{aligned}}$
We call the continuous linear operator $D_{P}:={}^{t}P_{*}:{\mathcal {D}}'(U)\to {\mathcal {D}}'(U)$ the differential operator on distributions extending $P$.[21] Its action on an arbitrary distribution $S$ is defined via:
$D_{P}(S)(\phi )=S\left(P_{*}(\phi )\right)\quad {\text{ for all }}\phi \in {\mathcal {D}}(U).$
If $(T_{i})_{i=1}^{\infty }$ converges to $T\in {\mathcal {D}}'(U)$ then for every multi-index $\alpha ,(\partial ^{\alpha }T_{i})_{i=1}^{\infty }$ converges to $\partial ^{\alpha }T\in {\mathcal {D}}'(U).$
Multiplication of distributions by smooth functions
A differential operator of order 0 is just multiplication by a smooth function. And conversely, if $f$ is a smooth function then $P:=f(x)$ is a differential operator of order 0, whose formal transpose is itself (that is, $P_{*}=P$). The induced differential operator $D_{P}:{\mathcal {D}}'(U)\to {\mathcal {D}}'(U)$ maps a distribution $T$ to a distribution denoted by $fT:=D_{P}(T).$ We have thus defined the multiplication of a distribution by a smooth function.
We now give an alternative presentation of the multiplication of a distribution $T$ on $U$ by a smooth function $m:U\to \mathbb {R} .$ The product $mT$ is defined by
$\langle mT,\phi \rangle =\langle T,m\phi \rangle \qquad {\text{ for all }}\phi \in {\mathcal {D}}(U).$
This definition coincides with the transpose definition since if $M:{\mathcal {D}}(U)\to {\mathcal {D}}(U)$ is the operator of multiplication by the function $m$ (that is, $(M\phi )(x)=m(x)\phi (x)$), then
$\int _{U}(M\phi )(x)\psi (x)\,dx=\int _{U}m(x)\phi (x)\psi (x)\,dx=\int _{U}\phi (x)m(x)\psi (x)\,dx=\int _{U}\phi (x)(M\psi )(x)\,dx,$
so that ${}^{t}M=M.$
Under multiplication by smooth functions, ${\mathcal {D}}'(U)$ is a module over the ring $C^{\infty }(U).$ With this definition of multiplication by a smooth function, the ordinary product rule of calculus remains valid. However, some unusual identities also arise. For example, if $\delta $ is the Dirac delta distribution on $\mathbb {R} ,$ then $m\delta =m(0)\delta ,$ and if $\delta ^{'}$ is the derivative of the delta distribution, then
$m\delta '=m(0)\delta '-m'\delta =m(0)\delta '-m'(0)\delta .$
The bilinear multiplication map $C^{\infty }(\mathbb {R} ^{n})\times {\mathcal {D}}'(\mathbb {R} ^{n})\to {\mathcal {D}}'\left(\mathbb {R} ^{n}\right)$ given by $(f,T)\mapsto fT$ is not continuous; it is however, hypocontinuous.[22]
Example. The product of any distribution $T$ with the function that is identically 1 on $U$ is equal to $T.$
Example. Suppose $(f_{i})_{i=1}^{\infty }$ is a sequence of test functions on $U$ that converges to the constant function $1\in C^{\infty }(U).$ For any distribution $T$ on $U,$ the sequence $(f_{i}T)_{i=1}^{\infty }$ converges to $T\in {\mathcal {D}}'(U).$[23]
If $(T_{i})_{i=1}^{\infty }$ converges to $T\in {\mathcal {D}}'(U)$ and $(f_{i})_{i=1}^{\infty }$ converges to $f\in C^{\infty }(U)$ then $(f_{i}T_{i})_{i=1}^{\infty }$ converges to $fT\in {\mathcal {D}}'(U).$
Problem of multiplying distributions
It is easy to define the product of a distribution with a smooth function, or more generally the product of two distributions whose singular supports are disjoint.[24] With more effort, it is possible to define a well-behaved product of several distributions provided their wave front sets at each point are compatible. A limitation of the theory of distributions (and hyperfunctions) is that there is no associative product of two distributions extending the product of a distribution by a smooth function, as has been proved by Laurent Schwartz in the 1950s. For example, if $\operatorname {p.v.} {\frac {1}{x}}$ is the distribution obtained by the Cauchy principal value
$\left(\operatorname {p.v.} {\frac {1}{x}}\right)(\phi )=\lim _{\varepsilon \to 0^{+}}\int _{|x|\geq \varepsilon }{\frac {\phi (x)}{x}}\,dx\quad {\text{ for all }}\phi \in {\mathcal {S}}(\mathbb {R} ).$
If $\delta $ is the Dirac delta distribution then
$(\delta \times x)\times \operatorname {p.v.} {\frac {1}{x}}=0$
but,
$\delta \times \left(x\times \operatorname {p.v.} {\frac {1}{x}}\right)=\delta $
so the product of a distribution by a smooth function (which is always well-defined) cannot be extended to an associative product on the space of distributions.
Thus, nonlinear problems cannot be posed in general and thus are not solved within distribution theory alone. In the context of quantum field theory, however, solutions can be found. In more than two spacetime dimensions the problem is related to the regularization of divergences. Here Henri Epstein and Vladimir Glaser developed the mathematically rigorous (but extremely technical) causal perturbation theory. This does not solve the problem in other situations. Many other interesting theories are non-linear, like for example the Navier–Stokes equations of fluid dynamics.
Several not entirely satisfactory theories of algebras of generalized functions have been developed, among which Colombeau's (simplified) algebra is maybe the most popular in use today.
Inspired by Lyons' rough path theory,[25] Martin Hairer proposed a consistent way of multiplying distributions with certain structures (regularity structures[26]), available in many examples from stochastic analysis, notably stochastic partial differential equations. See also Gubinelli–Imkeller–Perkowski (2015) for a related development based on Bony's paraproduct from Fourier analysis.
Composition with a smooth function
Let $T$ be a distribution on $U.$ Let $V$ be an open set in $\mathbb {R} ^{n}$ and $F:V\to U.$ If $F$ is a submersion then it is possible to define
$T\circ F\in {\mathcal {D}}'(V).$
This is the composition of the distribution $T$ with $F$, and is also called the pullback of $T$ along $F$, sometimes written
$F^{\sharp }:T\mapsto F^{\sharp }T=T\circ F.$
The pullback is often denoted $F^{*},$ although this notation should not be confused with the use of '*' to denote the adjoint of a linear mapping.
The condition that $F$ be a submersion is equivalent to the requirement that the Jacobian derivative $dF(x)$ of $F$ is a surjective linear map for every $x\in V.$ A necessary (but not sufficient) condition for extending $F^{\#}$ to distributions is that $F$ be an open mapping.[27] The Inverse function theorem ensures that a submersion satisfies this condition.
If $F$ is a submersion, then $F^{\#}$ is defined on distributions by finding the transpose map. The uniqueness of this extension is guaranteed since $F^{\#}$ is a continuous linear operator on ${\mathcal {D}}(U).$ Existence, however, requires using the change of variables formula, the inverse function theorem (locally), and a partition of unity argument.[28]
In the special case when $F$ is a diffeomorphism from an open subset $V$ of $\mathbb {R} ^{n}$ onto an open subset $U$ of $\mathbb {R} ^{n}$ change of variables under the integral gives:
$\int _{V}\phi \circ F(x)\psi (x)\,dx=\int _{U}\phi (x)\psi \left(F^{-1}(x)\right)\left|\det dF^{-1}(x)\right|\,dx.$
In this particular case, then, $F^{\#}$ is defined by the transpose formula:
$\left\langle F^{\sharp }T,\phi \right\rangle =\left\langle T,\left|\det d(F^{-1})\right|\phi \circ F^{-1}\right\rangle .$
Convolution
Under some circumstances, it is possible to define the convolution of a function with a distribution, or even the convolution of two distributions. Recall that if $f$ and $g$ are functions on $\mathbb {R} ^{n}$ then we denote by $f\ast g$ the convolution of $f$ and $g,$ defined at $x\in \mathbb {R} ^{n}$ to be the integral
$(f\ast g)(x):=\int _{\mathbb {R} ^{n}}f(x-y)g(y)\,dy=\int _{\mathbb {R} ^{n}}f(y)g(x-y)\,dy$
provided that the integral exists. If $1\leq p,q,r\leq \infty $ are such that $ {\frac {1}{r}}={\frac {1}{p}}+{\frac {1}{q}}-1$ then for any functions $f\in L^{p}(\mathbb {R} ^{n})$ and $g\in L^{q}(\mathbb {R} ^{n})$ we have $f\ast g\in L^{r}(\mathbb {R} ^{n})$ and $\|f\ast g\|_{L^{r}}\leq \|f\|_{L^{p}}\|g\|_{L^{q}}.$[29] If $f$ and $g$ are continuous functions on $\mathbb {R} ^{n},$ at least one of which has compact support, then $\operatorname {supp} (f\ast g)\subseteq \operatorname {supp} (f)+\operatorname {supp} (g)$ and if $A\subseteq \mathbb {R} ^{n}$ then the value of $f\ast g$ on $A$ do not depend on the values of $f$ outside of the Minkowski sum $A-\operatorname {supp} (g)=\{a-s:a\in A,s\in \operatorname {supp} (g)\}.$[29]
Importantly, if $g\in L^{1}(\mathbb {R} ^{n})$ has compact support then for any $0\leq k\leq \infty ,$ the convolution map $f\mapsto f\ast g$ is continuous when considered as the map $C^{k}(\mathbb {R} ^{n})\to C^{k}(\mathbb {R} ^{n})$ or as the map $C_{c}^{k}(\mathbb {R} ^{n})\to C_{c}^{k}(\mathbb {R} ^{n}).$[29]
Translation and symmetry
Given $a\in \mathbb {R} ^{n},$ the translation operator $\tau _{a}$ sends $f:\mathbb {R} ^{n}\to \mathbb {C} $ to $\tau _{a}f:\mathbb {R} ^{n}\to \mathbb {C} ,$ defined by $\tau _{a}f(y)=f(y-a).$ This can be extended by the transpose to distributions in the following way: given a distribution $T,$ the translation of $T$ by $a$ is the distribution $\tau _{a}T:{\mathcal {D}}(\mathbb {R} ^{n})\to \mathbb {C} $ defined by $\tau _{a}T(\phi ):=\left\langle T,\tau _{-a}\phi \right\rangle .$[30][31]
Given $f:\mathbb {R} ^{n}\to \mathbb {C} ,$ define the function ${\tilde {f}}:\mathbb {R} ^{n}\to \mathbb {C} $ by ${\tilde {f}}(x):=f(-x).$ Given a distribution $T,$ let ${\tilde {T}}:{\mathcal {D}}(\mathbb {R} ^{n})\to \mathbb {C} $ be the distribution defined by ${\tilde {T}}(\phi ):=T\left({\tilde {\phi }}\right).$ The operator $T\mapsto {\tilde {T}}$ is called the symmetry with respect to the origin.[30]
Convolution of a test function with a distribution
Convolution with $f\in {\mathcal {D}}(\mathbb {R} ^{n})$ defines a linear map:
${\begin{alignedat}{4}C_{f}:\,&{\mathcal {D}}(\mathbb {R} ^{n})&&\to \,&&{\mathcal {D}}(\mathbb {R} ^{n})\\&g&&\mapsto \,&&f\ast g\\\end{alignedat}}$
which is continuous with respect to the canonical LF space topology on ${\mathcal {D}}(\mathbb {R} ^{n}).$
Convolution of $f$ with a distribution $T\in {\mathcal {D}}'(\mathbb {R} ^{n})$ can be defined by taking the transpose of $C_{f}$ relative to the duality pairing of ${\mathcal {D}}(\mathbb {R} ^{n})$ with the space ${\mathcal {D}}'(\mathbb {R} ^{n})$ of distributions.[32] If $f,g,\phi \in {\mathcal {D}}(\mathbb {R} ^{n}),$ then by Fubini's theorem
$\langle C_{f}g,\phi \rangle =\int _{\mathbb {R} ^{n}}\phi (x)\int _{\mathbb {R} ^{n}}f(x-y)g(y)\,dy\,dx=\left\langle g,C_{\tilde {f}}\phi \right\rangle .$
Extending by continuity, the convolution of $f$ with a distribution $T$ is defined by
$\langle f\ast T,\phi \rangle =\left\langle T,{\tilde {f}}\ast \phi \right\rangle ,\quad {\text{ for all }}\phi \in {\mathcal {D}}(\mathbb {R} ^{n}).$
An alternative way to define the convolution of a test function $f$ and a distribution $T$ is to use the translation operator $\tau _{a}.$ The convolution of the compactly supported function $f$ and the distribution $T$ is then the function defined for each $x\in \mathbb {R} ^{n}$ by
$(f\ast T)(x)=\left\langle T,\tau _{x}{\tilde {f}}\right\rangle .$
It can be shown that the convolution of a smooth, compactly supported function and a distribution is a smooth function. If the distribution $T$ has compact support, and if $f$ is a polynomial (resp. an exponential function, an analytic function, the restriction of an entire analytic function on $\mathbb {C} ^{n}$ to $\mathbb {R} ^{n},$ the restriction of an entire function of exponential type in $\mathbb {C} ^{n}$ to $\mathbb {R} ^{n}$), then the same is true of $T\ast f.$[30] If the distribution $T$ has compact support as well, then $f\ast T$ is a compactly supported function, and the Titchmarsh convolution theorem Hörmander (1983, Theorem 4.3.3) implies that:
$\operatorname {ch} (\operatorname {supp} (f\ast T))=\operatorname {ch} (\operatorname {supp} (f))+\operatorname {ch} (\operatorname {supp} (T))$
where $\operatorname {ch} $ denotes the convex hull and $\operatorname {supp} $ denotes the support.
Convolution of a smooth function with a distribution
Let $f\in C^{\infty }(\mathbb {R} ^{n})$ and $T\in {\mathcal {D}}'(\mathbb {R} ^{n})$ and assume that at least one of $f$ and $T$ has compact support. The convolution of $f$ and $T,$ denoted by $f\ast T$ or by $T\ast f,$ is the smooth function:[30]
${\begin{alignedat}{4}f\ast T:\,&\mathbb {R} ^{n}&&\to \,&&\mathbb {C} \\&x&&\mapsto \,&&\left\langle T,\tau _{x}{\tilde {f}}\right\rangle \\\end{alignedat}}$
satisfying for all $p\in \mathbb {N} ^{n}$:
${\begin{aligned}&\operatorname {supp} (f\ast T)\subseteq \operatorname {supp} (f)+\operatorname {supp} (T)\\[6pt]&{\text{ for all }}p\in \mathbb {N} ^{n}:\quad {\begin{cases}\partial ^{p}\left\langle T,\tau _{x}{\tilde {f}}\right\rangle =\left\langle T,\partial ^{p}\tau _{x}{\tilde {f}}\right\rangle \\\partial ^{p}(T\ast f)=(\partial ^{p}T)\ast f=T\ast (\partial ^{p}f).\end{cases}}\end{aligned}}$
Let $M$ be the map $f\mapsto T\ast f$. If $T$ is a distribution, then $M$ is continuous as a map ${\mathcal {D}}(\mathbb {R} ^{n})\to C^{\infty }(\mathbb {R} ^{n})$. If $T$ also has compact support, then $M$ is also continuous as the map $C^{\infty }(\mathbb {R} ^{n})\to C^{\infty }(\mathbb {R} ^{n})$ and continuous as the map ${\mathcal {D}}(\mathbb {R} ^{n})\to {\mathcal {D}}(\mathbb {R} ^{n}).$[30]
If $L:{\mathcal {D}}(\mathbb {R} ^{n})\to C^{\infty }(\mathbb {R} ^{n})$ is a continuous linear map such that $L\partial ^{\alpha }\phi =\partial ^{\alpha }L\phi $ for all $\alpha $ and all $\phi \in {\mathcal {D}}(\mathbb {R} ^{n})$ then there exists a distribution $T\in {\mathcal {D}}'(\mathbb {R} ^{n})$ such that $L\phi =T\circ \phi $ for all $\phi \in {\mathcal {D}}(\mathbb {R} ^{n}).$[7]
Example.[7] Let $H$ be the Heaviside function on $\mathbb {R} .$ For any $\phi \in {\mathcal {D}}(\mathbb {R} ),$
$(H\ast \phi )(x)=\int _{-\infty }^{x}\phi (t)\,dt.$
Let $\delta $ be the Dirac measure at 0 and let $\delta '$ be its derivative as a distribution. Then $\delta '\ast H=\delta $ and $1\ast \delta '=0.$ Importantly, the associative law fails to hold:
$1=1\ast \delta =1\ast (\delta '\ast H)\neq (1\ast \delta ')\ast H=0\ast H=0.$
Convolution of distributions
It is also possible to define the convolution of two distributions $S$ and $T$ on $\mathbb {R} ^{n},$ provided one of them has compact support. Informally, to define $S\ast T$ where $T$ has compact support, the idea is to extend the definition of the convolution $\,\ast \,$ to a linear operation on distributions so that the associativity formula
$S\ast (T\ast \phi )=(S\ast T)\ast \phi $
continues to hold for all test functions $\phi .$[33]
It is also possible to provide a more explicit characterization of the convolution of distributions.[32] Suppose that $S$ and $T$ are distributions and that $S$ has compact support. Then the linear maps
${\begin{alignedat}{9}\bullet \ast {\tilde {S}}:\,&{\mathcal {D}}(\mathbb {R} ^{n})&&\to \,&&{\mathcal {D}}(\mathbb {R} ^{n})&&\quad {\text{ and }}\quad &&\bullet \ast {\tilde {T}}:\,&&{\mathcal {D}}(\mathbb {R} ^{n})&&\to \,&&{\mathcal {D}}(\mathbb {R} ^{n})\\&f&&\mapsto \,&&f\ast {\tilde {S}}&&&&&&f&&\mapsto \,&&f\ast {\tilde {T}}\\\end{alignedat}}$
are continuous. The transposes of these maps:
${}^{t}\left(\bullet \ast {\tilde {S}}\right):{\mathcal {D}}'(\mathbb {R} ^{n})\to {\mathcal {D}}'(\mathbb {R} ^{n})\qquad {}^{t}\left(\bullet \ast {\tilde {T}}\right):{\mathcal {E}}'(\mathbb {R} ^{n})\to {\mathcal {D}}'(\mathbb {R} ^{n})$
are consequently continuous and it can also be shown that[30]
${}^{t}\left(\bullet \ast {\tilde {S}}\right)(T)={}^{t}\left(\bullet \ast {\tilde {T}}\right)(S).$
This common value is called the convolution of $S$ and $T$ and it is a distribution that is denoted by $S\ast T$ or $T\ast S.$ It satisfies $\operatorname {supp} (S\ast T)\subseteq \operatorname {supp} (S)+\operatorname {supp} (T).$[30] If $S$ and $T$ are two distributions, at least one of which has compact support, then for any $a\in \mathbb {R} ^{n},$ $\tau _{a}(S\ast T)=\left(\tau _{a}S\right)\ast T=S\ast \left(\tau _{a}T\right).$[30] If $T$ is a distribution in $\mathbb {R} ^{n}$ and if $\delta $ is a Dirac measure then $T\ast \delta =T=\delta \ast T$;[30] thus $\delta $ is the identity element of the convolution operation. Moreover, if $f$ is a function then $f\ast \delta ^{\prime }=f^{\prime }=\delta ^{\prime }\ast f$ where now the associativity of convolution implies that $f^{\prime }\ast g=g^{\prime }\ast f$ for all functions $f$ and $g.$
Suppose that it is $T$ that has compact support. For $\phi \in {\mathcal {D}}(\mathbb {R} ^{n})$ consider the function
$\psi (x)=\langle T,\tau _{-x}\phi \rangle .$
It can be readily shown that this defines a smooth function of $x,$ which moreover has compact support. The convolution of $S$ and $T$ is defined by
$\langle S\ast T,\phi \rangle =\langle S,\psi \rangle .$
This generalizes the classical notion of convolution of functions and is compatible with differentiation in the following sense: for every multi-index $\alpha .$
$\partial ^{\alpha }(S\ast T)=(\partial ^{\alpha }S)\ast T=S\ast (\partial ^{\alpha }T).$
The convolution of a finite number of distributions, all of which (except possibly one) have compact support, is associative.[30]
This definition of convolution remains valid under less restrictive assumptions about $S$ and $T.$[34]
The convolution of distributions with compact support induces a continuous bilinear map ${\mathcal {E}}'\times {\mathcal {E}}'\to {\mathcal {E}}'$ defined by $(S,T)\mapsto S*T,$ where ${\mathcal {E}}'$ denotes the space of distributions with compact support.[22] However, the convolution map as a function ${\mathcal {E}}'\times {\mathcal {D}}'\to {\mathcal {D}}'$ is not continuous[22] although it is separately continuous.[35] The convolution maps ${\mathcal {D}}(\mathbb {R} ^{n})\times {\mathcal {D}}'\to {\mathcal {D}}'$ and ${\mathcal {D}}(\mathbb {R} ^{n})\times {\mathcal {D}}'\to {\mathcal {D}}(\mathbb {R} ^{n})$ given by $(f,T)\mapsto f*T$ both fail to be continuous.[22] Each of these non-continuous maps is, however, separately continuous and hypocontinuous.[22]
Convolution versus multiplication
In general, regularity is required for multiplication products, and locality is required for convolution products. It is expressed in the following extension of the Convolution Theorem which guarantees the existence of both convolution and multiplication products. Let $F(\alpha )=f\in {\mathcal {O}}'_{C}$ be a rapidly decreasing tempered distribution or, equivalently, $F(f)=\alpha \in {\mathcal {O}}_{M}$ be an ordinary (slowly growing, smooth) function within the space of tempered distributions and let $F$ be the normalized (unitary, ordinary frequency) Fourier transform.[36] Then, according to Schwartz (1951),
$F(f*g)=F(f)\cdot F(g)\qquad {\text{ and }}\qquad F(\alpha \cdot g)=F(\alpha )*F(g)$
hold within the space of tempered distributions.[37][38][39] In particular, these equations become the Poisson Summation Formula if $g\equiv \operatorname {\text{Ш}} $ is the Dirac Comb.[40] The space of all rapidly decreasing tempered distributions is also called the space of convolution operators ${\mathcal {O}}'_{C}$ and the space of all ordinary functions within the space of tempered distributions is also called the space of multiplication operators ${\mathcal {O}}_{M}.$ More generally, $F({\mathcal {O}}'_{C})={\mathcal {O}}_{M}$ and $F({\mathcal {O}}_{M})={\mathcal {O}}'_{C}.$[41][42] A particular case is the Paley-Wiener-Schwartz Theorem which states that $F({\mathcal {E}}')=\operatorname {PW} $ and $F(\operatorname {PW} )={\mathcal {E}}'.$ This is because ${\mathcal {E}}'\subseteq {\mathcal {O}}'_{C}$ and $\operatorname {PW} \subseteq {\mathcal {O}}_{M}.$ In other words, compactly supported tempered distributions ${\mathcal {E}}'$ belong to the space of convolution operators ${\mathcal {O}}'_{C}$ and Paley-Wiener functions $\operatorname {PW} ,$ better known as bandlimited functions, belong to the space of multiplication operators ${\mathcal {O}}_{M}.$[43]
For example, let $g\equiv \operatorname {\text{Ш}} \in {\mathcal {S}}'$ be the Dirac comb and $f\equiv \delta \in {\mathcal {E}}'$ be the Dirac delta;then $\alpha \equiv 1\in \operatorname {PW} $ is the function that is constantly one and both equations yield the Dirac-comb identity. Another example is to let $g$ be the Dirac comb and $f\equiv \operatorname {rect} \in {\mathcal {E}}'$ be the rectangular function; then $\alpha \equiv \operatorname {sinc} \in \operatorname {PW} $ is the sinc function and both equations yield the Classical Sampling Theorem for suitable $\operatorname {rect} $ functions. More generally, if $g$ is the Dirac comb and $f\in {\mathcal {S}}\subseteq {\mathcal {O}}'_{C}\cap {\mathcal {O}}_{M}$ is a smooth window function (Schwartz function), for example, the Gaussian, then $\alpha \in {\mathcal {S}}$ is another smooth window function (Schwartz function). They are known as mollifiers, especially in partial differential equations theory, or as regularizers in physics because they allow turning generalized functions into regular functions.
Tensor products of distributions
Let $U\subseteq \mathbb {R} ^{m}$ and $V\subseteq \mathbb {R} ^{n}$ be open sets. Assume all vector spaces to be over the field $\mathbb {F} ,$ where $\mathbb {F} =\mathbb {R} $ or $\mathbb {C} .$ For $f\in {\mathcal {D}}(U\times V)$ define for every $u\in U$ and every $v\in V$ the following functions:
${\begin{alignedat}{9}f_{u}:\,&V&&\to \,&&\mathbb {F} &&\quad {\text{ and }}\quad &&f^{v}:\,&&U&&\to \,&&\mathbb {F} \\&y&&\mapsto \,&&f(u,y)&&&&&&x&&\mapsto \,&&f(x,v)\\\end{alignedat}}$
Given $S\in {\mathcal {D}}^{\prime }(U)$ and $T\in {\mathcal {D}}^{\prime }(V),$ define the following functions:
${\begin{alignedat}{9}\langle S,f^{\bullet }\rangle :\,&V&&\to \,&&\mathbb {F} &&\quad {\text{ and }}\quad &&\langle T,f_{\bullet }\rangle :\,&&U&&\to \,&&\mathbb {F} \\&v&&\mapsto \,&&\langle S,f^{v}\rangle &&&&&&u&&\mapsto \,&&\langle T,f_{u}\rangle \\\end{alignedat}}$ :\,&V&&\to \,&&\mathbb {F} &&\quad {\text{ and }}\quad &&\langle T,f_{\bullet }\rangle :\,&&U&&\to \,&&\mathbb {F} \\&v&&\mapsto \,&&\langle S,f^{v}\rangle &&&&&&u&&\mapsto \,&&\langle T,f_{u}\rangle \\\end{alignedat}}}
where $\langle T,f_{\bullet }\rangle \in {\mathcal {D}}(U)$ and $\langle S,f^{\bullet }\rangle \in {\mathcal {D}}(V).$ These definitions associate every $S\in {\mathcal {D}}'(U)$ and $T\in {\mathcal {D}}'(V)$ with the (respective) continuous linear map:
${\begin{alignedat}{9}\,&&{\mathcal {D}}(U\times V)&\to \,&&{\mathcal {D}}(V)&&\quad {\text{ and }}\quad &&\,&{\mathcal {D}}(U\times V)&&\to \,&&{\mathcal {D}}(U)\\&&f\ &\mapsto \,&&\langle S,f^{\bullet }\rangle &&&&&f\ &&\mapsto \,&&\langle T,f_{\bullet }\rangle \\\end{alignedat}}$
Moreover, if either $S$ (resp. $T$) has compact support then it also induces a continuous linear map of $C^{\infty }(U\times V)\to C^{\infty }(V)$ (resp. $C^{\infty }(U\times V)\to C^{\infty }(U)$).[44]
Fubini's theorem for distributions[44] — Let $S\in {\mathcal {D}}'(U)$ and $T\in {\mathcal {D}}'(V).$ If $f\in {\mathcal {D}}(U\times V)$ then
$\langle S,\langle T,f_{\bullet }\rangle \rangle =\langle T,\langle S,f^{\bullet }\rangle \rangle .$
The tensor product of $S\in {\mathcal {D}}'(U)$ and $T\in {\mathcal {D}}'(V),$ denoted by $S\otimes T$ or $T\otimes S,$ is the distribution in $U\times V$ defined by:[44]
$(S\otimes T)(f):=\langle S,\langle T,f_{\bullet }\rangle \rangle =\langle T,\langle S,f^{\bullet }\rangle \rangle .$
Spaces of distributions
See also: Spaces of test functions and distributions
For all $0<k<\infty $ and all $1<p<\infty ,$ every one of the following canonical injections is continuous and has an image (also called the range) that is a dense subset of its codomain:
${\begin{matrix}C_{c}^{\infty }(U)&\to &C_{c}^{k}(U)&\to &C_{c}^{0}(U)&\to &L_{c}^{\infty }(U)&\to &L_{c}^{p}(U)&\to &L_{c}^{1}(U)\\\downarrow &&\downarrow &&\downarrow \\C^{\infty }(U)&\to &C^{k}(U)&\to &C^{0}(U)\\{}\end{matrix}}$
where the topologies on $L_{c}^{q}(U)$ ($1\leq q\leq \infty $) are defined as direct limits of the spaces $L_{c}^{q}(K)$ in a manner analogous to how the topologies on $C_{c}^{k}(U)$ were defined (so in particular, they are not the usual norm topologies). The range of each of the maps above (and of any composition of the maps above) is dense in its codomain.[45]
Suppose that $X$ is one of the spaces $C_{c}^{k}(U)$ (for $k\in \{0,1,\ldots ,\infty \}$) or $L_{c}^{p}(U)$ (for $1\leq p\leq \infty $) or $L^{p}(U)$ (for $1\leq p<\infty $). Because the canonical injection $\operatorname {In} _{X}:C_{c}^{\infty }(U)\to X$ is a continuous injection whose image is dense in the codomain, this map's transpose ${}^{t}\operatorname {In} _{X}:X'_{b}\to {\mathcal {D}}'(U)=\left(C_{c}^{\infty }(U)\right)'_{b}$ is a continuous injection. This injective transpose map thus allows the continuous dual space $X'$ of $X$ to be identified with a certain vector subspace of the space ${\mathcal {D}}'(U)$ of all distributions (specifically, it is identified with the image of this transpose map). This transpose map is continuous but it is not necessarily a topological embedding. A linear subspace of ${\mathcal {D}}'(U)$ carrying a locally convex topology that is finer than the subspace topology induced on it by ${\mathcal {D}}'(U)=\left(C_{c}^{\infty }(U)\right)'_{b}$ is called a space of distributions.[46] Almost all of the spaces of distributions mentioned in this article arise in this way (for example, tempered distribution, restrictions, distributions of order $\leq $ some integer, distributions induced by a positive Radon measure, distributions induced by an $L^{p}$-function, etc.) and any representation theorem about the continuous dual space of $X$ may, through the transpose ${}^{t}\operatorname {In} _{X}:X'_{b}\to {\mathcal {D}}'(U),$ be transferred directly to elements of the space $\operatorname {Im} \left({}^{t}\operatorname {In} _{X}\right).$
Radon measures
The inclusion map $\operatorname {In} :C_{c}^{\infty }(U)\to C_{c}^{0}(U)$ is a continuous injection whose image is dense in its codomain, so the transpose ${}^{t}\operatorname {In} :(C_{c}^{0}(U))'_{b}\to {\mathcal {D}}'(U)=(C_{c}^{\infty }(U))'_{b}$ :(C_{c}^{0}(U))'_{b}\to {\mathcal {D}}'(U)=(C_{c}^{\infty }(U))'_{b}} is also a continuous injection.
Note that the continuous dual space $(C_{c}^{0}(U))'_{b}$ can be identified as the space of Radon measures, where there is a one-to-one correspondence between the continuous linear functionals $T\in (C_{c}^{0}(U))'_{b}$ and integral with respect to a Radon measure; that is,
• if $T\in (C_{c}^{0}(U))'_{b}$ then there exists a Radon measure $\mu $ on U such that for all $ f\in C_{c}^{0}(U),T(f)=\int _{U}f\,d\mu ,$ and
• if $\mu $ is a Radon measure on U then the linear functional on $C_{c}^{0}(U)$ defined by sending $ f\in C_{c}^{0}(U)$ to $ \int _{U}f\,d\mu $ is continuous.
Through the injection ${}^{t}\operatorname {In} :(C_{c}^{0}(U))'_{b}\to {\mathcal {D}}'(U),$ :(C_{c}^{0}(U))'_{b}\to {\mathcal {D}}'(U),} every Radon measure becomes a distribution on U. If $f$ is a locally integrable function on U then the distribution $ \phi \mapsto \int _{U}f(x)\phi (x)\,dx$ is a Radon measure; so Radon measures form a large and important space of distributions.
The following is the theorem of the structure of distributions of Radon measures, which shows that every Radon measure can be written as a sum of derivatives of locally $L^{\infty }$ functions on U:
Theorem.[47] — Suppose $T\in {\mathcal {D}}'(U)$ is a Radon measure, where $U\subseteq \mathbb {R} ^{n},$ let $V\subseteq U$ be a neighborhood of the support of $T,$ and let $I=\{p\in \mathbb {N} ^{n}:|p|\leq n\}.$ There exists a family $f=(f_{p})_{p\in I}$ of locally $L^{\infty }$ functions on U such that $\operatorname {supp} f_{p}\subseteq V$ for every $p\in I,$ and
$T=\sum _{p\in I}\partial ^{p}f_{p}.$
Furthermore, $T$ is also equal to a finite sum of derivatives of continuous functions on $U,$ where each derivative has order $\leq 2n.$
Positive Radon measures
A linear function $T$ on a space of functions is called positive if whenever a function $f$ that belongs to the domain of $T$ is non-negative (that is, $f$ is real-valued and $f\geq 0$) then $T(f)\geq 0.$ One may show that every positive linear functional on $C_{c}^{0}(U)$ is necessarily continuous (that is, necessarily a Radon measure).[48] Lebesgue measure is an example of a positive Radon measure.
Locally integrable functions as distributions
One particularly important class of Radon measures are those that are induced locally integrable functions. The function $f:U\to \mathbb {R} $ is called locally integrable if it is Lebesgue integrable over every compact subset K of U. This is a large class of functions that includes all continuous functions and all Lp space $L^{p}$ functions. The topology on ${\mathcal {D}}(U)$ is defined in such a fashion that any locally integrable function $f$ yields a continuous linear functional on ${\mathcal {D}}(U)$ – that is, an element of ${\mathcal {D}}'(U)$ – denoted here by $T_{f},$ whose value on the test function $\phi $ is given by the Lebesgue integral:
$\langle T_{f},\phi \rangle =\int _{U}f\phi \,dx.$
Conventionally, one abuses notation by identifying $T_{f}$ with $f,$ provided no confusion can arise, and thus the pairing between $T_{f}$ and $\phi $ is often written
$\langle f,\phi \rangle =\langle T_{f},\phi \rangle .$
If $f$ and $g$ are two locally integrable functions, then the associated distributions $T_{f}$ and $T_{g}$ are equal to the same element of ${\mathcal {D}}'(U)$ if and only if $f$ and $g$ are equal almost everywhere (see, for instance, Hörmander (1983, Theorem 1.2.5)). Similarly, every Radon measure $\mu $ on $U$ defines an element of ${\mathcal {D}}'(U)$ whose value on the test function $\phi $ is $ \int \phi \,d\mu .$ As above, it is conventional to abuse notation and write the pairing between a Radon measure $\mu $ and a test function $\phi $ as $\langle \mu ,\phi \rangle .$ Conversely, as shown in a theorem by Schwartz (similar to the Riesz representation theorem), every distribution which is non-negative on non-negative functions is of this form for some (positive) Radon measure.
Test functions as distributions
The test functions are themselves locally integrable, and so define distributions. The space of test functions $C_{c}^{\infty }(U)$ is sequentially dense in ${\mathcal {D}}'(U)$ with respect to the strong topology on ${\mathcal {D}}'(U).$[49] This means that for any $T\in {\mathcal {D}}'(U),$ there is a sequence of test functions, $(\phi _{i})_{i=1}^{\infty },$ that converges to $T\in {\mathcal {D}}'(U)$ (in its strong dual topology) when considered as a sequence of distributions. Or equivalently,
$\langle \phi _{i},\psi \rangle \to \langle T,\psi \rangle \qquad {\text{ for all }}\psi \in {\mathcal {D}}(U).$
Distributions with compact support
The inclusion map $\operatorname {In} :C_{c}^{\infty }(U)\to C^{\infty }(U)$ is a continuous injection whose image is dense in its codomain, so the transpose map ${}^{t}\operatorname {In} :(C^{\infty }(U))'_{b}\to {\mathcal {D}}'(U)=(C_{c}^{\infty }(U))'_{b}$ :(C^{\infty }(U))'_{b}\to {\mathcal {D}}'(U)=(C_{c}^{\infty }(U))'_{b}} is also a continuous injection. Thus the image of the transpose, denoted by ${\mathcal {E}}'(U),$ forms a space of distributions.[13]
The elements of ${\mathcal {E}}'(U)=(C^{\infty }(U))'_{b}$ can be identified as the space of distributions with compact support.[13] Explicitly, if $T$ is a distribution on U then the following are equivalent,
• $T\in {\mathcal {E}}'(U).$
• The support of $T$ is compact.
• The restriction of $T$ to $C_{c}^{\infty }(U),$ when that space is equipped with the subspace topology inherited from $C^{\infty }(U)$ (a coarser topology than the canonical LF topology), is continuous.[13]
• There is a compact subset K of U such that for every test function $\phi $ whose support is completely outside of K, we have $T(\phi )=0.$
Compactly supported distributions define continuous linear functionals on the space $C^{\infty }(U)$; recall that the topology on $C^{\infty }(U)$ is defined such that a sequence of test functions $\phi _{k}$ converges to 0 if and only if all derivatives of $\phi _{k}$ converge uniformly to 0 on every compact subset of U. Conversely, it can be shown that every continuous linear functional on this space defines a distribution of compact support. Thus compactly supported distributions can be identified with those distributions that can be extended from $C_{c}^{\infty }(U)$ to $C^{\infty }(U).$
Distributions of finite order
Let $k\in \mathbb {N} .$ The inclusion map $\operatorname {In} :C_{c}^{\infty }(U)\to C_{c}^{k}(U)$ is a continuous injection whose image is dense in its codomain, so the transpose ${}^{t}\operatorname {In} :(C_{c}^{k}(U))'_{b}\to {\mathcal {D}}'(U)=(C_{c}^{\infty }(U))'_{b}$ :(C_{c}^{k}(U))'_{b}\to {\mathcal {D}}'(U)=(C_{c}^{\infty }(U))'_{b}} is also a continuous injection. Consequently, the image of ${}^{t}\operatorname {In} ,$ denoted by ${\mathcal {D}}'^{k}(U),$ forms a space of distributions. The elements of ${\mathcal {D}}'^{k}(U)$ are the distributions of order $\,\leq k.$[16] The distributions of order $\,\leq 0,$ which are also called distributions of order 0 are exactly the distributions that are Radon measures (described above).
For $0\neq k\in \mathbb {N} ,$ a distribution of order k is a distribution of order $\,\leq k$ that is not a distribution of order $\,\leq k-1$.[16]
A distribution is said to be of finite order if there is some integer $k$ such that it is a distribution of order $\,\leq k,$ and the set of distributions of finite order is denoted by ${\mathcal {D}}'^{F}(U).$ Note that if $k\leq l$ then ${\mathcal {D}}'^{k}(U)\subseteq {\mathcal {D}}'^{l}(U)$ so that ${\mathcal {D}}'^{F}(U):=\bigcup _{n=0}^{\infty }{\mathcal {D}}'^{n}(U)$ is a vector subspace of ${\mathcal {D}}'(U)$, and furthermore, if and only if ${\mathcal {D}}'^{F}(U)={\mathcal {D}}'(U).$[16]
Structure of distributions of finite order
Every distribution with compact support in U is a distribution of finite order.[16] Indeed, every distribution in U is locally a distribution of finite order, in the following sense:[16] If V is an open and relatively compact subset of U and if $\rho _{VU}$ is the restriction mapping from U to V, then the image of ${\mathcal {D}}'(U)$ under $\rho _{VU}$ is contained in ${\mathcal {D}}'^{F}(V).$
The following is the theorem of the structure of distributions of finite order, which shows that every distribution of finite order can be written as a sum of derivatives of Radon measures:
Theorem[16] — Suppose $T\in {\mathcal {D}}'(U)$ has finite order and $I=\{p\in \mathbb {N} ^{n}:|p|\leq k\}.$ Given any open subset V of U containing the support of $T,$ there is a family of Radon measures in U, $(\mu _{p})_{p\in I},$ such that for very $p\in I,\operatorname {supp} (\mu _{p})\subseteq V$ and
$T=\sum _{|p|\leq k}\partial ^{p}\mu _{p}.$
Example. (Distributions of infinite order) Let $U:=(0,\infty )$ and for every test function $f,$ let
$Sf:=\sum _{m=1}^{\infty }(\partial ^{m}f)\left({\frac {1}{m}}\right).$
Then $S$ is a distribution of infinite order on U. Moreover, $S$ can not be extended to a distribution on $\mathbb {R} $; that is, there exists no distribution $T$ on $\mathbb {R} $ such that the restriction of $T$ to U is equal to $S.$[50]
Tempered distributions and Fourier transform
Defined below are the tempered distributions, which form a subspace of ${\mathcal {D}}'(\mathbb {R} ^{n}),$ the space of distributions on $\mathbb {R} ^{n}.$ This is a proper subspace: while every tempered distribution is a distribution and an element of ${\mathcal {D}}'(\mathbb {R} ^{n}),$ the converse is not true. Tempered distributions are useful if one studies the Fourier transform since all tempered distributions have a Fourier transform, which is not true for an arbitrary distribution in ${\mathcal {D}}'(\mathbb {R} ^{n}).$
Schwartz space
The Schwartz space ${\mathcal {S}}(\mathbb {R} ^{n})$ is the space of all smooth functions that are rapidly decreasing at infinity along with all partial derivatives. Thus $\phi :\mathbb {R} ^{n}\to \mathbb {R} $ :\mathbb {R} ^{n}\to \mathbb {R} } is in the Schwartz space provided that any derivative of $\phi ,$ multiplied with any power of $|x|,$ converges to 0 as $|x|\to \infty .$ These functions form a complete TVS with a suitably defined family of seminorms. More precisely, for any multi-indices $\alpha $ and $\beta $ define:
$p_{\alpha ,\beta }(\phi )~=~\sup _{x\in \mathbb {R} ^{n}}\left|x^{\alpha }\partial ^{\beta }\phi (x)\right|.$
Then $\phi $ is in the Schwartz space if all the values satisfy:
$p_{\alpha ,\beta }(\phi )<\infty .$
The family of seminorms $p_{\alpha ,\beta }$ defines a locally convex topology on the Schwartz space. For $n=1,$ the seminorms are, in fact, norms on the Schwartz space. One can also use the following family of seminorms to define the topology:[51]
$|f|_{m,k}=\sup _{|p|\leq m}\left(\sup _{x\in \mathbb {R} ^{n}}\left\{(1+|x|)^{k}\left|(\partial ^{\alpha }f)(x)\right|\right\}\right),\qquad k,m\in \mathbb {N} .$
Otherwise, one can define a norm on ${\mathcal {S}}(\mathbb {R} ^{n})$ via
$\|\phi \|_{k}~=~\max _{|\alpha |+|\beta |\leq k}\sup _{x\in \mathbb {R} ^{n}}\left|x^{\alpha }\partial ^{\beta }\phi (x)\right|,\qquad k\geq 1.$
The Schwartz space is a Fréchet space (that is, a complete metrizable locally convex space). Because the Fourier transform changes $\partial ^{\alpha }$ into multiplication by $x^{\alpha }$ and vice versa, this symmetry implies that the Fourier transform of a Schwartz function is also a Schwartz function.
A sequence $\{f_{i}\}$ in ${\mathcal {S}}(\mathbb {R} ^{n})$ converges to 0 in ${\mathcal {S}}(\mathbb {R} ^{n})$ if and only if the functions $(1+|x|)^{k}(\partial ^{p}f_{i})(x)$ converge to 0 uniformly in the whole of $\mathbb {R} ^{n},$ which implies that such a sequence must converge to zero in $C^{\infty }(\mathbb {R} ^{n}).$[51]
${\mathcal {D}}(\mathbb {R} ^{n})$ is dense in ${\mathcal {S}}(\mathbb {R} ^{n}).$ The subset of all analytic Schwartz functions is dense in ${\mathcal {S}}(\mathbb {R} ^{n})$ as well.[52]
The Schwartz space is nuclear and the tensor product of two maps induces a canonical surjective TVS-isomorphisms
${\mathcal {S}}(\mathbb {R} ^{m})\ {\widehat {\otimes }}\ {\mathcal {S}}(\mathbb {R} ^{n})\to {\mathcal {S}}(\mathbb {R} ^{m+n}),$
where ${\widehat {\otimes }}$ represents the completion of the injective tensor product (which in this case is identical to the completion of the projective tensor product).[53]
Tempered distributions
The inclusion map $\operatorname {In} :{\mathcal {D}}(\mathbb {R} ^{n})\to {\mathcal {S}}(\mathbb {R} ^{n})$ :{\mathcal {D}}(\mathbb {R} ^{n})\to {\mathcal {S}}(\mathbb {R} ^{n})} is a continuous injection whose image is dense in its codomain, so the transpose ${}^{t}\operatorname {In} :({\mathcal {S}}(\mathbb {R} ^{n}))'_{b}\to {\mathcal {D}}'(\mathbb {R} ^{n})$ :({\mathcal {S}}(\mathbb {R} ^{n}))'_{b}\to {\mathcal {D}}'(\mathbb {R} ^{n})} is also a continuous injection. Thus, the image of the transpose map, denoted by ${\mathcal {S}}'(\mathbb {R} ^{n}),$ forms a space of distributions.
The space ${\mathcal {S}}'(\mathbb {R} ^{n})$ is called the space of tempered distributions. It is the continuous dual space of the Schwartz space. Equivalently, a distribution $T$ is a tempered distribution if and only if
$\left({\text{ for all }}\alpha ,\beta \in \mathbb {N} ^{n}:\lim _{m\to \infty }p_{\alpha ,\beta }(\phi _{m})=0\right)\Longrightarrow \lim _{m\to \infty }T(\phi _{m})=0.$
The derivative of a tempered distribution is again a tempered distribution. Tempered distributions generalize the bounded (or slow-growing) locally integrable functions; all distributions with compact support and all square-integrable functions are tempered distributions. More generally, all functions that are products of polynomials with elements of Lp space $L^{p}(\mathbb {R} ^{n})$ for $p\geq 1$ are tempered distributions.
