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Wythoff's game
Wythoff's game is a two-player mathematical subtraction game, played with two piles of counters. Players take turns removing counters from one or both piles; when removing counters from both piles, the numbers of counters removed from each pile must be equal. The game ends when one player removes the last counter or counters, thus winning.
An equivalent description of the game is that a single chess queen is placed somewhere on a large grid of squares, and each player can move the queen towards the lower left corner of the grid: south, west, or southwest, any number of steps. The winner is the player who moves the queen into the corner. The two Cartesian coordinates of the queen correspond to the sizes of two piles in the formulation of the game involving removing counters from piles.
Martin Gardner in his March 1977 "Mathematical Games column" in Scientific American claims that the game was played in China under the name 捡石子 jiǎn shízǐ ("picking stones").[1] The Dutch mathematician W. A. Wythoff published a mathematical analysis of the game in 1907.[2]
Optimal strategy
Any position in the game can be described by a pair of integers (n, m) with n ≤ m, describing the size of both piles in the position or the coordinates of the queen. The strategy of the game revolves around cold positions and hot positions: in a cold position, the player whose turn it is to move will lose with best play, while in a hot position, the player whose turn it is to move will win with best play. The optimal strategy from a hot position is to move to any reachable cold position.
The classification of positions into hot and cold can be carried out recursively with the following three rules:
1. (0,0) is a cold position.
2. Any position from which a cold position can be reached in a single move is a hot position.
3. If every move leads to a hot position, then a position is cold.
For instance, all positions of the form (0, m) and (m, m) with m > 0 are hot, by rule 2. However, the position (1,2) is cold, because the only positions that can be reached from it, (0,1), (0,2), (1,0) and (1,1), are all hot. The cold positions (n, m) with the smallest values of n and m are (0, 0), (1, 2), (3, 5), (4, 7), (6, 10) and (8, 13). (sequence A066096 and A090909 in OEIS) (Also see OEIS: A072061)
For misère game of this game, (0, 1) and (2, 2) are cold positions, and a position (n, m) with m, n > 2 is cold if and only if (n, m) in normal game is cold.
Formula for cold positions
Wythoff discovered that the cold positions follow a regular pattern determined by the golden ratio. Specifically, if k is any natural number and
$n_{k}=\lfloor k\phi \rfloor =\lfloor m_{k}\phi \rfloor -m_{k}\,$
$m_{k}=\lfloor k\phi ^{2}\rfloor =\lceil n_{k}\phi \rceil =n_{k}+k\,$
where φ is the golden ratio and we are using the floor function, then (nk, mk) is the kth cold position. These two sequences of numbers are recorded in the Online Encyclopedia of Integer Sequences as OEIS: A000201 and OEIS: A001950, respectively.
The two sequences nk and mk are the Beatty sequences associated with the equation
${\frac {1}{\phi }}+{\frac {1}{\phi ^{2}}}=1\,.$
As is true in general for pairs of Beatty sequences, these two sequences are complementary: each positive integer appears exactly once in either sequence.
See also
• Nim
• Grundy's game
• Subtract a square
• Wythoff array
References
1. Wythoff's game at Cut-the-knot, quoting Martin Gardner's book Penrose Tiles to Trapdoor Ciphers
2. Wythoff, W. A. (1907), "A modification of the game of nim", Nieuw Archief voor Wiskunde, 7 (2): 199–202
External links
• Weisstein, Eric W. "Wythoff's Game". MathWorld.
• Grime, James. "Wythoff's Game (Get Home)" (video). YouTube. Brady Haran. Archived from the original on 2021-12-15. Retrieved 21 August 2017.
| Wikipedia |
Wythoff array
In mathematics, the Wythoff array is an infinite matrix of integers derived from the Fibonacci sequence and named after Dutch mathematician Willem Abraham Wythoff. Every positive integer occurs exactly once in the array, and every integer sequence defined by the Fibonacci recurrence can be derived by shifting a row of the array.
The Wythoff array was first defined by Morrison (1980) using Wythoff pairs, the coordinates of winning positions in Wythoff's game. It can also be defined using Fibonacci numbers and Zeckendorf's theorem, or directly from the golden ratio and the recurrence relation defining the Fibonacci numbers.
Values
The Wythoff array has the values
${\begin{matrix}1&2&3&5&8&13&21&\cdots \\4&7&11&18&29&47&76&\cdots \\6&10&16&26&42&68&110&\cdots \\9&15&24&39&63&102&165&\cdots \\12&20&32&52&84&136&220&\cdots \\14&23&37&60&97&157&254&\cdots \\17&28&45&73&118&191&309&\cdots \\\vdots &\vdots &\vdots &\vdots &\vdots &\vdots &\vdots &\ddots \\\end{matrix}}$ (sequence A035513 in the OEIS).