The tempered distributions can also be characterized as slowly growing, meaning that each derivative of $T$ grows at most as fast as some polynomial. This characterization is dual to the rapidly falling behaviour of the derivatives of a function in the Schwartz space, where each derivative of $\phi $ decays faster than every inverse power of $|x|.$ An example of a rapidly falling function is $|x|^{n}\exp(-\lambda |x|^{\beta })$ for any positive $n,\lambda ,\beta .$
Fourier transform
To study the Fourier transform, it is best to consider complex-valued test functions and complex-linear distributions. The ordinary continuous Fourier transform $F:{\mathcal {S}}(\mathbb {R} ^{n})\to {\mathcal {S}}(\mathbb {R} ^{n})$ is a TVS-automorphism of the Schwartz space, and the Fourier transform is defined to be its transpose ${}^{t}F:{\mathcal {S}}'(\mathbb {R} ^{n})\to {\mathcal {S}}'(\mathbb {R} ^{n}),$ which (abusing notation) will again be denoted by $F.$ So the Fourier transform of the tempered distribution $T$ is defined by $(FT)(\psi )=T(F\psi )$ for every Schwartz function $\psi .$ $FT$ is thus again a tempered distribution. The Fourier transform is a TVS isomorphism from the space of tempered distributions onto itself. This operation is compatible with differentiation in the sense that
$F{\dfrac {dT}{dx}}=ixFT$
and also with convolution: if $T$ is a tempered distribution and $\psi $ is a slowly increasing smooth function on $\mathbb {R} ^{n},$ $\psi T$ is again a tempered distribution and
$F(\psi T)=F\psi *FT$
is the convolution of $FT$ and $F\psi .$ In particular, the Fourier transform of the constant function equal to 1 is the $\delta $ distribution.
Expressing tempered distributions as sums of derivatives
If $T\in {\mathcal {S}}'(\mathbb {R} ^{n})$ is a tempered distribution, then there exists a constant $C>0,$ and positive integers $M$ and $N$ such that for all Schwartz functions $\phi \in {\mathcal {S}}(\mathbb {R} ^{n})$
$\langle T,\phi \rangle \leq C\sum \nolimits _{|\alpha |\leq N,|\beta |\leq M}\sup _{x\in \mathbb {R} ^{n}}\left|x^{\alpha }\partial ^{\beta }\phi (x)\right|=C\sum \nolimits _{|\alpha |\leq N,|\beta |\leq M}p_{\alpha ,\beta }(\phi ).$
This estimate, along with some techniques from functional analysis, can be used to show that there is a continuous slowly increasing function $F$ and a multi-index $\alpha $ such that
$T=\partial ^{\alpha }F.$
Restriction of distributions to compact sets
If $T\in {\mathcal {D}}'(\mathbb {R} ^{n}),$ then for any compact set $K\subseteq \mathbb {R} ^{n},$ there exists a continuous function $F$compactly supported in $\mathbb {R} ^{n}$ (possibly on a larger set than K itself) and a multi-index $\alpha $ such that $T=\partial ^{\alpha }F$ on $C_{c}^{\infty }(K).$
Using holomorphic functions as test functions
The success of the theory led to an investigation of the idea of hyperfunction, in which spaces of holomorphic functions are used as test functions. A refined theory has been developed, in particular Mikio Sato's algebraic analysis, using sheaf theory and several complex variables. This extends the range of symbolic methods that can be made into rigorous mathematics, for example, Feynman integrals.
See also
• Cauchy principal value – Method for assigning values to certain improper integrals which would otherwise be undefined
• Gelfand triple – Construction linking the study of "bound" and continuous eigenvalues in functional analysisPages displaying short descriptions of redirect targets
• Gelfand–Shilov space
• Generalized function – Objects extending the notion of functions
• Hilbert transform – Integral transform and linear operator
• Homogeneous distribution
• Laplacian of the indicator – Limit of sequence of smooth functions
• Limit of distributions
• Mollifier – more narrowlyPages displaying wikidata descriptions as a fallback
• Vague topology
Differential equations related
• Fundamental solution – Concept in the solution of linear partial differential equations
• Pseudo-differential operator – Type of differential operator
• Weak solution – Mathematical solution
Generalizations of distributions
• Colombeau algebra – commutative associative differential algebra of generalized functions into which smooth functions (but not arbitrary continuous ones) embed as a subalgebra and distributions embed as a subspacePages displaying wikidata descriptions as a fallback
• Current (mathematics) – Distributions on spaces of differential forms
• Distribution (number theory) – function on finite sets which is analogous to an integralPages displaying wikidata descriptions as a fallback
• Distribution on a linear algebraic group – Linear function satisfying a support condition
• Green's function – Impulse response of an inhomogeneous linear differential operator
• Hyperfunction – Type of generalized function
• Malgrange–Ehrenpreis theorem
Notes
1. Note that $i$ being an integer implies $i\neq \infty .$ This is sometimes expressed as $0\leq i<k+1.$ Since $\infty +1=\infty ,$ the inequality "$0\leq i<k+1$" means: $0\leq i<\infty $ if $k=\infty ,$ while if $k\neq \infty $ then it means $0\leq i\leq k.$
2. The image of the compact set $K$ under a continuous $\mathbb {R} $-valued map (for example, $x\mapsto \left|\partial ^{p}f(x)\right|$ for $x\in U$) is itself a compact, and thus bounded, subset of $\mathbb {R} .$ If $K\neq \varnothing $ then this implies that each of the functions defined above is $\mathbb {R} $-valued (that is, none of the supremums above are ever equal to $\infty $).
3. Exactly as with $C^{k}(K;U),$ the space $C^{k}(K;U')$ is defined to be the vector subspace of $C^{k}(U')$ consisting of maps with support contained in $K$ endowed with the subspace topology it inherits from $C^{k}(U')$.
4. Even though the topology of $C_{c}^{\infty }(U)$ is not metrizable, a linear functional on $C_{c}^{\infty }(U)$ is continuous if and only if it is sequentially continuous.
5. A null sequence is a sequence that converges to the origin.
6. If ${\mathcal {P}}$ is also directed under the usual function comparison then we can take the finite collection to consist of a single element.
7. The extension theorem for mappings defined from a subspace S of a topological vector space E to the topological space E itself works for non-linear mappings as well, provided they are assumed to be uniformly continuous. But, unfortunately, this is not our case, we would desire to “extend” a linear continuous mapping A from a tvs E into another tvs F, in order to obtain a linear continuous mapping from the dual E’ to the dual F’ (note the order of spaces). In general, this is not even an extension problem, because (in general) E is not necessarily a subset of its own dual E’. Moreover, It is not a classic topological transpose problem, because the transpose of A goes from F’ to E’ and not from E’ to F’. Our case needs, indeed, a new order of ideas, involving the specific topological properties of the Laurent Schwartz spaces D(U) and D’(U), together with the fundamental concept of weak (or Schwartz) adjoint of the linear continuous operator A.
8. For example, let $U=\mathbb {R} $ and take $P$ to be the ordinary derivative for functions of one real variable and assume the support of $\phi $ to be contained in the finite interval $(a,b),$ then since $\operatorname {supp} (\phi )\subseteq (a,b)$
${\begin{aligned}\int _{\mathbb {R} }\phi '(x)f(x)\,dx&=\int _{a}^{b}\phi '(x)f(x)\,dx\\&=\phi (x)f(x){\big \vert }_{a}^{b}-\int _{a}^{b}f'(x)\phi (x)\,dx\\&=\phi (b)f(b)-\phi (a)f(a)-\int _{a}^{b}f'(x)\phi (x)\,dx\\&=-\int _{a}^{b}f'(x)\phi (x)\,dx\end{aligned}}$
where the last equality is because $\phi (a)=\phi (b)=0.$
References
1. Trèves 2006, pp. 222–223.
2. Grubb 2009, p. 14
3. Trèves 2006, pp. 85–89.
4. Trèves 2006, pp. 142–149.
5. Trèves 2006, pp. 356–358.
6. Trèves 2006, pp. 131–134.
7. Rudin 1991, pp. 149–181.
8. Trèves 2006, pp. 526–534.
9. Trèves 2006, p. 357.
10. See for example Grubb 2009, p. 14.
11. Trèves 2006, pp. 245–247.
12. Trèves 2006, pp. 253–255.
13. Trèves 2006, pp. 255–257.
14. Trèves 2006, pp. 264–266.
15. Rudin 1991, p. 165.
16. Trèves 2006, pp. 258–264.
17. Rudin 1991, pp. 169–170.
18. Strichartz, Robert (1993). A Guide to Distribution Theory and Fourier Transforms. USA. p. 17.{{cite book}}: CS1 maint: location missing publisher (link)
19. Strichartz 1994, §2.3; Trèves 2006.
20. Rudin 1991, p. 180.
21. Trèves 2006, pp. 247–252.
22. Trèves 2006, p. 423.
23. Trèves 2006, p. 261.
24. Per Persson (username: md2perpe) (Jun 27, 2017). "Multiplication of two distributions whose singular supports are disjoint". Stack Exchange Network.
25. Lyons, T. (1998). "Differential equations driven by rough signals". Revista Matemática Iberoamericana: 215–310. doi:10.4171/RMI/240.
26. Hairer, Martin (2014). "A theory of regularity structures". Inventiones Mathematicae. 198 (2): 269–504. arXiv:1303.5113. Bibcode:2014InMat.198..269H. doi:10.1007/s00222-014-0505-4. S2CID 119138901.
27. See for example Hörmander 1983, Theorem 6.1.1.
28. See Hörmander 1983, Theorem 6.1.2.
29. Trèves 2006, pp. 278–283.
30. Trèves 2006, pp. 284–297.
31. See for example Rudin 1991, §6.29.
32. Trèves 2006, Chapter 27.
33. Hörmander 1983, §IV.2 proves the uniqueness of such an extension.
34. See for instance Gel'fand & Shilov 1966–1968, v. 1, pp. 103–104 and Benedetto 1997, Definition 2.5.8.
35. Trèves 2006, p. 294.
36. Folland, G.B. (1989). Harmonic Analysis in Phase Space. Princeton, NJ: Princeton University Press.
37. Horváth, John (1966). Topological Vector Spaces and Distributions. Reading, MA: Addison-Wesley Publishing Company.
38. Barros-Neto, José (1973). An Introduction to the Theory of Distributions. New York, NY: Dekker.
39. Petersen, Bent E. (1983). Introduction to the Fourier Transform and Pseudo-Differential Operators. Boston, MA: Pitman Publishing.
40. Woodward, P.M. (1953). Probability and Information Theory with Applications to Radar. Oxford, UK: Pergamon Press.
41. Trèves 2006, pp. 318–319.
42. Friedlander, F.G.; Joshi, M.S. (1998). Introduction to the Theory of Distributions. Cambridge, UK: Cambridge University Press.
43. Schwartz 1951.
44. Trèves 2006, pp. 416–419.
45. Trèves 2006, pp. 150–160.
46. Trèves 2006, pp. 240–252.
47. Trèves 2006, pp. 262–264.
48. Trèves 2006, p. 218.
49. Trèves 2006, pp. 300–304.
50. Rudin 1991, pp. 177–181.
51. Trèves 2006, pp. 92–94.
52. Trèves 2006, pp. 160.
53. Trèves 2006, p. 531.
Bibliography
• Barros-Neto, José (1973). An Introduction to the Theory of Distributions. New York, NY: Dekker.
• Benedetto, J.J. (1997), Harmonic Analysis and Applications, CRC Press.
• Folland, G.B. (1989). Harmonic Analysis in Phase Space. Princeton, NJ: Princeton University Press.
• Friedlander, F.G.; Joshi, M.S. (1998). Introduction to the Theory of Distributions. Cambridge, UK: Cambridge University Press..
• Gårding, L. (1997), Some Points of Analysis and their History, American Mathematical Society.
• Gel'fand, I.M.; Shilov, G.E. (1966–1968), Generalized functions, vol. 1–5, Academic Press.
• Grubb, G. (2009), Distributions and Operators, Springer.
• Hörmander, L. (1983), The analysis of linear partial differential operators I, Grundl. Math. Wissenschaft., vol. 256, Springer, doi:10.1007/978-3-642-96750-4, ISBN 3-540-12104-8, MR 0717035.
• Horváth, John (1966). Topological Vector Spaces and Distributions. Addison-Wesley series in mathematics. Vol. 1. Reading, MA: Addison-Wesley Publishing Company. ISBN 978-0201029857.
• Kolmogorov, Andrey; Fomin, Sergei V. (1957). Elements of the Theory of Functions and Functional Analysis. Dover Books on Mathematics. New York: Dover Books. ISBN 978-1-61427-304-2. OCLC 912495626.
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Petersen, Bent E. (1983). Introduction to the Fourier Transform and Pseudo-Differential Operators. Boston, MA: Pitman Publishing..
• Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
• Schwartz, Laurent (1954), "Sur l'impossibilité de la multiplications des distributions", C. R. Acad. Sci. Paris, 239: 847–848.
• Schwartz, Laurent (1951), Théorie des distributions, vol. 1–2, Hermann.
• Sobolev, S.L. (1936), "Méthode nouvelle à résoudre le problème de Cauchy pour les équations linéaires hyperboliques normales", Mat. Sbornik, 1: 39–72.
• Stein, Elias; Weiss, Guido (1971), Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, ISBN 0-691-08078-X.
• Strichartz, R. (1994), A Guide to Distribution Theory and Fourier Transforms, CRC Press, ISBN 0-8493-8273-4.
• Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
• Woodward, P.M. (1953). Probability and Information Theory with Applications to Radar. Oxford, UK: Pergamon Press.
Further reading
• M. J. Lighthill (1959). Introduction to Fourier Analysis and Generalised Functions. Cambridge University Press. ISBN 0-521-09128-4 (requires very little knowledge of analysis; defines distributions as limits of sequences of functions under integrals)
• V.S. Vladimirov (2002). Methods of the theory of generalized functions. Taylor & Francis. ISBN 0-415-27356-0
• Vladimirov, V.S. (2001) [1994], "Generalized function", Encyclopedia of Mathematics, EMS Press.
• Vladimirov, V.S. (2001) [1994], "Generalized functions, space of", Encyclopedia of Mathematics, EMS Press.
• Vladimirov, V.S. (2001) [1994], "Generalized function, derivative of a", Encyclopedia of Mathematics, EMS Press.
• Vladimirov, V.S. (2001) [1994], "Generalized functions, product of", Encyclopedia of Mathematics, EMS Press.
• Oberguggenberger, Michael (2001) [1994], "Generalized function algebras", Encyclopedia of Mathematics, EMS Press.
Functional analysis (topics – glossary)
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| Wikipedia |
Theory of equations
In algebra, the theory of equations is the study of algebraic equations (also called "polynomial equations"), which are equations defined by a polynomial. The main problem of the theory of equations was to know when an algebraic equation has an algebraic solution. This problem was completely solved in 1830 by Évariste Galois, by introducing what is now called Galois theory.
Not to be confused with equational theory.
Before Galois, there was no clear distinction between the "theory of equations" and "algebra". Since then algebra has been dramatically enlarged to include many new subareas, and the theory of algebraic equations receives much less attention. Thus, the term "theory of equations" is mainly used in the context of the history of mathematics, to avoid confusion between old and new meanings of "algebra".
History
Until the end of the 19th century, "theory of equations" was almost synonymous with "algebra". For a long time, the main problem was to find the solutions of a single non-linear polynomial equation in a single unknown. The fact that a complex solution always exists is the fundamental theorem of algebra, which was proved only at the beginning of the 19th century and does not have a purely algebraic proof. Nevertheless, the main concern of the algebraists was to solve in terms of radicals, that is to express the solutions by a formula which is built with the four operations of arithmetics and with nth roots. This was done up to degree four during the 16th century. Scipione del Ferro and Niccolò Fontana Tartaglia discovered solutions for cubic equations. Gerolamo Cardano published them in his 1545 book Ars Magna, together with a solution for the quartic equations, discovered by his student Lodovico Ferrari. In 1572 Rafael Bombelli published his L'Algebra in which he showed how to deal with the imaginary quantities that could appear in Cardano's formula for solving cubic equations.
The case of higher degrees remained open until the 19th century, when Paolo Ruffini gave an incomplete proof in 1799 that some fifth degree equations cannot be solved in radicals followed by Niels Henrik Abel's complete proof in 1824 (now known as the Abel–Ruffini theorem). Évariste Galois later introduced a theory (presently called Galois theory) to decide which equations are solvable by radicals.
Further problems
Other classical problems of the theory of equations are the following:
• Linear equations: this problem was solved during antiquity.
• Simultaneous linear equations: The general theoretical solution was provided by Gabriel Cramer in 1750. However devising efficient methods (algorithms) to solve these systems remains an active subject of research now called linear algebra.
• Finding the integer solutions of an equation or of a system of equations. These problems are now called Diophantine equations, which are considered a part of number theory (see also integer programming).
• Systems of polynomial equations: Because of their difficulty, these systems, with few exceptions, have been studied only since the second part of the 19th century. They have led to the development of algebraic geometry.
See also
• Root-finding algorithm
• Properties of polynomial roots
• Quintic function
Further reading
• Uspensky, James Victor, Theory of Equations (McGraw-Hill), 1963
• Dickson, Leonard E., Elementary Theory of Equations (Internet Archive), originally 1914
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Invariant theory
Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit description of polynomial functions that do not change, or are invariant, under the transformations from a given linear group. For example, if we consider the action of the special linear group SLn on the space of n by n matrices by left multiplication, then the determinant is an invariant of this action because the determinant of A X equals the determinant of X, when A is in SLn.
Introduction
Let $G$ be a group, and $V$ a finite-dimensional vector space over a field $k$ (which in classical invariant theory was usually assumed to be the complex numbers). A representation of $G$ in $V$ is a group homomorphism $\pi :G\to GL(V)$, which induces a group action of $G$ on $V$. If $k[V]$ is the space of polynomial functions on $V$, then the group action of $G$ on $V$ produces an action on $k[V]$ by the following formula:
$(g\cdot f)(x):=f(g^{-1}(x))\qquad \forall x\in V,g\in G,f\in k[V].$
With this action it is natural to consider the subspace of all polynomial functions which are invariant under this group action, in other words the set of polynomials such that $g\cdot f=f$ for all $g\in G$. This space of invariant polynomials is denoted $k[V]^{G}$.
First problem of invariant theory:[1] Is $k[V]^{G}$ a finitely generated algebra over $k$?
For example, if $G=SL_{n}$ and $V=M_{n}$ the space of square matrices, and the action of $G$ on $V$ is given by left multiplication, then $k[V]^{G}$ is isomorphic to a polynomial algebra in one variable, generated by the determinant. In other words, in this case, every invariant polynomial is a linear combination of powers of the determinant polynomial. So in this case, $k[V]^{G}$ is finitely generated over $k$.
If the answer is yes, then the next question is to find a minimal basis, and ask whether the module of polynomial relations between the basis elements (known as the syzygies) is finitely generated over $k[V]$.
Invariant theory of finite groups has intimate connections with Galois theory. One of the first major results was the main theorem on the symmetric functions that described the invariants of the symmetric group $S_{n}$ acting on the polynomial ring $R[x_{1},\ldots ,x_{n}$] by permutations of the variables. More generally, the Chevalley–Shephard–Todd theorem characterizes finite groups whose algebra of invariants is a polynomial ring. Modern research in invariant theory of finite groups emphasizes "effective" results, such as explicit bounds on the degrees of the generators. The case of positive characteristic, ideologically close to modular representation theory, is an area of active study, with links to algebraic topology.
Invariant theory of infinite groups is inextricably linked with the development of linear algebra, especially, the theories of quadratic forms and determinants. Another subject with strong mutual influence was projective geometry, where invariant theory was expected to play a major role in organizing the material. One of the highlights of this relationship is the symbolic method. Representation theory of semisimple Lie groups has its roots in invariant theory.
David Hilbert's work on the question of the finite generation of the algebra of invariants (1890) resulted in the creation of a new mathematical discipline, abstract algebra. A later paper of Hilbert (1893) dealt with the same questions in more constructive and geometric ways, but remained virtually unknown until David Mumford brought these ideas back to life in the 1960s, in a considerably more general and modern form, in his geometric invariant theory. In large measure due to the influence of Mumford, the subject of invariant theory is seen to encompass the theory of actions of linear algebraic groups on affine and projective varieties. A distinct strand of invariant theory, going back to the classical constructive and combinatorial methods of the nineteenth century, has been developed by Gian-Carlo Rota and his school. A prominent example of this circle of ideas is given by the theory of standard monomials.
Examples
Simple examples of invariant theory come from computing the invariant monomials from a group action. For example, consider the $\mathbb {Z} /2\mathbb {Z} $-action on $\mathbb {C} [x,y]$ sending
${\begin{aligned}x\mapsto -x&&y\mapsto -y\end{aligned}}$
Then, since $x^{2},xy,y^{2}$ are the lowest degree monomials which are invariant, we have that
$\mathbb {C} [x,y]^{\mathbb {Z} /2\mathbb {Z} }\cong \mathbb {C} [x^{2},xy,y^{2}]\cong {\frac {\mathbb {C} [a,b,c]}{(ac-b^{2})}}$
This example forms the basis for doing many computations.
The nineteenth-century origins
The theory of invariants came into existence about the middle of the nineteenth century somewhat like Minerva: a grown-up virgin, mailed in the shining armor of algebra, she sprang forth from Cayley's Jovian head.
Weyl (1939b, p.489)
Cayley first established invariant theory in his "On the Theory of Linear Transformations (1845)." In the opening of his paper, Cayley credits an 1841 paper of George Boole, "investigations were suggested to me by a very elegant paper on the same subject... by Mr Boole." (Boole's paper was Exposition of a General Theory of Linear Transformations, Cambridge Mathematical Journal.)[2]
Classically, the term "invariant theory" refers to the study of invariant algebraic forms (equivalently, symmetric tensors) for the action of linear transformations. This was a major field of study in the latter part of the nineteenth century. Current theories relating to the symmetric group and symmetric functions, commutative algebra, moduli spaces and the representations of Lie groups are rooted in this area.
In greater detail, given a finite-dimensional vector space V of dimension n we can consider the symmetric algebra S(Sr(V)) of the polynomials of degree r over V, and the action on it of GL(V). It is actually more accurate to consider the relative invariants of GL(V), or representations of SL(V), if we are going to speak of invariants: that is because a scalar multiple of the identity will act on a tensor of rank r in S(V) through the r-th power 'weight' of the scalar. The point is then to define the subalgebra of invariants I(Sr(V)) for the action. We are, in classical language, looking at invariants of n-ary r-ics, where n is the dimension of V. (This is not the same as finding invariants of GL(V) on S(V); this is an uninteresting problem as the only such invariants are constants.) The case that was most studied was invariants of binary forms where n = 2.
Other work included that of Felix Klein in computing the invariant rings of finite group actions on $\mathbf {C} ^{2}$ (the binary polyhedral groups, classified by the ADE classification); these are the coordinate rings of du Val singularities.
Like the Arabian phoenix rising out of its ashes, the theory of invariants, pronounced dead at the turn of the century, is once again at the forefront of mathematics.
Kung & Rota (1984, p.27)
The work of David Hilbert, proving that I(V) was finitely presented in many cases, almost put an end to classical invariant theory for several decades, though the classical epoch in the subject continued to the final publications of Alfred Young, more than 50 years later. Explicit calculations for particular purposes have been known in modern times (for example Shioda, with the binary octavics).
Hilbert's theorems
Hilbert (1890) proved that if V is a finite-dimensional representation of the complex algebraic group G = SLn(C) then the ring of invariants of G acting on the ring of polynomials R = S(V) is finitely generated. His proof used the Reynolds operator ρ from R to RG with the properties
• ρ(1) = 1
• ρ(a + b) = ρ(a) + ρ(b)
• ρ(ab) = a ρ(b) whenever a is an invariant.
Hilbert constructed the Reynolds operator explicitly using Cayley's omega process Ω, though now it is more common to construct ρ indirectly as follows: for compact groups G, the Reynolds operator is given by taking the average over G, and non-compact reductive groups can be reduced to the case of compact groups using Weyl's unitarian trick.
Given the Reynolds operator, Hilbert's theorem is proved as follows. The ring R is a polynomial ring so is graded by degrees, and the ideal I is defined to be the ideal generated by the homogeneous invariants of positive degrees. By Hilbert's basis theorem the ideal I is finitely generated (as an ideal). Hence, I is finitely generated by finitely many invariants of G (because if we are given any – possibly infinite – subset S that generates a finitely generated ideal I, then I is already generated by some finite subset of S). Let i1,...,in be a finite set of invariants of G generating I (as an ideal). The key idea is to show that these generate the ring RG of invariants. Suppose that x is some homogeneous invariant of degree d > 0. Then
x = a1i1 + ... + anin
for some aj in the ring R because x is in the ideal I. We can assume that aj is homogeneous of degree d − deg ij for every j (otherwise, we replace aj by its homogeneous component of degree d − deg ij; if we do this for every j, the equation x = a1i1 + ... + anin will remain valid). Now, applying the Reynolds operator to x = a1i1 + ... + anin gives
x = ρ(a1)i1 + ... + ρ(an)in
We are now going to show that x lies in the R-algebra generated by i1,...,in.
First, let us do this in the case when the elements ρ(ak) all have degree less than d. In this case, they are all in the R-algebra generated by i1,...,in (by our induction assumption). Therefore, x is also in this R-algebra (since x = ρ(a1)i1 + ... + ρ(an)in).
In the general case, we cannot be sure that the elements ρ(ak) all have degree less than d. But we can replace each ρ(ak) by its homogeneous component of degree d − deg ij. As a result, these modified ρ(ak) are still G-invariants (because every homogeneous component of a G-invariant is a G-invariant) and have degree less than d (since deg ik > 0). The equation x = ρ(a1)i1 + ... + ρ(an)in still holds for our modified ρ(ak), so we can again conclude that x lies in the R-algebra generated by i1,...,in.
Hence, by induction on the degree, all elements of RG are in the R-algebra generated by i1,...,in.
Geometric invariant theory
The modern formulation of geometric invariant theory is due to David Mumford, and emphasizes the construction of a quotient by the group action that should capture invariant information through its coordinate ring. It is a subtle theory, in that success is obtained by excluding some 'bad' orbits and identifying others with 'good' orbits. In a separate development the symbolic method of invariant theory, an apparently heuristic combinatorial notation, has been rehabilitated.
One motivation was to construct moduli spaces in algebraic geometry as quotients of schemes parametrizing marked objects. In the 1970s and 1980s the theory developed interactions with symplectic geometry and equivariant topology, and was used to construct moduli spaces of objects in differential geometry, such as instantons and monopoles.
See also
• Gram's theorem
• Representation theory of finite groups
• Molien series
• Invariant (mathematics)
• Invariant of a binary form
• Invariant measure
• First and second fundamental theorems of invariant theory
References
1. Borel, Armand (2001). Essays in the History of Lie groups and algebraic groups. Vol. History of Mathematics, Vol. 21. American mathematical society and London mathematical society. ISBN 978-0821802885.
2. Wolfson, Paul R. (2008). "George Boole and the origins of invariant theory". Historia Mathematica. Elsevier BV. 35 (1): 37–46. doi:10.1016/j.hm.2007.06.004. ISSN 0315-0860.
• Dieudonné, Jean A.; Carrell, James B. (1970), "Invariant theory, old and new", Advances in Mathematics, 4: 1–80, doi:10.1016/0001-8708(70)90015-0, ISSN 0001-8708, MR 0255525 Reprinted as Dieudonné, Jean A.; Carrell, James B. (1971), "Invariant theory, old and new", Advances in Mathematics, Boston, MA: Academic Press, 4: 1–80, doi:10.1016/0001-8708(70)90015-0, ISBN 978-0-12-215540-6, MR 0279102
• Dolgachev, Igor (2003), Lectures on invariant theory, London Mathematical Society Lecture Note Series, vol. 296, Cambridge University Press, doi:10.1017/CBO9780511615436, ISBN 978-0-521-52548-0, MR 2004511
• Grace, J. H.; Young, Alfred (1903), The algebra of invariants, Cambridge: Cambridge University Press
• Grosshans, Frank D. (1997), Algebraic homogeneous spaces and invariant theory, New York: Springer, ISBN 3-540-63628-5
• Kung, Joseph P. S.; Rota, Gian-Carlo (1984), "The invariant theory of binary forms", Bulletin of the American Mathematical Society, New Series, 10 (1): 27–85, doi:10.1090/S0273-0979-1984-15188-7, ISSN 0002-9904, MR 0722856
• Hilbert, David (1890), "Ueber die Theorie der algebraischen Formen", Mathematische Annalen, 36 (4): 473–534, doi:10.1007/BF01208503, ISSN 0025-5831
• Hilbert, D. (1893), "Über die vollen Invariantensysteme (On Full Invariant Systems)", Math. Annalen, 42 (3): 313, doi:10.1007/BF01444162
• Neusel, Mara D.; Smith, Larry (2002), Invariant Theory of Finite Groups, Providence, RI: American Mathematical Society, ISBN 0-8218-2916-5 A recent resource for learning about modular invariants of finite groups.
• Olver, Peter J. (1999), Classical invariant theory, Cambridge: Cambridge University Press, ISBN 0-521-55821-2 An undergraduate level introduction to the classical theory of invariants of binary forms, including the Omega process starting at page 87.
• Popov, V.L. (2001) [1994], "Invariants, theory of", Encyclopedia of Mathematics, EMS Press
• Springer, T. A. (1977), Invariant Theory, New York: Springer, ISBN 0-387-08242-5 An older but still useful survey.
• Sturmfels, Bernd (1993), Algorithms in Invariant Theory, New York: Springer, ISBN 0-387-82445-6 A beautiful introduction to the theory of invariants of finite groups and techniques for computing them using Gröbner bases.
• Weyl, Hermann (1939), The Classical Groups. Their Invariants and Representations, Princeton University Press, ISBN 978-0-691-05756-9, MR 0000255
• Weyl, Hermann (1939b), "Invariants", Duke Mathematical Journal, 5 (3): 489–502, doi:10.1215/S0012-7094-39-00540-5, ISSN 0012-7094, MR 0000030
External links
• H. Kraft, C. Procesi, Classical Invariant Theory, a Primer
• V. L. Popov, E. B. Vinberg, ``Invariant Theory", in Algebraic geometry. IV. Encyclopaedia of Mathematical Sciences, 55 (translated from 1989 Russian edition) Springer-Verlag, Berlin, 1994; vi+284 pp.; ISBN 3-540-54682-0
| Wikipedia |
Theory of pure equality
In mathematical logic the theory of pure equality is a first-order theory. It has a signature consisting of only the equality relation symbol, and includes no non-logical axioms at all.[1]
This theory is consistent but incomplete, as a non-empty set with the usual equality relation provides an interpretation making certain sentences true. It is an example of a decidable theory and is a fragment of more expressive decidable theories, including monadic class of first-order logic (which also admits unary predicates and is, via Skolem normal form, related[2] to set constraints in program analysis) and monadic second-order theory of a pure set (which additionally permits quantification over predicates and whose signature extends to monadic second-order logic of k successors[3]).
Historical significance
The theory of pure equality was proven to be decidable by Leopold Löwenheim in 1915.
If an additional axiom is added saying that there are exactly m objects for a fixed natural number m, or an axiom scheme is added saying that there are infinitely many objects, then the resulting theory is complete.
Definition as FOL theory
The pure theory of equality contains formulas of first-order logic with equality, where the only predicate symbol is equality itself and there are no function symbols.
Consequently, the only form of an atomic formula is $x=y$ where $x,y$ are (possibly identical) variables. Syntactically more complex formulas can be built as usual in first-order logic using propositional connectives such as $\land ,\lor ,\lnot $ and quantifiers $\forall ,\exists $.
A first-order structure with equality interpreting such formulas is just a set with the equality relation on its elements. Isomorphic structures with such signature are thus sets of the same cardinality. Cardinality thus uniquely determines whether a sentence is true in the structure.
Example
The following formula:
$\forall x,y,z.\ (z=x\lor z=y)$
is true when the set interpreting the formula has at most two elements.
Expressive power
Further information: Spectrum of a sentence and Counting quantification
This theory can express the fact that the domain of interpretation has at least $k$ elements for a constant $k$ using the formula that we will denote $D_{k}$ for a constant $k$:
$\exists x_{1},\ldots ,x_{k}.\ \bigwedge _{1\leq i<j\leq k}x_{i}\neq x_{j}$
Using negation, it can then express that the domain has more than $k$ elements. More generally, it can constrain the domain to have a given finite set of finite cardinalities.
Definition of the theory
In terms of models, pure theory of equality can be defined as set of those first-order sentences that are true for all (non-empty) sets, regardless of their cardinality. For example, the following is a valid formula in the pure theory of equality:
$(\forall x,y,z.\ (z=x\lor z=y))\lor (\exists p,q,r.\ p\neq q\land p\neq r\land q\neq r)$
By completeness of first-order logic, all valid formulas are provable using axioms of first-order logic and the equality axioms (see also equational logic).
Decidability
Decidability can be shown by establishing that every sentence can be shown equivalent to a propositional combination of formulas about the cardinality of the domain.[4]
To obtain quantifier elimination, one can expand the signature of the language while preserving the definable relations (a technique that works more generally for monadic second-order formulas). Another approach to establish decidability is to use Ehrenfeucht–Fraïssé games.
See also
• List of first-order theories
• Equational logic
• Free theory
References
1. Monk, J. Donald (1976). "Chapter 13: Some Decidable Theories". Mathematical Logic. Graduate Texts in Mathematics. Berlin, New York: Springer-Verlag. p. 240. ISBN 978-0-387-90170-1.
2. Bachmair, L.; Ganzinger, H.; Waldmann, U. (1993). "Set constraints are the monadic class". [1993] Proceedings Eighth Annual IEEE Symposium on Logic in Computer Science. pp. 75–83. doi:10.1109/LICS.1993.287598. hdl:11858/00-001M-0000-0014-B322-4. ISBN 0-8186-3140-6. S2CID 2351050.
3. Rabin, Michael O. (July 1969). "Decidability of Second-Order Theories and Automata on Infinite Trees". Transactions of the American Mathematical Society. 141: 1–35. doi:10.2307/1995086. JSTOR 1995086.
4. Monk, J. Donald (1976). "Chapter 13: Some Decidable Theories, Lemma 13.11". Mathematical Logic. Graduate Texts in Mathematics. Berlin, New York: Springer-Verlag. p. 241. ISBN 978-0-387-90170-1.
| Wikipedia |
Existential quantification
In predicate logic, an existential quantification is a type of quantifier, a logical constant which is interpreted as "there exists", "there is at least one", or "for some". It is usually denoted by the logical operator symbol ∃, which, when used together with a predicate variable, is called an existential quantifier ("∃x" or "∃(x)" or "(∃x)"[1]). Existential quantification is distinct from universal quantification ("for all"), which asserts that the property or relation holds for all members of the domain.[2][3] Some sources use the term existentialization to refer to existential quantification.[4]
Existential quantification
TypeQuantifier
FieldMathematical logic
Statement$\exists xP(x)$ is true when $P(x)$ is true for at least one value of $x$.
Symbolic statement$\exists xP(x)$
Basics
Consider a formula that states that some natural number multiplied by itself is 25.
0·0 = 25, or 1·1 = 25, or 2·2 = 25, or 3·3 = 25, ...
This would seem to be a logical disjunction because of the repeated use of "or". However, the ellipses make this impossible to integrate and to interpret it as a disjunction in formal logic. Instead, the statement could be rephrased more formally as
For some natural number n, n·n = 25.
This is a single statement using existential quantification.
This statement is more precise than the original one, since the phrase "and so on" does not necessarily include all natural numbers and exclude everything else. And since the domain was not stated explicitly, the phrase could not be interpreted formally. In the quantified statement, however, the natural numbers are mentioned explicitly.
This particular example is true, because 5 is a natural number, and when we substitute 5 for n, we produce "5·5 = 25", which is true. It does not matter that "n·n = 25" is only true for a single natural number, 5; even the existence of a single solution is enough to prove this existential quantification as being true. In contrast, "For some even number n, n·n = 25" is false, because there are no even solutions.
The domain of discourse, which specifies the values the variable n is allowed to take, is therefore critical to a statement's trueness or falseness. Logical conjunctions are used to restrict the domain of discourse to fulfill a given predicate. For example:
For some positive odd number n, n·n = 25
is logically equivalent to
For some natural number n, n is odd and n·n = 25.
Here, "and" is the logical conjunction.
In symbolic logic, "∃" (a rotated letter "E", in a sans-serif font) is used to indicate existential quantification.[5] Thus, if P(a, b, c) is the predicate "a·b = c", and $\mathbb {N} $ is the set of natural numbers, then
$\exists {n}{\in }\mathbb {N} \,P(n,n,25)$
is the (true) statement
For some natural number n, n·n = 25.
Similarly, if Q(n) is the predicate "n is even", then
$\exists {n}{\in }\mathbb {N} \,{\big (}Q(n)\;\!\;\!{\wedge }\;\!\;\!P(n,n,25){\big )}$
is the (false) statement
For some natural number n, n is even and n·n = 25.
In mathematics, the proof of a "some" statement may be achieved either by a constructive proof, which exhibits an object satisfying the "some" statement, or by a nonconstructive proof, which shows that there must be such an object but without exhibiting one.
Properties
Negation
A quantified propositional function is a statement; thus, like statements, quantified functions can be negated. The $\lnot \ $ symbol is used to denote negation.
For example, if P(x) is the predicate "x is greater than 0 and less than 1", then, for a domain of discourse X of all natural numbers, the existential quantification "There exists a natural number x which is greater than 0 and less than 1" can be symbolically stated as:
$\exists {x}{\in }\mathbf {X} \,P(x)$
This can be demonstrated to be false. Truthfully, it must be said, "It is not the case that there is a natural number x that is greater than 0 and less than 1", or, symbolically:
$\lnot \ \exists {x}{\in }\mathbf {X} \,P(x)$.
If there is no element of the domain of discourse for which the statement is true, then it must be false for all of those elements. That is, the negation of
$\exists {x}{\in }\mathbf {X} \,P(x)$
is logically equivalent to "For any natural number x, x is not greater than 0 and less than 1", or:
$\forall {x}{\in }\mathbf {X} \,\lnot P(x)$
Generally, then, the negation of a propositional function's existential quantification is a universal quantification of that propositional function's negation; symbolically,
$\lnot \ \exists {x}{\in }\mathbf {X} \,P(x)\equiv \ \forall {x}{\in }\mathbf {X} \,\lnot P(x)$
(This is a generalization of De Morgan's laws to predicate logic.)
A common error is stating "all persons are not married" (i.e., "there exists no person who is married"), when "not all persons are married" (i.e., "there exists a person who is not married") is intended:
$\lnot \ \exists {x}{\in }\mathbf {X} \,P(x)\equiv \ \forall {x}{\in }\mathbf {X} \,\lnot P(x)\not \equiv \ \lnot \ \forall {x}{\in }\mathbf {X} \,P(x)\equiv \ \exists {x}{\in }\mathbf {X} \,\lnot P(x)$
Negation is also expressible through a statement of "for no", as opposed to "for some":
$\nexists {x}{\in }\mathbf {X} \,P(x)\equiv \lnot \ \exists {x}{\in }\mathbf {X} \,P(x)$
Unlike the universal quantifier, the existential quantifier distributes over logical disjunctions:
$\exists {x}{\in }\mathbf {X} \,P(x)\lor Q(x)\to \ (\exists {x}{\in }\mathbf {X} \,P(x)\lor \exists {x}{\in }\mathbf {X} \,Q(x))$
Rules of inference
Transformation rules
Propositional calculus
Rules of inference
• Implication introduction / elimination (modus ponens)
• Biconditional introduction / elimination
• Conjunction introduction / elimination
• Disjunction introduction / elimination
• Disjunctive / hypothetical syllogism
• Constructive / destructive dilemma
• Absorption / modus tollens / modus ponendo tollens
• Negation introduction
Rules of replacement
• Associativity
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• Distributivity
• Double negation
• De Morgan's laws
• Transposition
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• Tautology
Predicate logic
Rules of inference
• Universal generalization / instantiation
• Existential generalization / instantiation
A rule of inference is a rule justifying a logical step from hypothesis to conclusion. There are several rules of inference which utilize the existential quantifier.
Existential introduction (∃I) concludes that, if the propositional function is known to be true for a particular element of the domain of discourse, then it must be true that there exists an element for which the proposition function is true. Symbolically,
$P(a)\to \ \exists {x}{\in }\mathbf {X} \,P(x)$
Existential instantiation, when conducted in a Fitch style deduction, proceeds by entering a new sub-derivation while substituting an existentially quantified variable for a subject—which does not appear within any active sub-derivation. If a conclusion can be reached within this sub-derivation in which the substituted subject does not appear, then one can exit that sub-derivation with that conclusion. The reasoning behind existential elimination (∃E) is as follows: If it is given that there exists an element for which the proposition function is true, and if a conclusion can be reached by giving that element an arbitrary name, that conclusion is necessarily true, as long as it does not contain the name. Symbolically, for an arbitrary c and for a proposition Q in which c does not appear:
$\exists {x}{\in }\mathbf {X} \,P(x)\to \ ((P(c)\to \ Q)\to \ Q)$
$P(c)\to \ Q$ must be true for all values of c over the same domain X; else, the logic does not follow: If c is not arbitrary, and is instead a specific element of the domain of discourse, then stating P(c) might unjustifiably give more information about that object.
The empty set
The formula $\exists {x}{\in }\varnothing \,P(x)$ is always false, regardless of P(x). This is because $\varnothing $ denotes the empty set, and no x of any description – let alone an x fulfilling a given predicate P(x) – exist in the empty set. See also Vacuous truth for more information.
As adjoint
Main article: Universal quantification § As adjoint
In category theory and the theory of elementary topoi, the existential quantifier can be understood as the left adjoint of a functor between power sets, the inverse image functor of a function between sets; likewise, the universal quantifier is the right adjoint.[6]
Encoding
In Unicode and HTML, symbols are encoded U+2203 ∃ THERE EXISTS (∃, ∃ · as a mathematical symbol) and U+2204 ∄ THERE DOES NOT EXIST (∄, ∄, ∄).
In TeX, the symbol is produced with "\exists".
Origin
The symbol's first usage is thought to be by Giuseppe Peano in his book of mathematical logic and notation Formulario Mathematico 0f 1896. Afterwards, Bertrand Russell popularised its use as the existential quantifier. Through his research in set theory, Peano also introduced the symbols $\cap $ and $\cup $ to each denote the intersection and union of sets.[7]
See also
• Existential clause
• Existence theorem
• First-order logic
• Lindström quantifier
• List of logic symbols – for the unicode symbol ∃
• Quantifier variance
• Uniqueness quantification
Notes
1. Bergmann, Merrie (2014). The Logic Book. McGraw Hill. ISBN 978-0-07-803841-9.
2. "Predicates and Quantifiers". www.csm.ornl.gov. Retrieved 2020-09-04.
3. "1.2 Quantifiers". www.whitman.edu. Retrieved 2020-09-04.
4. Allen, Colin; Hand, Michael (2001). Logic Primer. MIT Press. ISBN 0262303965.
5. This symbol is also known as the existential operator. It is sometimes represented with V.
6. Saunders Mac Lane, Ieke Moerdijk, (1992): Sheaves in Geometry and Logic Springer-Verlag ISBN 0-387-97710-4. See p. 58.
7. Webb, Stephen (2018). Clash of Symbols. Cham: Springer International Publishing. doi:10.1007/978-3-319-71350-2. ISBN 978-3-319-71349-6.
References
• Hinman, P. (2005). Fundamentals of Mathematical Logic. A K Peters. ISBN 1-56881-262-0.
Common logical symbols
∧ or &
and
∨
or
¬ or ~
not
→
implies
⊃
implies,
superset
↔ or ≡
iff
|
nand
∀
universal
quantification
∃
existential
quantification
⊤
true,
tautology
⊥
false,
contradiction
⊢
entails,
proves
⊨
entails,
therefore
∴
therefore
∵
because
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| Wikipedia |
Therese Biedl
Therese Charlotte Biedl is an Austrian computer scientist known for her research in computational geometry and graph drawing. Currently she is a professor at the University of Waterloo in Canada.[1]
Therese Biedl
NationalityAustrian
Alma materRutgers University
Known forComputational geometry, planar graphs
AwardsRoss & Muriel Cheriton Faculty Fellow, 2011
Websitehttps://cs.uwaterloo.ca/~biedl/
Education
Biedl received her Diploma in Mathematics at the Technical University of Berlin, graduating in 1996 and earned a Ph.D. from Rutgers University in 1997 under the supervision of Endre Boros.[1][2][3]
Research
Biedl's research is in developing algorithms related to graphs and geometry. Planar graphs are graphs that can be drawn without crossings. Biedl develops algorithms that minimize or approximate the area and the height of such drawings.[A] With Alam, Felsner, Gerasch, Kaufmann, and Kobourov, Biedl found provably optimal linear time algorithms for proportional contact representation of a maximal planar graph.[C]
Awards
Biedl was named a Ross & Muriel Cheriton Faculty Fellow in 2011, a recognition of the reach and importance of her scholarly works.[4]
Selected publications
A.
Biedl, Therese (2014). "On Area-Optimal Planar Graph Drawings". Automata, Languages, and Programming: 41st International Colloquium, ICALP 2014, Copenhagen, Denmark, July 8–11, 2014, Proceedings, Part I. Lecture Notes in Computer Science. Vol. 8572. Springer. pp. 198–210. doi:10.1007/978-3-662-43948-7_17.
B.
Alam, Md Jawaherul; Biedl, Therese; Felsner, Stefan; Kaufmann, Michael; Kobourov, Stephen G.; Ueckerdt, Torsten (1 October 2013). "Computing Cartograms with Optimal Complexity". Discrete & Computational Geometry. 50 (3): 784–810. arXiv:1201.0066. doi:10.1007/s00454-013-9521-1. S2CID 47049050.
C.
Alam, Muhammad Jawaherul; Biedl, Therese; Felsner, Stefan; Gerasch, Andreas; Kaufmann, Michael; Kobourov, Stephen G. (2011). "Linear-Time Algorithms for Hole-Free Rectilinear Proportional Contact Graph Representations". Algorithms and Computation: 22nd International Symposium, ISAAC 2011, Yokohama, Japan, December 5–8, 2011, Proceedings. Lecture Notes in Computer Science. Vol. 7074. Springer. pp. 281–291. doi:10.1007/978-3-642-25591-5_30.
D.
Biedl, Therese (2002). "Drawing outer-planar graphs in O(n log n) area". Graph Drawing:10th International Symposium, GD 2002, Irvine, CA, USA, August 26–28, 2002, Revised Papers. Lecture Notes in Computer Science. Vol. 2528. Springer. pp. 54–65. doi:10.1007/3-540-36151-0_6. MR 2063411.
E.
Biedl, Therese C.; Bose, Prosenjit; Demaine, Erik D.; Lubiw, Anna (2000). "Efficient Algorithms for Petersen's Matching Theorem". Journal of Algorithms. 38 (1): 110–134. doi:10.1006/jagm.2000.1132. S2CID 287038.
F.