Equivalent definitions
Inspired by a similar Stolarsky array previously defined by Stolarsky (1977), Morrison (1980) defined the Wythoff array as follows. Let $\varphi ={\frac {1+{\sqrt {5}}}{2}}$ denote the golden ratio; then the $i$th winning position in Wythoff's game is given by the pair of positive integers $(\lfloor i\varphi \rfloor ,\lfloor i\varphi ^{2}\rfloor )$, where the numbers on the left and right sides of the pair define two complementary Beatty sequences that together include each positive integer exactly once. Morrison defines the first two numbers in row $m$ of the array to be the Wythoff pair given by the equation $i=\lfloor m\varphi \rfloor $, and where the remaining numbers in each row are determined by the Fibonacci recurrence relation. That is, if $A_{m,n}$ denotes the entry in row $m$ and column $n$ of the array, then
$A_{m,1}=\left\lfloor \lfloor m\varphi \rfloor \varphi \right\rfloor $,
$A_{m,2}=\left\lfloor \lfloor m\varphi \rfloor \varphi ^{2}\right\rfloor $, and
$A_{m,n}=A_{m,n-2}+A_{m,n-1}$ for $n>2$.
The Zeckendorf representation of any positive integer is a representation as a sum of distinct Fibonacci numbers, no two of which are consecutive in the Fibonacci sequence. As Kimberling (1995) describes, the numbers within each row of the array have Zeckendorf representation that differ by a shift operation from each other, and the numbers within each column have Zeckendorf representations that all use the same smallest Fibonacci number. In particular the entry $A_{m,n}$ of the array is the $m$th smallest number whose Zeckendorf representation begins with the $(n+1)$th Fibonacci number.
Properties
Each Wythoff pair occurs exactly once in the Wythoff array, as a consecutive pair of numbers in the same row, with an odd index for the first number and an even index for the second. Because each positive integer occurs in exactly one Wythoff pair, each positive integer occurs exactly once in the array (Morrison 1980).
Every sequence of positive integers satisfying the Fibonacci recurrence occurs, shifted by at most finitely many positions, in the Wythoff array. In particular, the Fibonacci sequence itself is the first row, and the sequence of Lucas numbers appears in shifted form in the second row (Morrison 1980).
References
• Kimberling, Clark (1995), "The Zeckendorf array equals the Wythoff array" (PDF), Fibonacci Quarterly, 33 (1): 3–8.
• Morrison, D. R. (1980), "A Stolarsky array of Wythoff pairs", A Collection of Manuscripts Related to the Fibonacci Sequence (PDF), Santa Clara, Calif: The Fibonacci Association, pp. 134–136.
• Stolarsky, K. B. (1977), "A set of generalized Fibonacci sequences such that each natural number belongs to exactly one" (PDF), Fibonacci Quarterly, 15 (3): 224.
External links
• Weisstein, Eric W. "Wythoff Array". MathWorld.
| Wikipedia |
Wythoff symbol
In geometry, the Wythoff symbol is a notation representing a Wythoff construction of a uniform polyhedron or plane tiling within a Schwarz triangle. It was first used by Coxeter, Longuet-Higgins and Miller in their enumeration of the uniform polyhedra. Later the Coxeter diagram was developed to mark uniform polytopes and honeycombs in n-dimensional space within a fundamental simplex.
A Wythoff symbol consists of three numbers and a vertical bar. It represents one uniform polyhedron or tiling, although the same tiling/polyhedron can have different Wythoff symbols from different symmetry generators. For example, the regular cube can be represented by 3 | 2 4 with Oh symmetry, and 2 4 | 2 as a square prism with 2 colors and D4h symmetry, as well as 2 2 2 | with 3 colors and D2h symmetry.
With a slight extension, Wythoff's symbol can be applied to all uniform polyhedra. However, the construction methods do not lead to all uniform tilings in Euclidean or hyperbolic space.
Description
The Wythoff construction begins by choosing a generator point on a fundamental triangle. This point must be chosen at equal distance from all edges that it does not lie on, and a perpendicular line is then dropped from it to each such edge.
The three numbers in Wythoff's symbol, p, q, and r, represent the corners of the Schwarz triangle used in the construction, which are π/p, π/q, and π/r radians respectively. The triangle is also represented with the same numbers, written (p q r). The vertical bar in the symbol specifies a categorical position of the generator point within the fundamental triangle according to the following:
• p | q r indicates that the generator lies on the corner p,
• p q | r indicates that the generator lies on the edge between p and q,
• p q r | indicates that the generator lies in the interior of the triangle.
In this notation the mirrors are labeled by the reflection-order of the opposite vertex. The p, q, r values are listed before the bar if the corresponding mirror is active.
A special use is the symbol | p q r which is designated for the case where all mirrors are active, but odd-numbered reflected images are ignored. The resulting figure has rotational symmetry only.
The generator point can either be on or off each mirror, activated or not. This distinction creates 8 (23) possible forms, but the one where the generator point is on all the mirrors is impossible. The symbol that would normally refer to that is reused for the snub tilings.
The Wythoff symbol is functionally similar to the more general Coxeter-Dynkin diagram, in which each node represents a mirror and the arcs between them – marked with numbers – the angles between the mirrors. (An arc representing a right angle is omitted.) A node is circled if the generator point is not on the mirror.
Example spherical, euclidean and hyperbolic tilings on right triangles
The fundamental triangles are drawn in alternating colors as mirror images. The sequence of triangles (p 3 2) change from spherical (p = 3, 4, 5), to Euclidean (p = 6), to hyperbolic (p ≥ 7). Hyperbolic tilings are shown as a Poincaré disk projection.
Wythoff symbol q | p 2 2 q | p 2 | p q 2 p | q p | q 2 p q | 2 p q 2 | | p q 2
Coxeter diagram
Vertex figure pq q.2p.2p p.q.p.q p.2q.2q qp p.4.q.4 4.2p.2q 3.3.p.3.q
Fund. triangles 7 forms and snub
(3 3 2)
3 | 3 2
33
2 3 | 3
3.6.6
2 | 3 3
3.3.3.3
2 3 | 3
3.6.6
3 | 3 2
33
3 3 | 2
3.4.3.4
3 3 2 |
4.6.6
| 3 3 2
3.3.3.3.3
(4 3 2)
3 | 4 2
43
2 3 | 4
3.8.8
2 | 4 3
3.4.3.4
2 4 | 3
4.6.6
4 | 3 2
34
4 3 | 2
3.4.4.4
4 3 2 |
4.6.8
| 4 3 2
3.3.3.3.4
(5 3 2)
3 | 5 2
53
2 3 | 5
3.10.10
2 | 5 3
3.5.3.5
2 5 | 3
5.6.6
5 | 3 2
35
5 3 | 2
3.4.5.4
5 3 2 |
4.6.10
| 5 3 2
3.3.3.3.5
(6 3 2)
3 | 6 2
63
2 3 | 6
3.12.12
2 | 6 3
3.6.3.6
2 6 | 3
6.6.6
6 | 3 2
36
6 3 | 2
3.4.6.4
6 3 2 |
4.6.12
| 6 3 2
3.3.3.3.6
(7 3 2)
3 | 7 2
73
2 3 | 7
3.14.14
2 | 7 3
3.7.3.7
2 7 | 3
7.6.6
7 | 3 2
37
7 3 | 2
3.4.7.4
7 3 2 |
4.6.14
| 7 3 2
3.3.3.3.7
(8 3 2)
3 | 8 2
83
2 3 | 8
3.16.16
2 | 8 3
3.8.3.8
2 8 | 3
8.6.6
8 | 3 2
38
8 3 | 2
3.4.8.4
8 3 2 |
4.6.16
| 8 3 2
3.3.3.3.8
(∞ 3 2)
3 | ∞ 2
∞3
2 3 | ∞
3.∞.∞
2 | ∞ 3
3.∞.3.∞
2 ∞ | 3
∞.6.6
∞ | 3 2
3∞
∞ 3 | 2
3.4.∞.4
∞ 3 2 |
4.6.∞
| ∞ 3 2
3.3.3.3.∞
See also
• Regular polytope
• Regular polyhedron
• List of uniform tilings
• Uniform tilings in hyperbolic plane
• List of uniform polyhedra
• List of uniform polyhedra by Schwarz triangle
• Lists of uniform tilings on the sphere, plane, and hyperbolic plane
References
• Coxeter Regular Polytopes, Third edition, (1973), Dover edition, ISBN 0-486-61480-8 (Chapter V: The Kaleidoscope, Section: 5.7 Wythoff's construction)
• Coxeter The Beauty of Geometry: Twelve Essays, Dover Publications, 1999, ISBN 0-486-40919-8 (Chapter 3: Wythoff's Construction for Uniform Polytopes)
• Coxeter, Longuet-Higgins, Miller, Uniform polyhedra, Phil. Trans. 1954, 246 A, 401–50.
• Wenninger, Magnus (1974). Polyhedron Models. Cambridge University Press. ISBN 0-521-09859-9. pp. 9–10.
External links
• Weisstein, Eric W. "Wythoff symbol". MathWorld.