Biedl, Therese; Kant, Goos (1998). "A better heuristic for orthogonal graph drawings". Computational Geometry. 9 (3): 159–180. doi:10.1016/s0925-7721(97)00026-6. hdl:1874/2715.
References
1. Faculty profile, Univ. of Waterloo, retrieved 2017-12-08.
2. Therese Biedl at the Mathematics Genealogy Project
3. Curriculum Vitae, Univ. of Waterloo, retrieved 2017-12-08.
4. "Ross & Muriel Cheriton Faculty Fellowship | Cheriton School of Computer Science". Cheriton School of Computer Science. 10 February 2017. Retrieved 9 December 2017.
External links
• Home page at University of Waterloo
• Therese Biedl publications indexed by Google Scholar
Authority control: Academics
• Association for Computing Machinery
• DBLP
• Google Scholar
• MathSciNet
• Mathematics Genealogy Project
| Wikipedia |
Theta*
Theta* is an any-angle path planning algorithm that is based on the A* search algorithm. It can find near-optimal paths with run times comparable to those of A*.[1]
Description
For the simplest version of Theta*, the main loop is much the same as that of A*. The only difference is the ${\text{update}}\_{\text{vertex}}()$function. Compared to A*, the parent of a node in Theta* does not have to be a neighbour of the node as long as there is a line-of-sight between the two nodes.
Pseudocode
Adapted from.[2]
function theta*(start, goal)
// This main loop is the same as A*
gScore(start) := 0
parent(start) := start
// Initializing open and closed sets. The open set is initialized
// with the start node and an initial cost
open := {}
open.insert(start, gScore(start) + heuristic(start))
// gScore(node) is the current shortest distance from the start node to node
// heuristic(node) is the estimated distance of node from the goal node
// there are many options for the heuristic such as Euclidean or Manhattan
closed := {}
while open is not empty
s := open.pop()
if s = goal
return reconstruct_path(s)
closed.push(s)
for each neighbor of s
// Loop through each immediate neighbor of s
if neighbor not in closed
if neighbor not in open
// Initialize values for neighbor if it is
// not already in the open list
gScore(neighbor) := infinity
parent(neighbor) := Null
update_vertex(s, neighbor)
return Null
function update_vertex(s, neighbor)
// This part of the algorithm is the main difference between A* and Theta*
if line_of_sight(parent(s), neighbor)
// If there is line-of-sight between parent(s) and neighbor
// then ignore s and use the path from parent(s) to neighbor
if gScore(parent(s)) + c(parent(s), neighbor) < gScore(neighbor)
// c(s, neighbor) is the Euclidean distance from s to neighbor
gScore(neighbor) := gScore(parent(s)) + c(parent(s), neighbor)
parent(neighbor) := parent(s)
if neighbor in open
open.remove(neighbor)
open.insert(neighbor, gScore(neighbor) + heuristic(neighbor))
else
// If the length of the path from start to s and from s to
// neighbor is shorter than the shortest currently known distance
// from start to neighbor, then update node with the new distance
if gScore(s) + c(s, neighbor) < gScore(neighbor)
gScore(neighbor) := gScore(s) + c(s, neighbor)
parent(neighbor) := s
if neighbor in open
open.remove(neighbor)
open.insert(neighbor, gScore(neighbor) + heuristic(neighbor))
function reconstruct_path(s)
total_path = {s}
// This will recursively reconstruct the path from the goal node
// until the start node is reached
if parent(s) != s
total_path.push(reconstruct_path(parent(s)))
else
return total_path
Variants
The following variants of the algorithm exist:
• Lazy Theta*[3] – Node expansions are delayed, resulting in fewer line-of-sight checks
• Incremental Phi* – A modification of Theta* that allows for dynamic path planning similar to D*
See also
• Any-angle path planning
• A*
References
1. An Empirical Comparison of Any-Angle Path-Planning Algorithms
2. Theta*: Any-Angle Path Planning of Grids
3. http://idm-lab.org/bib/abstracts/papers/aaai10b.pdf
| Wikipedia |
θ10
In representation theory, a branch of mathematics, θ10 is a cuspidal unipotent complex irreducible representation of the symplectic group Sp4 over a finite, local, or global field.
Srinivasan (1968) introduced θ10 for the symplectic group Sp4(Fq) over a finite field Fq of order q, and showed that in this case it is q(q – 1)2/2-dimensional. The subscript 10 in θ10 is a historical accident that has stuck: Srinivasan arbitrarily named some of the characters of Sp4(Fq) as θ1, θ2, ..., θ13, and the tenth one in her list happens to be the cuspidal unipotent character.
θ10 is the only cuspidal unipotent representation of Sp4(Fq). It is the simplest example of a cuspidal unipotent representation of a reductive group, and also the simplest example of a degenerate cuspidal representation (one without a Whittaker model). General linear groups have no cuspidal unipotent representations and no degenerate cuspidal representations, so θ10 exhibits properties of general reductive groups that do not occur for general linear groups.
Howe & Piatetski-Shapiro (1979) used the representations θ10 over local and global fields in their construction of counterexamples to the generalized Ramanujan conjecture for the symplectic group. Adams (2004) described the representation θ10 of the Lie group Sp4(R) over the local field R in detail.
References
• Adams, Jeffrey (2004), Hida, Haruzo; Ramakrishnan, Dinakar; Shahidi, Freydoon (eds.), "Theta-10", Contributions to automorphic forms, geometry, and number theory: a volume in honor of Joseph A. Shalika, American Journal of Mathematics, Supplement, Baltimore, MD: Johns Hopkins Univ. Press: 39–56, ISBN 978-0-8018-7860-2, MR 2058602
• Deshpande, Tanmay (2008). "An exceptional representation of Sp(4,Fq)". arXiv:0804.2722 [math.RT].
• Gol'fand, Ya. Yu. (1978), "An exceptional representation of Sp(4,Fq)", Functional Analysis and Its Applications, Institute of Problems in Management, Academy of Sciences of the USSR. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, 12 (4): 83–84, doi:10.1007/BF01076387, MR 0515634, S2CID 122223668.
• Howe, Roger; Piatetski-Shapiro, I. I. (1979), "A counterexample to the "generalized Ramanujan conjecture" for (quasi-) split groups", in Borel, Armand; Casselman, W. (eds.), Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, Proc. Sympos. Pure Math., XXXIII, Providence, R.I.: American Mathematical Society, pp. 315–322, ISBN 978-0-8218-1435-2, MR 0546605
• Kim, Ju-Lee; Piatetski-Shapiro, Ilya I. (2001), "Quadratic base change of θ10", Israel Journal of Mathematics, 123: 317–340, doi:10.1007/BF02784134, MR 1835303, S2CID 121587192
• Srinivasan, Bhama (1968), "The characters of the finite symplectic group Sp(4,q)", Transactions of the American Mathematical Society, 131 (2): 488–525, doi:10.2307/1994960, ISSN 0002-9947, JSTOR 1994960, MR 0220845
| Wikipedia |
Θ (set theory)
In set theory, Θ (pronounced like the letter theta) is the least nonzero ordinal α such that there is no surjection from the reals onto α.
If the axiom of choice (AC) holds (or even if the reals can be wellordered), then Θ is simply $(2^{\aleph _{0}})^{+}$, the cardinal successor of the cardinality of the continuum. However, Θ is often studied in contexts where the axiom of choice fails, such as models of the axiom of determinacy.
Θ is also the supremum of the lengths of all prewellorderings of the reals.
Proof of existence
It may not be obvious that it can be proven, without using AC, that there even exists a nonzero ordinal onto which there is no surjection from the reals (if there is such an ordinal, then there must be a least one because the ordinals are wellordered). However, suppose there were no such ordinal. Then to every ordinal α we could associate the set of all prewellorderings of the reals having length α. This would give an injection from the class of all ordinals into the set of all sets of orderings on the reals (which can to be seen to be a set via repeated application of the powerset axiom). Now the axiom of replacement shows that the class of all ordinals is in fact a set. But that is impossible, by the Burali-Forti paradox.
| Wikipedia |
Angle
In Euclidean geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.[1] Angles formed by two rays are also known as plane angles as they lie in the plane that contains the rays. Angles are also formed by the intersection of two planes; these are called dihedral angles. Two intersecting curves may also define an angle, which is the angle of the rays lying tangent to the respective curves at their point of intersection.
The magnitude of an angle is called an angular measure or simply "angle". Angle of rotation is a measure conventionally defined as the ratio of a circular arc length to its radius, and may be a negative number. In the case of a geometric angle, the arc is centered at the vertex and delimited by the sides. In the case of a rotation, the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation.
History and etymology
The word angle comes from the Latin word angulus, meaning "corner." Cognate words include the Greek ἀγκύλος (ankylοs) meaning "crooked, curved" and the English word "ankle." Both are connected with the Proto-Indo-European root *ank-, meaning "to bend" or "bow."[2]
Euclid defines a plane angle as the inclination to each other, in a plane, of two lines that meet each other and do not lie straight with respect to each other. According to the Neoplatonic metaphysician Proclus, an angle must be either a quality, a quantity, or a relationship. The first concept, angle as quality, was used by Eudemus of Rhodes, who regarded an angle as a deviation from a straight line; the second, angle as quality, by Carpus of Antioch, who regarded it as the interval or space between the intersecting lines; Euclid adopted the third: angle as a relationship.[3]
Identifying angles
In mathematical expressions, it is common to use Greek letters (α, β, γ, θ, φ, . . . ) as variables denoting the size of some angle[4] (to avoid confusion with its other meaning, the symbol π is typically not used for this purpose). Lower case Roman letters (a, b, c, . . . ) are also used. In contexts where this is not confusing, an angle may be denoted by the upper case Roman letter denoting its vertex. See the figures in this article for examples.
The three defining points may also identify angles in geometric figures. For example, the angle with vertex A formed by the rays AB and AC (that is, the half-lines from point A through points B and C) is denoted ∠BAC or ${\widehat {\rm {BAC}}}$. Where there is no risk of confusion, the angle may sometimes be referred to by a single vertex alone (in this case, "angle A").
Potentially, an angle denoted as, say, ∠BAC might refer to any of four angles: the clockwise angle from B to C about A, the anticlockwise angle from B to C about A, the clockwise angle from C to B about A, or the anticlockwise angle from C to B about A, where the direction in which the angle is measured determines its sign (see § Signed angles). However, in many geometrical situations, it is evident from the context that the positive angle less than or equal to 180 degrees is meant, and in these cases, no ambiguity arises. Otherwise, to avoid ambiguity, specific conventions may be adopted so that, for instance, ∠BAC always refers to the anticlockwise (positive) angle from B to C about A and ∠CAB the anticlockwise (positive) angle from C to B about A.
Types of angles
Individual angles
There is some common terminology for angles, whose measure is always non-negative (see § Signed angles):
• An angle equal to 0° or not turned is called a zero angle.[5]
• An angle smaller than a right angle (less than 90°) is called an acute angle[6] ("acute" meaning "sharp").
• An angle equal to 1/4 turn (90° or π/2 radians) is called a right angle. Two lines that form a right angle are said to be normal, orthogonal, or perpendicular.[7]
• An angle larger than a right angle and smaller than a straight angle (between 90° and 180°) is called an obtuse angle[6] ("obtuse" meaning "blunt").
• An angle equal to 1/2 turn (180° or π radians) is called a straight angle.[5]
• An angle larger than a straight angle but less than 1 turn (between 180° and 360°) is called a reflex angle.
• An angle equal to 1 turn (360° or 2π radians) is called a full angle, complete angle, round angle or perigon.
• An angle that is not a multiple of a right angle is called an oblique angle.
The names, intervals, and measuring units are shown in the table below:
Right angle
Acute (a), obtuse (b), and straight (c) angles. The acute and obtuse angles are also known as oblique angles.
Reflex angle
Name zero angle acute angle right angle obtuse angle straight angle reflex angle perigon
UnitInterval
turn 0 turn (0, 1/4) turn 1/4 turn (1/4, 1/2) turn 1/2 turn (1/2, 1) turn 1 turn
radian 0 rad (0, 1/2π) rad 1/2π rad (1/2π, π) rad π rad (π, 2π) rad 2π rad
degree 0° (0, 90)° 90° (90, 180)° 180° (180, 360)° 360°
gon 0g (0, 100)g 100g (100, 200)g 200g (200, 400)g 400g
Vertical and adjacent angle pairs
When two straight lines intersect at a point, four angles are formed. Pairwise, these angles are named according to their location relative to each other.
• A pair of angles opposite each other, formed by two intersecting straight lines that form an "X"-like shape, are called vertical angles or opposite angles or vertically opposite angles. They are abbreviated as vert. opp. ∠s.[8]
The equality of vertically opposite angles is called the vertical angle theorem. Eudemus of Rhodes attributed the proof to Thales of Miletus.[9][10] The proposition showed that since both of a pair of vertical angles are supplementary to both of the adjacent angles, the vertical angles are equal in measure. According to a historical note,[10] when Thales visited Egypt, he observed that whenever the Egyptians drew two intersecting lines, they would measure the vertical angles to make sure that they were equal. Thales concluded that one could prove that all vertical angles are equal if one accepted some general notions such as:
• All straight angles are equal.
• Equals added to equals are equal.
• Equals subtracted from equals are equal.
When two adjacent angles form a straight line, they are supplementary. Therefore, if we assume that the measure of angle A equals x, the measure of angle C would be 180° − x. Similarly, the measure of angle D would be 180° − x. Both angle C and angle D have measures equal to 180° − x and are congruent. Since angle B is supplementary to both angles C and D, either of these angle measures may be used to determine the measure of Angle B. Using the measure of either angle C or angle D, we find the measure of angle B to be 180° − (180° − x) = 180° − 180° + x = x. Therefore, both angle A and angle B have measures equal to x and are equal in measure.
• Adjacent angles, often abbreviated as adj. ∠s, are angles that share a common vertex and edge but do not share any interior points. In other words, they are angles side by side or adjacent, sharing an "arm". Adjacent angles which sum to a right angle, straight angle, or full angle are special and are respectively called complementary, supplementary, and explementary angles (see § Combining angle pairs below).
A transversal is a line that intersects a pair of (often parallel) lines and is associated with alternate interior angles, corresponding angles, interior angles, and exterior angles.[11]
Combining angle pairs
Three special angle pairs involve the summation of angles:
• Complementary angles are angle pairs whose measures sum to one right angle (1/4 turn, 90°, or π/2 radians).[12] If the two complementary angles are adjacent, their non-shared sides form a right angle. In Euclidean geometry, the two acute angles in a right triangle are complementary because the sum of internal angles of a triangle is 180 degrees, and the right angle accounts for 90 degrees.
The adjective complementary is from the Latin complementum, associated with the verb complere, "to fill up". An acute angle is "filled up" by its complement to form a right angle.
The difference between an angle and a right angle is termed the complement of the angle.[13]
If angles A and B are complementary, the following relationships hold:
${\begin{aligned}&\sin ^{2}A+\sin ^{2}B=1&&\cos ^{2}A+\cos ^{2}B=1\\[3pt]&\tan A=\cot B&&\sec A=\csc B\end{aligned}}$
(The tangent of an angle equals the cotangent of its complement, and its secant equals the cosecant of its complement.)
The prefix "co-" in the names of some trigonometric ratios refers to the word "complementary".
• Two angles that sum to a straight angle (1/2 turn, 180°, or π radians) are called supplementary angles.[14]
If the two supplementary angles are adjacent (i.e., have a common vertex and share just one side), their non-shared sides form a straight line. Such angles are called a linear pair of angles.[15] However, supplementary angles do not have to be on the same line and can be separated in space. For example, adjacent angles of a parallelogram are supplementary, and opposite angles of a cyclic quadrilateral (one whose vertices all fall on a single circle) are supplementary.
If a point P is exterior to a circle with center O, and if the tangent lines from P touch the circle at points T and Q, then ∠TPQ and ∠TOQ are supplementary.
The sines of supplementary angles are equal. Their cosines and tangents (unless undefined) are equal in magnitude but have opposite signs.
In Euclidean geometry, any sum of two angles in a triangle is supplementary to the third because the sum of the internal angles of a triangle is a straight angle.
• Two angles that sum to a complete angle (1 turn, 360°, or 2π radians) are called explementary angles or conjugate angles.
The difference between an angle and a complete angle is termed the explement of the angle or conjugate of an angle.
Polygon-related angles
• An angle that is part of a simple polygon is called an interior angle if it lies on the inside of that simple polygon. A simple concave polygon has at least one interior angle, that is, a reflex angle.
In Euclidean geometry, the measures of the interior angles of a triangle add up to π radians, 180°, or 1/2 turn; the measures of the interior angles of a simple convex quadrilateral add up to 2π radians, 360°, or 1 turn. In general, the measures of the interior angles of a simple convex polygon with n sides add up to (n − 2)π radians, or (n − 2)180 degrees, (n − 2)2 right angles, or (n − 2)1/2 turn.
• The supplement of an interior angle is called an exterior angle; that is, an interior angle and an exterior angle form a linear pair of angles. There are two exterior angles at each vertex of the polygon, each determined by extending one of the two sides of the polygon that meet at the vertex; these two angles are vertical and hence are equal. An exterior angle measures the amount of rotation one must make at a vertex to trace the polygon.[16] If the corresponding interior angle is a reflex angle, the exterior angle should be considered negative. Even in a non-simple polygon, it may be possible to define the exterior angle. Still, one will have to pick an orientation of the plane (or surface) to decide the sign of the exterior angle measure.
In Euclidean geometry, the sum of the exterior angles of a simple convex polygon, if only one of the two exterior angles is assumed at each vertex, will be one full turn (360°). The exterior angle here could be called a supplementary exterior angle. Exterior angles are commonly used in Logo Turtle programs when drawing regular polygons.
• In a triangle, the bisectors of two exterior angles and the bisector of the other interior angle are concurrent (meet at a single point).[17]: 149
• In a triangle, three intersection points, each of an external angle bisector with the opposite extended side, are collinear.[17]: p. 149
• In a triangle, three intersection points, two between an interior angle bisector and the opposite side, and the third between the other exterior angle bisector and the opposite side extended are collinear.[17]: 149
• Some authors use the name exterior angle of a simple polygon to mean the explement exterior angle (not supplement!) of the interior angle.[18] This conflicts with the above usage.
Plane-related angles
• The angle between two planes (such as two adjacent faces of a polyhedron) is called a dihedral angle.[13] It may be defined as the acute angle between two lines normal to the planes.
• The angle between a plane and an intersecting straight line is equal to ninety degrees minus the angle between the intersecting line and the line that goes through the point of intersection and is normal to the plane.
Measuring angles
The size of a geometric angle is usually characterized by the magnitude of the smallest rotation that maps one of the rays into the other. Angles of the same size are said to be equal congruent or equal in measure.
In some contexts, such as identifying a point on a circle or describing the orientation of an object in two dimensions relative to a reference orientation, angles that differ by an exact multiple of a full turn are effectively equivalent. In other contexts, such as identifying a point on a spiral curve or describing an object's cumulative rotation in two dimensions relative to a reference orientation, angles that differ by a non-zero multiple of a full turn are not equivalent.
To measure an angle θ, a circular arc centered at the vertex of the angle is drawn, e.g., with a pair of compasses. The ratio of the length s of the arc by the radius r of the circle is the number of radians in the angle:[19]
$\theta ={\frac {s}{r}}\,\mathrm {rad} .$
Conventionally, in mathematics and the SI, the radian is treated as being equal to the dimensionless unit 1, thus being normally omitted.
The angle expressed by another angular unit may then be obtained by multiplying the angle by a suitable conversion constant of the form k/2π, where k is the measure of a complete turn expressed in the chosen unit (for example, k = 360° for degrees or 400 grad for gradians):
$\theta ={\frac {k}{2\pi }}\cdot {\frac {s}{r}}.$
The value of θ thus defined is independent of the size of the circle: if the length of the radius is changed, then the arc length changes in the same proportion, so the ratio s/r is unaltered.[nb 1]
Angle addition postulate
The angle addition postulate states that if B is in the interior of angle AOC, then
$m\angle \mathrm {AOC} =m\angle \mathrm {AOB} +m\angle \mathrm {BOC} $
The measure of the angle AOC is the sum of the measure of angle AOB and the measure of angle BOC.
Units
Throughout history, angles have been measured in various units. These are known as angular units, with the most contemporary units being the degree ( ° ), the radian (rad), and the gradian (grad), though many others have been used throughout history.[21] Most units of angular measurement are defined such that one turn (i.e., the angle subtended by the circumference of a circle at its centre) is equal to n units, for some whole number n. Two exceptions are the radian (and its decimal submultiples) and the diameter part.
In the International System of Quantities, an angle is defined as a dimensionless quantity, and in particular, the radian unit is dimensionless. This convention impacts how angles are treated in dimensional analysis. See Radian § Dimensional analysis for a discussion.
The following table list some units used to represent angles.
namenumber in one turnin degreesdescription
radian2π≈57°17′The radian is determined by the circumference of a circle that is equal in length to the radius of the circle (n = 2π = 6.283...). It is the angle subtended by an arc of a circle that has the same length as the circle's radius. The symbol for radian is rad. One turn is 2π radians, and one radian is 180°/π, or about 57.2958 degrees. Often, particularly in mathematical texts, one radian is assumed to equal one, resulting in the unit rad being omitted. The radian is used in virtually all mathematical work beyond simple, practical geometry due, for example, to the pleasing and "natural" properties that the trigonometric functions display when their arguments are in radians. The radian is the (derived) unit of angular measurement in the SI.
degree3601°The degree, denoted by a small superscript circle (°), is 1/360 of a turn, so one turn is 360°. One advantage of this old sexagesimal subunit is that many angles common in simple geometry are measured as a whole number of degrees. Fractions of a degree may be written in normal decimal notation (e.g., 3.5° for three and a half degrees), but the "minute" and "second" sexagesimal subunits of the "degree–minute–second" system (discussed next) are also in use, especially for geographical coordinates and in astronomy and ballistics (n = 360)
arcminute21,6000°1′The minute of arc (or MOA, arcminute, or just minute) is 1/60 of a degree = 1/21,600 turn. It is denoted by a single prime ( ′ ). For example, 3° 30′ is equal to 3 × 60 + 30 = 210 minutes or 3 + 30/60 = 3.5 degrees. A mixed format with decimal fractions is sometimes used, e.g., 3° 5.72′ = 3 + 5.72/60 degrees. A nautical mile was historically defined as an arcminute along a great circle of the Earth. (n = 21,600).
arcsecond1,296,0000°0′1″The second of arc (or arcsecond, or just second) is 1/60 of a minute of arc and 1/3600 of a degree (n = 1,296,000). It is denoted by a double prime ( ″ ). For example, 3° 7′ 30″ is equal to 3 + 7/60 + 30/3600 degrees, or 3.125 degrees.
grad4000°54′The grad, also called grade, gradian, or gon. It is a decimal subunit of the quadrant. A right angle is 100 grads. A kilometre was historically defined as a centi-grad of arc along a meridian of the Earth, so the kilometer is the decimal analog to the sexagesimal nautical mile (n = 400). The grad is used mostly in triangulation and continental surveying.
turn1360°The turn is the angle subtended by the circumference of a circle at its centre. A turn is equal to 2π or tau radians.
hour angle2415°The astronomical hour angle is 1/24 turn. As this system is amenable to measuring objects that cycle once per day (such as the relative position of stars), the sexagesimal subunits are called minute of time and second of time. These are distinct from, and 15 times larger than, minutes and seconds of arc. 1 hour = 15° = π/12 rad = 1/6 quad = 1/24 turn = 16+2/3 grad.
(compass) point3211.25°The point or wind, used in navigation, is 1/32 of a turn. 1 point = 1/8 of a right angle = 11.25° = 12.5 grad. Each point is subdivided into four quarter points, so one turn equals 128.
milliradian2000π≈0.057°The true milliradian is defined as a thousandth of a radian, which means that a rotation of one turn would equal exactly 2000π mrad (or approximately 6283.185 mrad). Almost all scope sights for firearms are calibrated to this definition. In addition, three other related definitions are used for artillery and navigation, often called a 'mil,' which are approximately equal to a milliradian. Under these three other definitions, one turn makes up for exactly 6000, 6300, or 6400 mils, spanning the range from 0.05625 to 0.06 degrees (3.375 to 3.6 minutes). In comparison, the milliradian is approximately 0.05729578 degrees (3.43775 minutes). One "NATO mil" is defined as 1/6400 of a turn. Just like with the milliradian, each of the other definitions approximates the milliradian's useful property of subtensions, i.e. that the value of one milliradian approximately equals the angle subtended by a width of 1 meter as seen from 1 km away (2π/6400 = 0.0009817... ≈ 1/1000).
binary degree2561°33'45"The binary degree, also known as the binary radian or brad or binary angular measurement (BAM).[22] The binary degree is used in computing so that an angle can be efficiently represented in a single byte (albeit to limited precision). Other measures of the angle used in computing may be based on dividing one whole turn into 2n equal parts for other values of n.
[23] It is 1/256 of a turn.[22]
π radian2180°The multiples of π radians (MULπ) unit is implemented in the RPN scientific calculator WP 43S.[24][25][26] See also: IEEE 754 recommended operations
quadrant490°One quadrant is a 1/4 turn and also known as a right angle. The quadrant is the unit in Euclid's Elements. In German, the symbol ∟ has been used to denote a quadrant. 1 quad = 90° = π/2 rad = 1/4 turn = 100 grad.
sextant660°The sextant was the unit used by the Babylonians,[27][28] The degree, minute of arc and second of arc are sexagesimal subunits of the Babylonian unit. It is straightforward to construct with ruler and compasses. It is the angle of the equilateral triangle or is 1/6 turn. 1 Babylonian unit = 60° = π/3 rad ≈ 1.047197551 rad.
hexacontade606°The hexacontade is a unit used by Eratosthenes. It equals 6°, so a whole turn was divided into 60 hexacontades.
pechus144 to 1802° to 2+1/2°The pechus was a Babylonian unit equal to about 2° or 2+1/2°.
diameter part≈376.991≈0.95493°The diameter part (occasionally used in Islamic mathematics) is 1/60 radian. One "diameter part" is approximately 0.95493°. There are about 376.991 diameter parts per turn.
zam224≈1.607°In old Arabia, a turn was subdivided into 32 Akhnam, and each akhnam was subdivided into 7 zam so that a turn is 224 zam.
Signed angles
See also: Sign (mathematics) § Angles
Although the definition of the measurement of an angle does not support the concept of a negative angle, it is frequently helpful to impose a convention that allows positive and negative angular values to represent orientations and/or rotations in opposite directions relative to some reference.
In a two-dimensional Cartesian coordinate system, an angle is typically defined by its two sides, with its vertex at the origin. The initial side is on the positive x-axis, while the other side or terminal side is defined by the measure from the initial side in radians, degrees, or turns, with positive angles representing rotations toward the positive y-axis and negative angles representing rotations toward the negative y-axis. When Cartesian coordinates are represented by standard position, defined by the x-axis rightward and the y-axis upward, positive rotations are anticlockwise, and negative cycles are clockwise.
In many contexts, an angle of −θ is effectively equivalent to an angle of "one full turn minus θ". For example, an orientation represented as −45° is effectively equal to an orientation defined as 360° − 45° or 315°. Although the final position is the same, a physical rotation (movement) of −45° is not the same as a rotation of 315° (for example, the rotation of a person holding a broom resting on a dusty floor would leave visually different traces of swept regions on the floor).
In three-dimensional geometry, "clockwise" and "anticlockwise" have no absolute meaning, so the direction of positive and negative angles must be defined in terms of an orientation, which is typically determined by a normal vector passing through the angle's vertex and perpendicular to the plane in which the rays of the angle lie.
In navigation, bearings or azimuth are measured relative to north. By convention, viewed from above, bearing angles are positive clockwise, so a bearing of 45° corresponds to a north-east orientation. Negative bearings are not used in navigation, so a north-west orientation corresponds to a bearing of 315°.
Equivalent angles
• Angles that have the same measure (i.e., the same magnitude) are said to be equal or congruent. An angle is defined by its measure and is not dependent upon the lengths of the sides of the angle (e.g., all right angles are equal in measure).
• Two angles that share terminal sides, but differ in size by an integer multiple of a turn, are called coterminal angles.
• The reference angle (sometimes called related angle) for any angle θ in standard position is the positive acute angle between the terminal side of θ and the x-axis (positive or negative).[29][30] Procedurally, the magnitude of the reference angle for a given angle may determined by taking the angle's magnitude modulo 1/2 turn, 180°, or π radians, then stopping if the angle is acute, otherwise taking the supplementary angle, 180° minus the reduced magnitude. For example, an angle of 30 degrees is already a reference angle, and an angle of 150 degrees also has a reference angle of 30 degrees (180° − 150°). Angles of 210° and 510° correspond to a reference angle of 30 degrees as well (210° mod 180° = 30°, 510° mod 180° = 150° whose supplementary angle is 30°).
Related quantities
For an angular unit, it is definitional that the angle addition postulate holds. Some quantities related to angles where the angle addition postulate does not hold include:
• The slope or gradient is equal to the tangent of the angle; a gradient is often expressed as a percentage. For very small values (less than 5%), the slope of a line is approximately the measure in radians of its angle with the horizontal direction.
• The spread between two lines is defined in rational geometry as the square of the sine of the angle between the lines. As the sine of an angle and the sine of its supplementary angle are the same, any angle of rotation that maps one of the lines into the other leads to the same value for the spread between the lines.
• Although done rarely, one can report the direct results of trigonometric functions, such as the sine of the angle.
Angles between curves
The angle between a line and a curve (mixed angle) or between two intersecting curves (curvilinear angle) is defined to be the angle between the tangents at the point of intersection. Various names (now rarely, if ever, used) have been given to particular cases:—amphicyrtic (Gr. ἀμφί, on both sides, κυρτός, convex) or cissoidal (Gr. κισσός, ivy), biconvex; xystroidal or sistroidal (Gr. ξυστρίς, a tool for scraping), concavo-convex; amphicoelic (Gr. κοίλη, a hollow) or angulus lunularis, biconcave.[31]
Bisecting and trisecting angles
Main articles: Bisection § Angle bisector, and Angle trisection
The ancient Greek mathematicians knew how to bisect an angle (divide it into two angles of equal measure) using only a compass and straightedge but could only trisect certain angles. In 1837, Pierre Wantzel showed that this construction could not be performed for most angles.
Dot product and generalisations
In the Euclidean space, the angle θ between two Euclidean vectors u and v is related to their dot product and their lengths by the formula
$\mathbf {u} \cdot \mathbf {v} =\cos(\theta )\left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.$
This formula supplies an easy method to find the angle between two planes (or curved surfaces) from their normal vectors and between skew lines from their vector equations.
Inner product
To define angles in an abstract real inner product space, we replace the Euclidean dot product ( · ) by the inner product $\langle \cdot ,\cdot \rangle $, i.e.
$\langle \mathbf {u} ,\mathbf {v} \rangle =\cos(\theta )\ \left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.$
In a complex inner product space, the expression for the cosine above may give non-real values, so it is replaced with
$\operatorname {Re} \left(\langle \mathbf {u} ,\mathbf {v} \rangle \right)=\cos(\theta )\left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.$
or, more commonly, using the absolute value, with
$\left|\langle \mathbf {u} ,\mathbf {v} \rangle \right|=\left|\cos(\theta )\right|\left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.$
The latter definition ignores the direction of the vectors. It thus describes the angle between one-dimensional subspaces $\operatorname {span} (\mathbf {u} )$ and $\operatorname {span} (\mathbf {v} )$ spanned by the vectors $\mathbf {u} $ and $\mathbf {v} $ correspondingly.
Angles between subspaces
The definition of the angle between one-dimensional subspaces $\operatorname {span} (\mathbf {u} )$ and $\operatorname {span} (\mathbf {v} )$ given by
$\left|\langle \mathbf {u} ,\mathbf {v} \rangle \right|=\left|\cos(\theta )\right|\left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|$
in a Hilbert space can be extended to subspaces of finite dimensions. Given two subspaces ${\mathcal {U}}$, ${\mathcal {W}}$ with $\dim({\mathcal {U}}):=k\leq \dim({\mathcal {W}}):=l$, this leads to a definition of $k$ angles called canonical or principal angles between subspaces.
Angles in Riemannian geometry
In Riemannian geometry, the metric tensor is used to define the angle between two tangents. Where U and V are tangent vectors and gij are the components of the metric tensor G,
$\cos \theta ={\frac {g_{ij}U^{i}V^{j}}{\sqrt {\left|g_{ij}U^{i}U^{j}\right|\left|g_{ij}V^{i}V^{j}\right|}}}.$
Hyperbolic angle
A hyperbolic angle is an argument of a hyperbolic function just as the circular angle is the argument of a circular function. The comparison can be visualized as the size of the openings of a hyperbolic sector and a circular sector since the areas of these sectors correspond to the angle magnitudes in each case. Unlike the circular angle, the hyperbolic angle is unbounded. When the circular and hyperbolic functions are viewed as infinite series in their angle argument, the circular ones are just alternating series forms of the hyperbolic functions. This weaving of the two types of angle and function was explained by Leonhard Euler in Introduction to the Analysis of the Infinite.
Angles in geography and astronomy
In geography, the location of any point on the Earth can be identified using a geographic coordinate system. This system specifies the latitude and longitude of any location in terms of angles subtended at the center of the Earth, using the equator and (usually) the Greenwich meridian as references.
In astronomy, a given point on the celestial sphere (that is, the apparent position of an astronomical object) can be identified using any of several astronomical coordinate systems, where the references vary according to the particular system. Astronomers measure the angular separation of two stars by imagining two lines through the center of the Earth, each intersecting one of the stars. The angle between those lines and the angular separation between the two stars can be measured.
In both geography and astronomy, a sighting direction can be specified in terms of a vertical angle such as altitude /elevation with respect to the horizon as well as the azimuth with respect to north.
Astronomers also measure objects' apparent size as an angular diameter. For example, the full moon has an angular diameter of approximately 0.5° when viewed from Earth. One could say, "The Moon's diameter subtends an angle of half a degree." The small-angle formula can convert such an angular measurement into a distance/size ratio.
Other astronomical approximations include:
• 0.5° is the approximate diameter of the Sun and of the Moon as viewed from Earth.
• 1° is the approximate width of the little finger at arm's length.
• 10° is the approximate width of a closed fist at arm's length.
• 20° is the approximate width of a handspan at arm's length.
These measurements depend on the individual subject, and the above should be treated as rough rule of thumb approximations only.
In astronomy, right ascension and declination are usually measured in angular units, expressed in terms of time, based on a 24-hour day.
UnitSymbolDegreesRadiansTurnsOther
Hour h15°π⁄12 rad1⁄24 turn
Minute m0°15′π⁄720 rad1⁄1,440 turn1⁄60 hour
Second s0°0′15″π⁄43200 rad1⁄86,400 turn1⁄60 minute
See also
• Angle measuring instrument
• Angles between flats
• Angular statistics (mean, standard deviation)
• Angle bisector
• Angular acceleration
• Angular diameter
• Angular velocity
• Argument (complex analysis)
• Astrological aspect
• Central angle
• Clock angle problem
• Decimal degrees
• Dihedral angle
• Exterior angle theorem
• Golden angle
• Great circle distance
• Inscribed angle
• Irrational angle
• Phase (waves)
• Protractor
• Solid angle
• Spherical angle
• Transcendent angle
• Trisection
• Zenith angle
Notes
1. This approach requires, however, an additional proof that the measure of the angle does not change with changing radius r, in addition to the issue of "measurement units chosen". A smoother approach is to measure the angle by the length of the corresponding unit circle arc. Here "unit" can be chosen to be dimensionless in the sense that it is the real number 1 associated with the unit segment on the real line. See Radoslav M. Dimitrić, for instance.[20]
References
1. Sidorov 2001
2. Slocum 2007
3. Chisholm 1911; Heiberg 1908, pp. 177–178
4. Aboughantous 2010, p. 18.
5. Moser 1971, p. 41.
6. Godfrey & Siddons 1919, p. 9.
7. Moser 1971, p. 71.
8. Wong & Wong 2009, pp. 161–163
9. Euclid. The Elements. Proposition I:13.
10. Shute, Shirk & Porter 1960, pp. 25–27.
11. Jacobs 1974, p. 255.
12. "Complementary Angles". www.mathsisfun.com. Retrieved 2020-08-17.
13. Chisholm 1911
14. "Supplementary Angles". www.mathsisfun.com. Retrieved 2020-08-17.
15. Jacobs 1974, p. 97.
16. Henderson & Taimina 2005, p. 104.
17. Johnson, Roger A. Advanced Euclidean Geometry, Dover Publications, 2007.
18. D. Zwillinger, ed. (1995), CRC Standard Mathematical Tables and Formulae, Boca Raton, FL: CRC Press, p. 270 as cited in Weisstein, Eric W. "Exterior Angle". MathWorld.
19. International Bureau of Weights and Measures (20 May 2019), The International System of Units (SI) (PDF) (9th ed.), ISBN 978-92-822-2272-0, archived from the original on 18 October 2021
20. Dimitrić, Radoslav M. (2012). "On Angles and Angle Measurements" (PDF). The Teaching of Mathematics. XV (2): 133–140. Archived (PDF) from the original on 2019-01-17. Retrieved 2019-08-06.
21. "angular unit". TheFreeDictionary.com. Retrieved 2020-08-31.
22. "ooPIC Programmer's Guide - Chapter 15: URCP". ooPIC Manual & Technical Specifications - ooPIC Compiler Ver 6.0. Savage Innovations, LLC. 2007 [1997]. Archived from the original on 2008-06-28. Retrieved 2019-08-05.
23. Hargreaves, Shawn [in Polish]. "Angles, integers, and modulo arithmetic". blogs.msdn.com. Archived from the original on 2019-06-30. Retrieved 2019-08-05.
24. Bonin, Walter (2016-01-11). "RE: WP-32S in 2016?". HP Museum. Archived from the original on 2019-08-06. Retrieved 2019-08-05.
25. Bonin, Walter (2019) [2015]. WP 43S Owner's Manual (PDF). 0.12 (draft ed.). pp. 72, 118–119, 311. ISBN 978-1-72950098-9. Retrieved 2019-08-05. (314 pages)
26. Bonin, Walter (2019) [2015]. WP 43S Reference Manual (PDF). 0.12 (draft ed.). pp. iii, 54, 97, 128, 144, 193, 195. ISBN 978-1-72950106-1. Retrieved 2019-08-05. (271 pages)
27. Jeans, James Hopwood (1947). The Growth of Physical Science. CUP Archive. p. 7.
28. Murnaghan, Francis Dominic (1946). Analytic Geometry. p. 2.
29. "Mathwords: Reference Angle". www.mathwords.com. Archived from the original on 23 October 2017. Retrieved 26 April 2018.
30. McKeague, Charles P. (2008). Trigonometry (6th ed.). Belmont, CA: Thomson Brooks/Cole. p. 110. ISBN 0495382604.
31. Chisholm 1911; Heiberg 1908, p. 178
Bibliography
• Aboughantous, Charles H. (2010), A High School First Course in Euclidean Plane Geometry, Universal Publishers, ISBN 978-1-59942-822-2
• Godfrey, Charles; Siddons, A. W. (1919), Elementary geometry: practical and theoretical (3rd ed.), Cambridge University Press
• Henderson, David W.; Taimina, Daina (2005), Experiencing Geometry / Euclidean and Non-Euclidean with History (3rd ed.), Pearson Prentice Hall, p. 104, ISBN 978-0-13-143748-7
• Heiberg, Johan Ludvig (1908), Heath, T. L. (ed.), Euclid, The Thirteen Books of Euclid's Elements, vol. 1, Cambridge: Cambridge University Press.
• Sidorov, L. A. (2001) [1994], "Angle", Encyclopedia of Mathematics, EMS Press
• Jacobs, Harold R. (1974), Geometry, W. H. Freeman, pp. 97, 255, ISBN 978-0-7167-0456-0
• Moser, James M. (1971), Modern Elementary Geometry, Prentice-Hall
• Slocum, Jonathan (2007), Preliminary Indo-European lexicon — Pokorny PIE data, University of Texas research department: linguistics research center, archived from the original on 27 June 2010, retrieved 2 Feb 2010
• Shute, William G.; Shirk, William W.; Porter, George F. (1960), Plane and Solid Geometry, American Book Company, pp. 25–27
• Wong, Tak-wah; Wong, Ming-sim (2009), "Angles in Intersecting and Parallel Lines", New Century Mathematics, vol. 1B (1 ed.), Hong Kong: Oxford University Press, pp. 161–163, ISBN 978-0-19-800177-5
This article incorporates text from a publication now in the public domain: Chisholm, Hugh, ed. (1911), "Angle", Encyclopædia Britannica, vol. 2 (11th ed.), Cambridge University Press, p. 14
External links
Wikimedia Commons has media related to Angles (geometry).
The Wikibook Geometry has a page on the topic of: Unified Angles
• "Angle" , Encyclopædia Britannica, vol. 2 (9th ed.), 1878, pp. 29–30
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| Wikipedia |
Theta characteristic
In mathematics, a theta characteristic of a non-singular algebraic curve C is a divisor class Θ such that 2Θ is the canonical class. In terms of holomorphic line bundles L on a connected compact Riemann surface, it is therefore L such that L2 is the canonical bundle, here also equivalently the holomorphic cotangent bundle. In terms of algebraic geometry, the equivalent definition is as an invertible sheaf, which squares to the sheaf of differentials of the first kind. Theta characteristics were introduced by Rosenhain (1851)
History and genus 1
The importance of this concept was realised first in the analytic theory of theta functions, and geometrically in the theory of bitangents. In the analytic theory, there are four fundamental theta functions in the theory of Jacobian elliptic functions. Their labels are in effect the theta characteristics of an elliptic curve. For that case, the canonical class is trivial (zero in the divisor class group) and so the theta characteristics of an elliptic curve E over the complex numbers are seen to be in 1-1 correspondence with the four points P on E with 2P = 0; this is counting of the solutions is clear from the group structure, a product of two circle groups, when E is treated as a complex torus.
Higher genus
For C of genus 0 there is one such divisor class, namely the class of -P, where P is any point on the curve. In case of higher genus g, assuming the field over which C is defined does not have characteristic 2, the theta characteristics can be counted as
22g
in number if the base field is algebraically closed.
This comes about because the solutions of the equation on the divisor class level will form a single coset of the solutions of
2D = 0.
In other words, with K the canonical class and Θ any given solution of
2Θ = K,
any other solution will be of form
Θ + D.
This reduces counting the theta characteristics to finding the 2-rank of the Jacobian variety J(C) of C. In the complex case, again, the result follows since J(C) is a complex torus of dimension 2g. Over a general field, see the theory explained at Hasse-Witt matrix for the counting of the p-rank of an abelian variety. The answer is the same, provided the characteristic of the field is not 2.
A theta characteristic Θ will be called even or odd depending on the dimension of its space of global sections $H^{0}(C,\Theta )$. It turns out that on C there are $2^{g-1}(2^{g}+1)$ even and $2^{g-1}(2^{g}-1)$ odd theta characteristics.
Classical theory
Classically the theta characteristics were divided into these two kinds, odd and even, according to the value of the Arf invariant of a certain quadratic form Q with values mod 2. Thus in case of g = 3 and a plane quartic curve, there were 28 of one type, and the remaining 36 of the other; this is basic in the question of counting bitangents, as it corresponds to the 28 bitangents of a quartic. The geometric construction of Q as an intersection form is with modern tools possible algebraically. In fact the Weil pairing applies, in its abelian variety form. Triples (θ1, θ2, θ3) of theta characteristics are called syzygetic and asyzygetic depending on whether Arf(θ1)+Arf(θ2)+Arf(θ3)+Arf(θ1+θ2+θ3) is 0 or 1.
Spin structures
Atiyah (1971) showed that, for a compact complex manifold, choices of theta characteristics correspond bijectively to spin structures.
References
• Atiyah, Michael Francis (1971), "Riemann surfaces and spin structures", Annales Scientifiques de l'École Normale Supérieure, Série 4, 4: 47–62, ISSN 0012-9593, MR 0286136
• Dolgachev, Lectures on Classical Topics, Ch. 5 (PDF)
• Farkas, Gavril (2012), Theta characteristics and their moduli, arXiv:1201.2557, Bibcode:2012arXiv1201.2557F
• Mumford, David (1971), "Theta characteristics of an algebraic curve", Annales Scientifiques de l'École Normale Supérieure, Série 4, 4 (2): 181–192, MR 0292836
• Rosenhain, Johann Georg (1851), Mémoire sur les fonctions de deux variables, qui sont les inverses des intégrales ultra-elliptiques de la première classe, Paris
| Wikipedia |
Theta function of a lattice
In mathematics, the theta function of a lattice is a function whose coefficients give the number of vectors of a given norm.
Definition
One can associate to any (positive-definite) lattice Λ a theta function given by
$\Theta _{\Lambda }(\tau )=\sum _{x\in \Lambda }e^{i\pi \tau \|x\|^{2}}\qquad \mathrm {Im} \,\tau >0.$
The theta function of a lattice is then a holomorphic function on the upper half-plane. Furthermore, the theta function of an even unimodular lattice of rank n is actually a modular form of weight n/2. The theta function of an integral lattice is often written as a power series in $q=e^{2i\pi \tau }$ so that the coefficient of qn gives the number of lattice vectors of norm 2n.
References
• Deconinck, Bernard (2010), "Multidimensional Theta Functions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.
| Wikipedia |
Theta operator
In mathematics, the theta operator is a differential operator defined by[1][2]
$\theta =z{d \over dz}.$
This is sometimes also called the homogeneity operator, because its eigenfunctions are the monomials in z:
$\theta (z^{k})=kz^{k},\quad k=0,1,2,\dots $
In n variables the homogeneity operator is given by
$\theta =\sum _{k=1}^{n}x_{k}{\frac {\partial }{\partial x_{k}}}.$
As in one variable, the eigenspaces of θ are the spaces of homogeneous functions. (Euler's homogeneous function theorem)
See also
• Difference operator
• Delta operator
• Elliptic operator
• Fractional calculus
• Invariant differential operator
• Differential calculus over commutative algebras
References
1. Weisstein, Eric W. "Theta Operator". MathWorld. Retrieved 2013-02-16.
2. Weisstein, Eric W. (2002). CRC Concise Encyclopedia of Mathematics (2nd ed.). Hoboken: CRC Press. pp. 2976–2983. ISBN 1420035223.
Further reading
• Watson, G.N. (1995). A treatise on the theory of Bessel functions (Cambridge mathematical library ed., [Nachdr. der] 2. ed.). Cambridge: Univ. Press. ISBN 0521483913.
| Wikipedia |
Theta representation
In mathematics, the theta representation is a particular representation of the Heisenberg group of quantum mechanics. It gains its name from the fact that the Jacobi theta function is invariant under the action of a discrete subgroup of the Heisenberg group. The representation was popularized by David Mumford.
Construction
The theta representation is a representation of the continuous Heisenberg group $H_{3}(\mathbb {R} )$ over the field of the real numbers. In this representation, the group elements act on a particular Hilbert space. The construction below proceeds first by defining operators that correspond to the Heisenberg group generators. Next, the Hilbert space on which these act is defined, followed by a demonstration of the isomorphism to the usual representations.
Group generators
Let f(z) be a holomorphic function, let a and b be real numbers, and let $\tau $ be fixed, but arbitrary complex number in the upper half-plane; that is, so that the imaginary part of $\tau $ is positive. Define the operators Sa and Tb such that they act on holomorphic functions as
$(S_{a}f)(z)=f(z+a)=\exp(a\partial _{z})f(z)$
and
$(T_{b}f)(z)=\exp(i\pi b^{2}\tau +2\pi ibz)f(z+b\tau )=\exp(i\pi b^{2}\tau +2\pi ibz+b\tau \partial _{z})f(z).$
It can be seen that each operator generates a one-parameter subgroup:
$S_{a_{1}}\left(S_{a_{2}}f\right)=\left(S_{a_{1}}\circ S_{a_{2}}\right)f=S_{a_{1}+a_{2}}f$
and
$T_{b_{1}}\left(T_{b_{2}}f\right)=\left(T_{b_{1}}\circ T_{b_{2}}\right)f=T_{b_{1}+b_{2}}f.$
However, S and T do not commute:
$S_{a}\circ T_{b}=\exp(2\pi iab)T_{b}\circ S_{a}.$
Thus we see that S and T together with a unitary phase form a nilpotent Lie group, the (continuous real) Heisenberg group, parametrizable as $H=U(1)\times \mathbb {R} \times \mathbb {R} $ where U(1) is the unitary group.
A general group element $U_{\tau }(\lambda ,a,b)\in H$ then acts on a holomorphic function f(z) as
$U_{\tau }(\lambda ,a,b)f(z)=\lambda (S_{a}\circ T_{b}f)(z)=\lambda \exp(i\pi b^{2}\tau +2\pi ibz)f(z+a+b\tau )$
where $\lambda \in U(1).$ $U(1)=Z(H)$ is the center of H, the commutator subgroup $[H,H]$. The parameter $\tau $ on $U_{\tau }(\lambda ,a,b)$ serves only to remind that every different value of $\tau $ gives rise to a different representation of the action of the group.
Hilbert space
The action of the group elements $U_{\tau }(\lambda ,a,b)$ is unitary and irreducible on a certain Hilbert space of functions. For a fixed value of τ, define a norm on entire functions of the complex plane as
$\Vert f\Vert _{\tau }^{2}=\int _{\mathbb {C} }\exp \left({\frac {-\pi y^{2}}{\Im \tau }}\right)|f(x+iy)|^{2}\ dx\ dy.$
Here, $\Im \tau $ is the imaginary part of $\tau $ and the domain of integration is the entire complex plane.
Mumford sets the norm as $\int _{\mathbb {C} }\exp \left({\frac {-2\pi y^{2}}{\Im \tau }}\right)|f(x+iy)|^{2}\ dx\ dy$, but in this way $T_{b}$ is not unitary.
Let ${\mathcal {H}}_{\tau }$ be the set of entire functions f with finite norm. The subscript $\tau $ is used only to indicate that the space depends on the choice of parameter $\tau $. This ${\mathcal {H}}_{\tau }$ forms a Hilbert space. The action of $U_{\tau }(\lambda ,a,b)$ given above is unitary on ${\mathcal {H}}_{\tau }$, that is, $U_{\tau }(\lambda ,a,b)$ preserves the norm on this space. Finally, the action of $U_{\tau }(\lambda ,a,b)$ on ${\mathcal {H}}_{\tau }$ is irreducible.
This norm is closely related to that used to define Segal–Bargmann space.
Isomorphism
The above theta representation of the Heisenberg group is isomorphic to the canonical Weyl representation of the Heisenberg group. In particular, this implies that ${\mathcal {H}}_{\tau }$ and $L^{2}(\mathbb {R} )$ are isomorphic as H-modules. Let
$M(a,b,c)={\begin{bmatrix}1&a&c\\0&1&b\\0&0&1\end{bmatrix}}$
stand for a general group element of $H_{3}(\mathbb {R} ).$ In the canonical Weyl representation, for every real number h, there is a representation $\rho _{h}$ acting on $L^{2}(\mathbb {R} )$ as
$\rho _{h}(M(a,b,c))\psi (x)=\exp(ibx+ihc)\psi (x+ha)$
for $x\in \mathbb {R} $ and $\psi \in L^{2}(\mathbb {R} ).$
Here, h is Planck's constant. Each such representation is unitarily inequivalent. The corresponding theta representation is:
$M(a,0,0)\to S_{ah}$
$M(0,b,0)\to T_{b/2\pi }$
$M(0,0,c)\to e^{ihc}$
Discrete subgroup
Define the subgroup $\Gamma _{\tau }\subset H_{\tau }$ as
$\Gamma _{\tau }=\{U_{\tau }(1,a,b)\in H_{\tau }:a,b\in \mathbb {Z} \}.$
The Jacobi theta function is defined as
$\vartheta (z;\tau )=\sum _{n=-\infty }^{\infty }\exp(\pi in^{2}\tau +2\pi inz).$
It is an entire function of z that is invariant under $\Gamma _{\tau }.$ This follows from the properties of the theta function:
$\vartheta (z+1;\tau )=\vartheta (z;\tau )$
and
$\vartheta (z+a+b\tau ;\tau )=\exp(-\pi ib^{2}\tau -2\pi ibz)\vartheta (z;\tau )$ ;\tau )=\exp(-\pi ib^{2}\tau -2\pi ibz)\vartheta (z;\tau )}
when a and b are integers. It can be shown that the Jacobi theta is the unique such function.
See also
• Segal–Bargmann space
• Hardy space
References
• David Mumford, Tata Lectures on Theta I (1983), Birkhäuser, Boston ISBN 3-7643-3109-7
| Wikipedia |
Theta constant
In mathematics, a theta constant or Thetanullwert' (German for theta zero value; plural Thetanullwerte) is the restriction θm(τ) = θm(τ,0) of a theta function θm(τ,z) with rational characteristic m to z = 0. The variable τ may be a complex number in the upper half-plane in which case the theta constants are modular forms, or more generally may be an element of a Siegel upper half plane in which case the theta constants are Siegel modular forms. The theta function of a lattice is essentially a special case of a theta constant.
Definition
The theta function θm(τ,z) = θa,b(τ,z)is defined by
$\theta _{a,b}(\tau ,z)=\sum _{\xi \in Z^{n}}\exp \left[\pi {\rm {i}}(\xi +a)\tau (\xi +a)^{t}+2\pi i(\xi +a)(z+b)^{t}\right]$
where
• n is a positive integer, called the genus or rank.
• m = (a,b) is called the characteristic
• a,b are in Rn
• τ is a complex n by n matrix with positive definite imaginary part
• z is in Cn
• t means the transpose of a row vector.
If a,b are in Qn then θa,b(τ,0) is called a theta constant.
Examples
If n = 1 and a and b are both 0 or 1/2, then the functions θa,b(τ,z) are the four Jacobi theta functions, and the functions θa,b(τ,0) are the classical Jacobi theta constants. The theta constant θ1/2,1/2(τ,0) is identically zero, but the other three can be nonzero.
References
• Igusa, Jun-ichi (1972), Theta functions., Die Grundlehren der mathematischen Wissenschaften, vol. 194, New York-Heidelberg: Springer-Verlag, MR 0325625
| Wikipedia |
Margulis lemma
In differential geometry, the Margulis lemma (named after Grigory Margulis) is a result about discrete subgroups of isometries of a non-positively curved Riemannian manifold (e.g. the hyperbolic n-space). Roughly, it states that within a fixed radius, usually called the Margulis constant, the structure of the orbits of such a group cannot be too complicated. More precisely, within this radius around a point all points in its orbit are in fact in the orbit of a nilpotent subgroup (in fact a bounded finite number of such).
The Margulis lemma for manifolds of non-positive curvature
Formal statement
The Margulis lemma can be formulated as follows.[1]
Let $X$ be a simply-connected manifold of non-positive bounded sectional curvature. There exist constants $C,\varepsilon >0$ with the following property. For any discrete subgroup $\Gamma $ of the group of isometries of $X$ and any $x\in X$, if $F_{x}$ is the set:
$F_{x}=\{g\in \Gamma :d(x,gx)<\varepsilon \}$
then the subgroup generated by $F_{x}$ contains a nilpotent subgroup of index less than $C$. Here $d$ is the distance induced by the Riemannian metric.
An immediately equivalent statement can be given as follows: for any subset $F$ of the isometry group, if it satisfies that:
• there exists a $x\in X$ such that $\forall g\in F:d(x,gx)<\varepsilon $;
• the group $\langle F\rangle $ generated by $F$ is discrete
then $\langle F\rangle $ contains a nilpotent subgroup of index $\leq C$.
Margulis constants
The optimal constant $\varepsilon $ in the statement can be made to depend only on the dimension and the lower bound on the curvature; usually it is normalised so that the curvature is between -1 and 0. It is usually called the Margulis constant of the dimension.
One can also consider Margulis constants for specific spaces. For example, there has been an important effort to determine the Margulis constant of the hyperbolic spaces (of constant curvature -1). For example:
• the optimal constant for the hyperbolic plane is equal to $2\operatorname {arsinh} \left({\sqrt {\frac {2\cos(2\pi /7)-1}{8\cos(\pi /7)+7}}}\right)\simeq 0.2629$;[2]
• In general the Margulis constant $\varepsilon _{n}$ for the hyperbolic $n$-space is known to satisfy the bounds:
$c^{-n^{2}}<\varepsilon _{n}<K/{\sqrt {n}}$
for some $0<c<1,K>0$.[3]
Zassenhaus neighbourhoods
A particularly studied family of examples of negatively curved manifolds are given by the symmetric spaces associated to semisimple Lie groups. In this case the Margulis lemma can be given the following, more algebraic formulation which dates back to Hans Zassenhaus.[4]
If $G$ is a semisimple Lie group there exists a neighbourhood $\Omega $ of the identity in $G$ and a $C>0$ such that any discrete subgroup $\Gamma $ which is generated by $\Gamma \cap \Omega $ contains a nilpotent subgroup of index $\leq C$.
Such a neighbourhood $\Omega $ is called a Zassenhaus neighbourhood in $G$. If $G$ is compact this theorem amounts to Jordan's theorem on finite linear groups.
Thick-thin decomposition
Let $M$ be a Riemannian manifold and $\varepsilon >0$. The thin part of $M$ is the subset of points $x\in M$ where the injectivity radius of $M$ at $x$ is less than $\varepsilon $, usually denoted $M_{<\varepsilon }$, and the thick part its complement, usually denoted $M_{\geq \varepsilon }$. There is a tautological decomposition into a disjoint union $M=M_{<\varepsilon }\cup M_{\geq \varepsilon }$.
When $M$ is of negative curvature and $\varepsilon $ is smaller than the Margulis constant for the universal cover ${\widetilde {M}}$, the structure of the components of the thin part is very simple. Let us restrict to the case of hyperbolic manifolds of finite volume. Suppose that $\varepsilon $ is smaller than the Margulis constant for $\mathbb {H} ^{n}$ and let $M$ be a hyperbolic $n$-manifold of finite volume. Then its thin part has two sorts of components:[5]
• Cusps: these are the unbounded components, they are diffeomorphic to a flat $(n-1)$-manifold times a line;
• Margulis tubes: these are neighbourhoods of closed geodesics of length $<\varepsilon $ on $M$. They are bounded and (if $M$ is orientable) diffeomorphic to a circle times a $(n-1)$-disc.
In particular, a complete finite-volume hyperbolic manifold is always diffeomorphic to the interior of a compact manifold (possibly with empty boundary).
Other applications
The Margulis lemma is an important tool in the study of manifolds of negative curvature. Besides the thick-thin decomposition some other applications are:
• The collar lemma: this is a more precise version of the description of the compact components of the thin parts. It states that any closed geodesic of length $\ell <\varepsilon $ on an hyperbolic surface is contained in an embedded cylinder of diameter of order $\ell ^{-1}$.
• The Margulis lemma gives an immediate qualitative solution to the problem of minimal covolume among hyperbolic manifolds: since the volume of a Margulis tube can be seen to be bounded below by a constant depending only on the dimension, it follows that there exists a positive infimum to the volumes of hyperbolic n-manifolds for any n.[6]
• The existence of Zassenhaus neighbourhoods is a key ingredient in the proof of the Kazhdan–Margulis theorem.
• One can recover the Jordan–Schur theorem as a corollary to the existence of Zassenhaus neighbourhoods.
See also
• Jorgensen's inequality gives a quantitative statement for discrete subgroups of the isometry group $\mathrm {PGL} _{2}(\mathbb {C} )$ of the 3-dimensional hyperbolic space.
Notes
1. Ballmann, Gromov & Schroeder, Theorem 9.5. sfn error: no target: CITEREFBallmannGromovSchroeder (help)
2. Yamada, A. (1981). "On Marden's universal constant of Fuchsian groups". Kodai Math. J. 4 (2): 266–277. doi:10.2996/kmj/1138036373.
3. Belolipetsky, Mikhail (2014). "Hyperbolic orbifolds of small volume". Proceedings of ICM 2014. Kyung Moon SA. arXiv:1402.5394.
4. Raghunatan, 1972 & Definition 8.22. sfn error: no target: CITEREFRaghunatan1972Definition_8.22 (help)
5. Thurston 1998, Chapter 4.5. sfn error: no target: CITEREFThurston1998 (help)
6. Ratcliffe 2006, p. 666.
References
• Ballmann, Werner; Gromov, Mikhail; Schroeder, Viktor (1985). Manifolds of Nonpositive Curvature. Birkhâuser.
• Raghunathan, M. S. (1972). Discrete subgroups of Lie groups. Ergebnisse de Mathematik und ihrer Grenzgebiete. Springer-Verlag. MR 0507234.
• Ratcliffe, John (2006). Foundations of hyperbolic manifolds, Second edition. Springer. pp. xii+779. ISBN 978-0387-33197-3.
• Thurston, William (1997). Three-dimensional geometry and topology. Vol. 1. Princeton University Press.
| Wikipedia |
Thickness (graph theory)
In graph theory, the thickness of a graph G is the minimum number of planar graphs into which the edges of G can be partitioned. That is, if there exists a collection of k planar graphs, all having the same set of vertices, such that the union of these planar graphs is G, then the thickness of G is at most k.[1][2] In other words, the thickness of a graph is the minimum number of planar subgraphs whose union equals to graph G.[3]
Thus, a planar graph has thickness 1. Graphs of thickness 2 are called biplanar graphs. The concept of thickness originates in the Earth–Moon problem on the chromatic number of biplanar graphs, posed in 1959 by Gerhard Ringel,[4] and on a related 1962 conjecture of Frank Harary: For any graph on 9 points, either itself or its complementary graph is non-planar. The problem is equivalent to determining whether the complete graph K9 is biplanar (it is not, and the conjecture is true).[5] A comprehensive[3] survey on the state of the arts of the topic as of 1998 was written by Petra Mutzel, Thomas Odenthal and Mark Scharbrodt.[2]
Specific graphs
The thickness of the complete graph on n vertices, Kn, is
$\left\lfloor {\frac {n+7}{6}}\right\rfloor ,$
except when n = 9, 10 for which the thickness is three.[6][7]
With some exceptions, the thickness of a complete bipartite graph Ka,b is generally:[8][9]
$\left\lceil {\frac {ab}{2(a+b-2)}}\right\rceil .$
Properties
Every forest is planar, and every planar graph can be partitioned into at most three forests. Therefore, the thickness of any graph G is at most equal to the arboricity of the same graph (the minimum number of forests into which it can be partitioned) and at least equal to the arboricity divided by three.[2][10]
The graphs of maximum degree $d$ have thickness at most $\lceil d/2\rceil $.[11] This cannot be improved: for a $d$-regular graph with girth at least $2d$, the high girth forces any planar subgraph to be sparse, causing its thickness to be exactly $\lceil d/2\rceil $.[12]
Graphs of thickness $t$ with $n$ vertices have at most $t(3n-6)$ edges. Because this gives them average degree less than $6t$, their degeneracy is at most $6t-1$ and their chromatic number is at most $6t$. Here, the degeneracy can be defined as the maximum, over subgraphs of the given graph, of the minimum degree within the subgraph. In the other direction, if a graph has degeneracy $D$ then its arboricity and thickness are at most $D$. One can find an ordering of the vertices of the graph in which each vertex has at most $D$ neighbors that come later than it in the ordering, and assigning these edges to $D$ distinct subgraphs produces a partition of the graph into $D$ trees, which are planar graphs.
Even in the case $t=2$, the precise value of the chromatic number is unknown; this is Gerhard Ringel's Earth–Moon problem. An example of Thom Sulanke shows that, for $t=2$, at least 9 colors are needed.[13]
Related problems
Thickness is closely related to the problem of simultaneous embedding.[14] If two or more planar graphs all share the same vertex set, then it is possible to embed all these graphs in the plane, with the edges drawn as curves, so that each vertex has the same position in all the different drawings. However, it may not be possible to construct such a drawing while keeping the edges drawn as straight line segments.
A different graph invariant, the rectilinear thickness or geometric thickness of a graph G, counts the smallest number of planar graphs into which G can be decomposed subject to the restriction that all of these graphs can be drawn simultaneously with straight edges. The book thickness adds an additional restriction, that all of the vertices be drawn in convex position, forming a circular layout of the graph. However, in contrast to the situation for arboricity and degeneracy, no two of these three thickness parameters are always within a constant factor of each other.[15]
Computational complexity
It is NP-hard to compute the thickness of a given graph, and NP-complete to test whether the thickness is at most two.[16] However, the connection to arboricity allows the thickness to be approximated to within an approximation ratio of 3 in polynomial time.
References
1. Tutte, W. T. (1963), "The thickness of a graph", Indag. Math., 66: 567–577, doi:10.1016/S1385-7258(63)50055-9, MR 0157372.
2. Mutzel, Petra; Odenthal, Thomas; Scharbrodt, Mark (1998), "The thickness of graphs: a survey" (PDF), Graphs and Combinatorics, 14 (1): 59–73, CiteSeerX 10.1.1.34.6528, doi:10.1007/PL00007219, MR 1617664, S2CID 31670574.
3. Christian A. Duncan, On Graph Thickness, Geometric Thickness, and Separator Theorems, CCCG 2009, Vancouver, BC, August 17–19, 2009
4. Ringel, Gerhard (1959), Färbungsprobleme auf Flächen und Graphen, Mathematische Monographien, vol. 2, Berlin: VEB Deutscher Verlag der Wissenschaften, MR 0109349
5. Mäkinen, Erkki; Poranen, Timo (2012), "An annotated bibliography on the thickness, outerthickness, and arboricity of a graph", Missouri J. Math. Sci., 24 (1): 76–87, doi:10.35834/mjms/1337950501, S2CID 117703458
6. Mutzel, Odenthal & Scharbrodt (1998), Theorem 3.2.
7. Alekseev, V. B.; Gončakov, V. S. (1976), "The thickness of an arbitrary complete graph", Mat. Sb., New Series, 101 (143): 212–230, Bibcode:1976SbMat..30..187A, doi:10.1070/SM1976v030n02ABEH002267, MR 0460162.
8. Mutzel, Odenthal & Scharbrodt (1998), Theorem 3.4.
9. Beineke, Lowell W.; Harary, Frank; Moon, John W. (1964), "On the thickness of the complete bipartite graph", Proc. Cambridge Philos. Soc., 60 (1): 1–5, Bibcode:1964PCPS...60....1B, doi:10.1017/s0305004100037385, MR 0158388, S2CID 122829092.
10. Dean, Alice M.; Hutchinson, Joan P.; Scheinerman, Edward R. (1991), "On the thickness and arboricity of a graph", Journal of Combinatorial Theory, Series B, 52 (1): 147–151, doi:10.1016/0095-8956(91)90100-X, MR 1109429.
11. Halton, John H. (1991), "On the thickness of graphs of given degree", Information Sciences, 54 (3): 219–238, doi:10.1016/0020-0255(91)90052-V, MR 1079441
12. Sýkora, Ondrej; Székely, László A.; Vrt'o, Imrich (2004), "A note on Halton's conjecture", Information Sciences, 164 (1–4): 61–64, doi:10.1016/j.ins.2003.06.008, MR 2076570
13. Gethner, Ellen (2018), "To the Moon and beyond", in Gera, Ralucca; Haynes, Teresa W.; Hedetniemi, Stephen T. (eds.), Graph Theory: Favorite Conjectures and Open Problems, II, Problem Books in Mathematics, Springer International Publishing, pp. 115–133, doi:10.1007/978-3-319-97686-0_11, MR 3930641
14. Brass, Peter; Cenek, Eowyn; Duncan, Christian A.; Efrat, Alon; Erten, Cesim; Ismailescu, Dan P.; Kobourov, Stephen G.; Lubiw, Anna; Mitchell, Joseph S. B. (2007), "On simultaneous planar graph embeddings", Computational Geometry, 36 (2): 117–130, doi:10.1016/j.comgeo.2006.05.006, MR 2278011.
15. Eppstein, David (2004), "Separating thickness from geometric thickness", Towards a theory of geometric graphs, Contemp. Math., vol. 342, Amer. Math. Soc., Providence, RI, pp. 75–86, arXiv:math/0204252, doi:10.1090/conm/342/06132, MR 2065254.
16. Mansfield, Anthony (1983), "Determining the thickness of graphs is NP-hard", Mathematical Proceedings of the Cambridge Philosophical Society, 93 (1): 9–23, Bibcode:1983MPCPS..93....9M, doi:10.1017/S030500410006028X, MR 0684270, S2CID 122028023.
| Wikipedia |
Thiele's interpolation formula
In mathematics, Thiele's interpolation formula is a formula that defines a rational function $f(x)$ from a finite set of inputs $x_{i}$ and their function values $f(x_{i})$. The problem of generating a function whose graph passes through a given set of function values is called interpolation. This interpolation formula is named after the Danish mathematician Thorvald N. Thiele. It is expressed as a continued fraction, where ρ represents the reciprocal difference:
$f(x)=f(x_{1})+{\cfrac {x-x_{1}}{\rho (x_{1},x_{2})+{\cfrac {x-x_{2}}{\rho _{2}(x_{1},x_{2},x_{3})-f(x_{1})+{\cfrac {x-x_{3}}{\rho _{3}(x_{1},x_{2},x_{3},x_{4})-\rho (x_{1},x_{2})+\cdots }}}}}}$
Be careful that the $n$-th level in Thiele's interpolation formula is
$\rho _{n}(x_{1},x_{2},\cdots ,x_{n+1})-\rho _{n-2}(x_{1},x_{2},\cdots ,x_{n-1})+{\cfrac {x-x_{n}}{\rho _{n+1}(x_{1},x_{2},\cdots ,x_{n+2})-\rho _{n-1}(x_{1},x_{2},\cdots ,x_{n})+\cdots }},$
while the $n$-th reciprocal difference is defined to be
$\rho _{n}(x_{1},x_{2},\ldots ,x_{n+1})={\frac {x_{1}-x_{n+1}}{\rho _{n-1}(x_{1},x_{2},\ldots ,x_{n})-\rho _{n-1}(x_{2},x_{3},\ldots ,x_{n+1})}}+\rho _{n-2}(x_{2},\ldots ,x_{n})$.
The two $\rho _{n-2}$ terms are different and can not be cancelled!
References
• Weisstein, Eric W. "Thiele's Interpolation Formula". MathWorld.
| Wikipedia |
Thierry Aubin
Thierry Aubin (6 May 1942 – 21 March 2009) was a French mathematician who worked at the Centre de Mathématiques de Jussieu, and was a leading expert on Riemannian geometry and non-linear partial differential equations. His fundamental contributions to the theory of the Yamabe equation led, in conjunction with results of Trudinger and Schoen, to a proof of the Yamabe Conjecture: every compact Riemannian manifold can be conformally rescaled to produce a manifold of constant scalar curvature. Along with Yau, he also showed that Kähler manifolds with negative first Chern classes always admit Kähler–Einstein metrics, a result closely related to the Calabi conjecture. The latter result, established by Yau, provides the largest class of known examples of compact Einstein manifolds. Aubin was the first mathematician to propose the Cartan–Hadamard conjecture.
Thierry Aubin
Thierry Aubin in 1976
(photo by George Bergman)
Born(1942-05-06)6 May 1942
Died21 March 2009(2009-03-21) (aged 66)
Nationality France
Scientific career
FieldsMathematics
InstitutionsPierre and Marie Curie University
Doctoral advisorAndré Lichnerowicz
Aubin was a visiting scholar at the Institute for Advanced Study in 1979.[1] He was elected to the Académie des sciences in 2003.
Research
In 1970, Aubin established that any closed smooth manifold of dimension larger than two has a Riemannian metric of negative scalar curvature. Furthermore, he proved that a Riemannian metric of nonnegative Ricci curvature can be deformed to positive Ricci curvature, provided that its Ricci curvature is strictly positive at one point.
In the same year, Aubin introduced an approach to the Calabi conjecture, in the field of Kähler geometry, via the calculus of variations. Later, in 1976, Aubin established the existence of Kähler–Einstein metrics on Kähler manifolds whose first Chern class is negative.[2] Independently, Shing-Tung Yau proved the more powerful Calabi conjecture, which concerns the general problem of prescribing the Ricci curvature of a Kähler metric, via non-variational methods. As such, the existence of Kähler–Einstein metrics with negative first Chern class is often called the Aubin–Yau theorem. After learning Yau's techniques from Jerry Kazdan, Aubin found some simplifications and modifications of his work, along with Kazdan and Jean-Pierre Bourguignon.[3]
Aubin made a number of fundamental contributions to the study of Sobolev spaces on Riemannian manifolds. He established Riemannian formulations of many classical results for Sobolev spaces, such as the equivalence of various definitions, the density of various subclasses of functions, and the standard embedding theorems.[4] In one of Aubin's best-known works, the analysis of the optimal constant in the Sobolev embedding theorem was carried out. Along with similar results for the Moser–Trudinger inequality, Aubin later proved improvements of the optimal constants when the functions are assumed to satisfy certain orthogonality constraints.[5]
Such results are naturally applicable to many problems in the field of geometric analysis. Aubin considered the Yamabe problem on conformal deformation to constant scalar curvature, which Yamabe had reduced to a problem in the calculus of variations. Following prior work of Neil Trudinger, Aubin was able to resolve the problem in high dimensions under the condition that the Weyl curvature is nonzero at some point. The key of Aubin's analysis is essentially local, with an estimate on the geometry of the Green's function based on the Weyl curvature. The more subtle case of locally conformally flat manifolds, along with the low-dimensional case, was later established by Richard Schoen as an application of Schoen and Yau's positive mass theorem.
All of the results outlined here, along with many others, were absorbed into Aubin's book Some Nonlinear Problems in Riemannian Geometry, which has become a basic part of the research literature.[6]
Major Publications
Articles. Aubin was the author of around sixty research papers. The following, among the best-known, are outlined above.
• Aubin, Thierry (1970). "Métriques riemanniennes et courbure". Journal of Differential Geometry. 4 (4): 383–424. doi:10.4310/jdg/1214429638. MR 0279731. Zbl 0212.54102.
• Aubin, Thierry (1976a). "Espaces de Sobolev sur les variétés riemanniennes". Bulletin des Sciences Mathématiques. 2e Série. 100 (2): 149–173. MR 0488125. Zbl 0328.46030.
• Aubin, Thierry (1976b). "Problèmes isopérimétriques et espaces de Sobolev". Journal of Differential Geometry. 11 (4): 573–598. doi:10.4310/jdg/1214433725. MR 0448404. Zbl 0371.46011.
• Aubin, Thierry (1976c). "Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire". Journal de Mathématiques Pures et Appliquées. Neuvième Série. 55 (3): 269–296. MR 0431287. Zbl 0336.53033.
• Aubin, Thierry (1976d). "Équations du type Monge–Ampère sur les variétés kähleriennes compactes". Comptes Rendus de l'Académie des Sciences, Série A. 283 (3): 119–121. MR 0433520. Zbl 0333.53040.
• Aubin, Thierry (1978). "Équations du type Monge–Ampère sur les variétés kählériennes compactes". Bulletin des Sciences Mathématiques. 2e Série. 102 (1): 63–95. MR 0494932. Zbl 0374.53022.
• Aubin, Thierry (1979). "Meilleures constantes dans le théorème d'inclusion de Sobolev et un théorème de Fredholm non linéaire pour la transformation conforme de la courbure scalaire". Journal of Functional Analysis. 32 (2): 148–174. doi:10.1016/0022-1236(79)90052-1. MR 0534672. Zbl 0411.46019.
Books
• Aubin, Thierry (1998). Some nonlinear problems in Riemannian geometry. Springer Monographs in Mathematics. Berlin: Springer-Verlag. doi:10.1007/978-3-662-13006-3. ISBN 3-540-60752-8. MR 1636569. Zbl 0896.53003.
Expansion of:
Aubin, Thierry (1982). Nonlinear analysis on manifolds. Monge–Ampère equations. Grundlehren der mathematischen Wissenschaften. Vol. 252. New York: Springer-Verlag. doi:10.1007/978-1-4612-5734-9. ISBN 0-387-90704-1. MR 0681859. Zbl 0512.53044.
• Aubin, Thierry (2001). A course in differential geometry. Graduate Studies in Mathematics. Vol. 27. Providence, RI: American Mathematical Society. doi:10.1090/gsm/027. ISBN 0-8218-2709-X. MR 1799532. Zbl 0966.53001.
References
1. Institute for Advanced Study: A Community of Scholars Archived 2013-01-06 at the Wayback Machine
2. Aubin 1976d.
3. Aubin 1978.
4. Aubin 1976a.
5. Aubin 1979.
6. This book is an expansion of Aubin's prior book Nonlinear analysis on manifolds. Monge–Ampère equations.
External links
• Thierry Aubin at the Mathematics Genealogy Project
• Obituary on the SMF Gazette
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| Wikipedia |
Delaunay triangulation
In mathematics and computational geometry, a Delaunay triangulation (also known as a Delone triangulation) for a given set P of discrete points in a general position is a triangulation DT(P) such that no point in P is inside the circumcircle of any triangle in DT(P). Delaunay triangulations maximize the minimum of all the angles of the triangles in the triangulation; they tend to avoid sliver triangles. The triangulation is named after Boris Delaunay for his work on this topic from 1934.[1]
For broader coverage of this topic, see Triangulation (geometry).
For a set of points on the same line there is no Delaunay triangulation (the notion of triangulation is degenerate for this case). For four or more points on the same circle (e.g., the vertices of a rectangle) the Delaunay triangulation is not unique: each of the two possible triangulations that split the quadrangle into two triangles satisfies the "Delaunay condition", i.e., the requirement that the circumcircles of all triangles have empty interiors.
By considering circumscribed spheres, the notion of Delaunay triangulation extends to three and higher dimensions. Generalizations are possible to metrics other than Euclidean distance. However, in these cases a Delaunay triangulation is not guaranteed to exist or be unique.
Relationship with the Voronoi diagram
The Delaunay triangulation with all the circumcircles and their centers (in red).
Connecting the centers of the circumcircles produces the Voronoi diagram (in red).
The Delaunay triangulation of a discrete point set P in general position corresponds to the dual graph of the Voronoi diagram for P. The circumcenters of Delaunay triangles are the vertices of the Voronoi diagram. In the 2D case, the Voronoi vertices are connected via edges, that can be derived from adjacency-relationships of the Delaunay triangles: If two triangles share an edge in the Delaunay triangulation, their circumcenters are to be connected with an edge in the Voronoi tesselation.
Special cases where this relationship does not hold, or is ambiguous, include cases like:
• Three or more collinear points, where the circumcircles are of infinite radii.
• Four or more points on a perfect circle, where the triangulation is ambiguous and all circumcenters are trivially identical.
• Edges of the Voronoi diagram going to infinity are not defined by this relation in case of a finite set P. If the Delaunay triangulation is calculated using the Bowyer–Watson algorithm then the circumcenters of triangles having a common vertex with the "super" triangle should be ignored. Edges going to infinity start from a circumcenter and they are perpendicular to the common edge between the kept and ignored triangle.
d-dimensional Delaunay
For a set P of points in the (d-dimensional) Euclidean space, a Delaunay triangulation is a triangulation DT(P) such that no point in P is inside the circum-hypersphere of any d-simplex in DT(P). It is known[1] that there exists a unique Delaunay triangulation for P if P is a set of points in general position; that is, the affine hull of P is d-dimensional and no set of d + 2 points in P lie on the boundary of a ball whose interior does not intersect P.
The problem of finding the Delaunay triangulation of a set of points in d-dimensional Euclidean space can be converted to the problem of finding the convex hull of a set of points in (d + 1)-dimensional space. This may be done by giving each point p an extra coordinate equal to |p|2, thus turning it into a hyper-paraboloid (this is termed "lifting"); taking the bottom side of the convex hull (as the top end-cap faces upwards away from the origin, and must be discarded); and mapping back to d-dimensional space by deleting the last coordinate. As the convex hull is unique, so is the triangulation, assuming all facets of the convex hull are simplices. Nonsimplicial facets only occur when d + 2 of the original points lie on the same d-hypersphere, i.e., the points are not in general position.[2]
Properties
Let n be the number of points and d the number of dimensions.
• The union of all simplices in the triangulation is the convex hull of the points.
• The Delaunay triangulation contains $O\left(n^{\lceil d/2\rceil }\right)$ simplices.[3]
• In the plane (d = 2), if there are b vertices on the convex hull, then any triangulation of the points has at most 2n – 2 – b triangles, plus one exterior face (see Euler characteristic).
• If points are distributed according to a Poisson process in the plane with constant intensity, then each vertex has on average six surrounding triangles. More generally for the same process in d dimensions the average number of neighbors is a constant depending only on d.[4]
• In the plane, the Delaunay triangulation maximizes the minimum angle. Compared to any other triangulation of the points, the smallest angle in the Delaunay triangulation is at least as large as the smallest angle in any other. However, the Delaunay triangulation does not necessarily minimize the maximum angle.[5] The Delaunay triangulation also does not necessarily minimize the length of the edges.
• A circle circumscribing any Delaunay triangle does not contain any other input points in its interior.
• If a circle passing through two of the input points doesn't contain any other input points in its interior, then the segment connecting the two points is an edge of a Delaunay triangulation of the given points.
• Each triangle of the Delaunay triangulation of a set of points in d-dimensional spaces corresponds to a facet of convex hull of the projection of the points onto a (d + 1)-dimensional paraboloid, and vice versa.
• The closest neighbor b to any point p is on an edge bp in the Delaunay triangulation since the nearest neighbor graph is a subgraph of the Delaunay triangulation.
• The Delaunay triangulation is a geometric spanner: In the plane (d = 2), the shortest path between two vertices, along Delaunay edges, is known to be no longer than 1.998 times the Euclidean distance between them.[6]
Visual Delaunay definition: Flipping
From the above properties an important feature arises: Looking at two triangles △ABD, △BCD with the common edge BD (see figures), if the sum of the angles α + γ ≤ 180°, the triangles meet the Delaunay condition.
This is an important property because it allows the use of a flipping technique. If two triangles do not meet the Delaunay condition, switching the common edge BD for the common edge AC produces two triangles that do meet the Delaunay condition:
• This triangulation does not meet the Delaunay condition (the sum of α and γ is bigger than 180°).
• This pair of triangles does not meet the Delaunay condition (there is a point within the interior of the circumcircle).
• Flipping the common edge produces a valid Delaunay triangulation for the four points.
This operation is called a flip, and can be generalised to three and higher dimensions.[7]
Algorithms
Many algorithms for computing Delaunay triangulations rely on fast operations for detecting when a point is within a triangle's circumcircle and an efficient data structure for storing triangles and edges. In two dimensions, one way to detect if point D lies in the circumcircle of A, B, C is to evaluate the determinant:[8]
${\begin{aligned}&{\begin{vmatrix}A_{x}&A_{y}&A_{x}^{2}+A_{y}^{2}&1\\B_{x}&B_{y}&B_{x}^{2}+B_{y}^{2}&1\\C_{x}&C_{y}&C_{x}^{2}+C_{y}^{2}&1\\D_{x}&D_{y}&D_{x}^{2}+D_{y}^{2}&1\end{vmatrix}}\\[8pt]={}&{\begin{vmatrix}A_{x}-D_{x}&A_{y}-D_{y}&(A_{x}-D_{x})^{2}+(A_{y}-D_{y})^{2}\\B_{x}-D_{x}&B_{y}-D_{y}&(B_{x}-D_{x})^{2}+(B_{y}-D_{y})^{2}\\C_{x}-D_{x}&C_{y}-D_{y}&(C_{x}-D_{x})^{2}+(C_{y}-D_{y})^{2}\end{vmatrix}}>0\end{aligned}}$
When A, B, C are sorted in a counterclockwise order, this determinant is positive only if D lies inside the circumcircle.
Flip algorithms
As mentioned above, if a triangle is non-Delaunay, we can flip one of its edges. This leads to a straightforward algorithm: construct any triangulation of the points, and then flip edges until no triangle is non-Delaunay. Unfortunately, this can take Ω(n2) edge flips.[9] While this algorithm can be generalised to three and higher dimensions, its convergence is not guaranteed in these cases, as it is conditioned to the connectedness of the underlying flip graph: this graph is connected for two-dimensional sets of points, but may be disconnected in higher dimensions.[7]
Incremental
The most straightforward way of efficiently computing the Delaunay triangulation is to repeatedly add one vertex at a time, retriangulating the affected parts of the graph. When a vertex v is added, we split in three the triangle that contains v, then we apply the flip algorithm. Done naïvely, this will take O(n) time: we search through all the triangles to find the one that contains v, then we potentially flip away every triangle. Then the overall runtime is O(n2).
If we insert vertices in random order, it turns out (by a somewhat intricate proof) that each insertion will flip, on average, only O(1) triangles – although sometimes it will flip many more.[10] This still leaves the point location time to improve. We can store the history of the splits and flips performed: each triangle stores a pointer to the two or three triangles that replaced it. To find the triangle that contains v, we start at a root triangle, and follow the pointer that points to a triangle that contains v, until we find a triangle that has not yet been replaced. On average, this will also take O(log n) time. Over all vertices, then, this takes O(n log n) time.[11] While the technique extends to higher dimension (as proved by Edelsbrunner and Shah[12]), the runtime can be exponential in the dimension even if the final Delaunay triangulation is small.
The Bowyer–Watson algorithm provides another approach for incremental construction. It gives an alternative to edge flipping for computing the Delaunay triangles containing a newly inserted vertex.
Unfortunately the flipping-based algorithms are generally hard to parallelize, since adding some certain point (e.g. the center point of a wagon wheel) can lead to up to O(n) consecutive flips. Blelloch et al.[13] proposed another version of incremental algorithm based on rip-and-tent, which is practical and highly parallelized with polylogarithmic span.
Divide and conquer
A divide and conquer algorithm for triangulations in two dimensions was developed by Lee and Schachter and improved by Guibas and Stolfi[14][15] and later by Dwyer.[16] In this algorithm, one recursively draws a line to split the vertices into two sets. The Delaunay triangulation is computed for each set, and then the two sets are merged along the splitting line. Using some clever tricks, the merge operation can be done in time O(n), so the total running time is O(n log n).[17]
For certain types of point sets, such as a uniform random distribution, by intelligently picking the splitting lines the expected time can be reduced to O(n log log n) while still maintaining worst-case performance.
A divide and conquer paradigm to performing a triangulation in d dimensions is presented in "DeWall: A fast divide and conquer Delaunay triangulation algorithm in Ed" by P. Cignoni, C. Montani, R. Scopigno.[18]
The divide and conquer algorithm has been shown to be the fastest DT generation technique sequentially.[19][20]
Sweephull
Sweephull[21] is a hybrid technique for 2D Delaunay triangulation that uses a radially propagating sweep-hull, and a flipping algorithm. The sweep-hull is created sequentially by iterating a radially-sorted set of 2D points, and connecting triangles to the visible part of the convex hull, which gives a non-overlapping triangulation. One can build a convex hull in this manner so long as the order of points guarantees no point would fall within the triangle. But, radially sorting should minimize flipping by being highly Delaunay to start. This is then paired with a final iterative triangle flipping step.
Applications
See also: Voronoi diagram § Applications
The Euclidean minimum spanning tree of a set of points is a subset of the Delaunay triangulation of the same points,[22] and this can be exploited to compute it efficiently.
For modelling terrain or other objects given a point cloud, the Delaunay triangulation gives a nice set of triangles to use as polygons in the model. In particular, the Delaunay triangulation avoids narrow triangles (as they have large circumcircles compared to their area). See triangulated irregular network.
Delaunay triangulations can be used to determine the density or intensity of points samplings by means of the Delaunay tessellation field estimator (DTFE).
Delaunay triangulations are often used to generate meshes for space-discretised solvers such as the finite element method and the finite volume method of physics simulation, because of the angle guarantee and because fast triangulation algorithms have been developed. Typically, the domain to be meshed is specified as a coarse simplicial complex; for the mesh to be numerically stable, it must be refined, for instance by using Ruppert's algorithm.
The increasing popularity of finite element method and boundary element method techniques increases the incentive to improve automatic meshing algorithms. However, all of these algorithms can create distorted and even unusable grid elements. Fortunately, several techniques exist which can take an existing mesh and improve its quality. For example, smoothing (also referred to as mesh refinement) is one such method, which repositions nodes to minimize element distortion. The stretched grid method allows the generation of pseudo-regular meshes that meet the Delaunay criteria easily and quickly in a one-step solution.
Constrained Delaunay triangulation has found applications in path planning in automated driving and topographic surveying. [23]
See also
• Beta skeleton
• Centroidal Voronoi tessellation
• Convex hull algorithms
• Delaunay refinement
• Delone set – also known as a Delaunay set
• Disordered hyperuniformity
• Farthest-first traversal – incremental Voronoi insertion
• Gabriel graph
• Giant's Causeway
• Gradient pattern analysis
• Hamming bound – sphere-packing bound
• Linde–Buzo–Gray algorithm
• Lloyd's algorithm – Voronoi iteration
• Meyer set
• Pisot–Vijayaraghavan number
• Pitteway triangulation
• Plesiohedron
• Quasicrystal
• Quasitriangulation
• Salem number
• Steiner point (triangle)
• Triangle mesh
• Urquhart graph
• Voronoi diagram
References
1. Delaunay, Boris (1934). "Sur la sphère vide". Bulletin de l'Académie des Sciences de l'URSS, Classe des Sciences Mathématiques et Naturelles. 6: 793–800.
2. Fukuda, Komei. "Frequently Asked Questions in Polyhedral Computation". www.cs.mcgill.ca. Retrieved 29 October 2018.
3. Seidel, Raimund (1995). "The upper bound theorem for polytopes: an easy proof of its asymptotic version". Computational Geometry. 5 (2): 115–116. doi:10.1016/0925-7721(95)00013-Y.
4. Meijering, J. L. (1953), "Interface area, edge length, and number of vertices in crystal aggregates with random nucleation" (PDF), Philips Research Reports, 8: 270–290, archived from the original (PDF) on 2017-03-08. As cited by Dwyer, Rex A. (1991), "Higher-dimensional Voronoĭ diagrams in linear expected time", Discrete and Computational Geometry, 6 (4): 343–367, doi:10.1007/BF02574694, MR 1098813.
5. Edelsbrunner, Herbert; Tan, Tiow Seng; Waupotitsch, Roman (1992), "An O(n2 log n) time algorithm for the minmax angle triangulation" (PDF), SIAM Journal on Scientific and Statistical Computing, 13 (4): 994–1008, CiteSeerX 10.1.1.66.2895, doi:10.1137/0913058, MR 1166172, archived from the original (PDF) on 2017-02-09, retrieved 2017-10-24.
6. Xia, Ge (2013), "The stretch factor of the Delaunay triangulation is less than 1.998", SIAM Journal on Computing, 42 (4): 1620–1659, arXiv:1103.4361, doi:10.1137/110832458, MR 3082502, S2CID 6646528
7. De Loera, Jesús A.; Rambau, Jörg; Santos, Francisco (2010). Triangulations, Structures for Algorithms and Applications. Algorithms and Computation in Mathematics. Vol. 25. Springer.
8. Guibas, Leonidas; Stolfi, Jorge (1985). "Primitives for the manipulation of general subdivisions and the computation of Voronoi". ACM Transactions on Graphics. 4 (2): 74–123. doi:10.1145/282918.282923. S2CID 52852815.
9. Hurtado, F.; Noy, M.; Urrutia, J. (1999). "Flipping Edges in Triangulations". Discrete & Computational Geometry. 22 (3): 333–346. doi:10.1007/PL00009464.
10. Guibas, Leonidas J.; Knuth, Donald E.; Sharir, Micha (1992). "Randomized incremental construction of Delaunay and Voronoi diagrams". Algorithmica. 7 (1–6): 381–413. doi:10.1007/BF01758770. S2CID 3770886.
11. de Berg, Mark; Otfried Cheong; Marc van Kreveld; Mark Overmars (2008). Computational Geometry: Algorithms and Applications (PDF). Springer-Verlag. ISBN 978-3-540-77973-5. Archived from the original (PDF) on 2009-10-28. Retrieved 2010-02-23.
12. Edelsbrunner, Herbert; Shah, Nimish (1996). "Incremental Topological Flipping Works for Regular Triangulations". Algorithmica. 15 (3): 223–241. doi:10.1007/BF01975867. S2CID 12976796.
13. Blelloch, Guy; Gu, Yan; Shun, Julian; and Sun, Yihan. Parallelism in Randomized Incremental Algorithms Archived 2018-04-25 at the Wayback Machine. SPAA 2016. doi:10.1145/2935764.2935766.
14. Guibas, Leonidas; Stolfi, Jorge (April 1985). "Primitives for the manipulation of general subdivisions and the computation of Voronoi". ACM Transactions on Graphics. 4 (2): 74–123. doi:10.1145/282918.282923. S2CID 52852815.
15. "COMPUTING CONSTRAINED DELAUNAY TRIANGULATIONS IN THE PLANE". www.geom.uiuc.edu. Archived from the original on 22 September 2017. Retrieved 25 April 2018.
16. Dwyer, Rex A. (November 1987). "A faster divide-and-conquer algorithm for constructing delaunay triangulations". Algorithmica. 2 (1–4): 137–151. doi:10.1007/BF01840356. S2CID 10828441.
17. Leach, G. (June 1992). "Improving Worst-Case Optimal Delaunay Triangulation Algorithms". 4th Canadian Conference on Computational Geometry. CiteSeerX 10.1.1.56.2323.
18. Cignoni, P.; C. Montani; R. Scopigno (1998). "DeWall: A fast divide and conquer Delaunay triangulation algorithm in Ed". Computer-Aided Design. 30 (5): 333–341. doi:10.1016/S0010-4485(97)00082-1.
19. A Comparison of Sequential Delaunay Triangulation Algorithms "Archived copy" (PDF). Archived from the original (PDF) on 2012-03-08. Retrieved 2010-08-18.{{cite web}}: CS1 maint: archived copy as title (link)
20. "Triangulation Algorithms and Data Structures". www.cs.cmu.edu. Archived from the original on 10 October 2017. Retrieved 25 April 2018.
21. "S-hull" (PDF). s-hull.org. Archived (PDF) from the original on 2013-10-27. Retrieved 25 April 2018.
22. Franz Aurenhammer; Rolf Klein; Der-tsai Lee (26 June 2013). Voronoi Diagrams And Delaunay Triangulations. World Scientific Publishing Company. pp. 197–. ISBN 978-981-4447-65-2.
23. Sterling J Anderson; Sisir B. Karumanchi; Karl Iagnemma (5 July 2012). "Constraint-based planning and control for safe, semi-autonomous operation of vehicles" (PDF). 2012 IEEE Intelligent Vehicles Symposium. IEEE. doi:10.1109/IVS.2012.6232153. Archived from the original (PDF) on 28 February 2019. Retrieved 27 February 2019.
External links
See also: Voronoi diagram § Software
• Delaunay triangulation in CGAL, the Computational Geometry Algorithms Library:
• Mariette Yvinec. 2D Triangulation. Retrieved April 2010.
• Pion, Sylvain; Teillaud, Monique. 3D Triangulations. Retrieved April 2010.
• Hornus, Samuel; Devillers, Olivier; Jamin, Clément. dD Triangulations.
• Hert, Susan; Seel, Michael. dD Convex Hulls and Delaunay Triangulations. Retrieved April 2010.
• "Poly2Tri: Incremental constrained Delaunay triangulation. Open source C++ implementation. Retrieved April 2019.
• "Divide & Conquer Delaunay triangulation construction". Open source C99 implementation. Retrieved April 2019.
• "CDT: Constrained Delaunay Triangulation in C++". Open source C++ implementation. Retrieved August 2022.
| Wikipedia |
Posetal category
In mathematics, specifically category theory, a posetal category, or thin category,[1] is a category whose homsets each contain at most one morphism.[2] As such, a posetal category amounts to a preordered class (or a preordered set, if its objects form a set). As suggested by the name, the further requirement that the category be skeletal is often assumed for the definition of "posetal"; in the case of a category that is posetal, being skeletal is equivalent to the requirement that the only isomorphisms are the identity morphisms, equivalently that the preordered class satisfies antisymmetry and hence, if a set, is a poset.
All diagrams commute in a posetal category. When the commutative diagrams of a category are interpreted as a typed equational theory whose objects are the types, a codiscrete posetal category corresponds to an inconsistent theory understood as one satisfying the axiom x = y at all types.
Viewing a 2-category as an enriched category whose hom-objects are categories, the hom-objects of any extension of a posetal category to a 2-category having the same 1-cells are monoids.
Some lattice-theoretic structures are definable as posetal categories of a certain kind, usually with the stronger assumption of being skeletal. For example, under this assumption, a poset may be defined as a small posetal category, a distributive lattice as a small posetal distributive category, a Heyting algebra as a small posetal finitely cocomplete cartesian closed category, and a Boolean algebra as a small posetal finitely cocomplete *-autonomous category. Conversely, categories, distributive categories, finitely cocomplete cartesian closed categories, and finitely cocomplete *-autonomous categories can be considered the respective categorifications of posets, distributive lattices, Heyting algebras, and Boolean algebras.
References
1. Thin category at the nLab
2. Roman, Steven (2017). An Introduction to the Language of Category Theory. Compact Textbooks in Mathematics. Cham: Springer International Publishing. p. 5. doi:10.1007/978-3-319-41917-6. ISBN 978-3-319-41916-9.
| Wikipedia |
Thin group (finite group theory)
In the mathematical classification of finite simple groups, a thin group is a finite group such that for every odd prime number p, the Sylow p-subgroups of the 2-local subgroups are cyclic. Informally, these are the groups that resemble rank 1 groups of Lie type over a finite field of characteristic 2.
Janko (1972) defined thin groups and classified those of characteristic 2 type in which all 2-local subgroups are solvable. The thin simple groups were classified by Aschbacher (1976, 1978). The list of finite simple thin groups consists of:
• The projective special linear groups PSL2(q) and PSL3(p) for p = 1 + 2a3b and PSL3(4)
• The projective special unitary groups PSU3(p) for p =−1 + 2a3b and b = 0 or 1 and PSU3(2n)
• The Suzuki groups Sz(2n)
• The Tits group 2F4(2)'
• The Steinberg group 3D4(2)
• The Mathieu group M11
• The Janko group J1
See also
• Quasithin group
References
• Aschbacher, Michael (1976), "Thin finite simple groups", Bulletin of the American Mathematical Society, 82 (3): 484, doi:10.1090/S0002-9904-1976-14063-3, ISSN 0002-9904, MR 0396735
• Aschbacher, Michael (1978), "Thin finite simple groups", Journal of Algebra, 54 (1): 50–152, doi:10.1016/0021-8693(78)90022-4, ISSN 0021-8693, MR 0511458
• Janko, Zvonimir (1972), "Nonsolvable finite groups all of whose 2-local subgroups are solvable. I", Journal of Algebra, 21: 458–517, doi:10.1016/0021-8693(72)90009-9, ISSN 0021-8693, MR 0357584
| Wikipedia |
Thin group (algebraic group theory)
In algebraic group theory, a thin group is a discrete Zariski-dense subgroup of G(R) that has infinite covolume, where G is a semisimple algebraic group over the reals. This is in contrast to a lattice, which is a discrete subgroup of finite covolume.
The theory of "group expansion" (expander graph properties of related Cayley graphs) for particular thin groups has been applied to arithmetic properties of Apollonian circles and in Zaremba's conjecture.[1]
References
1. "Archived copy" (PDF). Archived from the original (PDF) on 2014-07-29. Retrieved 2014-07-24.{{cite web}}: CS1 maint: archived copy as title (link)
• Breuillard, Emmanuel; Oh, Hee, eds. (2014), Thin Groups and Superstrong Approximation, Cambridge University Press, ISBN 978-1-107-03685-7
| Wikipedia |
Thin plate energy functional
The exact thin plate energy functional (TPEF) for a function $f(x,y)$ is
$\int _{y_{0}}^{y_{1}}\int _{x_{0}}^{x_{1}}(\kappa _{1}^{2}+\kappa _{2}^{2}){\sqrt {g}}\,dx\,dy$
where $\kappa _{1}$ and $\kappa _{2}$ are the principal curvatures of the surface mapping $f$ at the point $(x,y).$[1][2] This is the surface integral of $\kappa _{1}^{2}+\kappa _{2}^{2},$ hence the ${\sqrt {g}}$ in the integrand.
Minimizing the exact thin plate energy functional would result in a system of non-linear equations. So in practice, an approximation that results in linear systems of equations is often used.[1][3][4] The approximation is derived by assuming that the gradient of $f$ is 0. At any point where $f_{x}=f_{y}=0,$ the first fundamental form $g_{ij}$ of the surface mapping $f$ is the identity matrix and the second fundamental form $b_{ij}$ is
${\begin{pmatrix}f_{xx}&f_{xy}\\f_{xy}&f_{yy}\end{pmatrix}}$.
We can use the formula for mean curvature $H=b_{ij}g^{ij}/2$[5] to determine that $H=(f_{xx}+f_{yy})/2$ and the formula for Gaussian curvature $K=b/g$[5] (where $b$ and $g$ are the determinants of the second and first fundamental forms, respectively) to determine that $K=f_{xx}f_{yy}-(f_{xy})^{2}.$ Since $H=(k_{1}+k_{2})/2$ and $K=k_{1}k_{2},$[5] the integrand of the exact TPEF equals $4H^{2}-2K.$ The expressions we just computed for the mean curvature and Gaussian curvature as functions of partial derivatives of $f$ show that the integrand of the exact TPEF is
$4H^{2}-2K=(f_{xx}+f_{yy})^{2}-2(f_{xx}f_{yy}-f_{xy}^{2})=f_{xx}^{2}+2f_{xy}^{2}+f_{yy}^{2}.$
So the approximate thin plate energy functional is
$J[f]=\int _{y_{0}}^{y_{1}}\int _{x_{0}}^{x_{1}}f_{xx}^{2}+2f_{xy}^{2}+f_{yy}^{2}\,dx\,dy.$
Rotational invariance
The TPEF is rotationally invariant. This means that if all the points of the surface $z(x,y)$ are rotated by an angle $\theta $ about the $z$-axis, the TPEF at each point $(x,y)$ of the surface equals the TPEF of the rotated surface at the rotated $(x,y).$ The formula for a rotation by an angle $\theta $ about the $z$-axis is
${\binom {X}{Y}}={\begin{pmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \end{pmatrix}}{\binom {x}{y}}=R{\binom {x}{y}}.$
(1)
The fact that the $z$ value of the surface at $(x,y)$ equals the $z$ value of the rotated surface at the rotated $(x,y)$ is expressed mathematically by the equation
$Z(X,Y)=z(x,y)=(z\circ xy)(X,Y)$
where $xy$ is the inverse rotation, that is, $xy(X,Y)=R^{-1}(X,Y)^{\text{T}}=R^{\text{T}}(X,Y)^{\text{T}}.$ So $Z=z\circ xy$ and the chain rule implies
$Z_{i}=z_{j}R_{ij}.$
(2)
In equation (2), $Z_{0}$ means $Z_{X},$ $Z_{1}$ means $Z_{Y},$ $z_{0}$ means $z_{x},$ and $z_{1}$ means $z_{y}.$ Equation (2) and all subsequent equations in this section use non-tensor summation convention, that is, sums are taken over repeated indices in a term even if both indices are subscripts. The chain rule is also needed to differentiate equation (2) since $z_{j}$ is actually the composition $z_{j}\circ xy:$
$Z_{ik}=z_{jl}R_{kl}R_{ij}$.
Swapping the index names $j$ and $k$ yields
$Z_{ij}=z_{kl}R_{jl}R_{ik}.$
(3)
Expanding the sum for each pair $i,j$ yields
${\begin{array}{lcl}Z_{XX}&=&R_{00}^{2}z_{xx}+2R_{00}R_{01}z_{xy}+R_{01}^{2}z_{yy},\\Z_{XY}&=&R_{00}R_{10}z_{xx}+(R_{00}R_{11}+R_{01}R_{10})z_{xy}+R_{01}R_{11}z_{yy},\\Z_{YY}&=&R_{10}^{2}z_{xx}+2R_{10}R_{11}z_{xy}+R_{11}^{2}z_{yy}.\end{array}}$
Computing the TPEF for the rotated surface yields
${\begin{aligned}Z_{XX}^{2}+2Z_{XY}^{2}+Z_{YY}^{2}&=(R_{11}^{2}+R_{01}^{2})z_{yy}^{2}\\&+2(R_{10}R_{11}+R_{00}R_{01})^{2}z_{xx}z_{yy}\\&+2(2R_{10}^{2}R_{11}^{2}+R_{00}^{2}R_{11}^{2}+2R_{00}R_{01}R_{10}R_{11}\\&\qquad +R_{01}^{2}R_{10}^{2}+2R_{00}^{2}R_{01}^{2})z_{xy}^{2}\\&+4(R_{10}R_{11}+R_{00}R_{01})(R_{11}^{2}+R_{01}^{2})z_{xy}z_{yy}\\&+4(R_{10}^{2}+R_{00}^{2})(R_{10}R_{11}+R_{00}R_{01})z_{xx}z_{xy}\\&+(R_{10}^{2}+R_{00}^{2})z_{xx}^{2}.\\\end{aligned}}$
(4)
Inserting the coefficients of the rotation matrix $R$ from equation (1) into the right-hand side of equation (4) simplifies it to $z_{xx}^{2}+2z_{xy}^{2}+z_{yy}^{2}.$
Data fitting
The approximate thin plate energy functional can be used to fit B-spline surfaces to scattered 1D data on a 2D grid (for example, digital terrain model data).[6][3] Call the grid points $(x_{i},y_{i})$ for $i=1\dots N$ (with $x_{i}\in [a,b]$ and $y_{i}\in [c,d]$) and the data values $z_{i}.$ In order to fit a uniform B-spline $f(x,y)$ to the data, the equation
$\sum _{i=1}^{N}(f(x_{i},y_{i})-z_{i})^{2}+\lambda \int _{c}^{d}\int _{a}^{b}(f_{xx}^{2}+2f_{xy}^{2}+f_{yy}^{2})\,dx\,dy$
(5)
(where $\lambda $ is the "smoothing parameter") is minimized. Larger values of $\lambda $ result in a smoother surface and smaller values result in a more accurate fit to the data. The following images illustrate the results of fitting a B-spline surface to some terrain data using this method.
• Original terrain data
• Fitted B-spline surface with large lambda and more smoothing
• Fitted B-spline surface with smaller lambda and less smoothing
The thin plate smoothing spline also minimizes equation (5), but it is much more expensive to compute than a B-spline and not as smooth (it is only $C^{1}$ at the "centers" and has unbounded second derivatives there).
References
1. Greiner, Günther (1994). "Variational Design and Fairing of Spline Surfaces" (PDF). Eurographics '94. Retrieved January 3, 2016.
2. Moreton, Henry P. (1992). "Functional Optimization for Fair Surface Design" (PDF). Computer Graphics. Retrieved January 4, 2016.
3. Eck, Matthias (1996). "Automatic reconstruction of B-splines surfaces of arbitrary topological type" (PDF). Proceedings of SIGGRAPH 96, Computer Graphics Proceedings, Annual Conference Series. Retrieved January 3, 2016.
4. Halstead, Mark (1993). "Efficient, Fair Interpolation using Catmull-Clark Surfaces" (PDF). Proceedings of the 20th annual conference on Computer graphics and interactive techniques. Retrieved January 4, 2016.
5. Kreyszig, Erwin (1991). Differential Geometry. Mineola, New York: Dover. pp. 131. ISBN 0-486-66721-9.
6. Hjelle, Oyvind (2005). "Multilevel Least Squares Approximation of Scattered Data over Binary Triangulations" (PDF). Computing and Visualization in Science. Retrieved January 14, 2016.
| Wikipedia |
Thin set (analysis)
In mathematical analysis, a thin set is a subset of n-dimensional complex space Cn with the property that each point has a neighbourhood on which some non-zero holomorphic function vanishes. Since the set on which a holomorphic function vanishes is closed and has empty interior (by the Identity theorem), a thin set is nowhere dense, and the closure of a thin set is also thin.
The fine topology was introduced in 1940 by Henri Cartan to aid in the study of thin sets.
References
• Gunning, Robert C.; Rossi, Hugo (1965), Analytic functions of several complex variables, Prentice–Hall
| Wikipedia |
Think-a-Dot
The Think-a-Dot was a mathematical toy invented by Joseph Weisbecker[1] and manufactured by E.S.R., Inc. during the 1960s that demonstrated automata theory. It had eight coloured disks on its front, and three holes on its top – left, right, and center – through which a ball bearing could be dropped. Each disk would display either a yellow or blue face, depending on whether the mechanism behind it was tipped to the right or the left. The Think-a-Dot thus had 28=256 internal states. When the ball fell to the bottom it would exit either to a hole on the left or the right of the device.
As the ball passed through the Think-a-Dot, it would flip the disk mechanisms that it passed, and they in turn would determine whether the ball would be deflected to the left or to the right. Various puzzles and games were possible with the Think-a-Dot, such as flipping the colours of all cells in the minimum number of moves, or reaching a given state from a monochrome state or vice versa.
See also
• Digi-Comp I
• Digi-Comp II
• Dr. NIM
References
1. "Object of the Week: Think-a-Dot". The Sarnoff Collection. Retrieved 2020-03-01.
• Schwartz, Benjamin L. (1967), "Mathematical theory of Think-a-Dot", Mathematics Magazine, 40 (4): 187–193, doi:10.2307/2688674, MR 1571696.
• Beidler, John A. (1973), "Think-a-Dot revisited", Mathematics Magazine, 46: 128–136, doi:10.2307/2687967, MR 0379077.
• Gemignani, Michael (1979), "Think-a-Dot: a useful generalization", Mathematics Magazine, 52 (2): 110–112, doi:10.2307/2689850, MR 1572295.
External links
• Picture of a Think-a-Dot
• Think-a-Dot instruction leaflet
• 3-D printable Think-a-Dot Replica
| Wikipedia |
Thinning (morphology)
Thinning is the transformation of a digital image into a simplified, but topologically equivalent image. It is a type of topological skeleton, but computed using mathematical morphology operators.
Example
Let $E=Z^{2}$, and consider the eight composite structuring elements, composed by:
$C_{1}=\{(0,0),(-1,-1),(0,-1),(1,-1)\}$ and $D_{1}=\{(-1,1),(0,1),(1,1)\}$,
$C_{2}=\{(-1,0),(0,0),(-1,-1),(0,-1)\}$ and $D_{2}=\{(0,1),(1,1),(1,0)\}$
and the three rotations of each by $90^{o}$, $180^{o}$, and $270^{o}$. The corresponding composite structuring elements are denoted $B_{1},\ldots ,B_{8}$.
For any i between 1 and 8, and any binary image X, define
$X\otimes B_{i}=X\setminus (X\odot B_{i})$,
where $\setminus $ denotes the set-theoretical difference and $\odot $ denotes the hit-or-miss transform.
The thinning of an image A is obtained by cyclically iterating until convergence:
$A\otimes B_{1}\otimes B_{2}\otimes \ldots \otimes B_{8}\otimes B_{1}\otimes B_{2}\otimes \ldots $.
Thickening
Thickening is the dual of thinning that is used to grow selected regions of foreground pixels. In most cases in image processing thickening is performed by thinning the background [1] ${\text{thicken}}(X,B_{i})=X\cup (X\odot B_{i})$
where $\cup $ denotes the set-theoretical difference and $\odot $ denotes the hit-or-miss transform, and $B_{i}$ is the structural element and $X$ is the image being operated on.
References
1. Gonzalez, Rafael C. (2002). Digital image processing. Woods, Richard E. (Richard Eugene), 1954- (2nd ed.). Upper Saddle River, N.J. ISBN 0-201-18075-8. OCLC 48944550.{{cite book}}: CS1 maint: location missing publisher (link)
| Wikipedia |
Lie's third theorem
In the mathematics of Lie theory, Lie's third theorem states that every finite-dimensional Lie algebra ${\mathfrak {g}}$ over the real numbers is associated to a Lie group $G$. The theorem is part of the Lie group–Lie algebra correspondence.
Historically, the third theorem referred to a different but related result. The two preceding theorems of Sophus Lie, restated in modern language, relate to the infinitesimal transformations of a group action on a smooth manifold. The third theorem on the list stated the Jacobi identity for the infinitesimal transformations of a local Lie group. Conversely, in the presence of a Lie algebra of vector fields, integration gives a local Lie group action. The result now known as the third theorem provides an intrinsic and global converse to the original theorem.
Historical notes
The equivalence between the category of simply connected real Lie groups and finite-dimensional real Lie algebras is usually called (in the literature of the second half of 20th century) Cartan's or the Cartan-Lie theorem as it was proved by Élie Cartan. Sophus Lie had previously proved the infinitesimal version: local solvability of the Maurer-Cartan equation, or the equivalence between the category of finite-dimensional Lie algebras and the category of local Lie groups.
Lie listed his results as three direct and three converse theorems. The infinitesimal variant of Cartan's theorem was essentially Lie's third converse theorem. In an influential book[1] Jean-Pierre Serre called it the third theorem of Lie. The name is historically somewhat misleading, but often used in connection to generalizations.
Serre provided two proofs in his book: one based on Ado's theorem and another recounting the proof by Élie Cartan.
Proofs
There are several proofs of Lie's third theorem, each of them employing different algebraic and/or geometric techniques.
Algebraic proof
The classical proof is straightforward but relies on Ado's theorem, whose proof is algebraic and highly non-trivial.[2] Ado's theorem states that any finite-dimensional Lie algebra can be represented by matrices. As a consequence, integrating such algebra of matrices via the matrix exponential yields a Lie group integrating the original Lie algebra.
Cohomological proof
A more geometric proof is due to Élie Cartan and was published by Willem van Est.[3] This proof uses induction on the dimension of the center and it involves the Chevalley-Eilenberg complex.[4]
Geometric proof
A different geometric proof was discovered in 2000 by Duistermaat and Kolk.[5] Unlike the previous ones, it is a constructive proof: the integrating Lie group is built as the quotient of the (infinite-dimensional) Banach Lie group of paths on the Lie algebra by a suitable subgroup. This proof was influential for Lie theory[6] since it paved the way to the generalisation of Lie third theorem for Lie groupoids and Lie algebroids.[7]
See also
• Lie group integrator
References
1. Jean-Pierre Serre (1992)[1965] Lie Algebras and Lie Groups: 1964 Lectures Given at Harvard University, page 152, Springer ISBN 978-3-540-55008-2
2. Tao, Terence (2011-05-10). "Ado's theorem". What's new. Retrieved 2022-09-18.
3. Van Est, Willem (1987). "Une démonstration de E. Cartan du troisième théorème de Lie" [A proof of Elie Cartan of Lie's third theorem]. Actions Hamiltoniennes des groupes, troisième théorème de Lie, travaux en cours (in French). Paris: Hermann. 27: 83–96.
4. Ebert, Johannes. "Van Est's exposition of Cartan's proof of Lie's third theorem" (PDF).
5. Duistermaat, J. J.; Kolk, J. A. C. (2000). Lie Groups. Universitext. Berlin, Heidelberg: Springer Berlin Heidelberg. doi:10.1007/978-3-642-56936-4. ISBN 978-3-540-15293-4.
6. Sjamaar, Reyer (2011-10-25). "Hans Duistermaat's contributions to Poisson geometry". arXiv:1110.5627 [math.HO].
7. Crainic, Marius; Fernandes, Rui (2003-03-01). "Integrability of Lie brackets". Annals of Mathematics. 157 (2): 575–620. doi:10.4007/annals.2003.157.575. ISSN 0003-486X. S2CID 6992408.
• Cartan, Élie (1930), "La théorie des groupes finis et continus et l'Analysis Situs", Mémorial Sc. Math., vol. XLII, pp. 1–61
• Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, doi:10.1007/978-3-319-13467-3, ISBN 978-3319134666
• Helgason, Sigurdur (2001), Differential geometry, Lie groups, and symmetric spaces, Graduate Studies in Mathematics, vol. 34, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-2848-9, MR 1834454
External links
• Encyclopaedia of Mathematics (EoM) article
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| Wikipedia |
Third Vote
The Third Vote (German: Drittstimme) is an election method proposed within the mathematical theory of democracy by Andranik Tangian. It is aimed at emphasizing issues of policy over personality[1] The Third Vote has to-date only been used experimentally in student elections.
Description
The aim of the Third Vote method is to draw the voters' attention from personalities of politicians to policy issues, that is, from the question "Who should be elected?" to "What do we choose?". Therefore, the electors do not cast votes by candidate name, but give Yes/No-answers to several policy questions as raised in the candidates' manifestos. The same procedure is inherent in voting advice applications (VAA) but the results are evaluated in a different way. In contrast to VAAs, answering the questionnaire implies no voting recommendation for the individual user. Instead, the answers of all voters are processed, and the political profile of the entire electorate is built with the balance of public opinion of pros/cons percentages for each issue. The election winner is the candidate whose policy profile (constituted by his/her Yes/No-answers to the same questions) best matches with that of the electorate.
If the candidates are political parties competing for parliament seats, the proximity of the party profiles to the electorate profile is indexed, and the parliament seats are allocated proportionally to the party indices. When considering decision options instead of candidates (whan a commission has to agree on a certain business plan, technical proposal, etc.), the questions focus on their specific characteristics.
Since the electorate is considered a single body with a single policy profile, no multiple-voter paradox like that of Borda, Condorcet or Arrow can arise.
History
The Third Vote method was developed in the 2010s at the Hans Böckler Foundation and the Karlsruhe Institute of Technology as part of the mathematical theory of democracy[2][3] in order to improve policy representation and surmount the voters' irrationality. Although this voting method is completely self-sufficient, it was first tested as a complement to the two-vote system. The name "Third Vote" emphasizes the complementarity to the two-vote system of mixed-member proportional representation, which is used in Germany, New Zealand, Bolivia, Lesotho, Thailand, South Africa, South Korea, United Kingdom (Scotland, Wales, and the London assembly) and Ethiopia.
The Third Vote was implemented during the annual student parliament elections (StuPa elections) 2016–2019 at the Karlsruhe Institute of Technology.[4][5][6][7] The experiments were monitored on the Third Vote website[8] and the results were discussed in the German media[9][10][11][12][13] as well as at the 2016 and 2019 World Forums for Democracy.[14][15]
Example
Table 1 shows five dichotomous questions (assuming Yes/No answers) and the answers from three parties - Conservatives, Socialists and Greens - and from three equal groups of voters - A, B and C. (Questions 1 to 5 are the Questions 1, 2, 7, 28 and 32 from the 2017 German VAA Wahl-O-Mat, and the answers to these questions are that of the German conservative party CDU/CSU, the Social Democrats SPD and the greens GRÜNE.[16]
Table 1: Policy profiles of three parties and three voter groups, the party representativeness indices, and allocation of parliament seats
Questions Parties Voter group Balance of public opinion
Conservatives Socialists Greens A B C Yes No
1. Counter-terrorism domestic deployment of the army YesNoNoNoYesNo1/32/3
2. Higher taxes for diesel fuel NoNoYesNoYesYes2/31/3
3. Extending video surveillance YesYesNoYesNoNo1/32/3
4. Statutory health insurance for all NoYesYesYesNoYes2/31/3
5. Allow cannabis sales NoNoYesNoNoNo03/3
Popularity (average size of the group represented), in% 476053
Parliament faction size, in % 293833
Table 1 contains the party representation index "Popularity", which is the average size of the group represented. For example, the Conservatives answer Questions 1 to 4 as 1/3 of all voters and Question 5 as all voters (3/3). That leads to
${\text{P}}_{\text{Conservatives}}={\frac {1/3+1/3+1/3+1/3+3/3}{5}}={\frac {7}{15}}\approx 47\%.$
The popularity of the Socialists and Greens is calculated in the same way, giving 60% and 53%, respectively. The election winners are therefore the Socialists.
The parliamentary seats are allocated in proportion to the parties’ indices of Popularity:
${\text{Conservatives : Socialists : Greens}}\approx {\frac {47:60:53}{47+60+53}}\approx 29\%:38\%:33\%.$ : Socialists : Greens}}\approx {\frac {47:60:53}{47+60+53}}\approx 29\%:38\%:33\%.}
The Third Vote is also applicable in the context of collective multiple-criteria decision making.[17]
Implementation
Formulation of the questions. The questions are proposed by the candidates themselves - as part of the election campaign. After that, each candidate answers all questions, including the questions from other candidates. Thereby, complete policy profiles of all candidates are defined.
Final selection of questions. The final selection of a reasonable number of questions for the ballot, which best emphasize the contrast between the candidates, is done either by a special commission, as in the case of VAAs, or by a computer program that analyzes the parties' answers.
Unequal importance of questions. The voters can optionally assign question weights (e.g. from 0 - unimportant to 5 - very important). The sum of the voter weights for each question is then used to determine the "average" public opinion about the relative importance of each issue, which is then used in calculations.
Taking into account the candidates’ credibility. To account for voter confidence, the Third Vote is combined with traditional voting by candidate name. The final rating of the candidates is then based on the average of the candidate's Popularity index and the percentage of votes the candidate received.
Ballot. An exemplary ballot form, which is virtually filled in by a voter from Group A, is shown in Table 2.
Table 2: Ballot form virtually completed by a voter from Group A
Question Yes No Weight (optional)
1. Counter-terrorism domestic deployment of the army X1
2. Higher taxes for diesel fuel X1
3. Extending video surveillance X1
4. Statutory health insurance for all X1
5. Allow cannabis sales X1
Vote for a party (optional)
Conservatives
Socialists X
Greens
The Third Vote versus plurality vote, Borda count and Condorcet count
In the above example, the Third Vote finds a single winner, whereas no single winner is found by the plurality vote (the electors cast votes for the favorite candidate) and the Condorcet and Borda counts, which use the electors' preference orders shown in Table 3. In parentheses, the VAA-ratings are indicated, that is, the number of coincidences in the voter and party profiles. For example, Voter Group 1 coincides with the Socialists in 5 issues, with the Conservatives in 3 issues and the Greens in 2 issues. Therefore, Group 1's preference ordering is Socialists > Conservatives > Greens.
Table 3: Preference orderings of the voters (with VAA-assisted ratings of the parties)
Rank Preference orderings of Voter Groups
ABC
1 Socialists (5)Conservatives (3)Greens (4)
2 Conservatives (3)Greens (2)Socialists (3)
3 Greens (2)Socialists (1)Conservatives (1)
Table 4 shows that neither plurality vote nor the Borda count (sum of ranks) nor Condorcet count (pairwise vote) results in a single winner.
Table 4: Plurality vote, Borda scores and pairwise vote, resulting in a Condorcet cycle
Conservatives Socialists Greens
Plurality votes 111
Borda score (sum of ranks) 2+1+3=61+3+2=63+2+1=6
Pairwise vote Vote ratios
Conservatives1:22:1
Socialists2:11:2
Greens1:22:1
Indeed, each voter group has its own favorite candidate, the sums of ranks (Borda count) are the same for all three candidates, and the pairwise vote (Condorcet count) leads to a Condorcet cycle without the weakest link to be cut:
${\text{Conservatives}}{\stackrel {2:1}{\succ }}{\text{Greens}}{\stackrel {2:1}{\succ }}{\text{Socialists}}{\stackrel {2:1}{\succ }}{\text{Conservatives}}.$
References
1. Budge, Ian; McDonald, Michael D (2007). "Election and party system effects on policy representation: Bringing time into a comparative perspective". Electoral Studies. 26 (1): 168–179. doi:10.1016/j.electstud.2006.02.001.
2. Tangian, Andranik (2014). Mathematical theory of democracy. Studies in Choice and Welfare. Berlin-Heidelberg: Springer. doi:10.1007/978-3-642-38724-1. ISBN 978-3-642-38723-4.
3. Tangian, Andranik (2020). Analytical theory of democracy. Vols. 1 and 2. Studies in Choice and Welfare. Cham, Switzerland: Springer. doi:10.1007/978-3-030-39691-6. ISBN 978-3-030-39690-9. S2CID 216190330.
4. "Drittstimmenaktion: Eine neue Idee zur Umsetzung direkter Demokratie (Third vote action: A new idea for implementing direct democracy)". AStA Ventil, 1 July 2016. Karlsruhe: Karlsruhe Institute of Technology, AStA. 134: 6. Retrieved 30 December 2019.
5. "The Third Vote: Eine neue Idee zur Umsetzung direkter Demokratie (The Third Vote: A new idea for implementing direct democracy)" (PDF). Ventil StuPa-Wahl, 23 June 2017. Karlsruhe: Karlsruhe Institute of Technology, AStA. 136. Retrieved 30 December 2019.
6. "The Third Vote: Improving our Democracy" (PDF). Wahlventil 15 June 2018 – Ventil (in German). Karlsruhe: Karlsruhe Institute of Technology, AStA. 141: 7–8. Retrieved 15 February 2021.
7. "The Third Vote: Eine neue Idee zur Umsetzung direkter Demokratie (The Third Vote: A new idea to implement direct democracy)" (PDF). Wahlventil 28 June 2019 (in German). Karlsruhe: Karlsruhe Institute of Technology, AStA. 143: 12–13. Retrieved 15 February 2021.
8. Amrhein, Marius; Diemer, Antonia; Eßwein, Bastian; Waldeck, Maximilian; Schäfer, Sebastian. "The Third Vote (web page)". Karlsruhe: Karlsruhe Institute of Technology, Institute ECON. Retrieved 15 December 2020.
9. Klein, Manuel T. (21 March 2011). "Von Wahlen, Wählern und Gewählten (Of elections, voters and elected)" (PDF). Die Rheinpfalz (in German). 67. Retrieved 19 February 2021.
10. Kinkel, Ekart (6 May 2014). "Plädoyer für die Drittstimme: KIT-Forscher für sanfte Reform des Wahlrechts (Plea for the third vote: KIT researcher for soft reform of the electoral law)" (PDF). Badische Neueste Nachrichten (in German). 103: 15. Retrieved 19 February 2021.
11. Schmidt, Nicola (7 October 2016). "Denn sie wissen nicht, was sie wählen (Then they do not know what they elect)". Autopilot An! Perspective-Daily (in German). Retrieved 19 February 2021.
12. Dittrich, Tobias; Tangian, Andranik (2017). "Politische Technologien gegen Populismus (Political technologies against populism)" (PDF). Karlsruhe Transfer (KT) (in German). 52: 8–11. Retrieved 20 February 2021.
13. Kinkel, Ekart (2017). "Die Drittstimme: Volkswirtschaftsprofessor Andranik S. Tangian hat am KIT ein alternatives Wahlverfahren zur Repräsentanz des Bürgerwillens entwickelt (The Third Vote: Economics professor Andranik S. Tangian has developed an alternative election method to represent citizens' wishes)" (PDF). LooKIT (in German and English) (2): 74–76. Retrieved 19 February 2021.
14. "Turning a political education instrument (voting advice application) in a new election method", World Forum for Democracy 2016, Lab 7: Reloading Elections, Strasbourg: Council of Europe, 7–9 November 2016, retrieved 15 December 2020
15. "Well Informed Vote", World Forum for Democracy 2019, Lab 5: Voting under the Influence, Strasbourg: Council of Europe, 6–8 November 2019, retrieved 15 December 2020
16. Bundeszentrale für politische Bildung. "Wahl-O-Mat". Berlin. Retrieved 15 December 2020.
17. Tangian, Andranik (2021). "MCDM application of the Third Vote" (PDF). Group Decision and Negotiation. 30 (4): 775–787. doi:10.1007/s10726-021-09733-2. S2CID 235571433.
| Wikipedia |
Third derivative
In calculus, a branch of mathematics, the third derivative or third-order derivative is the rate at which the second derivative, or the rate of change of the rate of change, is changing. The third derivative of a function $y=f(x)$ can be denoted by
${\frac {d^{3}y}{dx^{3}}},\quad f'''(x),\quad {\text{or }}{\frac {d^{3}}{dx^{3}}}[f(x)].$
Part of a series of articles about
Calculus
• Fundamental theorem
• Limits
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Other notations can be used, but the above are the most common.
Mathematical definitions
Let $f(x)=x^{4}$. Then $f'(x)=4x^{3}$ and $f''(x)=12x^{2}$. Therefore, the third derivative of f is, in this case,
$f'''(x)=24x$
or, using Leibniz notation,
${\frac {d^{3}}{dx^{3}}}[x^{4}]=24x.$
Now for a more general definition. Let f be any function of x such that f ′′ is differentiable. Then the third derivative of f is given by
${\frac {d^{3}}{dx^{3}}}[f(x)]={\frac {d}{dx}}[f''(x)].$
The third derivative is the rate at which the second derivative (f′′(x)) is changing.
Applications in geometry
In differential geometry, the torsion of a curve — a fundamental property of curves in three dimensions — is computed using third derivatives of coordinate functions (or the position vector) describing the curve.[1]
Applications in physics
In physics, particularly kinematics, jerk is defined as the third derivative of the position function of an object. It is, essentially, the rate at which acceleration changes. In mathematical terms:
$\mathbf {j} (t)={\frac {d^{3}\mathbf {r} }{dt^{3}}}$
where j(t) is the jerk function with respect to time, and r(t) is the position function of the object with respect to time.
Economic examples
When campaigning for a second term in office, U.S. President Richard Nixon announced that the rate of increase of inflation was decreasing, which has been noted as "the first time a sitting president used the third derivative to advance his case for reelection."[2] Since inflation is itself a derivative—the rate at which the purchasing power of money decreases—then the rate of increase of inflation is the derivative of inflation, opposite in sign to the second time derivative of the purchasing power of money. Stating that a function is decreasing is equivalent to stating that its derivative is negative, so Nixon's statement is that the second derivative of inflation is negative, and so the third derivative of purchasing power is positive.
Nixon's statement allowed for the rate of inflation to increase, however, so his statement was not as indicative of stable prices as it sounds.
See also
• Aberrancy (geometry)
• Derivative (mathematics)
• Second derivative
References
1. do Carmo, Manfredo (1976). Differential Geometry of Curves and Surfaces. ISBN 0-13-212589-7.
2. Rossi, Hugo (October 1996). "Mathematics Is an Edifice, Not a Toolbox" (PDF). Notices of the American Mathematical Society. 43 (10): 1108. Retrieved 13 November 2012.
| Wikipedia |
Third fundamental form
In differential geometry, the third fundamental form is a surface metric denoted by $\mathrm {I\!I\!I} $. Unlike the second fundamental form, it is independent of the surface normal.
Definition
Let S be the shape operator and M be a smooth surface. Also, let up and vp be elements of the tangent space Tp(M). The third fundamental form is then given by
$\mathrm {I\!I\!I} (\mathbf {u} _{p},\mathbf {v} _{p})=S(\mathbf {u} _{p})\cdot S(\mathbf {v} _{p})\,.$
Properties
The third fundamental form is expressible entirely in terms of the first fundamental form and second fundamental form. If we let H be the mean curvature of the surface and K be the Gaussian curvature of the surface, we have
$\mathrm {I\!I\!I} -2H\mathrm {I\!I} +K\mathrm {I} =0\,.$
As the shape operator is self-adjoint, for u,v ∈ Tp(M), we find
$\mathrm {I\!I\!I} (u,v)=\langle Su,Sv\rangle =\langle u,S^{2}v\rangle =\langle S^{2}u,v\rangle \,.$
See also
• Metric tensor
• First fundamental form
• Second fundamental form
• Tautological one-form
Various notions of curvature defined in differential geometry
Differential geometry
of curves
• Curvature
• Torsion of a curve
• Frenet–Serret formulas
• Radius of curvature (applications)
• Affine curvature
• Total curvature
• Total absolute curvature
Differential geometry
of surfaces
• Principal curvatures
• Gaussian curvature
• Mean curvature
• Darboux frame
• Gauss–Codazzi equations
• First fundamental form
• Second fundamental form
• Third fundamental form
Riemannian geometry
• Curvature of Riemannian manifolds
• Riemann curvature tensor
• Ricci curvature
• Scalar curvature
• Sectional curvature
Curvature of connections
• Curvature form
• Torsion tensor
• Cocurvature
• Holonomy
| Wikipedia |
Unknot
In the mathematical theory of knots, the unknot, not knot, or trivial knot, is the least knotted of all knots. Intuitively, the unknot is a closed loop of rope without a knot tied into it, unknotted. To a knot theorist, an unknot is any embedded topological circle in the 3-sphere that is ambient isotopic (that is, deformable) to a geometrically round circle, the standard unknot.
Unknot
Common nameCircle
Arf invariant0
Braid no.1
Bridge no.0
Crossing no.0
Genus0
Linking no.0
Stick no.3
Tunnel no.0
Unknotting no.0
Conway notation-
A–B notation01
Dowker notation-
Next31
Other
torus, fibered, prime, slice, fully amphichiral
The unknot is the only knot that is the boundary of an embedded disk, which gives the characterization that only unknots have Seifert genus 0. Similarly, the unknot is the identity element with respect to the knot sum operation.
Unknotting problem
Main article: Unknotting problem
Deciding if a particular knot is the unknot was a major driving force behind knot invariants, since it was thought this approach would possibly give an efficient algorithm to recognize the unknot from some presentation such as a knot diagram. Unknot recognition is known to be in both NP and co-NP.
It is known that knot Floer homology and Khovanov homology detect the unknot, but these are not known to be efficiently computable for this purpose. It is not known whether the Jones polynomial or finite type invariants can detect the unknot.
Examples
It can be difficult to find a way to untangle string even though the fact it started out untangled proves the task is possible. Thistlethwaite and Ochiai provided many examples of diagrams of unknots that have no obvious way to simplify them, requiring one to temporarily increase the diagram's crossing number.
• Thistlethwaite unknot
• One of Ochiai's unknots
While rope is generally not in the form of a closed loop, sometimes there is a canonical way to imagine the ends being joined together. From this point of view, many useful practical knots are actually the unknot, including those that can be tied in a bight.[1]
Every tame knot can be represented as a linkage, which is a collection of rigid line segments connected by universal joints at their endpoints. The stick number is the minimal number of segments needed to represent a knot as a linkage, and a stuck unknot is a particular unknotted linkage that cannot be reconfigured into a flat convex polygon.[2] Like crossing number, a linkage might need to be made more complex by subdividing its segments before it can be simplified.
Invariants
The Alexander–Conway polynomial and Jones polynomial of the unknot are trivial:
$\Delta (t)=1,\quad \nabla (z)=1,\quad V(q)=1.$
No other knot with 10 or fewer crossings has trivial Alexander polynomial, but the Kinoshita–Terasaka knot and Conway knot (both of which have 11 crossings) have the same Alexander and Conway polynomials as the unknot. It is an open problem whether any non-trivial knot has the same Jones polynomial as the unknot.
The unknot is the only knot whose knot group is an infinite cyclic group, and its knot complement is homeomorphic to a solid torus.
See also
• Knot (mathematics) – Embedding of the circle in three dimensional Euclidean space
• Unlink – Link that consists of finitely many unlinked unknots
References
1. Volker Schatz. "Knotty topics". Archived from the original on 2011-07-17. Retrieved 2007-04-23.
2. Godfried Toussaint (2001). "A new class of stuck unknots in Pol-6" (PDF). Contributions to Algebra and Geometry. 42 (2): 301–306. Archived from the original (PDF) on 2003-05-12.
External links
• "Unknot", The Knot Atlas. Accessed: May 7, 2013.
• Weisstein, Eric W. "Unknot". MathWorld.
Knot theory (knots and links)
Hyperbolic
• Figure-eight (41)
• Three-twist (52)
• Stevedore (61)
• 62
• 63
• Endless (74)
• Carrick mat (818)
• Perko pair (10161)
• (−2,3,7) pretzel (12n242)
• Whitehead (52
1
)
• Borromean rings (63
2
)
• L10a140
• Conway knot (11n34)
Satellite
• Composite knots
• Granny
• Square
• Knot sum
Torus
• Unknot (01)
• Trefoil (31)
• Cinquefoil (51)
• Septafoil (71)
• Unlink (02
1
)
• Hopf (22
1
)
• Solomon's (42
1
)
Invariants
• Alternating
• Arf invariant
• Bridge no.
• 2-bridge
• Brunnian
• Chirality
• Invertible
• Crosscap no.
• Crossing no.
• Finite type invariant
• Hyperbolic volume
• Khovanov homology
• Genus
• Knot group
• Link group
• Linking no.
• Polynomial
• Alexander
• Bracket
• HOMFLY
• Jones
• Kauffman
• Pretzel
• Prime
• list
• Stick no.
• Tricolorability
• Unknotting no. and problem
Notation
and operations
• Alexander–Briggs notation
• Conway notation
• Dowker–Thistlethwaite notation
• Flype
• Mutation
• Reidemeister move
• Skein relation
• Tabulation
Other
• Alexander's theorem
• Berge
• Braid theory
• Conway sphere
• Complement
• Double torus
• Fibered
• Knot
• List of knots and links
• Ribbon
• Slice
• Sum
• Tait conjectures
• Twist
• Wild
• Writhe
• Surgery theory
• Category
• Commons
| Wikipedia |
George Thom
George Thom (1842–1916) was a Scottish mathematician and pedagogue who was principal at Dollar Academy from 1878 to 1902.
George Thom
Born(1842-07-02)2 July 1842
Forgue (Huntly), Aberdeenshire, Scotland
Died20 December 1916(1916-12-20) (aged 74)
Aberdeen, Scotland
Resting placeDollar Churchyard (Clackmannanshire, Scotland)
56.163675°N 3.669141°W / 56.163675; -3.669141
Alma materUniversity of Aberdeen
Scientific career
FieldsMathematics
InstitutionsDollar Academy
Life and work
Thom graduated from the University of Aberdeen in 1863. In 1867 he became Principal of Doveton College in Madras, India and he remained there till 1876, when he returned to Scotland as Vice-Principal of Chanonry School Aberdeen. In 1878 he was appointed Rector of Dollar Institution (later to become Dollar Academy). He held this post for 24 years, till his retirement in 1902. He was a founder member of the Edinburgh Mathematical Society in 1883 and became its fifth President in 1886. In 1887 the University of St Andrews conferred on him the degree of Doctor of Laws[1]
He wrote a number of standard class-books on mathematics, botany, physiology, and other subjects.[2]
References
1. O'Connor & Robertson, MacTutor History of Mathematics.
2. Anonymous 1917, p. 186.
Bibliography
• Anonymous (1917). "Obituary of Gorge Thom". The Aberdeen University Review. 4: 186.
External links
• O'Connor, John J.; Robertson, Edmund F., "George Thom", MacTutor History of Mathematics Archive, University of St Andrews
Authority control
International
• VIAF
Other
• IdRef
| Wikipedia |
Thom space
In mathematics, the Thom space, Thom complex, or Pontryagin–Thom construction (named after René Thom and Lev Pontryagin) of algebraic topology and differential topology is a topological space associated to a vector bundle, over any paracompact space.
Construction of the Thom space
One way to construct this space is as follows. Let
$p:E\to B$
be a rank n real vector bundle over the paracompact space B. Then for each point b in B, the fiber $E_{b}$ is an $n$-dimensional real vector space. Choose an orthogonal structure on E, a smoothly varying inner product on the fibers; we can do this using partitions of unity. Let $D(E)$ be the unit ball bundle with respect to our orthogonal structure, and let $S(E)$ be the unit sphere bundle, then the Thom space $T(E)$ is the quotient $T(E):=D(E)/S(E)$ of topological spaces. $T(E)$ is a pointed space with the image of $S(E)$ in the quotient as basepoint. If B is compact, then $T(E)$ is the one-point compactification of E.
For example, if E is the trivial bundle $B\times \mathbb {R} ^{n}$, then $D(E)=B\times D^{n}$ and $S(E)=B\times S^{n-1}$. Writing $B_{+}$ for B with a disjoint basepoint, $T(E)$ is the smash product of $B_{+}$ and $S^{n}$; that is, the n-th reduced suspension of $B_{+}$.
The Thom isomorphism
The significance of this construction begins with the following result, which belongs to the subject of cohomology of fiber bundles. (We have stated the result in terms of $\mathbb {Z} _{2}$ coefficients to avoid complications arising from orientability; see also Orientation of a vector bundle#Thom space.)
Let $p:E\to B$ be a real vector bundle of rank n. Then there is an isomorphism, now called a Thom isomorphism
$\Phi :H^{k}(B;\mathbb {Z} _{2})\to {\widetilde {H}}^{k+n}(T(E);\mathbb {Z} _{2}),$
for all k greater than or equal to 0, where the right hand side is reduced cohomology.
This theorem was formulated and proved by René Thom in his famous 1952 thesis.
We can interpret the theorem as a global generalization of the suspension isomorphism on local trivializations, because the Thom space of a trivial bundle on B of rank k is isomorphic to the kth suspension of $B_{+}$, B with a disjoint point added (cf. #Construction of the Thom space.) This can be more easily seen in the formulation of the theorem that does not make reference to Thom space:
Thom isomorphism — Let $\Lambda $ be a ring and $p:E\to B$ be an oriented real vector bundle of rank n. Then there exists a class
$u\in H^{n}(E,E\setminus B;\Lambda ),$
where B is embedded into E as a zero section, such that for any fiber F the restriction of u
$u|_{(F,F\setminus 0)}\in H^{n}(F,F\setminus 0;\Lambda )$
is the class induced by the orientation of F. Moreover,
${\begin{cases}H^{k}(E;\Lambda )\to H^{k+n}(E,E\setminus B;\Lambda )\\x\longmapsto x\smile u\end{cases}}$
is an isomorphism.
In concise terms, the last part of the theorem says that u freely generates $H^{*}(E,E\setminus B;\Lambda )$ as a right $H^{*}(E;\Lambda )$-module. The class u is usually called the Thom class of E. Since the pullback $p^{*}:H^{*}(B;\Lambda )\to H^{*}(E;\Lambda )$ is a ring isomorphism, $\Phi $ is given by the equation:
$\Phi (b)=p^{*}(b)\smile u.$
In particular, the Thom isomorphism sends the identity element of $H^{*}(B)$ to u. Note: for this formula to make sense, u is treated as an element of (we drop the ring $\Lambda $)
${\tilde {H}}^{n}(T(E))=H^{n}(\operatorname {Sph} (E),B)\simeq H^{n}(E,E\setminus B).$[1]
Significance of Thom's work
In his 1952 paper, Thom showed that the Thom class, the Stiefel–Whitney classes, and the Steenrod operations were all related. He used these ideas to prove in the 1954 paper Quelques propriétés globales des variétés differentiables that the cobordism groups could be computed as the homotopy groups of certain Thom spaces MG(n). The proof depends on and is intimately related to the transversality properties of smooth manifolds—see Thom transversality theorem. By reversing this construction, John Milnor and Sergei Novikov (among many others) were able to answer questions about the existence and uniqueness of high-dimensional manifolds: this is now known as surgery theory. In addition, the spaces MG(n) fit together to form spectra MG now known as Thom spectra, and the cobordism groups are in fact stable. Thom's construction thus also unifies differential topology and stable homotopy theory, and is in particular integral to our knowledge of the stable homotopy groups of spheres.
If the Steenrod operations are available, we can use them and the isomorphism of the theorem to construct the Stiefel–Whitney classes. Recall that the Steenrod operations (mod 2) are natural transformations
$Sq^{i}:H^{m}(-;\mathbb {Z} _{2})\to H^{m+i}(-;\mathbb {Z} _{2}),$
defined for all nonnegative integers m. If $i=m$, then $Sq^{i}$ coincides with the cup square. We can define the ith Stiefel–Whitney class $w_{i}(p)$ of the vector bundle $p:E\to B$ by:
$w_{i}(p)=\Phi ^{-1}(Sq^{i}(\Phi (1)))=\Phi ^{-1}(Sq^{i}(u)).$
Consequences for differentiable manifolds
If we take the bundle in the above to be the tangent bundle of a smooth manifold, the conclusion of the above is called the Wu formula, and has the following strong consequence: since the Steenrod operations are invariant under homotopy equivalence, we conclude that the Stiefel–Whitney classes of a manifold are as well. This is an extraordinary result that does not generalize to other characteristic classes. There exists a similar famous and difficult result establishing topological invariance for rational Pontryagin classes, due to Sergei Novikov.
Thom spectrum
Real cobordism
There are two ways to think about bordism: one as considering two $n$-manifolds $M,M'$ are cobordant if there is an $(n+1)$-manifold with boundary $W$ such that
$\partial W=M\coprod M'$
Another technique to encode this kind of information is to take an embedding $M\hookrightarrow \mathbb {R} ^{N+n}$ and considering the normal bundle
$\nu :N_{\mathbb {R} ^{N+n}/M}\to M$
The embedded manifold together with the isomorphism class of the normal bundle actually encodes the same information as the cobordism class $[M]$. This can be shown[2] by using a cobordism $W$ and finding an embedding to some $\mathbb {R} ^{N_{W}+n}\times [0,1]$ which gives a homotopy class of maps to the Thom space $MO(n)$ defined below. Showing the isomorphism of
$\pi _{n}MO\cong \Omega _{n}^{O}$
requires a little more work.[3]
Definition of Thom spectrum
By definition, the Thom spectrum[4] is a sequence of Thom spaces
$MO(n)=T(\gamma ^{n})$
where we wrote $\gamma ^{n}\to BO(n)$ for the universal vector bundle of rank n. The sequence forms a spectrum.[5] A theorem of Thom says that $\pi _{*}(MO)$ is the unoriented cobordism ring;[6] the proof of this theorem relies crucially on Thom’s transversality theorem.[7] The lack of transversality prevents from computing cobordism rings of, say, topological manifolds from Thom spectra.
See also
• Cobordism
• Cohomology operation
• Steenrod problem
• Hattori–Stong theorem
Notes
1. Proof of the isomorphism. We can embed B into $\operatorname {Sph} (E)$ either as the zero section; i.e., a section at zero vector or as the infinity section; i.e., a section at infinity vector (topologically the difference is immaterial.) Using two ways of embedding we have the triple:
$(\operatorname {Sph} (E),\operatorname {Sph} (E)\setminus B,B)$.
Clearly, $\operatorname {Sph} (E)\setminus B$ deformation-retracts to B. Taking the long exact sequence of this triple, we then see:
$H^{n}(Sph(E),B)\simeq H^{n}(\operatorname {Sph} (E),\operatorname {Sph} (E)\setminus B),$
the latter of which is isomorphic to:
$H^{n}(E,E\setminus B)$
by excision.
2. "Thom's theorem" (PDF). Archived (PDF) from the original on 17 Jan 2021.
3. "Transversality" (PDF). Archived (PDF) from the original on 17 Jan 2021.
4. See pp. 8-9 in Greenlees, J. P. C. (2006-09-15). "Spectra for commutative algebraists". arXiv:math/0609452.
5. http://math.northwestern.edu/~jnkf/classes/mflds/2cobordism.pdf
6. Stong 1968, p. 18
7. http://math.northwestern.edu/~jnkf/classes/mflds/4transversality.pdf
References
• Sullivan, Dennis (2004). "René Thom's Work on Geometric Homology and Bordism". Bulletin of the American Mathematical Society. 41 (3): 341–350. doi:10.1090/S0273-0979-04-01026-2.
• Bott, Raoul; Tu, Loring (1982). Differential Forms in Algebraic Topology. New York: Springer. ISBN 0-387-90613-4. A classic reference for differential topology, treating the link to Poincaré duality and the Euler class of Sphere bundles
• May, J. Peter (1999). A Concise Course in Algebraic Topology. University of Chicago Press. pp. 183–198. ISBN 0-226-51182-0.
• "Explanation for the Pontryagin–Thom construction". MathOverflow.
• Stong, Robert E. (1968). Notes on cobordism theory. Princeton University Press.
• Thom, René (1954). "Quelques propriétés globales des variétés différentiables". Commentarii Mathematici Helvetici. 28: 17–86. doi:10.1007/BF02566923. S2CID 120243638.
• Ando, Matthew; Blumberg, Andrew J.; Gepner, David J.; Hopkins, Michael J.; Rezk, Charles (2014). "Units of ring spectra and Thom spectra". Journal of Topology. 7 (4): 1077–1117. arXiv:0810.4535. doi:10.1112/jtopol/jtu009. MR 0286898. S2CID 119613530.
External links
• http://ncatlab.org/nlab/show/Thom+spectrum
• "Thom space", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Akhil Mathew's blog posts: https://amathew.wordpress.com/tag/thom-space/
| Wikipedia |
Thom–Mather stratified space
In topology, a branch of mathematics, an abstract stratified space, or a Thom–Mather stratified space is a topological space X that has been decomposed into pieces called strata; these strata are manifolds and are required to fit together in a certain way. Thom–Mather stratified spaces provide a purely topological setting for the study of singularities analogous to the more differential-geometric theory of Whitney. They were introduced by René Thom, who showed that every Whitney stratified space was also a topologically stratified space, with the same strata. Another proof was given by John Mather in 1970, inspired by Thom's proof.
Basic examples of Thom–Mather stratified spaces include manifolds with boundary (top dimension and codimension 1 boundary) and manifolds with corners (top dimension, codimension 1 boundary, codimension 2 corners), real or complex analytic varieties, or orbit spaces of smooth transformation groups.
Definition
A Thom–Mather stratified space is a triple $(V,{\mathcal {S}},{\mathfrak {J}})$ where $V$ is a topological space (often we require that it is locally compact, Hausdorff, and second countable), ${\mathcal {S}}$ is a decomposition of $V$ into strata,
$V=\bigsqcup _{X\in {\mathcal {S}}}X,$
and ${\mathfrak {J}}$ is the set of control data $\{(T_{X}),(\pi _{X}),(\rho _{X})\ |X\in S\}$ where $T_{X}$ is an open neighborhood of the stratum $X$ (called the tubular neighborhood), $\pi _{X}:T_{X}\to X$ is a continuous retraction, and $\rho _{X}:T_{X}\to [0,+\infty )$ is a continuous function. These data need to satisfy the following conditions.
1. Each stratum $X$ is a locally closed subset and the decomposition $S$ is locally finite.
2. The decomposition $S$ satisfies the axiom of the frontier: if $X,Y\in {\mathcal {S}}$ and $Y\cap {\overline {X}}\neq \emptyset $, then $Y\subseteq {\overline {X}}$. This condition implies that there is a partial order among strata: $Y<X$ if and only if $Y\subset {\overline {X}}$ and $Y\neq X$.
3. Each stratum $X$ is a smooth manifold.
4. $X=\{v\in T_{X}\ |\ \rho _{X}(v)=0\}$. So $\rho _{X}$ can be viewed as the distance function from the stratum $X$.
5. For each pair of strata $Y<X$, the restriction $(\pi _{Y},\rho _{Y}):T_{Y}\cap X\to Y\times (0,+\infty )$ is a submersion.
6. For each pair of strata $Y<X$, there holds $\pi _{Y}\circ \pi _{X}=\pi _{Y}$ and $\rho _{Y}\circ \pi _{X}=\rho _{Y}$ (both over the common domain of both sides of the equation).
Examples
One of the original motivations for stratified spaces were decomposing singular spaces into smooth chunks. For example, given a singular variety $X$, there is a naturally defined subvariety, $\mathrm {Sing} (X)$, which is the singular locus. This may not be a smooth variety, so taking the iterated singularity locus $\mathrm {Sing} (\mathrm {Sing} (X))$ will eventually give a natural stratification. A simple algebreo-geometric example is the singular hypersurface
${\text{Spec}}\left(\mathbb {C} [x,y,z]/\left(x^{4}+y^{4}+z^{4}\right)\right)\xleftarrow {(0,0,0)} {\text{Spec}}(\mathbb {C} )$
where ${\text{Spec}}(-)$ is the prime spectrum.
See also
• Singularity theory
• Whitney conditions
• Stratifold
• Intersection homology
• Thom's first isotopy lemma
• stratified space
References
• Goresky, Mark; MacPherson, Robert Stratified Morse theory, Springer-Verlag, Berlin, 1988.
• Goresky, Mark; MacPherson, Robert Intersection homology II, Invent. Math. 72 (1983), no. 1, 77--129.
• Mather, J. Notes on topological stability, Harvard University, 1970.
• Thom, R. Ensembles et morphismes stratifiés, Bulletin of the American Mathematical Society 75 (1969), pp.240-284.
• Weinberger, Shmuel (1994). The topological classification of stratified spaces. Chicago Lectures in Mathematics. Chicago, IL: University of Chicago Press. ISBN 9780226885667.
| Wikipedia |
Thom conjecture
In mathematics, a smooth algebraic curve $C$ in the complex projective plane, of degree $d$, has genus given by the genus–degree formula
$g=(d-1)(d-2)/2$.
The Thom conjecture, named after French mathematician René Thom, states that if $\Sigma $ is any smoothly embedded connected curve representing the same class in homology as $C$, then the genus $g$ of $\Sigma $ satisfies the inequality
$g\geq (d-1)(d-2)/2$.
In particular, C is known as a genus minimizing representative of its homology class. It was first proved by Peter Kronheimer and Tomasz Mrowka in October 1994,[1] using the then-new Seiberg–Witten invariants.
Assuming that $\Sigma $ has nonnegative self intersection number this was generalized to Kähler manifolds (an example being the complex projective plane) by John Morgan, Zoltán Szabó, and Clifford Taubes,[2] also using the Seiberg–Witten invariants.
There is at least one generalization of this conjecture, known as the symplectic Thom conjecture (which is now a theorem, as proved for example by Peter Ozsváth and Szabó in 2000[3]). It states that a symplectic surface of a symplectic 4-manifold is genus minimizing within its homology class. This would imply the previous result because algebraic curves (complex dimension 1, real dimension 2) are symplectic surfaces within the complex projective plane, which is a symplectic 4-manifold.
See also
• Adjunction formula
• Milnor conjecture (topology)
References
1. Kronheimer, Peter B.; Mrowka, Tomasz S. (1994). "The Genus of Embedded Surfaces in the Projective Plane". Mathematical Research Letters. 1 (6): 797–808. doi:10.4310/mrl.1994.v1.n6.a14.
2. Morgan, John; Szabó, Zoltán; Taubes, Clifford (1996). "A product formula for the Seiberg-Witten invariants and the generalized Thom conjecture". Journal of Differential Geometry. 44 (4): 706–788. doi:10.4310/jdg/1214459408. MR 1438191.
3. Ozsváth, Peter; Szabó, Zoltán (2000). "The symplectic Thom conjecture". Annals of Mathematics. 151 (1): 93–124. arXiv:math.DG/9811087. doi:10.2307/121113. JSTOR 121113. S2CID 5283657.
| Wikipedia |
Thomae's function
Thomae's function is a real-valued function of a real variable that can be defined as:[1]: 531
$f(x)={\begin{cases}{\frac {1}{q}}&{\text{if }}x={\tfrac {p}{q}}\quad (x{\text{ is rational), with }}p\in \mathbb {Z} {\text{ and }}q\in \mathbb {N} {\text{ coprime}}\\0&{\text{if }}x{\text{ is irrational.}}\end{cases}}$
It is named after Carl Johannes Thomae, but has many other names: the popcorn function, the raindrop function, the countable cloud function, the modified Dirichlet function, the ruler function,[2] the Riemann function, or the Stars over Babylon (John Horton Conway's name).[3] Thomae mentioned it as an example for an integrable function with infinitely many discontinuities in an early textbook on Riemann's notion of integration.[4]
Since every rational number has a unique representation with coprime (also termed relatively prime) $p\in \mathbb {Z} $ and $q\in \mathbb {N} $, the function is well-defined. Note that $q=+1$ is the only number in $\mathbb {N} $ that is coprime to $p=0.$
It is a modification of the Dirichlet function, which is 1 at rational numbers and 0 elsewhere.
Properties
• Thomae's function $f$ is bounded and maps all real numbers to the unit interval:$f:\mathbb {R} \to [0,1].$
• $f$ is periodic with period $1:\;f(x+n)=f(x)$ for all integers n and all real x.
Proof of periodicity
For all $x\in \mathbb {R} \setminus \mathbb {Q} ,$ we also have $x+n\in \mathbb {R} \setminus \mathbb {Q} $ and hence $f(x+n)=f(x)=0,$
For all $x\in \mathbb {Q} ,\;$ there exist $p\in \mathbb {Z} $ and $q\in \mathbb {N} $ such that $\;x=p/q,\;$ and $\gcd(p,\;q)=1.$ Consider $x+n=(p+nq)/q$. If $d$ divides $p$ and $q$, it divides $p+nq$ and $q$. Conversely, if $d$ divides $p+nq$ and $q$, it divides $(p+nq)-nq=p$ and $q$. So $\gcd(p+nq,q)=\gcd(p,q)=1$, and $f(x+n)=1/q=f(x)$.
• $f$ is discontinuous at all rational numbers, dense within the real numbers.
Proof of discontinuity at rational numbers
Let $x_{0}=p/q$ be an arbitrary rational number, with $\;p\in \mathbb {Z} ,\;q\in \mathbb {N} ,$ and $p$ and $q$ coprime.
This establishes $f(x_{0})=1/q.$
Let $\;\alpha \in \mathbb {R} \setminus \mathbb {Q} \;$ be any irrational number and define $x_{n}=x_{0}+{\frac {\alpha }{n}}$ for all $n\in \mathbb {N} .$
These $x_{n}$ are all irrational, and so $f(x_{n})=0$ for all $n\in \mathbb {N} .$
This implies $|x_{0}-x_{n}|={\frac {\alpha }{n}},$ and $|f(x_{0})-f(x_{n})|={\frac {1}{q}}.$
Let $\;\varepsilon =1/q\;$, and given $\delta >0$ let $n=1+\left\lceil {\frac {\alpha }{\delta }}\right\rceil .$ For the corresponding $\;x_{n}$ we have
$|f(x_{0})-f(x_{n})|=1/q\geq \varepsilon $
and
$|x_{0}-x_{n}|={\frac {\alpha }{n}}={\frac {\alpha }{1+\left\lceil {\frac {\alpha }{\delta }}\right\rceil }}<{\frac {\alpha }{\left\lceil {\frac {\alpha }{\delta }}\right\rceil }}\leq \delta ,$
which is exactly the definition of discontinuity of $f$ at $x_{0}$.
• $f$ is continuous at all irrational numbers, also dense within the real numbers.
Proof of continuity at irrational arguments
Since $f$ is periodic with period $1$ and $0\in \mathbb {Q} ,$ it suffices to check all irrational points in $I=(0,1).\;$ Assume now $\varepsilon >0,\;i\in \mathbb {N} $ and $x_{0}\in I\setminus \mathbb {Q} .$ According to the Archimedean property of the reals, there exists $r\in \mathbb {N} $ with $1/r<\varepsilon ,$ and there exist $\;k_{i}\in \mathbb {N} ,$ such that
for $i=1,\ldots ,r$ we have $0<{\frac {k_{i}}{i}}<x_{0}<{\frac {k_{i}+1}{i}}.$
The minimal distance of $x_{0}$ to its i-th lower and upper bounds equals
$d_{i}:=\min \left\{\left|x_{0}-{\frac {k_{i}}{i}}\right|,\;\left|x_{0}-{\frac {k_{i}+1}{i}}\right|\right\}.$
We define $\delta $ as the minimum of all the finitely many $d_{i}.$
$\delta :=\min _{1\leq i\leq r}\{d_{i}\},\;$ :=\min _{1\leq i\leq r}\{d_{i}\},\;}
so that
for all $i=1,\dots ,r,$ $|x_{0}-k_{i}/i|\geq \delta $ and $|x_{0}-(k_{i}+1)/i|\geq \delta .$
This is to say, all these rational numbers $k_{i}/i,\;(k_{i}+1)/i,\;$ are outside the $\delta $-neighborhood of $x_{0}.$
Now let $x\in \mathbb {Q} \cap (x_{0}-\delta ,x_{0}+\delta )$ with the unique representation $x=p/q$ where $p,q\in \mathbb {N} $ are coprime. Then, necessarily, $q>r,\;$ and therefore,
$f(x)=1/q<1/r<\varepsilon .$
Likewise, for all irrational $x\in I,\;f(x)=0=f(x_{0}),\;$ and thus, if $\varepsilon >0$ then any choice of (sufficiently small) $\delta >0$ gives
$|x-x_{0}|<\delta \implies |f(x_{0})-f(x)|=f(x)<\varepsilon .$
Therefore, $f$ is continuous on $\mathbb {R} \setminus \mathbb {Q} .$
• $f$ is nowhere differentiable.
Proof of being nowhere differentiable
• For rational numbers, this follows from non-continuity.
• For irrational numbers:
For any sequence of irrational numbers $(a_{n})_{n=1}^{\infty }$ with $a_{n}\neq x_{0}$ for all $n\in \mathbb {N} _{+}$ that converges to the irrational point $x_{0},\;$ the sequence $(f(a_{n}))_{n=1}^{\infty }$ is identically $0,\;$ and so $\lim _{n\to \infty }\left|{\frac {f(a_{n})-f(x_{0})}{a_{n}-x_{0}}}\right|=0.$
According to Hurwitz's theorem, there also exists a sequence of rational numbers $(b_{n})_{n=1}^{\infty }=(k_{n}/n)_{n=1}^{\infty },\;$ converging to $x_{0},\;$ with $k_{n}\in \mathbb {Z} $ and $n\in \mathbb {N} $ coprime and $|k_{n}/n-x_{0}|<{\frac {1}{{\sqrt {5}}\cdot n^{2}}}.\;$
Thus for all $n,$ $\left|{\frac {f(b_{n})-f(x_{0})}{b_{n}-x_{0}}}\right|>{\frac {1/n-0}{1/({\sqrt {5}}\cdot n^{2})}}={\sqrt {5}}\cdot n\neq 0\;$ and so $f$ is not differentiable at all irrational $x_{0}.$
• $f$ has a strict local maximum at each rational number.
See the proofs for continuity and discontinuity above for the construction of appropriate neighbourhoods, where $f$ has maxima.
• $f$ is Riemann integrable on any interval and the integral evaluates to $0$ over any set.
The Lebesgue criterion for integrability states that a bounded function is Riemann integrable if and only if the set of all discontinuities has measure zero.[5] Every countable subset of the real numbers - such as the rational numbers - has measure zero, so the above discussion shows that Thomae's function is Riemann integrable on any interval. The function's integral is equal to $0$ over any set because the function is equal to zero almost everywhere.
• If $G=\{\,(x,f(x)):x\in (0,1)\,\}\subset \mathbb {R} ^{2}$ is the graph of the restriction of $f$ to $(0,1)$, then the box-counting dimension of $G$ is $4/3$.[6]
Related probability distributions
Empirical probability distributions related to Thomae's function appear in DNA sequencing.[7] The human genome is diploid, having two strands per chromosome. When sequenced, small pieces ("reads") are generated: for each spot on the genome, an integer number of reads overlap with it. Their ratio is a rational number, and typically distributed similarly to Thomae's function.
If pairs of positive integers $m,n$ are sampled from a distribution $f(n,m)$ and used to generate ratios $q=n/(n+m)$, this gives rise to a distribution $g(q)$ on the rational numbers. If the integers are independent the distribution can be viewed as a convolution over the rational numbers, $ g(a/(a+b))=\sum _{t=1}^{\infty }f(ta)f(tb)$. Closed form solutions exist for power-law distributions with a cut-off. If $f(k)=k^{-\alpha }e^{-\beta k}/\mathrm {Li} _{\alpha }(e^{-\beta })$ (where $\mathrm {Li} _{\alpha }$ is the polylogarithm function) then $g(a/(a+b))=(ab)^{-\alpha }\mathrm {Li} _{2\alpha }(e^{-(a+b)\beta })/\mathrm {Li} _{\alpha }^{2}(e^{-\beta })$. In the case of uniform distributions on the set $\{1,2,\ldots ,L\}$ $g(a/(a+b))=(1/L^{2})\lfloor L/\max(a,b)\rfloor $, which is very similar to Thomae's function.[7]
The ruler function
Main article: Ruler function
For integers, the exponent of the highest power of 2 dividing $n$ gives 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, ... (sequence A007814 in the OEIS). If 1 is added, or if the 0s are removed, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, ... (sequence A001511 in the OEIS). The values resemble tick-marks on a 1/16th graduated ruler, hence the name. These values correspond to the restriction of the Thomae function to the dyadic rationals: those rational numbers whose denominators are powers of 2.
Related functions
A natural follow-up question one might ask is if there is a function which is continuous on the rational numbers and discontinuous on the irrational numbers. This turns out to be impossible. The set of discontinuities of any function must be an Fσ set. If such a function existed, then the irrationals would be an Fσ set. The irrationals would then be the countable union of closed sets $ \bigcup _{i=0}^{\infty }C_{i}$, but since the irrationals do not contain an interval, neither can any of the $C_{i}$. Therefore, each of the $C_{i}$ would be nowhere dense, and the irrationals would be a meager set. It would follow that the real numbers, being the union of the irrationals and the rationals (which, as a countable set, is evidently meager), would also be a meager set. This would contradict the Baire category theorem: because the reals form a complete metric space, they form a Baire space, which cannot be meager in itself.
A variant of Thomae's function can be used to show that any Fσ subset of the real numbers can be the set of discontinuities of a function. If $ A=\bigcup _{n=1}^{\infty }F_{n}$ is a countable union of closed sets $F_{n}$, define
$f_{A}(x)={\begin{cases}{\frac {1}{n}}&{\text{if }}x{\text{ is rational and }}n{\text{ is minimal so that }}x\in F_{n}\\-{\frac {1}{n}}&{\text{if }}x{\text{ is irrational and }}n{\text{ is minimal so that }}x\in F_{n}\\0&{\text{if }}x\notin A\end{cases}}$
Then a similar argument as for Thomae's function shows that $f_{A}$ has A as its set of discontinuities.
See also
• Blumberg theorem
• Cantor function
• Dirichlet function
• Euclid's orchard – Thomae's function can be interpreted as a perspective drawing of Euclid's orchard
• Volterra's function
References
1. Beanland, Kevin; Roberts, James W.; Stevenson, Craig (2009), "Modifications of Thomae's Function and Differentiability", The American Mathematical Monthly, 116 (6): 531–535, doi:10.4169/193009709x470425, JSTOR 40391145
2. Dunham, William (2008), The Calculus Gallery: Masterpieces from Newton to Lebesgue (Paperback ed.), Princeton: Princeton University Press, page 149, chapter 10, ISBN 978-0-691-13626-4, ...the so-called ruler function, a simple but provocative example that appeared in a work of Johannes Karl Thomae ... The graph suggests the vertical markings on a ruler—hence the name.
3. John Conway. "Topic: Provenance of a function". The Math Forum. Archived from the original on 13 June 2018.
4. Thomae, J. (1875), Einleitung in die Theorie der bestimmten Integrale (in German), Halle a/S: Verlag von Louis Nebert, p. 14, §20
5. Spivak, M. (1965), Calculus on manifolds, Perseus Books, page 53, Theorem 3-8, ISBN 978-0-8053-9021-6
6. Chen, Haipeng; Fraser, Jonathan M.; Yu, Han (2022). "Dimensions of the popcorn graph". Proceedings of the American Mathematical Society. 150 (11): 4729–4742. arXiv:2007.08407. doi:10.1090/proc/15729.
7. Trifonov, Vladimir; Pasqualucci, Laura; Dalla-Favera, Riccardo; Rabadan, Raul (2011). "Fractal-like Distributions over the Rational Numbers in High-throughput Biological and Clinical Data". Scientific Reports. 1 (191): 191. arXiv:1010.4328. Bibcode:2011NatSR...1E.191T. doi:10.1038/srep00191. PMC 3240948. PMID 22355706.
• Abbott, Stephen (2016), Understanding Analysis (Softcover reprint of the original 2nd ed.), New York: Springer, ISBN 978-1-4939-5026-3
• Bartle, Robert G.; Sherbert, Donald R. (1999), Introduction to Real Analysis (3rd ed.), Wiley, ISBN 978-0-471-32148-4 (Example 5.1.6 (h))
External links
• "Dirichlet-function", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Weisstein, Eric W. "Dirichlet Function". MathWorld.
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Thomae's formula
In mathematics, Thomae's formula is a formula introduced by Carl Johannes Thomae (1870) relating theta constants to the branch points of a hyperelliptic curve (Mumford 1984, section 8).
History
In 1824 the Abel–Ruffini theorem established that polynomial equations of a degree of five or higher could have no solutions in radicals. It became clear to mathematicians since then that one needed to go beyond radicals in order to express the solutions to equations of the fifth and higher degrees. In 1858, Charles Hermite, Leopold Kronecker, and Francesco Brioschi independently discovered that the quintic equation could be solved with elliptic transcendents. This proved to be a generalization of the radical, which can be written as:
${\sqrt[{n}]{x}}=\exp \left({{\frac {1}{n}}\ln x}\right)=\exp \left({\frac {1}{n}}\int _{1}^{x}{\frac {dt}{t}}\right).$
With the restriction to only this exponential, as shown by Galois theory, only compositions of Abelian extensions may be constructed, which suffices only for equations of the fourth degree and below. Something more general is required for equations of higher degree, so to solve the quintic, Hermite, et al. replaced the exponential by an elliptic modular function and the integral (logarithm) by an elliptic integral. Kronecker believed that this was a special case of a still more general method.[1] Camille Jordan showed[2] that any algebraic equation may be solved by use of modular functions. This was accomplished by Thomae in 1870.[3] Thomae generalized Hermite's approach by replacing the elliptic modular function with even more general Siegel modular forms and the elliptic integral by a hyperelliptic integral. Hiroshi Umemura[4] expressed these modular functions in terms of higher genus theta functions.
Formula
If we have a polynomial function:
$f(x)=a_{0}x^{n}+a_{1}x^{n-1}+\cdots +a_{n}$
with $a_{0}\neq 0$ irreducible over a certain subfield of the complex numbers, then its roots $x_{k}$ may be expressed by the following equation involving theta functions of zero argument (theta constants):
${\begin{aligned}x_{k}={}&\left[\theta \left({\begin{matrix}1&0&\cdots &0\\0&\cdots &0&0\end{matrix}}\right)(\Omega )\right]^{4}\left[\theta \left({\begin{matrix}1&1&0&\cdots &0\\0&\cdots &0&0&0\end{matrix}}\right)(\Omega )\right]^{4}\\[6pt]&{}+\left[\theta \left({\begin{matrix}0&\cdots &0\\0&\cdots &0\end{matrix}}\right)(\Omega )\right]^{4}\left[\theta \left({\begin{matrix}0&1&0&\cdots &0\\0&0&0&\cdots &0\end{matrix}}\right)(\Omega )\right]^{4}\\[6pt]&{}-{\frac {\left[\theta \left({\begin{matrix}0&0&\cdots &0\\1&0&\cdots &0\end{matrix}}\right)(\Omega )\right]^{4}\left[\theta \left({\begin{matrix}0&1&0&\cdots &0\\1&0&\cdots &0&0\end{matrix}}\right)(\Omega )\right]^{4}}{2\left[\theta \left({\begin{matrix}1&0&\cdots &0\\0&\cdots &0&0\end{matrix}}\right)(\Omega )\right]^{4}\left[\theta \left({\begin{matrix}1&1&0&\cdots &0\\0&0&\cdots &0&0\end{matrix}}\right)(\Omega )\right]^{4}}}\end{aligned}}$
where $\Omega $ is the period matrix derived from one of the following hyperelliptic integrals. If $f(x)$ is of odd degree, then,
$u(a)=\int _{1}^{a}{\frac {dx}{\sqrt {x(x-1)f(x)}}}$
Or if $f(x)$ is of even degree, then,
$u(a)=\int _{1}^{a}{\frac {dx}{\sqrt {x(x-1)(x-2)f(x)}}}$
This formula applies to any algebraic equation of any degree without need for a Tschirnhaus transformation or any other manipulation to bring the equation into a specific normal form, such as the Bring–Jerrard form for the quintic. However, application of this formula in practice is difficult because the relevant hyperelliptic integrals and higher genus theta functions are very complex.
References
1. Kronecker, Leopold (1858). "Sur la résolution de l'equation du cinquème degré". Comptes rendus de l'Académie des Sciences. 46: 1150–1152.
2. Jordan, Camille (1870). Traité des substitutions et des équations algébriques. Paris: Gauthier-Villars.
3. Thomae, Carl Johannes (1870). "Beitrag zur Bestimmung von θ(0,0,...0) durch die Klassenmoduln algebraischer Funktionen". Journal für die reine und angewandte Mathematik. 71: 201–222.
4. Umemura, Hiroshi (1984). "Resolution of algebraic equations by theta constants". In David Mumford (ed.). Tata Lectures on Theta II. Birkhäuser. pp. 3.261–3.272. ISBN 3-7643-3109-7.
• Mumford, David (1984), Tata lectures on theta. II, Progress in Mathematics, vol. 43, Boston, MA: Birkhäuser Boston, ISBN 978-0-8176-3110-9, MR 0742776
• Thomae, Carl Johannes (1870), "Beitrag zur Bestimmung von θ(0,0,...0) durch die Klassenmoduln algebraischer Funktionen", Journal für die reine und angewandte Mathematik, 71: 201–222
| Wikipedia |
Thomas A. Garrity
Thomas Anthony Garrity (born 1959) is an American mathematician. He teaches at Williams College, where he is the Webster Atwell Class of 1921 Professor of Mathematics.[1]
Thomas A. Garrity
Born1959 (age 63–64)
Academic background
Education
• University of Texas at Austin
• Brown University
ThesisOn Ample Vector Bundles and Negative Curvature (1986)
Doctoral advisorWilliam Fulton
Early life and education
Thomas Anthony Garrity born in 1959 in the United States. He completed his bachelor's degree in mathematics at the University of Texas at Austin in 1981.[1] He attended Brown University for doctoral studies, completing a PhD in mathematics in 1986 under the supervision of professor William Fulton. Garrity's doctoral thesis was titled On Ample Vector Bundles and Negative Curvature.[2]
Career
Garrity is currently a professor of mathematics at Williams College, where he has taught since 1989.[3]
Bibliography
His notable books include:
• Algebraic Geometry: A Problem Solving Approach
• All the Mathematics You Missed
• The United States of Mathematics Presidential Debate
References
1. "Thomas Garrity". Mathematics & Statistics. Retrieved February 20, 2023.
2. Garrity, Thomas (December 2021). "Curriculum Vitae of Thomas A. Garrity" (PDF).
3. https://math.williams.edu/thomas-garrity/
External links
• Website
• Mathematics Genealogy Project
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Thomas Baker (mathematician)
Thomas Baker FRS (1625?–1689) was an English mathematician notable for producing a solution of biquadratic equations.
Biography
Baker is said to have been fifteen years old when he became a battler at Magdalen Hall, Oxford, in 1640. In spite of the puritanical education which, according to Wood, he received at the hall, 'he did some little petite service for his majesty within the garrison of Oxon.' It does not appear what was the nature of the 'little employments' through which, according to the same authority, he became 'minister' of Bishop's Nympton, in Devonshire. He was collated to the vicarage of Bishop's Nympton in 1681; but he seems to have lived for some years previously in that retired spot (perhaps as curate). His secluded life—as much of it at least as could be spared from professional occupations and the cares of a family—was devoted to mathematical studies. He speaks of himself as one 'who pretend(s) not to learning nor to the profession of the mathematic art, but one who(m) at some subcisive hours for diversion sake its study much delights.' He published in 1684 the 'Geometrical Key, or Gate of Equations Unlocked.' Montucla remembers having 'read somewhere' that Baker was imprisoned for debt at Newgate; upon which it was facetiously remarked that it would have been better for him to have had the key of Newgate than that of equations. That same year he was elected a Fellow of the Royal Society.[1][2]
The leading idea of Baker's work is the solution of biquadratic equations (and those of a lower degree) by a geometrical construction, a parabola intersected by a circle. The method is distinguished from that of Descartes by not requiring the equation to be previously deprived of its second term. The general principle is worked out in great detail; the author being of opinion that conciseness, like 'a watch contrived within the narrow sphere of the signet of a ring,' is rather admirable than useful. Some account of the work is given in the 'Transactions of the Royal Society' (referred to below).[1]
There exists a 'catalogue of the mathematical works of the learned Mr. Thomas Baker, with a proposal about printing the same.' The proposal was 'approved and agreed to by the council of the Royal Society,' but was not carried out.[1]
References
1. Edgeworth 1885, p. 17.
2. "Fellows Details". Royal Society. Retrieved 17 January 2017.
Attribution:
• This article incorporates text from a publication now in the public domain: Edgeworth, Francis Ysidro (1885). "Baker, Thomas (1625?–1689)". In Stephen, Leslie (ed.). Dictionary of National Biography. Vol. 3. London: Smith, Elder & Co. p. 17.
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Thomas Barker (mathematician)
Thomas Barker (1838–1907) was a Scottish mathematician, professor of pure mathematics at Owens College.
Life
Born 9 September 1838, he was son of Thomas Barker, farmer, of Murcar, Balgonie, near Aberdeen, and of his wife Margaret. Three other children died in infancy. He was educated at Aberdeen Grammar School, and at King's College in the same town, where he graduated in 1857 with distinction in mathematics.[1]
Barker entered Trinity College, Cambridge as minor scholar and subsizar in 1858.[2] He became foundation scholar in 1860, Sheepshanks astronomical exhibitioner in 1861, and came out in the Mathematical Tripos of 1862 as senior wrangler; he was also first Smith's prizeman. He was elected to a fellowship in the autumn of 1862, and was assistant tutor of Trinity till 1865, when he was appointed professor of pure mathematics in Owens College, Manchester. He held this post for twenty years.[1]
Barker was a follower of Augustus De Morgan and George Boole. He was interested in the logical basis rather than in the applications of mathematics, and was an austere teacher. He disliked publication.[1]
After resigning his chair in 1885, Barker lived in retirement, first at Whaley Bridge and then at Buxton. He pursued the study of cryptogamic botany. He died unmarried at Buxton on 20 November 1907, and was buried in Southern Cemetery, Manchester.[1]
Pupils
Barker had a number of distinguished mathematicians and physicists as pupils: they included John Walton Capstick,[3] John Hopkinson, John Henry Poynting, Arthur Schuster, and Joseph John Thomson.[1]
Legacy
By his will Barker provided for the foundation in the University of Manchester of a professorship of cryptogamic botany, and for the endowment of bursaries in mathematics and botany.[1]
Notes
1. Lee, Sidney, ed. (1912). "Barker, Thomas" . Dictionary of National Biography (2nd supplement). Vol. 1. London: Smith, Elder & Co.
2. "Barker, Thomas (BRKR858T)". A Cambridge Alumni Database. University of Cambridge.
3. Falconer, Isobel. "Barker, Thomas". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/30591. (Subscription or UK public library membership required.)
Attribution
This article incorporates text from a publication now in the public domain: Lee, Sidney, ed. (1912). "Barker, Thomas". Dictionary of National Biography (2nd supplement). Vol. 1. London: Smith, Elder & Co.
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Thomas Bartholin
Thomas Bartholin (/bɑːrˈtoʊlɪn, ˈbɑːrtəlɪn/; Latinized as Thomas Bartholinus; 20 October 1616 – 4 December 1680) was a Danish physician, mathematician, and theologian. He discovered the lymphatic system in humans and advanced the theory of refrigeration anesthesia, being the first to describe it scientifically.
Thomas Bartholin
Thomas Bartholin
Born20 October 1616
Malmø, Denmark-Norway
Died4 December 1680 (aged 64)
Copenhagen
NationalityDanish
Alma materUniversity of Padua
Known forLymphatic system
Scientific career
FieldsMedicine
InstitutionsUniversity of Copenhagen
Academic advisorsJohannes Walaeus
Thomas Bartholin came from a family that has become famous for its pioneering scientists, twelve of whom became professors at the University of Copenhagen. Three generations of the Bartholin family made significant contributions to anatomical science and medicine in the 17th and 18th centuries: Thomas Bartholin's father, Caspar Bartholin the Elder (1585–1629), his brother Rasmus Bartholin (1625–1698), and his son Caspar Bartholin the Younger (1655–1738).[1] Thomas Bartholin's son Thomas Bartholin the Younger (1659–1690) became a professor of history at the University of Copenhagen and was later appointed royal antiquarian and secretary to the Royal Archives.[2]
Personal life
Thomas Bartholin was the second of the six sons of Caspar Bartholin the Elder, a physician born in Malmø, Scania, and his spouse Anne Fincke. Bartholin the Elder published the first collected anatomical work in 1611. This work was later augmented, illustrated and revised by Thomas Bartholin, becoming the standard reference on anatomy; the son notably added updates on William Harvey's theory of blood circulation and on the lymphatic system.
Bartholin visited the Italian botanist Pietro Castelli at Messina in 1644. In 1663 Bartholin bought Hagestedgård, which burned down in 1670 including his library, with the loss of many manuscripts. King Christian V of Denmark appointed Bartholin as his physician with a substantial salary and freed the farm from taxation as recompense for the loss. In 1680 Bartholin's health failed, the farm was sold, and he moved back to Copenhagen, where he died. He was buried in Vor Frue Kirke (Church of Our Lady).
The Bartholinsgade, a street in Copenhagen, is named for the family. Nearby is the Bartholin Institute (Bartholin Institutet). One of the buildings of the University of Aarhus is named after him.
Contributions to medical research
In December 1652, Bartholin published the first full description of the human lymphatic system. Jean Pecquet had previously noted the lymphatic system in animals in 1651, and Pecquet's discovery of the thoracic duct and its entry into the veins made him the first person to describe the correct route of the lymphatic fluid into the blood.[3] Shortly after the publication of Pecquet's and Bartholin's findings, a similar discovery of the human lymphatic system was published by Olof Rudbeck in 1653, although Rudbeck had presented his findings at the court of Queen Christina of Sweden in April–May 1652, before Bartholin, but delayed in writing about it until 1653 (after Bartholin). As a result, an intense priority dispute ensued.[4] Niels Stensen or Steno became Bartholin's most famous pupil.
Thomas' publication De nivis usu medico observationes variae Chapter XXII, contains the first known mention of refrigeration anaesthesia, a technique whose invention Thomas Bartholin credits to the Italian Marco Aurelio Severino of Naples.[5] According to Bartholin, Severino was the first to present the use of freezing mixtures of snow and ice (1646), and Thomas Bartholin initially learnt about the technique from him during a visit to Naples.
Bartholin–Patau syndrome, a congenital syndrome of multiple abnormalities produced by trisomy 13, was first described by Bartholin in 1656.[6][7]
Caspar Bartholin the Elder, Thomas Bartholin's father; his brother Rasmus Bartholin; and his son Caspar Bartholin the Younger (who first described "Bartholin's glands"), all contributed to the practice of modern medicine through their discoveries of important anatomical structures and phenomena.[1] Bartholin the Elder started his tenure as professor at Copenhagen University in 1613, and over the next 125 years, the scientific accomplishments of the Bartholins while serving on the medical faculty of the University of Copenhagen won international acclaim and contributed to the reputation of the institution.
Selected works
• Historiarum anatomicarum rariorum [...] (Case histories of unusual anatomical and clinical structures, including descriptions and illustrations of anomalies and normal structures)
• ... centuria I et II at Google Books, Amsterdam, 1654.
• ... centuria III et IV at Google Books. The Hague: Vlacq, 1657.
• ... centuria V et VI at Google Books, Copenhagen: P. Haubold, 1661 (with Mantissa anatomica, by Johannes Rodius).
• De unicornu. Padua, 1645.
• De Angina Puerorum Campaniae Siciliaeque Epidemica Exercitationes. Paris, 1646.
• De lacteis thoracicis in homine brutisque nuperrime observatis historia anatomica at Google Books, Copenhagen: M. Martzan, 1652 (Bartholin's discovery of the thoracic duct).
• Vasa lymphatica nuper Hafniae in animalibus inventa et hepatis exsequiae. Hafniae (Copenhagen), Petrus Hakius, 1653.
• Vasa lymphatica in homine nuper inventa. Hafniae (Copenhagen), 1654.
• Historarium anatomicarum rariorum centuria I-VI. Copenhagen, 1654–1661.
• Anatomia. The Hague. Ex typographia Adriani Vlacq, 1655.
• Dispensarium hafniense. Copenhagen, 1658.
• De nivis usu medico observationes variae. Accessit D. Erasmi Bartholini de figura nivis dissertatio. With a book by Rasmus Bartholin. Copenhagen: Typis Matthiase Godichii, sumptibus Petri Haubold, 1661. (Contains the first known mention of refrigeration anaesthesia)
• Cista medica hafniensis. Copenhagen, 1662.
• De pulmonum substantia et motu. Copenhagen, 1663.
• De insolitis partus humani viis. Copenhagen, 1664.
• De medicina danorum domestica. Copenhagen, 1666.
• De flammula cordis epistola. Copenhagen, 1667.
• Orationes et dissertationes omnino varii argumenti. Copenhagen, 1668.
• Carmina varii argumenti. Copenhagen, 1669.
• De medicis poetis dissertatio. Hafinae, apud D. Paulli, 1669.
• De bibliothecae incendio. Copenhagen, 1670.
• De morbis biblicis miscellanea medica. Francofurti, D. Paulli, 1672.
• De cruce Christi hypomnemata IV, Typis Andreae ab Hoogenhuysen, Vesaliae (Wesel), 1673.
• Acta medica et philosophica. 1673–1680.
References
1. Hill, Robert V. (2007) "A Glimpse of Our Past – The contributions of the Bartholin family to the study and practice of clinical anatomy". Clinical Anatomy, Volume 20, Issue 2 (March 2007), pp. 113 – 115. Retrieved 22 February 2007.
2. Jónsson, Már (2012). Arnas Magnæus Philologus (1663–1730). [Odense]: University Press of Southern Denmark. pp. 48–49.
3. Detmar, Michael and Mihaela Skobe (2000). "Structure, Function, and Molecular Control of the Skin Lymphatic System". Journal of Investigative Dermatology Symposium Proceedings (2000) 5, 14–19. Retrieved 22 February 2007.
4. Eriksson, G. (2004). Svensk medicinhistorisk tidskrift, 2004;8(1):39-44. In Swedish. English abstract at Olaus Rudbeck as scientist and professor of medicine, U.S. National Library of Medicine. Retrieved 22 February 2007.
5. De nivis, p. 132, p. 132, at Google Books : " nix affricata induit stuporem. Id me docuit Marcus Aurelius Severinus in Gymnasio Neapolitano ".
6. synd/1024 at Who Named It?
7. Bartholinus, Thomas (1656). Historiarum anatomicarum rariorum centuria III et IV. Ejusdem cura accessere observationes anatomicae. The Hague: Vlacq. p. 95.
External links
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• View digitized titles by Thomas Bartholin in Botanicus.org
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• MyNDIR (My Norse Digital Image repository) illustrations from Thomas Bartholin's works. Clicking on the thumbnail will give you the full image and information concerning it.
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Thomas Baxter (mathematician)
Thomas Baxter (fl. 1732–1740), was a schoolmaster and mathematician who published an erroneous method of squaring the circle. He was derided as a "pseudo-mathematician" by F. Y. Edgeworth, writing for the Dictionary of National Biography.[1]
When he was master of a private school at Crathorne, North Yorkshire,[2] Baxter composed a book entitled The Circle squared (London: 1732), published in octavo.[3] The mathematical book begins with the untrue assertion that "if the diameter of a circle be unity or one, the circumference of that circle will be 3.0625", where the value should correctly be pi.[1] From this incorrect assumption, Baxter proves fourteen geometric theorems on circles, alongside some others on cones and ellipses, which Edgeworth refers to as of "equal absurdity" to Baxter's other assertions.[1] Thomas Gent, who published the work, wrote in his reminisces, in The Life of Mr. Thomas Gent, that "as it never proved of any effect, it was converted to waste paper, to the great mortification of the author".[4][5]
This book has received harsh reviews from modern mathematicians and scholars. Antiquary Edward Peacock referred to it as "no doubt, great rubbish".[6] Mathematician Augustus De Morgan included Baxter's proof among his Budget of Paradoxes (1872), dismissing it as an absurd work.[3] The work was the reason Edgeworth gave Baxter the epithet, "pseudo-mathematician".[1]
Baxter published another work, Matho, or the Principles of Astronomy and Natural Philosophy accommodated to the Use of Younger Persons (London: 1740). Unlike Baxter's other work, this volume enjoyed considerable popularity in its time.[7][8]
References
1. Edgeworth, F. Y. (1885). "Baxter, Thomas (fl.1732)" . In Stephen, Leslie (ed.). Dictionary of National Biography. Vol. 3. London: Smith, Elder & Co.
2. McConnell, Anita (2004). "Baxter, Thomas (fl.1732)". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/1737. (Subscription or UK public library membership required.)
3. De Morgan, Augustus (1872). A Budget of Paradoxes. London: Longmans, Green, & Co. p. 87.
4. Gent, Thomas (1832). The life of Mr. Thomas Gent. London: Thomas Thorpe.
5. Notes and Queries, 3rd Series, Vol. 5. 1849. p. 258.
6. Notes and Queries, 3rd Series, Vol. 5. 1849. p. 348.
7. Rose, Rev. Hugh James (1841). "Baxter, Thomas". A New General Biographical Dictionary. Vol. 3. p. 391.
8. Allibone, S. Austin (1871). "Baxter, Thomas". A Critical Dictionary of English Literature and British and American Authors. Vol. 1. p. 143.
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Thomas Bedwell
Thomas Bedwell (died April 1595) was an English mathematician and military engineer.
Bedwell matriculated as a sizar of Trinity College, Cambridge in November 1562. He became a scholar in the same year; in 1566–7 he took the degree of B.A.; he was subsequently elected fellow; and in 1570 commenced M.A.[1] He was appointed to the office of keeper of the ordnance stores in the Tower. He is said to have been the first to project 'the bringing of the waters of the Lea from Ware to London.' In conjunction with Frederico Genebelli he was employed as a military engineer in strengthening the works at Tilbury and Gravesend at the time of the Spanish Armada. He died in April 1595.
Thomas Bedwell was uncle of William Bedwell, the Arabic scholar, who speaks of him as 'our English Tycho.' The two are sometimes confounded, chiefly, it would appear, on account of an ambiguity on the title-page of the first of two works published by the nephew in explanation of a 'ruler' or mesolabium architectonicum which the uncle had devised to facilitate carpenters' calculations.[2]
References
1. "Bedwell, Thomas (BDWL562T)". A Cambridge Alumni Database. University of Cambridge.
2. See the Macclesfield collection of Corresp. of Scient. Men, Oxford, 1841, p. 1ff.
• "Bedwell, Thomas" . Dictionary of National Biography. London: Smith, Elder & Co. 1885–1900.
Attribution
This article incorporates text from a publication now in the public domain: "Bedwell, Thomas". Dictionary of National Biography. London: Smith, Elder & Co. 1885–1900.
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Brooke Benjamin
Thomas Brooke Benjamin, FRS[1] (15 April 1929 – 16 August 1995) was an English mathematical physicist and mathematician, best known for his work in mathematical analysis and fluid mechanics, especially in applications of nonlinear differential equations.[2]
T. Brooke Benjamin
Brooke Benjamin
Born(1929-04-15)15 April 1929
Wallasey, England
Died16 August 1995(1995-08-16) (aged 66)
Oxford, England
NationalityBritish
Alma materUniversity of Liverpool
Yale University
University of Cambridge
Known forBenjamin–Bona–Mahony equation
Benjamin–Ono equation
Benjamin–Feir instability
Scientific career
FieldsFluid dynamics
Mathematical analysis
InstitutionsUniversity of Cambridge
University of Essex
University of Oxford
Doctoral studentsJohn Dwyer
Alan Champneys
Education and career
Benjamin was educated at Wallasey Grammar School on the Wirral, the University of Liverpool (BEng. 1950) and Yale University (MEng. 1952), before being awarded his doctorate at King's College, Cambridge in 1955.[3][4] He was a fellow of King's from 1955 to 1964.[5]
From 1979 until his death in 1995 he was Sedleian Professor of Natural Philosophy at the Mathematical Institute, University of Oxford, and a fellow of The Queen's College, Oxford.[6]
Contributions
The Benjamin–Ono equation describes one-dimensional internal waves in deep water. It was introduced by Benjamin in 1967, and later studied also by Hiroaki Ono. Another equation named after Benjamin, the Benjamin–Bona–Mahony equation, models long surface gravity waves of small amplitude. Benjamin studied it with Jerry L. Bona and J. J. Mahony in a 1972 paper.
References
1. Hunt, J. C. R. (2003). "Thomas Brooke Benjamin. 15 April 1929 – 16 August 1995 Elected FRS 1966". Biographical Memoirs of Fellows of the Royal Society. 49: 39–67. doi:10.1098/rsbm.2003.0003.
2. Hunt, J. C. R. (2006). "Nonlinear and Wave Theory Contributions of T. Brooke Benjamin (1929–1995)". Annual Review of Fluid Mechanics. 38 (1): 1–25. Bibcode:2006AnRFM..38....1H. doi:10.1146/annurev.fluid.38.050304.092028.
3. Brooke Benjamin at the Mathematics Genealogy Project
4. O'Connor, John J.; Robertson, Edmund F., "Brooke Benjamin", MacTutor History of Mathematics Archive, University of St Andrews
5. "BENJAMIN, Prof. (Thomas) Brooke". Who's Who & Who Was Who. Vol. 2018 (online ed.). A & C Black. (Subscription or UK public library membership required.)
6. Longuet-Higgins, M. S. (2004). "Benjamin, (Thomas) Brooke". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/60105. Retrieved 15 April 2015. (Subscription or UK public library membership required.)
External links
• "The Brooke Benjamin Lecture in Fluid Dynamics". Mathematical Institute at the University of Oxford. Archived from the original on 25 June 2009. Retrieved 8 March 2009.
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Thomas Bloom
Thomas F. Bloom is a mathematician, who is a Royal Society University Research Fellow at the University of Oxford.[1][2] He works in arithmetic combinatorics and analytic number theory.
Thomas Bloom
NationalityBritish
Alma materUniversity of Oxford
University of Bristol
AwardsRoyal Society University Research Fellowship
Scientific career
InstitutionsUniversity of Cambridge
University of Oxford
University of Bristol
Doctoral advisorTrevor Wooley
Other academic advisorsTimothy Gowers
Education and career
Thomas did his undergraduate degree in Mathematics and Philosophy at Merton College, Oxford. He then went on to do his PhD in mathematics at the University of Bristol under the supervision of Trevor Wooley. After finishing his PhD, he was a Heilbronn Research Fellow at the University of Bristol. In 2018, he became a postdoctoral research fellow at the University of Cambridge with Timothy Gowers. In 2021, he joined the University of Oxford as a Research Fellow.[3]
Research
In July 2020, Bloom and Sisask[4] proved that any set such that $\sum _{n\in A}{\frac {1}{n}}$ diverges must contain arithmetic progressions of length 3. This is the first non-trivial case of a conjecture of Erdős postulating that any such set must in fact contain arbitrarily long arithmetic progressions.[5][6]
In November 2020, in joint work with James Maynard,[7] he improved the best-known bound for square-difference-free sets, showing that a set $A\subset [N]$ with no square difference has size at most ${\frac {N}{(\log N)^{c\log \log \log N}}}$ for some $c>0$.
In December 2021, he proved [8] that any set $A\subset \mathbb {N} $ of positive upper density contains a finite $S\subset A$ such that $\sum _{n\in S}{\frac {1}{n}}=1$. This answered a question of Erdős and Graham.[9]
References
1. "Thomas Bloom | Mathematical Institute". www.maths.ox.ac.uk. Retrieved 2022-07-28.
2. Cepelewicz, Jordana (2022-03-09). "Math's 'Oldest Problem Ever' Gets a New Answer". Quanta Magazine. Retrieved 2022-07-28.
3. "Thomas Bloom". thomasbloom.org. Retrieved 2022-07-28.
4. Bloom, Thomas F.; Sisask, Olof (2021-09-01). "Breaking the logarithmic barrier in Roth's theorem on arithmetic progressions". arXiv:2007.03528. {{cite journal}}: Cite journal requires |journal= (help)
5. Spalding, Katie (11 March 2022). "Math Problem 3,500 Years In The Making Finally Gets A Solution". IFLScience. Retrieved 28 July 2022.
6. Klarreich, Erica (3 August 2020). "Landmark Math Proof Clears Hurdle in Top Erdős Conjecture". Quanta Magazine. Retrieved 28 July 2022.
7. Bloom, Thomas F.; Maynard, James (24 February 2021). "A new upper bound for sets with no square differences". arXiv:2011.13266 [math.NT].
8. Bloom, Thomas F. (2021-12-07). "On a density conjecture about unit fractions". arXiv:2112.03726. {{cite journal}}: Cite journal requires |journal= (help)
9. Erdos, P.; Graham, R. (1980). "Old and new problems and results in combinatorial number theory". S2CID 117960481. {{cite journal}}: Cite journal requires |journal= (help)
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Thomas Bond Sprague Prize
The Thomas Bond Sprague Prize is a prize awarded annually to the student or students showing the greatest distinction in actuarial science, finance, insurance, mathematics of operational research, probability, risk and statistics in the Master of Mathematics/Master of Advanced Studies examinations of the University of Cambridge, also known as Part III of the Mathematical Tripos.[1] The prize is named after Thomas Bond Sprague, the only person to have been president of both the Institute of Actuaries in London and the Faculty of Actuaries in Edinburgh. It is awarded by the Rollo Davidson Trust[2] of Churchill College, Cambridge, following a donation by D. O. Forfar, MA, FFA, FRSE (alumnus of Trinity College, Cambridge), former Appointed Actuary of Scottish Widows.[3]
List of recipients
• 2012 P. J. S. Lowth (Clare)
• 2013 P. Parmar (Trinity) and S. E. Penington (Clare)
• 2014 T. Assiotis (Trinity) and T. B. Berrett (Gonville and Caius)
• 2015 P. Gurican (Trinity) and A. Q. Wang (St John's)
• 2016 I. Spasojevic (Trinity) and S. M. Thomas (St John's)
• 2017 O. Feng (Trinity) and D. Heydecker (Queens')
• 2018 B. B. He (Queens') and M. Lehmkuehler (Girton)
• 2019 P. Bevan (Queens')
• 2022 E. Katiyar (Corpus Christi) and S. McInerney (St. John's) and D. Yue (Emmanuel)
• 2023 L. J. Hill (Trinity) and M. Augustynowicz (Trinity)
See also
• List of mathematics awards
References
1. "External Notices - Cambridge University Reporter 6273". www.admin.cam.ac.uk. Retrieved 30 July 2019.
2. "Statistical Laboratory". www.statslab.cam.ac.uk. Retrieved 30 July 2019.
3. "Social news round-up - August 2012 | The Actuary, the official magazine of the Institute and Faculty of Actuaries". Retrieved 30 July 2019.
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| Wikipedia |
Thomas Brattle
Thomas Brattle (June 20, 1658 – May 18, 1713) was an American merchant who served as treasurer of Harvard College and member of the Royal Society. He is known for his involvement in the Salem Witch Trials and the formation of the Brattle Street Church.
Brattle was also a mathematician, astronomer, and an experienced traveler.[1]
Early life
Thomas Brattle was born on June 20, 1658, in Boston, Massachusetts Bay Colony, to Elizabeth Brattle née Tyng and Captain Thomas Brattle. He was the couple's second child, and the first son to survive past infancy. He had eight siblings, including William Brattle and Catherine Winthrop. Brattle's date of birth is often confused with the first-born son of the Brattle family (also named Thomas Brattle) – who was born on, and died on, September 5, 1657.[2]
As a child, Brattle was exposed to radical forms of the Puritan faith, primarily through his father's participation in the controversial founding of the Third (South) Church, which advocated for ecclesiastical reforms.[3] The church's membership included many notable members such as Samuel Sewall, Samuel Adams, and Benjamin Franklin. At one point in time, Thomas' father, Captain Brattle, was named the wealthiest man in the colony. After the death of his father, Thomas was appointed administrator of the estate on April 12, 1683, leaving him with a large sum of money and a healthy plot of land.[4]
Before attending Harvard University in 1676, he attended the Boston Latin School.[5] This school was open to all boys regardless of class, and served to educate and prepare the young men for university.[6] The Boston Latin School is where Brattle met influential Puritan leader Cotton Mather. Although the two men agreed on many social and political ideologies later in life, they did not see eye to eye during their time at the Boston Latin School. It is documented that Thomas Brattle and other schoolmates enjoyed picking on Cotton Mather (to the point where he wrote to his father, Increase Mather, and requested to come home early).[7] After his time at the Boston Latin School, Brattle attended Harvard University and received an AB in mathematics and science.
Education
In 1676, Brattle graduated from Harvard College with an A.B., equivalent to our modern B.A. At Harvard, Brattle developed marked skill in mathematics and science.[8] Though he is most well known for The Witchcraft Delusion, which was written to argue against the Salem Witch Trials, Brattle was interested in many areas including mathematics, architecture, and astronomy.[9]
Brattle gained most of his education on his own due to the bad leadership at Harvard in his undergraduate years. He used whatever books that were available and studied with John Foster and Dr. William Avery. Brattle wrote a letter to John Flamsteed, a mentor of his, stating that no one at Harvard could teach him mathematics so he took it upon himself to do so. Thomas and a group of other prominent colonists studied several comets that appeared in the late seventeenth century. He wrote several essays on these comets.
Brattle later travelled abroad and then settled in Boston in 1693, where he pursued a short business career and gave several gifts to Harvard.[10] That same year, he was appointed as the Harvard College treasurer and he served in that position for twenty years until his death. During his time as treasurer, the finances of the college grew exponentially. Brattle was a member of the intellectually elite Royal Society. The Royal Society was a new group of scientific thinkers that practiced a more intense and rational thought process. This group grew much larger in the eighteenth century when it was headed by Sir Isaac Newton. Sir Isaac Newton was so impressed with Brattle's work that he planned to procure his papers on astronomy and math after Brattle's death in order to benefit the Royal Society. In an attempt to obtain them, Newton tried to make his brother, William Brattle, a member of the Society, however William declined.[11] In 1711, Brattle attempted to use a mathematical algorithm in order to end smallpox. Although he failed, it can be seen that Brattle was heavily involved in education and scientific discovery.[3] Brattle made more substantial contributions to science than any other American of the day.
Personal life
Brattle's mother was Elizabeth Brattle and his father Captain Thomas Brattle, who was one of Boston's wealthy maritime merchants.[12] He was one of four children, including his brother, William Brattle (and nephew William Brattle), and his two sisters, Elizabeth Oliver and Katherine Winthrop Eyre.[13]
Brattle was an accomplished amateur mathematician and astronomer which eventually led him to becoming the unofficial professor of mathematics and astronomy at Harvard. There he taught and trained students in return for their assistance in his research. His work was directly influenced by the ideas of Robert Boyle and John Flamsteed, which he communicated to his students. In addition to being a professor, he became the treasurer of Harvard College after numerous donations. Circumstantial evidence indicates that he designed Stoughton Hall at Harvard and his own Brattle Street Meeting House.[14]
Brattle lived in London from 1682 to 1689 in order to study science. There he was involved in both scientific communities which can "help us understand a good deal about the progress of scientific expertise in colonial New England," since he was able to communicate information to both communities.[15] Among his accomplishments, he was also a member of Royal Society, and Ancient and Honorable Artillery Company of Massachusetts.[16]
Brattle was also the principal founder of the Brattle Street Church,[17] which broke away from the Congregational church. This sparked an intense dispute between Brattle and famous Puritan minister, Cotton Mather. Rather than being similar to the Puritans, his church was more like the Church of England.[9] He was eulogized by the Reverend Benjamin Colman as "worth Christian philosopher, who was also the glory of his country in respect to his excelling knowledge of mathematics".[15]
Death
Brattle died in Boston, but his death date is still questioned to this day. He was buried in the city.
Legacy
Salem Witch Trials
Brattle participated in the Salem Witch Trials as one of the observers and commentators.[12] Later, he was one of personages who became more open about their criticisms of the trials. Along with Robert Calef and Thomas Maule, he was particularly critical of the procedures adopted.[1]
On October 8, 1692, Brattle wrote a letter to an unnamed English clergyman containing his sentiments.[1] The letter was circulated widely in Boston at the time, and it continues to be studied for its reasoned attack on the witchcraft trials in Salem. The "highly literate" and "satirical tone" of the letter was seen as writing beyond its time, leading Perry Miller to call it a "milestone in American literature."[18] Brattle denounced the persecution of suspected witches, and his letter revealed a "chink in the armor" of Puritan ideology. Brattle presents a compelling argument against the legal premises and procedures involved in the afflictions, accusations, and executions, with a particular focus on the invalidity of spectral evidence in proceedings. He argued that the procedures were so contrary to established practice and were dire in their consequences.[19]
Brattle's letter was designed to illustrate the wrongful convictions that the Court of Oyer and Terminer made during this time as they based their evidence on witchcraft from intangible evidence. He was careful to not critique the "Salem Gentlemen", which he referred to as the judges and ministers, but rather focused on critiquing the methods they used.[20] After Governor Phips read Brattle's letter, he ordered that the courts could no longer use intangible evidence as a source to convict individuals of witchcraft. Phips dissolved the court entirely within the same month. Six months later, the Superior Court of Massachusetts took over the remaining witchcraft cases and no one else was found guilty thereafter. Although Brattle's letter was written after 20 people were already wrongfully convicted, his powerful letter helped shape the future of Salem.
Church Reformation
As a result of the reaction toward theological, political, and cultural transformations that affected the whole of New England in the later half of the 17th century, the Brattle Street Church was formed as a result of radical development in the evolution of colonial congregationalism - bringing reason and religion together in a new church. The Congregational Church was broadly catholic, but used conservative principles of congregationalism (that just liberty and privilege should be allowed to all, while imposing nothing upon an individual).[21] Although it did not make any radical changes from contemporary theological consensus - its foundation did represent the first concrete fragmentation of a previously united New England Congregational Community.
Outside of his involvement in the Salem witch trials, Brattle and his younger brother William provided new radical ideas that the Puritan Church did not agree with. Brattle preached some of these more liberal ideas in the church he founded, the Brattle Street Church, which led to an argument with Puritan minister Cotton Mather. Also, both Thomas and William improved Harvard College. Thomas donated money many times, served as treasurer of the college, and was an unofficial professor of astronomy and mathematics.[22] When Brattle died, he left the New World with a new rational approach towards thought.
Other achievements
Brattle is also credited as being the first person to import an organ to the colonies.[23]
References
1. Beau, Bryan F. Le (2016). The Story of the Salem Witch Trials. Oxon: Routledge. ISBN 978-1-315-50903-7.
2. Harris, Edward-Doubleday (1867). An Account of Some of the Descendants of Capt. Thomas Brattle. Boston: Clapp and Son. ISBN 978-1-332-09602-2.
3. Kennedy, Rick (Dec 1990). "Brattle, William". The New England Quarterly. 63 (4): 584–600. JSTOR 365919.
4. Harris, Edward Doubleday (1867). An account of some of the descendants of Capt. Thomas Brattle. Boston.{{cite book}}: CS1 maint: location missing publisher (link)
5. Wright, Conrad Edick (2004). ""Thomas Brattle (1658-1713)" in Oxford Dictionary of National Biography". Oxford Dictionary of National Biography (online ed.). Oxford: OUP. doi:10.1093/ref:odnb/71069. (Subscription or UK public library membership required.)
6. "Boston Latin School" in Encyclopædia Britannica. Encyclopædia Britannica Inc.
7. Kennedy, Rick (1958). The First American Evangelical: A Short Life of Cotton Mather. Michigan: Wm. B. Eerdmans Publishing Co.
8. Spencer, Mark G. (2015). The Bloomsbury Encyclopedia of the American Enlightenment. New York: Bloomsbury.
9. "Salem Witch Trials Notable Persons". salem.lib.virginia.edu. Retrieved 2016-04-01.
10. "Thomas Brattle | North American entrepreneur". Encyclopædia Britannica. Retrieved 2016-04-01.
11. Miller, Perry; Thomas Herbert Johnson, eds. (1963). The Puritans: A Sourcebook of Their Writings. New York: Harper & Row.
12. Goss, David (2018). Documents of the Salem Witch Trials. Santa Barbara, CA: ABC-CLIO. p. 126. ISBN 978-1-4408-5320-3.
13. "Thomas Brattle". geni_family_tree. Retrieved 2016-04-01.
14. Kennedy, Rick (2000). Brattle, Thomas (1658-1713), astronomer and architect. doi:10.1093/anb/9780198606697.article.0100100. ISBN 978-0-19-860669-7. {{cite book}}: |website= ignored (help)
15. Kennedy, Rick (1990-01-01). "Thomas Brattle and the Scientific Provincialism of New England, 1680-1713". The New England Quarterly. 63 (4): 584–600. doi:10.2307/365919. JSTOR 365919.
16. "Salem Witch Trials Documentary Archive". salem.lib.virginia.edu. Retrieved 2016-04-01.
17. Bridgman, Thomas; Everett, Edward (1856). The Pilgrims of Boston and their descendants. New York: Appleton & Co. p. 317. Retrieved April 30, 2009.
18. Madden, Matthew. "Salem Witch Trials Notable Persons". Salem Witch Trials Documentary Archive and Transcription Project. University of Virginia. Retrieved 1 April 2016.
19. Adams, Gretchen A. (2008). The Specter of Salem: Remembering the Witch Trials in Nineteenth-Century America. Chicago: University of Chicago Press. pp. 35. ISBN 978-0-226-00541-6.
20. "Thomas Brattle". March 28, 2016.
21. Elliot, Cara. "This House which I have built: The Foundation of the Brattle Street Church in Boston and Transformations in Colonial Congregationalism". Gettysburg College.edu. The Gettysburg Historical Journal. Retrieved 30 March 2016.
22. "Thomas Brattle: North American Entrepreneur". Encyclopædia Britannica. ©2016 Encyclopædia Britannica, Inc. Retrieved 29 March 2016.
23. Wilder Foote, Henry. Musical Life in Boston (PDF). American Antiquarian Society. p. 307. Retrieved 1 April 2016.
External links
• Burr, George Lincoln, 1857–1938. "Letter of Thomas Brattle", F.R.S., 1692", Narratives of the Witchcraft Cases, 1648–1706. Electronic Text Center, University of Virginia Library.
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| Wikipedia |
Thomas Callister Hales
Thomas Callister Hales (born June 4, 1958) is an American mathematician working in the areas of representation theory, discrete geometry, and formal verification. In representation theory he is known for his work on the Langlands program and the proof of the fundamental lemma over the group Sp(4) (many of his ideas were incorporated into the final proof of the fundamental lemma, due to Ngô Bảo Châu). In discrete geometry, he settled the Kepler conjecture on the density of sphere packings and the honeycomb conjecture. In 2014, he announced the completion of the Flyspeck Project, which formally verified the correctness of his proof of the Kepler conjecture.
Thomas Hales
Born (1958-06-04) June 4, 1958
San Antonio, Texas
NationalityAmerican
Alma materPrinceton University
Known forProving Kepler conjecture
Awards
• Chauvenet Prize (2003)
• David P. Robbins Prize (2007)
• Fulkerson Prize (2009)
• Tarski Lectures (2019)
Scientific career
FieldsMathematics
InstitutionsUniversity of Pittsburgh[1]
University of Michigan
Doctoral advisorRobert Langlands
Doctoral studentsJulia Gordon
Websitesites.google.com/site/thalespitt/
Biography
He received his Ph.D. from Princeton University in 1986 with a dissertation titled The Subregular Germ of Orbital Integrals.[2][3] Hales taught at Harvard University and the University of Chicago,[4] and from 1993 and 2002 he worked at the University of Michigan.[5]
In 1998, Hales submitted his paper on the computer-aided proof of the Kepler conjecture; a centuries-old problem in discrete geometry which states that the most space-efficient way to pack spheres is in a tetrahedron shape. He was aided by graduate student Samuel Ferguson.[6] In 1999, Hales proved the honeycomb conjecture, he also stated that the conjecture may have been present in the minds of mathematicians before Marcus Terentius Varro.
After 2002, Hales became the University of Pittsburgh's Mellon Professor of Mathematics. In 2003, Hales started work on Flyspeck to vindicate his proof of the Kepler conjecture. His proof relied on computer calculation to verify conjectures. The project used two proof assistants; HOL Light and Isabelle.[7][8][9][10] Annals of Mathematics accepted the proof in 2005; but was only 99% sure of the proof.[10] In August 2014, the Flyspeck team's software finally verified the proof to be correct.[10]
In 2017, he initiated the Formal Abstracts project which aims to provide formalised statements of the main results of each mathematical research paper in the language of an interactive theorem prover. The goal of this project is to benefit from the increased precision and interoperability that computer formalisation provides while circumventing the effort that a full-scale formalisation of all published proofs currently entails. In the long term, the project hopes to build a corpus of mathematical facts which would allow for the application of machine learning techniques in interactive and automated theorem proving.[11]
Awards and memberships
Hales won the Chauvenet Prize in 2003[12] and a Lester R. Ford Award in 2008.[13] In 2012 he became a fellow of the American Mathematical Society.[14] He was invited to give the Tarski Lectures in 2019; his three lectures were titled "A formal proof of the Kepler conjecture", "Formalizing mathematics", and "Integrating with Logic".[15][16]
Publications
• Hales, Thomas C. (1994). "The status of the Kepler conjecture". The Mathematical Intelligencer. 16 (3): 47–58. doi:10.1007/BF03024356. ISSN 0343-6993. MR 1281754. S2CID 123375854.
• Hales, Thomas C. (2001). "The Honeycomb Conjecture". Discrete and Computational Geometry. 25 (1): 1–22. arXiv:math/9906042. doi:10.1007/s004540010071. MR 1797293. S2CID 14849112.
• Hales, Thomas C. (2005). "A proof of the Kepler conjecture". Annals of Mathematics. 162 (3): 1065–1185. arXiv:math/9811078. doi:10.4007/annals.2005.162.1065.
• Hales, Thomas C. (2006). "Historical overview of the Kepler conjecture". Discrete & Computational Geometry. 36 (1): 5–20. doi:10.1007/s00454-005-1210-2. ISSN 0179-5376. MR 2229657.
• Hales, Thomas C.; Ferguson, Samuel P. (2006). "A formulation of the Kepler conjecture". Discrete & Computational Geometry. 36 (1): 21–69. arXiv:math/9811078. doi:10.1007/s00454-005-1211-1. ISSN 0179-5376. MR 2229658. S2CID 6529590.
• Hales, Thomas C.; Ferguson, Samuel P. (2011), The Kepler Conjecture: The Hales-Ferguson Proof, New York: Springer, ISBN 978-1-4614-1128-4
• Hales, Thomas C.; Adams, Mark; Bauer, Gertrud; Dang, Tat Dat; Harrison, John; Hoang, Truong Le; Kaliszyk, Cezary; Magron, Victor; McLaughlin, Sean; Nguyen, Tat Thang; Nguyen, Quang Truong; Nipkow, Tobias; Obua, Steven; Pleso, Joseph; Rute, Jason; Solovyev, Alexey; An Hoai Thi Ta; Tran, Nam Trung; Trieu, Thi Diep; Urban, Josef; Vu, Ky; Zumkeller, Roland (2017). "A formal proof of the Kepler conjecture". Forum of Mathematics, Pi. 5: e2. arXiv:1501.02155. doi:10.1017/fmp.2017.1.
Notes
1. "Thomas Hales | Department of Mathematics | University of Pittsburgh".
2. "Thomas Hales - the Mathematics Genealogy Project".
3. Hales, Thomas C. (1992). "The subregular germ of orbital integrals" (PDF). Memoirs of the American Mathematical Society. 99 (476). doi:10.1090/MEMO/0476. S2CID 121175826. Archived from the original (PDF) on 2020-02-29.
4. "Brief Bio of Thomas C. Hales - thalespitt". Archived from the original on 2020-12-27.
5. http://um2017.org/faculty-history/faculty/thomas-c-hales
6. "University of Pittsburgh: Department of Mathematics".
7. "Thalespitt".
8. Flyspeck Project
9. Hales solves oldest problem in discrete geometry The University Record (University of Michigan), September 16, 1998
10. Aron, Jacob (August 12, 2014). "Proof confirmed of 400-year-old fruit-stacking problem". New Scientist. Retrieved May 10, 2017.
11. Project website https://formalabstracts.github.io/, retrieved 2020-01-10.
12. Hales, Thomas C. (2000). "Cannonballs and Honeycombs". Notices of the AMS. 47 (4): 440–449.
13. Hales, Thomas C. (2007). "The Jordan Curve Theorem, Formally and Informally". American Mathematical Monthly. 114 (10): 882–894. doi:10.1080/00029890.2007.11920481. JSTOR 27642361. S2CID 887392.
14. List of Fellows of the American Mathematical Society, retrieved 2013-01-19.
15. "2019 Tarski Lectures | Department of Mathematics at University of California Berkeley". math.berkeley.edu. Retrieved 2021-11-02.
16. "Group in Logic and the Methodology of Science - Tarski Lectures". logic.berkeley.edu. Retrieved 2021-11-02.
External links
• Thomas Callister Hales at the Mathematics Genealogy Project
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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
| Wikipedia |
Thomas Digges
Thomas Digges (/dɪɡz/; c. 1546 – 24 August 1595) was an English mathematician and astronomer. He was the first to expound the Copernican system in English but discarded the notion of a fixed shell of immoveable stars to postulate infinitely many stars at varying distances.[1] He was also first to postulate the "dark night sky paradox".[2]
Thomas Digges
Bornc. 1546
Wootton, Kent, England
Died(1595-08-24)24 August 1595
London, England
NationalityEnglish
Known forProposing that the universe is infinite in extent
Scientific career
FieldsAstronomer and mathematician
Academic advisorsJohn Dee
Notes
Son of Leonard Digges, and father of Dudley Digges and Leonard Digges (II)
Life
Thomas Digges, born about 1546, was the son of Leonard Digges (c. 1515 – c. 1559), the mathematician and surveyor, and Bridget Wilford, the daughter of Thomas Wilford, esquire, of Hartridge in Cranbrook, Kent, by his first wife, Elizabeth Culpeper, the daughter of Walter Culpeper, esquire. Digges had two brothers, James and Daniel, and three sisters, Mary, who married a man with the surname of Barber; Anne, who married William Digges; and Sarah, whose first husband was surnamed Martin, and whose second husband was John Weston.[3]
After the death of his father, Digges grew up under the guardianship of John Dee,[4] a typical Renaissance natural philosopher. In 1583, Lord Burghley appointed Digges, with John Chamber and Henry Savile, to sit on a commission to consider whether England should adopt the Gregorian calendar, as proposed by Dee.[5]
Digges served as a member of parliament for Wallingford and also had a military career as a Muster-Master General to the English forces from 1586 to 1594 during the war with the Spanish Netherlands. In his capacity of Master-Muster General he was instrumental in promoting improvements at the Port of Dover.[6]
Digges died on 24 August 1595. His last will, in which he specifically excluded both his brother, James Digges, and William Digges, was proved on 1 September. Digges was buried in the chancel of the church of St Mary Aldermanbury, London.[7]
Marriage and issue
Digges married Anne St Leger (1555–1636), daughter of Sir Warham St Leger and his first wife, Ursula Neville (d. 1575), the fifth daughter of George Neville, 5th Baron Bergavenny, by his third wife, Mary Stafford.[8] In his will he named two surviving sons, Sir Dudley Digges (1583–1639), politician and statesman, and Leonard Digges (1588–1635), poet, and two surviving daughters, Margaret and Ursula. After Digges's death, his widow, Anne, married Thomas Russell of Alderminster in Warwickshire, "whom in 1616 William Shakespeare named as an overseer of his will".[9]
Work
Digges attempted to determine the parallax of the 1572 supernova observed by Tycho Brahe, and concluded it had to be beyond the orbit of the Moon. This contradicted Aristotle's view of the universe, according to which no change could take place among the fixed stars.
In 1576, he published a new edition of his father's perpetual almanac, A Prognostication everlasting. The text written by Leonard Digges for the third edition of 1556 was left unchanged, but Thomas added new material in several appendices. The most important of these was A Perfit Description of the Caelestiall Orbes according to the most aunciente doctrine of the Pythagoreans, latelye revived by Copernicus and by Geometricall Demonstrations approved. Contrary to the Ptolemaic cosmology of the original book by his father, the appendix featured a detailed discussion of the controversial and still poorly known Copernican heliocentric model of the Universe. This was the first publication of that model in English, and a milestone in the popularisation of science.
For the most part, the appendix was a loose translation into English of chapters from Copernicus' book De revolutionibus orbium coelestium. Thomas Digges went further than Copernicus, however, by proposing that the universe is infinite, containing infinitely many stars, and may have been the first person to do so, predating Giordano Bruno's (1584)[10] and William Gilbert's (1600)[11] same views. According to Harrison:[12]
Copernicus had said little or nothing about what lay beyond the sphere of fixed stars. Digges's original contribution to cosmology consisted of dismantling the starry sphere, and scattering the stars throughout endless space. ...
By grafting endless space onto the Copernican system and scattering the stars throughout this endless space, Digges pioneered... the idea of an unlimited universe filled with the mingling rays of countless stars.
An illustration of the Copernican universe can be seen above right. The outer inscription on the map reads (after spelling adjustments from Elizabethan to Modern English):
This orb of stars fixed infinitely up extends itself in altitude spherically, and therefore immovable the palace of felicity garnished with perpetual shining glorious lights innumerable, far excelling our sun both in quantity and quality the very court of celestial angels, devoid of grief and replenished with perfect endless joy, the habitacle for the elect.
In 1583, Lord Burghley appointed Digges, along with Henry Savile (Bible translator) and John Chamber, to sit on a commission to consider whether England should adopt the Gregorian calendar, as proposed by John Dee; in fact Britain did not adopt the calendar until 1752.[13]
References
1. Johnston 2004b.
2. Al-Khalili, Jim, Everything and Nothing – 1. Everything, BBC Four, 9:00PM Mon, 21 March 2011
3. Richardson_I 2011, p. 81; Johnston 2004a.
4. Johnston 2004b.
5. Mosley 2004
6. Lane, Anthony (2011). Front Line Harbour: A History of the Port of Dover. Stroud: Amberley Publishing Limited. ISBN 9781445620084.
7. Johnston 2004b.
8. Edwards 2004.
9. Lee 2004.
10. Bruno, Giordano. "Third Dialogue". On the infinite universe and worlds. Archived from the original on 27 April 2012.
11. Gilbert, William (1893). "Book 6, Chapter III". De Magnete. Translated by Mottelay, P. Fleury. (Facsimile). New York: Dover Publications. ISBN 0-486-26761-X.
12. Harrison (1987), p. 35, 37.
13. Adam Mosley, 'Chamber, John (1546–1604), in Oxford Dictionary of National Biography (Oxford University Press, 2004)
Sources and further reading
• Edwards, David (2004). "St Leger, Sir Warham (1525?–1597)". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/24514. (Subscription or UK public library membership required.) The first edition of this text is available at Wikisource: "St. Leger, Warham" . Dictionary of National Biography. London: Smith, Elder & Co. 1885–1900.
• Johnston, Stephen (2004a). "Digges, Leonard (c.1515–c.1559)". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/7637. (Subscription or UK public library membership required.) The first edition of this text is available at Wikisource: "Digges, Leonard (d.1571?)" . Dictionary of National Biography. London: Smith, Elder & Co. 1885–1900.
• Johnston, Stephen (2004b). "Digges, Thomas (c.1546–1595)". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/7639. (Subscription or UK public library membership required.) The first edition of this text is available at Wikisource: "Digges, Thomas" . Dictionary of National Biography. London: Smith, Elder & Co. 1885–1900.
• Lee, Sidney, rev. Haresnape, Elizabeth (2004). "Digges, Leonard (1588–1635)". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/7638.{{cite encyclopedia}}: CS1 maint: multiple names: authors list (link) (Subscription or UK public library membership required.) The first edition of this text is available at Wikisource: "Digges, Leonard (1588-1635)" . Dictionary of National Biography. London: Smith, Elder & Co. 1885–1900.
• Mosley, Adam (2004). "Chamber, John (1546–1604)". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/5044. (Subscription or UK public library membership required.)
• Richardson, Douglas (2011). Everingham, Kimball G. (ed.). Magna Carta Ancestry: A Study in Colonial and Medieval Families. Vol. I (2nd ed.). Salt Lake City. ISBN 978-1449966379. Retrieved 27 February 2013.{{cite book}}: CS1 maint: location missing publisher (link)
• Text of the Perfit Description:
• Johnson, Francis R. and Larkey, Sanford V., "Thomas Digges, the Copernican System and the idea of the Infinity of the Universe in 1576," Huntington Library Bulletin 5 (1934): 69–117.
• Harrison, Edward Robert (1987) Darkness at Night. Harvard University Press: 211–17. An abridgement of the preceding.
• Internet version at Dartmouth retrieved on 2 November 2013
• Gribbin, John, 2002. Science: A History. Penguin.
• Johnson, Francis R., Astronomical Thought in Renaissance England: A Study of the English Scientific Writings from 1500 to 1645, Johns Hopkins Press, 1937.
• Kugler, Martin Astronomy in Elizabethan England, 1558 to 1585: John Dee, Thomas Digges, and Giordano Bruno, Montpellier: Université Paul Valéry, 1982.
• Vickers, Brian (ed.), Occult & Scientific Mentalities in the Renaissance. Cambridge: Cambridge University Press, 1984. ISBN 0-521-25879-0
External links
• Digges, Thomas
• Thomas Digges, Gentleman and Mathematician
• John Dee, Thomas Digges and the identity of the mathematician
• Digges' Mactutor biography
• Digges, Thomas (1546–1595), History of Parliament
• Hutchinson, John (1892). "Thomas Digges" . Men of Kent and Kentishmen (Subscription ed.). Canterbury: Cross & Jackman. p. 40.
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| Wikipedia |
Thomas F. Coleman
Thomas F. Coleman is a Canadian mathematician and computer scientist who is a Professor in the Department of Combinatorics and Optimization at the University of Waterloo,[1] where he holds the Ophelia Lazaridis University Research Chair. In addition, Coleman is the director of WatRISQ,[2] an institute composed of quantitative and computational finance researchers spanning several Faculties at the University of Waterloo.
Thomas F. Coleman
Born
1950 (1950)
Montreal, Quebec
Died20 April 2021 (aged 70–71)
OccupationProfessor
Academic background
EducationUniversity of Waterloo, Ph.D., Mathematics, 1979
Academic work
InstitutionsDepartment of Combinatorics and Optimization, University of Waterloo
Education
Coleman earned his PhD from University of Waterloo in 1979 with the dissertation A Superlinear Penalty Function Method to Solve the Nonlinear Programming Problem supervised by Andrew Conn.[3] He followed that up with a two-year postdoctoral appointment in the Applied Mathematics Division at Argonne National Laboratory.
Career
From 1981 to 2005, Coleman was a professor of computer science at Cornell University. From 1998 to 2005 he served as the director of Cornell Theory Center, now Cornell University Center for Advanced Computing[4]
From 2005 to 2010, Coleman served as the dean of the Faculty of Mathematics at the University of Waterloo. During his tenure as Cornell Theory Center director, Coleman founded and directed a computational finance academic-industry-government venture located at 55 Broad Street in New York, which shaped into Cornell Financial Engineering Manhattan.[5]
Research
Coleman's research is concerned with the design and understanding of practical and efficient numerical algorithms for continuous optimization problems. His work has been applied in many scientific & industrial areas that include finance and risk-management, structural design, logistics and planning, protein structure, data-mining, medical imaging and informatics.[6]
Awards and honors
Coleman was selected a SIAM Fellow in 2016[7] for his contributions to financial optimization, sparse numerical optimization and leadership in mathematical education and industry engagement.
Coleman has published over 80 journal articles in the areas of parallel computing, optimization, automatic differentiation, computational finance, and optimization applications and is the author of three books on computational mathematics.
References
1. "Thomas F. Coleman". University of Waterloo. Retrieved 2021-03-23.
2. "About Waterloo Research Institute in Insurance, Securities and Quantitative Finance (WatRISQ)". 23 April 2014.
3. Thomas F. Coleman at the Mathematics Genealogy Project
4. "Cornell University Center for Advanced Computing". Cornell University Center for Advanced Computing. 2011-01-25. Retrieved 2021-03-23.
5. "Financial Engineering Concentration and Cornell Financial Engineering Manhattan (CFEM) | Operations Research and Information Engineering". www.orie.cornell.edu.
6. "thomas coleman list". thomas coleman. Retrieved 2021-03-23.
7. "Tom Coleman Selected SIAM Fellow". Waterloo Research Institute in Insurance, Securities and Quantitative Finance. April 22, 2016.
External links
• Thomas F. Coleman publications indexed by Google Scholar
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Thomas Muirhead Flett
Thomas Muirhead Flett (28 July 1923 – 13 February 1976) was an English mathematician at Sheffield University working on analysis.[1][2]
Thomas Muirhead Flett
Born(1923-07-28)28 July 1923
London, England
Died13 February 1976(1976-02-13) (aged 52)
NationalityBritish
Scientific career
FieldsMathematics
Biography
Thomas Muirhead Flett was born on 28 July 1923, in London, England, when his parents moved from Scotland to London. At age 11, he won a scholarship by the County of Middlesex from a state primary school to University College School (U.C.S.), Hampstead.[2]
Career
Flett was first employed as a laboratory assistant in Post Office Research Station at Dollis Hill. While working, he studied part-time at Acton Technical College and obtained first class Honours in the London University B.Sc. general Degree in Mathematics and Physics. After working in the Post Office Research Station for three years, Flett worked as a research physicist at Simmonds Aerocessories in Brentford for two years. In 1945, pursuing his interest in Mathematics at Chelsea Polytecnic, he obtained class Honours in the B.Sc. Special Degree in Mathematics. In 1946, he obtained a mark in optional subjects, and in 1947, he was awarded for M.Sc. in Mathematics of London University.[2]
References
1. Thomas Muirhead Flett at the Mathematics Genealogy Project
2. Taylor, S.J. (1977). "Thomas Muirhead Flett". Bulletin of the London Mathematical Society. London Mathematical Society. 9 (3): 330–339. doi:10.1112/blms/9.3.330. ISSN 1469-2120.
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Thomas Forster (mathematician)
Thomas Edward Forster (born 12 April 1948) is a British set theorist and philosopher. His work has focused on Quine's New Foundations, the theory of well-quasi-orders and better-quasi-orders, and various topics in philosophy.[1]
Dr
Thomas Forster
Born (1948-04-12) 12 April 1948
Alma materCambridge
Scientific career
InstitutionsDPMMS, Cambridge
ThesisNF (1977)
Doctoral advisorAdrian Mathias, Maurice Boffa
Forster is an Affiliated Lecturer at DPMMS, Cambridge,[2] a bye-fellow at Queens' College,[3] and holds honorary appointments for many other organisations worldwide, including the Center for Philosophy of Science in Pittsburgh, the Centre National de Recherches de Logique in Belgium, and the Centre for Discrete Mathematics and Theoretical Computer Science at the University of Auckland.[1] Amongst his undergraduate supervisees are Phebe Mann, Rosi Sexton, Richard Taylor, Rebecca Kitteridge, Doug Gurr, Sarah Flannery and Ursula Martin.
Forster was awarded the J.T. Knight Prize as a PhD student at Cambridge in 1974.[1] His article "The Iterative Conception of Set"[4] was recognised by the Philosophers' Annual as one of the ten best philosophy articles of 2008.[5]
References
1. Forster, Thomas. "CV" (PDF).{{cite web}}: CS1 maint: url-status (link)
2. "Dr Thomas Forster | Department of Pure Mathematics and Mathematical Statistics". www.dpmms.cam.ac.uk. Retrieved 29 November 2020.
3. "Dr Thomas Forster | Queens' College". www.queens.cam.ac.uk. Retrieved 29 November 2020.
4. FORSTER, THOMAS (June 2008). "The Iterative Conception of Set". The Review of Symbolic Logic. 1 (1): 97–110. doi:10.1017/s1755020308080064. ISSN 1755-0203. S2CID 15231169.
5. "The Philosopher's Annual". www.pgrim.org. Retrieved 29 November 2020.
External links
• Forster's website at DPMMS, Cambridge.
• Forster's CV.
• Thomas Edward Forster at the Mathematics Genealogy Project.
• T.E. Forster. Set Theory with a Universal Set: Exploring an Untyped Universe. 2nd ed. Oxford Logic Guides, Clarendon Press, 1995.
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Thomas G. Kurtz
Thomas G. Kurtz (born 14 July 1941 in Kansas City, Missouri, United States) is an American emeritus professor of Mathematics[1] and Statistics[2] at University of Wisconsin-Madison known for his research contributions to many areas of probability theory and stochastic processes. In particular, Kurtz’s research focuses on convergence, approximation and representation of several important classes of Markov processes. His findings appear in scientific disciplines such as systems biology, population genetics, telecommunications networks and mathematical finance.
Education
Kurtz obtained his Ph.D. from Stanford University in 1967 under the supervision of James L. McGregor.[3] As an undergraduate student he attended University of Missouri where he graduated in 1963 with a Bachelor's degree in Mathematics. Kurtz is also an alumnus of La Plata High School in La Plata, Missouri.
Academic career
After completing his Ph.D. in 1967, Kurtz joined the Department of Mathematics at the University of Wisconsin–Madison where he remained for his entire career. He received a joint appointment in the Statistics department in 1985. In 1996, he was awarded the WARF-University Houses Professorship, which he chose to identify as the Paul Lévy Professorship to honor one of the founders of modern probability theory. At UW Madison, Kurtz served as the Mathematics Department Chair from 1985 to 1988 and as the Director of the Center for Mathematical Sciences from 1990 to 1996. He retired from active teaching in 2008 but he continues to work as an emeritus professor. During his academic career Kurtz has supervised twenty-nine Ph.D. students[4] and lectured extensively at UW Madison and elsewhere. For almost a decade, he organized a Summer Internship Program in Madison, which helped in grooming the next generation of probabilists.
Kurtz has given several invited seminars and tutorials around the world. Over the years he has also held many visiting positions, which include:
• Nelder Visiting Fellow, Imperial College, London, U.K. April–May 2016.[5]
• Guest Professor, Goethe University, Frankfurt, Germany, May–June, 2013.
• Visiting Fellow, Isaac Newton Institute for Mathematical Sciences, Cambridge, U.K, February–May, 2010.
• Very Important Visitor (VIV), Institute for Mathematics and its Applications, Minneapolis, 2003–2004.
• Visiting Professor, Stanford University, April–June, 1989
• Visiting Professor, University of Utah, Salt Lake City, January–March, 1989
• Visiting Professor, Université de Strasbourg, France, 1977–1978.
• Visiting Fellow, Australian National University, Canberra, 1973.
Prof. Kurtz has served in many scientific committees and Editorial boards of academic journals. He is currently a trustee of the Mathematical Biosciences Institute in Columbus, Ohio.
Awards and honors
Kurtz is a former president of the Institute of Mathematical Statistics (2005–2006) and a former editor of the Annals of Probability (2000–2002). He is a Fellow of the Institute of Mathematical Statistics (IMS) and of the American Academy of Arts and Sciences. He was chosen to give the Wald Memorial Lectures in 2014 at the IMS Annual Meeting in Sydney, Australia.[6]
He was elected as a Fellow of the American Mathematical Society in the 2020 Class, for "research in probability and its applications, especially for contributions to the study of Markov processes".[7]
Publications
In his five decades of research, Kurtz has published more than 100 peer-reviewed articles[8] in prominent mathematical journals. He has also authored four books, which are as follows:
1. Markov Processes: Characterization and Convergence (John Wiley & Sons Inc. 1986):[9] This book with his former Ph.D. student Stewart Ethier is one of Kurtz’s most well-known works and it is a standard reference for advanced theory of Markov processes. This book develops an intricate, yet elegant mathematical framework for establishing the convergence of Markov processes and characterising the limiting process.
2. Stochastic Analysis of Biochemical Systems (Springer 2015): This book with David Anderson provides a timely survey of the wide array of methods and techniques that can be employed to analyze stochastic models of chemical reaction networks. Such models are frequently encountered in the rapidly growing field of Systems Biology. Mindful of the interdisciplinary nature of the research community in this field, the authors present the material in such a way that it is accessible to anyone who is familiar with the standard undergraduate mathematics curriculum.
3. Large Deviations for Stochastic Processes (American Mathematical Society 2006): This book with his former Ph.D. student Jin Feng, presents a general theory for obtaining large deviation results for a large class of stochastic processes. This theory is based on the idea that the large deviation principle for a sequence of Markov processes can be obtained by proving the convergence of an associated family of nonlinear semigroups. To overcome the formidable theoretical challenge of proving this convergence of nonlinear semigroups, the authors employ tools from the modern theory of viscosity solutions that has been developed for solving partial differential equation. With these viscosity methods, the authors demonstrate that large deviation results can be readily derived using their approach for a range of interesting examples.
4. Approximation of Population Processes (Society of Industrial and Applied Mathematics 1981): This book provides a self-contained treatment of the limiting behavior of a wide class of population processes as the population-size approaches infinity. Diffusion approximation results are developed for population processes on a very general state-space, thereby allowing the results to be applied in a variety of examples from Branching Processes, Population Genetics, Epidemics and Chemical Reaction Networks. Furthermore, random time-change formulas are introduced that provide sample path representations of complicated stochastic processes in terms of its simpler counterparts such as Poisson processes or Brownian motions. These formulas are used to derive approximation results and also examine the relationship between the stochastic process and the corresponding deterministic process in the “law of large numbers” limit.
References
1. "Department of Mathematics | Van Vleck Hall, 480 Lincoln Drive, Madison, WI". Math.wisc.edu. Retrieved 2016-07-27.
2. "Home | Department of Statistics". Stat.wisc.edu. Retrieved 2016-07-27.
3. "James McGregor - The Mathematics Genealogy Project". Genealogy.math.ndsu.nodak.edu. Retrieved 12 February 2022.
4. "Former Students". People.math.wisc.edu. Retrieved 12 February 2022.
5. "Nelder Visiting Fellowships | Imperial College London". Archived from the original on 2016-08-14. Retrieved 2016-07-27.
6. "Honored Special Lecturers (recipient list)". Archived from the original on 10 August 2016. Retrieved 12 February 2022.
7. 2020 Class of the Fellows of the AMS, American Mathematical Society, retrieved 2019-11-03
8. "Thomas G. Kurtz – Google Scholar Citations". Scholar.google.com. Retrieved 2016-07-27.
9. Aldous, David J. (1987). "Book Review: Markov processes: Characterization and convergence". Bulletin of the American Mathematical Society. 16 (2): 315–319. doi:10.1090/S0273-0979-1987-15533-9. ISSN 0273-0979.
External links
• Thomas G. Kurtz University of Wisconsin
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Thomas Geisser
Thomas Hermann Geisser (born February 28, 1966, in Wuppertal) is a German mathematician working at Rikkyo University (Tokyo, Japan). He works in the field of arithmetic geometry, motivic cohomology and algebraic K-theory.
Thomas Geisser
Thomas Geisser in 2005
Born (1966-02-28) 28 February 1966
Wuppertal
NationalityGerman
Alma materUniversity of Münster
AwardsSloan Fellowship 2000, Humboldt Prize 2021
Scientific career
FieldsMathematics
InstitutionsRikkyo University
ThesisA p-adic analog of Beilinson's conjectures for Hecke characters of imaginary quadratic fields (1994)
Doctoral advisorChristopher Deninger
Websitehttps://www2.rikkyo.ac.jp/web/geisser/
Education
From 1985 Geisser studied at Bonn University under the supervision of Günther Harder and obtained a master's degree in 1990. He continued to obtain a Ph.D. under the supervision of Christopher Deninger at the University of Münster; the title of his thesis is A p-adic analog of Beilinson's conjecture for Hecke characters of imaginary quadratic fields.[1]
Career
Geisser spent three years at Harvard University as a visiting scholar and visiting fellow, respectively. After further stays in Essen, University of Illinois, Urbana-Champaign and Tokyo University, he became assistant professor at the University of Southern California, and was promoted to associate professor in 2002 and professor in 2006.
After visiting Tokyo University again he became professor at Nagoya University in 2010, and moved to Rikkyo University in 2015[2]
He received a Sloan Research Fellowship (2000) and a Humboldt Prize (2021).[3]
He is editor for Documenta Mathematica[4] and managing editor for Commentarii Mathematici Universitatis St.Pauli.[5]
Selected publications
• "Algebraic 𝐾-Theory". Proceedings of Symposia in Pure Mathematics. Vol. 67. Providence, Rhode Island: American Mathematical Society. 1999. doi:10.1090/pspum/067. hdl:10852/39333. ISBN 978-0-8218-0927-3. ISSN 2324-707X.
• Geisser, Thomas; Levine, Marc (1 March 2000). "The K -theory of fields in characteristic p". Inventiones Mathematicae. Springer Science and Business Media LLC. 139 (3): 459–493. Bibcode:2000InMat.139..459G. doi:10.1007/s002220050014. ISSN 0020-9910. S2CID 5675613.
• Geisser, T.; Levine, M. (12 January 2001). "The Bloch-Kato conjecture and a theorem of Suslin-Voevodsky". Journal für die reine und angewandte Mathematik (Crelle's Journal). Walter de Gruyter GmbH. 2001 (530). doi:10.1515/crll.2001.006. ISSN 0075-4102.
• Geisser, Thomas (15 May 2006). "Arithmetic cohomology over finite fields and special values of ζ-functions". Duke Mathematical Journal. Duke University Press. 133 (1). arXiv:math/0405164. doi:10.1215/s0012-7094-06-13312-4. ISSN 0012-7094. S2CID 119678716.
• Geisser, Thomas; Hesselholt, Lars (13 May 2006). "Bi-relative algebraic K-theory and topological cyclic homology". Inventiones Mathematicae. Springer Science and Business Media LLC. 166 (2): 359–395. arXiv:math/0409122. Bibcode:2006InMat.166..359G. doi:10.1007/s00222-006-0515-y. ISSN 0020-9910. S2CID 12507964.
• Geisser, Thomas (9 August 2010). "Duality via cycle complexes". Annals of Mathematics. 172 (2): 1095–1127. doi:10.4007/annals.2010.172.1095. ISSN 0003-486X. S2CID 59500496.
• Geisser, Thomas H.; Schmidt, Alexander (23 August 2018). "Poitou–Tate duality for arithmetic schemes". Compositio Mathematica. 154 (9): 2020–2044. arXiv:1709.06913. doi:10.1112/s0010437x18007340. ISSN 0010-437X. S2CID 119735104.
• Geisser, Thomas H. (19 July 2018). "COMPARING THE BRAUER GROUP TO THE TATE–SHAFAREVICH GROUP". Journal of the Institute of Mathematics of Jussieu. Cambridge University Press (CUP). 19 (3): 965–970. doi:10.1017/s1474748018000294. ISSN 1474-7480. S2CID 119152180.
References
1. Thomas Geisser at the Mathematics Genealogy Project
2. "立教大学理学部数学科：研究室の紹介".
3. https://www.humboldt-foundation.de/en/connect/explore-the-humboldt-network/singleview?tx_rsmavhsolr_solrview%5BpPersonId%5D=1058906&cHash=f7c3d8df94082f66cc50542a436dfc45
4. "ELibM – Documenta Mathematica".
5. "立教大学学術リポジトリ - 立教大学学術リポジトリ".
External links
• Homepage Rikkyo
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Thomas Goodwillie (mathematician)
Thomas G. Goodwillie (born 1954) is an American mathematician and professor at Brown University who has made fundamental contributions to algebraic and geometric topology. He is especially famous for developing the concept of calculus of functors, often also named Goodwillie calculus.
Thomas G. Goodwillie
NationalityAmerican
Alma materHarvard University (A.B., M.A., 1976)
Princeton University (Ph.D., 1982)
Known forCalculus of functors
Scientific career
FieldsTopology, K-Theory
InstitutionsHarvard University, Brown University
Doctoral advisorWu-Chung Hsiang
Life
While studying at Harvard University, Goodwillie became a Putnam Fellow in 1974 and 1975.[1] He then studied at Princeton University, where he completed his PhD at in 1982, under the supervision of Wu-Chung Hsiang.[2] He returned to Harvard as Junior Fellow in 1979, and was an associate professor (without tenure) at Harvard from 1982 to 1987. In 1987 he was hired with tenure by Brown University, where he was promoted to full professor in 1991.[3]
He developed the calculus of functors in a series of three papers in the 1990s and 2000s,[4][5][6] which has since been expanded and applied in a number of areas, including the theory of smooth manifolds, algebraic K-theory, and homotopy theory.[7]
He has advised 11 PhD students.[2]
Goodwillie is interested in issues of racial and gender equality and has taught a course on this topic.[8] He is an active user on MathOverflow.[9]
Recognition
Goodwillie received a Sloan Fellowship and the Harriet S. Sheridan Award. He is a Fellow of the American Mathematical Society.[10]
A conference with leading topologists as speakers was organized on the occasion of his 60th birthday.[11]
References
1. "Putnam Competition Individual and Team Winners". Mathematical Association of America. Retrieved December 13, 2021.
2. Thomas Goodwillie at the Mathematics Genealogy Project
3. "Goodwillie's CV" (PDF).
4. T. Goodwillie, Calculus I: The first derivative of pseudoisotopy theory, K-theory 4 (1990), 1-27.
5. T. Goodwillie, Calculus II: Analytic functors, K-theory 5 (1992), 295-332.
6. T. Goodwillie, Calculus III: Taylor series, Geom. Topol. 7 (2003), 645-711.
7. "Workshop at berwolfach about Goodwillie Calculus".
8. "Goodwillie's class on race and gender".
9. "Goodwillie's profile page on Mathoverflow".
10. "List of fellows of the AMS".
11. "Goodwillie's birthday conference".
External links
• Website at Brown University
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Philip J. Davis
Philip J. Davis (January 2, 1923[3] – March 14, 2018) was an American academic applied mathematician.
Philip J. Davis
Born(1923-01-02)January 2, 1923
Lawrence, Massachusetts, U.S.
DiedMarch 14, 2018(2018-03-14) (aged 95)
NationalityAmerican
Alma materHarvard University
AwardsChauvenet Prize (1963)
Lester R. Ford Award (1982)[1][2]
Scientific career
FieldsMathematics
InstitutionsBrown University
Doctoral advisorRalph Philip Boas, Jr.
Doctoral studentsFrank Deutsch
Jeffery J. Leader
Davis was born in Lawrence, Massachusetts. He was known for his work in numerical analysis and approximation theory, as well as his investigations in the history and philosophy of mathematics. He earned his degrees in mathematics from Harvard University (SB, 1943; PhD, 1950, advisor Ralph P. Boas, Jr.), and his final position was Professor Emeritus at the Division of Applied Mathematics at Brown University.
He served briefly in an aerodynamics research position in the Air Force in World War II before joining the National Bureau of Standards (now the National Institute of Standards and Technology). He became Chief of Numerical Analysis there and worked on the well-known Abramowitz and Stegun Handbook of Mathematical Functions before joining Brown in 1963.
He was awarded the Chauvenet Prize for mathematical writing in 1963 for an article on the gamma function,[4] and won numerous other prizes, including being chosen to deliver the 1991 Hendrick Lectures of the MAA (which became the basis for his book Spirals: From Theodorus to Chaos). He was a frequent invited lecturer and authored several books. Among the best known are The Mathematical Experience (with Reuben Hersh), a popular survey of modern mathematics and its history and philosophy; Methods of Numerical Integration (with Philip Rabinowitz),[5] long the standard work on the subject of quadrature; and Interpolation and Approximation, still an important reference in this area.
For The Mathematical Experience (1981), Davis and Hersh won a National Book Award in Science.[6][lower-alpha 1]
Davis also wrote an autobiography, The Education of a Mathematician; some of his other books include autobiographical sections as well. In addition, he published works of fiction. His best-known book outside the field of mathematics is The Thread: A Mathematical Yarn (1983, 2nd ed. 1989), which "has raised Digression into a literary form" (Gerard Piel); it takes off from the name of the Russian mathematician Tschebyscheff, and in the course of explaining why he insists on that "barbaric, Teutonic, non-standard orthography" (in the words of a reader of Interpolation and Approximation who wrote him to complain), he digresses in many amusing directions.
Davis died on March 14, 2018, at the age of 95.[7]
Publications (Books Only)
• Ancient Loons: Stories Pingree Told Me (2016)
• Circulant matrices
• Descartes' Dream: The World According to Mathematics by Philip J. Davis and Reuben Hersh
• Interpolation and approximation
• Mathematical Encounters of the Second Kind
• Mathematics & Common Sense: A Case of Creative Tension (2006)
• Mathematics, Substance and Surmise: Views on the Meaning and Ontology of Mathematics by Ernest Davis and Philip J. Davis
• Methods of numerical integration
• Numerical Integration by Philip Davis, Philip J & Rabinowitz
• Spirals: From Theodorus to Chaos
• The Companion Guide to the Mathematical Experience: by Philip J. Davis and Reuben
• The Education of a Mathematician (2000)
• The Lore of Large Numbers (1975)
• The Mathematical Experience (Modern Birkhäuser Classics) (2011)
• The mathematics of matrices: A first book of matrix theory and linear algebra
• The Schwarz Function and Its Applications (Carus Mathematical Monographs #17) (1974)
• The Thread: A Mathematical Yarn
• Thomas Gray in Copenhagen: In Which the Philosopher Cat Meets the Ghost of Hans Christian Andersen (1995)
• Unity and Disunity and Other Mathematical Essays, American Math Society, (2015)
Notes
1. This was the 1983 award for paperback Science.
From 1980 to 1983 in National Book Award history there were dual hardcover and paperback awards in most categories, and several nonfiction subcategories including General Nonfiction. Most of the paperback award-winners were reprints, including this one.
References
1. Paul R. Halmos – Lester R. Ford Awards, Mathematical Association of America
2. "Are There Coincidences in Mathematics?" by Philip Davis
3. Gazette - Australian Mathematical Society, Vols. 25-26 (1998), p. 141.
4. Davis, Philip J. (1959). "Leonhard Euler's Integral: An Historical Profile of the Gamma Function". Amer. Math. Monthly. 66 (10): 849–869. doi:10.2307/2309786. JSTOR 2309786.
5. Barnhill, Robert E. (1976). "Review: Methods of numerical integration, by Philip J. Davis and Philip Rabinowitz" (PDF). Bull. Amer. Math. Soc. 82 (4): 538–539. doi:10.1090/s0002-9904-1976-14087-6.
6. "National Book Awards – 1983". National Book Foundation. Retrieved 2012-03-07.
7. "Philip J. Davis, Professor Emeritus". Brown University. Archived from the original on 2018-03-15. Retrieved 2018-03-14.
External links
• Personal web site at Brown University.
• Official web site at Brown University.
• Interview at SIAM
• Philip J. Davis at the Mathematics Genealogy Project
• Bibliography
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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Thomas H. Brylawski
Thomas Henry Brylawski (June 17, 1944 – July 18, 2007) was an American mathematician and professor at the University of North Carolina, Chapel Hill. He worked primarily in matroid theory.
Thomas H. Brylawski
Tom Brylawski (from the Oberwolfach Photo Collection)
Born
Thomas Henry Brylawski
(1944-06-17)June 17, 1944
Washington, D.C., U.S.
DiedJuly 18, 2007(2007-07-18) (aged 63)
Hillsborough, North Carolina, U.S.
NationalityAmerican
Other namesTom Brylawski
Alma mater
• Massachusetts Institute of Technology (BS)
• Dartmouth College (PhD)
Known forMatroid theory
Scientific career
FieldsMathematics
InstitutionsUniversity of North Carolina, Chapel Hill
ThesisThe Tutte–Grothendieck Ring (1970)
Doctoral advisors
• Gian-Carlo Rota
• Robert Norman
Doctoral studentsJenny McNulty
Education and career
Brylawski was born in 1944, and grew up in Washington, D.C.[1][2] He attended the Massachusetts Institute of Technology for his undergraduate degree, finishing with a Bachelor of Science in 1966. He then went on to Dartmouth College for his graduate work. He completed his PhD under the direction of Gian-Carlo Rota and Robert Norman in 1970.[3] After his PhD, he moved to the University of North Carolina, Chapel Hill, where he spent the rest of his career.[4][1]
Brylawski was an editor for the Proceedings of the American Mathematical Society from 1977 until 1989.[5] Brylawski wrote 40 mathematical publications,[6] and advised 6 PhD students.[3]
He died in 2007 of esophageal cancer at the Duke Hospice inpatient facility in Hillsborough, North Carolina.[2]
Work
Brylawski's early work used ideas and tools from category theory to understand the Tutte polynomial of a matroid. Indeed, this idea already appeared in his thesis, which made constructions in matroid theory similar to the Grothendieck group.[7][8] He developed similar ideas in two papers in the Transactions of the American Mathematical Society.[1][9][10] Another influential early paper of Brylawski's, published in the same journal, described the influence of a modular element in the lattice of flats on the characteristic polynomial of a matroid.[1][11]
Brylawski also contributed expository chapters to several matroid theory books that appeared in the Encyclopedia of Mathematics and its Applications series published by Cambridge University Press.[1] The Tutte polynomial chapter (written jointly with James Oxley) has around 500 citations.[12]
In addition to his work in matroid theory, Brylawski also had an interest in mathematics in art, particularly in the role of symmetry in art.[1] He gave lectures on mathematics in art on two occasions at the National Gallery of Art in Washington, D.C.[13][14]
Awards and honors
A memorial conference was held in honor of Brylawski in October 2008 at the University of North Carolina, Chapel Hill,[1] and a special issue of the European Journal of Combinatorics in 2011 was dedicated as a tribute to the work of Brylawski.[15]
References
1. Gordon, Gary; McNulty, Jennifer (2011). "Thomas H. Brylawski (1944–2007)". European Journal of Combinatorics. 32 (6): 712–721. doi:10.1016/j.ejc.2011.02.009. ISSN 0195-6698. MR 2821546.
2. "Obituaries for August 10, 2007". The Washington Post. Retrieved November 9, 2019.
3. Thomas Henry Brylawski at the Mathematics Genealogy Project
4. "Memorial page for Professor Thomas H. Brylawski" (PDF). University of North Carolina, Chapel Hill Department of Mathematics.
5. "Past Editorial Board Members". Proceedings of the American Mathematical Society. Retrieved November 9, 2019.
6. "Thomas H. Brylawski author profile". MathSciNet. American Mathematical Society.
7. Brylawski, Thomas Henry (1970). The Tutte-Grothendieck Ring (PhD). Dartmouth College.
8. Brylawski, T. H. (1972). "The Tutte-Grothendieck Ring". Algebra Universalis. 2 (1): 375–388. doi:10.1007/BF02945050. ISSN 0002-5240. MR 0330004. S2CID 121150927.
9. Brylawski, Thomas (1971). "A Combinatorial Model for Series-Parallel Networks". Transactions of the American Mathematical Society. 154: 1. doi:10.1090/S0002-9947-1971-0288039-7. ISSN 0002-9947. MR 0288039.
10. Brylawski, Thomas H. (1972). "A Decomposition for Combinatorial Geometries". Transactions of the American Mathematical Society. 171: 235–282. doi:10.1090/S0002-9947-1972-0309764-6. ISSN 0002-9947. MR 0309764.
11. Brylawski, Tom (1975). "Modular Constructions for Combinatorial Geometries". Transactions of the American Mathematical Society. 203: 1–44. doi:10.1090/S0002-9947-1975-0357163-6. ISSN 0002-9947. MR 0357163.
12. "The Tutte Polynomials and its Applications (citations)". Google Scholar. Retrieved March 15, 2021.
13. "Calendar of Events for February 1973" (PDF). National Gallery of Art. Retrieved November 9, 2019.
14. "Calendar of Events for April 1988" (PDF). National Gallery of Art. Retrieved November 9, 2019.
15. Gordon, Gary; Oxley, James (2011). "Preface". European Journal of Combinatorics. 32 (6): 709–711. doi:10.1016/j.ejc.2011.02.006. ISSN 0195-6698. MR 2821545.
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Thomas Harriot
Thomas Harriot (/ˈhæriət/;[2] c. 1560 – 2 July 1621), also spelled Harriott, Hariot or Heriot, was an English astronomer, mathematician, ethnographer and translator to whom the theory of refraction is attributed. Thomas Harriot was also recognized for his contributions in navigational techniques,[3] working closely with John White to create advanced maps for navigation.[3] While Harriot worked extensively on numerous papers on the subjects of astronomy, mathematics and navigation, he remains obscure because he published little of it,[4] namely only The Briefe and True Report of the New Found Land of Virginia (1588).[3] This book includes descriptions of English settlements and financial issues in Virginia at the time.[3] He is sometimes credited with the introduction of the potato to the British Isles.[5] Harriot invented binary notation and arithmetic several decades before Gottfried Wilhelm Leibniz, but this remained unknown until the 1920's.[6] He was also the first person to make a drawing of the Moon through a telescope, on 5 August 1609, about four months before Galileo Galilei.[7]
Thomas Harriot
Portrait often claimed to be Thomas Harriot (1602), which hangs in Trinity College, Oxford. The provenance of this portrait is not known, and there is little evidence to link it to Harriott.[1]
Bornc. 1560
Oxford, England
Died2 July 1621(1621-07-02) (aged 60–61)
London, England
NationalityEnglish
Alma materSt Mary Hall, Oxford
Known for
• Introducing symbols for "is less than" [<] and "is greater than" [>].
• Translation of the Carolina Algonquian language into English
Scientific career
FieldsAstronomy, mathematics, ethnography
After graduating from St Mary Hall, Oxford, Harriot traveled to the Americas, accompanying the 1585 expedition to Roanoke island funded by Sir Walter Raleigh and led by Sir Ralph Lane. He learned the Carolina Algonquian language from two Native Americans, Wanchese and Manteo, and could translate it, making him a vital member of the expedition. On his return to England, he worked for the 9th Earl of Northumberland.
Biography
Early life and education
Born in 1560 in Oxford, England, Thomas Harriot attended St Mary Hall, Oxford. His name appears in the hall's registry dating from 1577.[8]
Harriot started to study navigation shortly after receiving a bachelor's degree from Oxford University.[4] The study of navigation that Harriot studied concentrated on the idea of the open seas and how to cross to the New World from the Atlantic Ocean.[3] He used instruments such as the astrolabe and sextants to aide his studies of navigation.[3] After educating himself by incorporating ideals from his astronomic and nautical studies, Harriot taught other captains his navigational techniques in Raleigh.[4] His findings were recorded in the Articon but were later never found.[3]
Roanoke
After his graduation from Oxford in 1580, Harriot was first hired by Sir Walter Raleigh as a mathematics tutor; he used his knowledge of astronomy/astrology to provide navigational expertise, help design Raleigh's ships, and serve as his accountant. Prior to his expedition with Raleigh, Harriot wrote a treatise on navigation.[9] He made efforts to communicate with Manteo and Wanchese, two Native Americans who had been brought to England. Harriot devised a phonetic alphabet to transcribe their Carolina Algonquian language.
Harriot and Manteo spent many days in one another's company; Harriot interrogated Manteo closely about life in the New World and learned much that was to the advantage of the English settlers.[10] In addition, he recorded the sense of awe with which the Native Americans viewed European technology:
"Many things they sawe with us...as mathematical instruments, sea compasses...[and] spring clocks that seemed to goe of themselves – and many other things we had – were so strange unto them, and so farre exceeded their capacities to comprehend the reason and meanes how they should be made and done, that they thought they were rather the works of gods than men."[10]
He made only one expedition, around 1585–86, and spent some time in the New World visiting Roanoke Island off the coast of North Carolina, expanding his knowledge by improving his understanding of the Carolina Algonquian language. As the only Englishman who had learned Algonquin prior to the voyage, Harriot was vital to the success of the expedition.[11] Hariot smoked tobacco before Raleigh, and may have taught him to do so.[12]
His account of the voyage, named A Briefe and True Report of the New Found Land of Virginia, was published in 1588 (probably written a year before). The True Report contains an early account of the Native American population encountered by the expedition; it proved very influential upon later English explorers and colonists. He wrote: "Whereby it may be hoped, if means of good government be used, that they may in short time be brought to civility and the embracing of true religion."[13] At the same time, his views of Native Americans' industry and capacity to learn were later largely ignored in favor of the parts of the "True Report" about extractable minerals and resources.
As a scientific adviser during the voyage, Harriot was asked by Raleigh to find the most efficient way to stack cannonballs on the deck of the ship. His ensuing theory about the close-packing of spheres shows a striking resemblance to atomism and modern atomic theory, which he was later accused of believing. His correspondence about optics with Johannes Kepler, in which he described some of his ideas, later influenced Kepler's conjecture.
Later years
Harriot was employed for many years by Henry Percy, 9th Earl of Northumberland, with whom he resided at Syon House, which was run by Henry Percy's cousin Thomas Percy.
The Duke was surrounded by many scholars and learned men and provided a more stable form of patronage than Raleigh, In 1595 the Duke of Northumberland made property in Durham over to Harriot, moving him up the social ladder into 'the landed gentry'. Not long after the Durham transactions, the Duke gave Harriot the use of one of the houses on the estate at Syon, to work on optics and the sine law of refraction.[14]
Harriot's sponsors began to fall from favor: Raleigh was the first, and Harriot's other patron Henry Percy, the Earl of Northumberland, was imprisoned in 1605 in connection with the Gunpowder Plot as he was closely connected to one of the conspirators, Thomas Percy. Harriot himself was interrogated and briefly imprisoned but was soon released.[3] Walter Warner, Robert Hues, William Lower, and other scientists were present around the Earl of Northumberland's mansion as they worked for him and assisted in the teaching of the family's children.[8]
While this was occurring, Harriot continued his work involving mainly astronomy, and in 1607 Harriot used his notes from the observations of Halley's Comet to elaborate on his understanding of its orbit.[3] Soon after, in 1609 and 1610 respectively, Harriot turned his attention towards the physical aspects of the Moon and his observations of the first sightings of sunspots.[4]
In early 1609, he bought a "Dutch trunke" (telescope), invented in 1608, and his observations were among the first uses of a telescope for astronomy. Harriot is now credited as the first astronomer to draw an astronomical object after viewing it through a telescope: he drew a map of the Moon on 5 August 1609 [O.S. 26 July 1609], preceding Galileo by several months. By 1613, Harriot had created two maps of the whole Moon, with many identifiable features such as lunar craters depicted in their correct relative positions that were not to be improved upon for several decades.[15][16] He also observed sunspots in December 1610.[17]
Death
From 1614 Harriot was consulting Theodore de Mayerne, who was among James I's doctors, for an apparent cancer of the left nostril that was gradually eating away the septum[18] and was apparently linked to a cancerous ulcer on his lip. This progressed until 1621, when he was living with a friend named Thomas Buckner on Threadneedle Street. There he died, apparently from skin cancer. It was suspected that Harriot's cancer was due to excessive tobacco consumption.[3]
He died on 2 July 1621, three days after writing his will (discovered by Henry Stevens).[19] His executors posthumously published his Artis Analyticae Praxis on algebra in 1631; Nathaniel Torporley was the intended executor of Harriot's wishes, but Walter Warner in the end pulled the book into shape.[20] It may be a compendium of some of his works but does not represent all that he left unpublished (more than 400 sheets of annotated writing). It is not directed in a way that follows the manuscripts and it fails to give the full significance of Harriot's writings.[8]
Thomas Harriot was buried in the church of St Christopher le Stocks in Threadneedle Street, near where he died. The church was subsequently damaged in the Great Fire of London, and demolished in 1781 to enable expansion of the Bank of England.
Legacy
Harriott also studied optics and refraction, and apparently discovered Snell's law 20 years before Snellius did; like so many of his works, this remained unpublished. In Virginia he learned the local Algonquian language, which may have had some effect on his mathematical thinking. He founded the "English school" of algebra. Around 1600, he introduced an algebraic symbolism close to modern notation; thus, computation with unknowns became as easy as with numbers.[21] He is also credited with discovering Girard's theorem, although the formula bears Girard's name as he was the first to publish it.[22]
His algebra book Artis Analyticae Praxis[23] (1631) was published posthumously in Latin. Unfortunately, the editors did not understand much of his reasoning and removed the parts they did not comprehend such as the negative and complex roots of equations. Because of the dispersion of Harriot's writings the full annotated English translation of the Praxis was not completed until 2007.[24] A more complete manuscript, De numeris triangularibus et inde de progressionibus arithmeticis: Magisteria magna, was finally published in facsimile form with commentary by Janet Beery and Jackie Stedall in 2009.[25]
The first biography of Harriot was written in 1876 by Henry Stevens of Vermont but not published until 1900[19] fourteen years after his death. The publication was limited to 167 copies and so the work was not widely known until 1972 when a reprint edition appeared.[26] Prominent American poet, novelist and biographer Muriel Rukeyser wrote an extended literary inquiry into the life and significance of Hariot (her preferred spelling), The Traces of Thomas Hariot (1970, 1971). Interest in Harriot continued to revive with the convening of a symposium at the University of Delaware in April 1971 with the proceedings published by the Oxford University Press in 1974.[27] John W. Shirley the editor (1908-1988) went on to publish A Sourcebook for the Study of Thomas Harriot[28] and his Harriot biography (1983).[29] The papers of John Shirley are held in Special Collections at the University of Delaware.[30]
Harriot's accomplishments remain relatively obscure because he did not publish any of his results and also because many of his manuscripts have been lost; those that survive are in the British Museum and in the archives of the Percy family at Petworth House (Sussex) and Alnwick Castle (Northumberland). He was frequently accused of being an atheist, and it has been suggested that he deliberately refrained from publishing for fear of intensifying such attacks; as the literary historian Stephen Greenblatt writes "... he preferred life to fame. And who can blame him?"[31]
An event was held at Syon House, West London, to celebrate the 400th anniversary of Harriot's first observations of the Moon on 26 July 2009. This event, Telescope400,[32] included the unveiling of a plaque to commemorate Harriot by Lord Egremont. The plaque can now be seen by visitors to Syon House, the location of Harriot's historic observations. His drawing made 400 years earlier is believed to be based on the first observations of the Moon through a telescope. The event (sponsored by the Royal Astronomical Society) was run as part of the International Year of Astronomy (IYA).
The original documents showing Harriot's Moon map of c. 1611, observations of Jupiter's satellites, and first observations of sunspots were on display at the Science Museum, London, from 23 July 2009 until the end of the IYA.[33]
The observatory in the campus of the College of William & Mary is named in Harriot's honor. A crater on the Moon was named after him in 1970; it is on the Moon's far side and hence unobservable from Earth.
In July 2014 the International Astronomical Union launched NameExoWorlds, a process for giving proper names to certain exoplanets and their host stars. The process involved public nomination and voting for the new names. In December 2015, the IAU announced the winning name was Harriot for this planet. (55 Cancri in the constellation Cancer). The winning name was submitted by the Royal Netherlands Association for Meteorology and Astronomy of the Netherlands. It honors the astronomer.
The Thomas Harriot College of Arts and Sciences at East Carolina University in Greenville, NC is named in recognition of this Harriot's scientific contributions to the New World such as his work A Briefe and True Report of the New Found Land of Virginia.[10]
In fiction
An alternate history short story, "Harriott Publishes", depicts the consequences of Harriott publishing his observations before Galileo. It appears in anthology of similar stories, Altered Times, pages 13–15.
Telescope and Moon mapping
Harriot's 5 August [O.S. 26 July] 1609 drawings of his observations of the Moon have been noted as the first recorded telescopic observations ever made, predating Galileo Galilei's 30 November 1609 observation by almost four months.[34][35] Galileo's drawings, which were the first such observations to be published, contained greater detail such as identifying previously unknown features including mountains and craters.[34] Harriot inaccurately drew how far the crescent Moon would be illuminated around its limb, inaccurately drew the position of the craters, and did not draw the relief details that one would see along the Moon's light/dark terminator.[36] Critics, such as Terrie Bloom, accused Harriot of plagiarizing depictions directly from Galileo's works and argued that Harriot's representation of the Moon was an inadequate representation that needed to be improved.[36] However, both descriptions were also deemed valuable due to the scientists focusing on different specific observations.[34] Galileo describes the arrangement in a topographical way while Harriot used cartographical concepts to illustrate his views of the Moon.[34] Harriot used a 6X Dutch telescope for his observations of the Moon.[34] Harriot's recordings and descriptions were very simple with minimal detail causing his sketches to be difficult to analyze by later scientists.[36] Galileo's astronomical observations regarding the Moon were published in his book Sidereus Nuncius in 1610 and Harriot's observations were published in 1784 with some not coming to light until 1965.[34] Harriot's lack of publication is presumed to be connected to the issues with the Ninth Earl of Northumberland and the Gunpowder Plot.[34] Harriot was also known to have read and admired the work of Galileo in Sidereus Nuncius. Harriot continued his observations of the Moon until 1612.[34]
Sunspots
Thomas Harriot is recognized as the first person to observe sunspots in 1610 with the use of a telescope.[37] Harriot observed the sunspot with the use of a telescope in a direct and hazardous way.[38] Even though Harriot observed the Sun directly through his telescope, there were no recorded injuries to his eyes.[4] Harriot's depiction of the sunspots were documented in 199 drawings that provided details about the solar rotation and its acceleration.[38] Like many of Harriot's other notes, depictions of the sunspots were not published.[38] Similar to the early observation of the Moon, Galileo was also known to contribute his observations of sunspots and published his findings in 1613.[37] The specifics as to how Harriot's telescope was set up remains largely unknown.[38] However, it is known that Harriot used different magnifications of telescopes with 10X and 20X power being used most often.[38] Harriot chose to observe the sunspots after sunrise because it made the vertical easier to analyze.[38] According to Harriot's notes there was a total of 690 observations of sunspots recorded.[38] Harriot's findings challenged the idea of the unchanging heavens by explaining the Sun's axial rotation and provided further support for the heliocentric theory.[4]
Compounding
Around 1620, Harriot's unpublished papers include the early basis of continuous compounding.[39] Harriot uses modern mathematical concepts to explain the process behind continuous compounding.[39] The concept of compounded interest occurs when the more times interest is added within the year assuming the rate stays the same then the final interest will be larger.[39] Based on this observation, Harriot created mathematical equations that included logarithms and series calculations to illustrate his concepts.[39]
See also
• The School of Night
References
1. "A Tale of Two Portraits. A Note on Two Alleged Images of Thomas Harriot". April 2000.
2. "Pronunciation Guide for Mathematics". ceadserv1.nku.edu. Retrieved 17 July 2022.
3. Moran, Michael (2014). "Thomas Hariot (ca. 1560–1621)". Encyclopedia Virginia. Retrieved 28 November 2018.
4. Chapman, A. (2008). "Thomas Harriot: the first telescopic astronomer". Journal of the British Astronomical Association. 118: 315–325. Bibcode:2008JBAA..118..315C.
5. "Sir Walter Raleigh – American colonies". Archived from the original on 26 May 2012.
6. Strickland, Lloyd (2023). "Why Did Thomas Harriot Invent Binary?". The Mathematical Intelligencer. doi:10.1007/s00283-023-10271-9. Retrieved 17 May 2023.
7. "Celebrating Thomas Harriot, the world's first telescopic astronomer (RAS PN 09/47)". Royal Astronomical Society. 2011. Archived from the original on 27 June 2013. Retrieved 4 March 2011.
8. Stedall, Jacqueline (2003) The Greate Invention of Algebra, Oxford University Press. p.3, ISBN 0-19-852602-4.
9. Jehlen, Myra; Warner, Michael (1997). The English Literatures of America, 1500–1800. Routledge. p. 64. ISBN 0-415-91903-7.
10. Milton 2000, p. 73.
11. Milton 2000, p. 89.
12. Ley, Willy (December 1965). "The Healthfull Aromatick Herbe". For Your Information. Galaxy Science Fiction. pp. 88–98.
13. Hariot, Thomas (1588). A Brief and True Report of the New Found Land of Virginia (1588). University of Nebraska-Lincoln.
14. "Thomas Harriot - Biography". Maths History. Retrieved 3 November 2022.
15. McGourty, Christine (14 January 2009). "English Galileo' maps on display". BBC.
16. Van Helden, Al (1995). "Thomas Harriot's Moon Drawings". The Galileo Project. Retrieved 14 May 2019.
17. Van Helden, Al (1995). "Thomas Harriot (1560–1621)". The Galileo Project. Retrieved 14 May 2019.
18. H. R. Trevor-Roper; Blair Worden, ed., Europe's physician: the various life of Sir Theodore de Mayerne (New Haven: Yale University Press, 2006) pp. 206-207 and note 1
19. Stevens, Henry (1900). Thomas Hariot, the Mathematician, the Philosopher and the Scholar. London: Privately printed at the Chiswick press. p. 142.
20. Pycior, Helena M. (1997). Symbols, Impossible Numbers, and Geometric Entanglements: British Algebra Through the Commentaries on Newton's Universal Arithmetick. Cambridge University Press. pp. 55–56. ISBN 978-0-521-02740-3.
21. Katz, Victor J.; Parshall, Karen Hunger (2014). Taming the Unknown: A History of Algebra from Antiquity to the Early Twentieth Century. Princeton University Press. p. 248. ISBN 978-1-4008-5052-5.
22. Richeson, David (2008). Euler's gem : the polyhedron formula and the birth of topology. Princeton University Press. p. 91. ISBN 9780691126777. LCCN 2008062108.
23. Harrriot, Thomas (1631). Artis analyticae praxis. LCCN 43022232.
24. Harriot, Thomas (2007). Seltman, Muriel; Goulding, Robert (eds.). Thomas Harriot's Artis Analyticae Praxis: An English Translation with Commentary. Springer. ISBN 978-0-387-49511-8. LCCN 2006938536.
25. Schemmel, Matthias (September 2010). "Before calculus". Notes and Records of the Royal Society of London. 64 (3): 303–304. doi:10.1098/rsnr.2010.0016. JSTOR 20753908. S2CID 202575019.
26. Stevens, Henry (1972) [1900]. Thomas Hariot, the mathematician, the philosopher and the scholar: developed chiefly from dormant materials, with notices of his associates, including biographical and bibliographical disquisitions upon the materials of the history of 'Ould Virginia'. New York: B. Franklin. LCCN 72082483.
27. Shirley, John W., ed. (1974). Thomas Harriot; Renaissance scientist. Oxford: Clarendon Press. ISBN 0198581408. LCCN 74176704.
28. Shirley, John W. (1981). A Sourcebook for the Study of Thomas Harriot. New York: Arno Press. ISBN 0405138318. LCCN 80002111.
29. Shirley, John W. (1983). Thomas Harriot, a Biography. Oxford: Clarendon Press. LCCN 83003961.
30. Guide to the John Shirley papers related to Thomas Harriot, Special Collections, University of Delaware Library, Newark, Delaware. Retrieved 10 May 2020.
31. — (2011). The Swerve: How the World Became Modern. New York: W. W. Norton. ISBN 978-0-393-06447-6.
32. "Telescope400 – celebrating Thomas Harriot's first ever use of the Telescope in Astronomy".
33. Devlin, Hannah (24 July 2009). "Galileo was beaten to the Moon by a shy Englishman". The Times.
34. Pumfrey, Stephen (2009). "Harriot's Maps of the Moon: New Interpretations". Notes and Records of the Royal Society of London. 63 (2): 163–168. doi:10.1098/rsnr.2008.0062. JSTOR 40647255.
35. Bloom, Terrie F. (2016). "Borrowed Perceptions: Harriot's Maps of the Moon". Journal for the History of Astronomy. 9 (2): 117–122. doi:10.1177/002182867800900203. ISSN 0021-8286. S2CID 125625494.
36. Alexander, Amir (1998). "Lunar Maps and Coastal Outlines: Thomas Hariot's Mapping of the Moon". Stud. Hist. Philos. Sci. 29 (3): 345–368. Bibcode:1998SHPSA..29..345A. doi:10.1016/S0039-3681(98)00011-9.
37. Voss, David (2015). "March 9, 1611: Dutch astronomer Johannes Fabricius observes sunspots". APS News. 24.
38. Herr, Richard B. (1978). "Solar Rotation Determined from Thomas Harriot's Sunspot Observations of 1611 to 1613". Science. New Series. 202 (4372): 1079–1081. Bibcode:1978Sci...202.1079H. doi:10.1126/science.202.4372.1079. JSTOR 1747843. PMID 17777957. S2CID 39401037.
39. Biggs, Norman (2013). "Thomas Harriot on continuous compounding". Journal of the British Society for the History of Mathematics. 28 (2): 66–74. doi:10.1080/17498430.2013.721331. ISSN 1749-8430. S2CID 53586313.
Sources
• Kupperman, Karen Ordahl (2000). Indians and English: Facing Off in Early America. Ithaca: Cornell University Press. ISBN 0801482828.
• Mancall, Peter C. (2007). Hakluyt's Promise: An Elizabethan's Obsession for an English America. New Haven: Yale University Press. ISBN 9780300164220.
• Milton, Giles (2000). Big Chief Elizabeth – How England's Adventurers Gambled and Won the New World'. London: Hodder & Stoughton. ISBN 9780312420185.
• Rukeyser, Muriel (1971). The Traces of Thomas Hariot. NY: Random House.
• Vaughan, Alden T. (2002). "Sir Walter Ralegh's Indian Interpreters, 1584–1618". The William and Mary Quarterly. 59 (2): 341–376. doi:10.2307/3491741. ISSN 0043-5597. JSTOR 3491741.
External links
Wikisource has original works by or about:
Thomas Harriot
• Thomas Harriot: Trumpeter of Roanoke
• Thomas Harriot
• John Shirley papers related to Thomas Harriot, Special Collections, University of Delaware Library, Newark, Delaware.
Works by Thomas Harriot
• Works by Thomas Harriot at Project Gutenberg
• Works by or about Thomas Harriot at Internet Archive
• A Brief and True Report of the New Found Land of Virginia by Thomas Hariot; Reproduced in Facsimile from the First Edition of 1688; with an Introduction by Luther S. Livingston. New York: Dodd, Mead & Company. 1903. Retrieved 30 April 2018 – via Internet Archive.
• A Brief and True Report online pdf text edition
• A Briefe and True Report of the New Found Land of Virginia from American Studies at the University of Virginia.
• Annotated Translation of Harriot's Praxis by Ian Bruce
Works or sites about Thomas Harriot
• O'Connor, John J.; Robertson, Edmund F., "Thomas Harriot", MacTutor History of Mathematics Archive, University of St Andrews
• Harriot, Thomas
• The Englishman who beat Galileo Archived 16 February 2015 at the Wayback Machine
• The Soft Logic of Thomas Harriot
• The Thomas Harriot Seminar
• Searching for the Lost Colony Blog
• The Harriot Voyages of Discovery Lecture Series at East Carolina University
• The Thomas Harriot College of Arts and Sciences at East Carolina University, Greenville, NC
• Thomas Harriot Quintessential Renaissance Scholar
• Account of the Roanoke settlements Retrieved April 2011
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