• The Wythoff symbol
• Wythoff symbol
• Greg Egan's applet to display uniform polyhedra using Wythoff's construction method
• A Shadertoy renderization of Wythoff's construction method
• KaleidoTile 3 Free educational software for Windows by Jeffrey Weeks that generated many of the images on the page.
• Hatch, Don. "Hyperbolic Planar Tessellations".
| Wikipedia |
Hugo Steinhaus
Hugo Dyonizy Steinhaus (Polish: [ˈxuɡɔ ˈʃtaɪ̯nˌhaʊ̯s]; English: /ˈhjuːɡoʊ ˈstaɪnˌhaʊs/; January 14, 1887 – February 25, 1972) was a Polish mathematician and educator. Steinhaus obtained his PhD under David Hilbert at Göttingen University in 1911 and later became a professor at the Jan Kazimierz University in Lwów (now Lviv, Ukraine), where he helped establish what later became known as the Lwów School of Mathematics. He is credited with "discovering" mathematician Stefan Banach, with whom he gave a notable contribution to functional analysis through the Banach–Steinhaus theorem. After World War II Steinhaus played an important part in the establishment of the mathematics department at Wrocław University and in the revival of Polish mathematics from the destruction of the war.
Hugo Steinhaus
Hugo Steinhaus (1968)
Born
Hugo Dyonizy Steinhaus
(1887-01-14)January 14, 1887
Jasło, Austria-Hungary (now Poland)
DiedFebruary 25, 1972(1972-02-25) (aged 85)
Wrocław, Poland
NationalityPolish
Alma materLemberg University
Göttingen University
Known forBanach–Steinhaus theorem, many others, see section below.
Scientific career
FieldsMathematician and mathematics populariser
InstitutionsJan Kazimierz University
University of Wrocław
University of Notre Dame
University of Sussex
Doctoral advisorDavid Hilbert
Doctoral studentsStefan Banach
Z. W. (Bill) Birnbaum
Mark Kac
Władysław Orlicz
Aleksander Rajchman
Juliusz Schauder
Stanisław Trybula
Author of around 170 scientific articles and books, Steinhaus has left his legacy and contribution in many branches of mathematics, such as functional analysis, geometry, mathematical logic, and trigonometry. Notably he is regarded as one of the early founders of game theory and probability theory which led to later development of more comprehensive approaches by other scholars.
Early life and studies
Steinhaus was born on January 14, 1887, in Jasło,[1] Austria-Hungary to a family with Jewish roots. His father, Bogusław, was a local industrialist, owner of a brick factory and a merchant. His mother was Ewelina, née Lipschitz. Hugo's uncle, Ignacy Steinhaus, was a lawyer and an activist in the Koło Polskie (Polish Circle), and a deputy to the Galician Diet, the regional assembly of the Kingdom of Galicia and Lodomeria.[2]
Hugo finished his studies at the gymnasium in Jasło in 1905. His family wanted him to become an engineer but he was drawn to abstract mathematics and began to study the works of famous contemporary mathematicians on his own. In the same year he began studying philosophy and mathematics at the University of Lemberg.[2] In 1906 he transferred to Göttingen University.[1] At that University he received his Ph.D. in 1911, having written his doctoral dissertation under the supervision of David Hilbert. The title of his thesis was Neue Anwendungen des Dirichlet'schen Prinzips ("New applications to Dirichlet's principle").[3]
At the start of World War I Steinhaus returned to Poland and served in Józef Piłsudski's Polish Legion, after which he lived in Kraków.[4]
He was an atheist.[5]
Academic career
Interwar Poland
During the 1916-1917 period and before Poland had regained its full independence, which occurred in 1918, Steinhaus worked in Kraków for the Ministry of the Interior in the ephemeral puppet state of Kingdom of Poland.[6]
In 1917 he started to work at the University of Lemberg (later Jan Kazimierz University in Poland) and acquired his habilitation qualification in 1920.[6] In 1921 he became a profesor nadzwyczajny (associate professor) and in 1925 profesor zwyczajny (full professor) at the same university.[6] During this time he taught a course on the then cutting edge theory of Lebesgue integration, one of the first such courses offered outside of France.[3]
While in Lwów, Steinhaus co-founded the Lwów School of Mathematics[7] and was active in the circle of mathematicians associated with the Scottish cafe, although, according to Stanislaw Ulam, for the circle's gatherings, Steinhaus would have generally preferred a more upscale tea shop down the street.[4]
World War II
In September 1939 after Nazi Germany and the Soviet Union both invaded and occupied Poland, as a fulfillment of the Molotov–Ribbentrop Pact they had signed earlier, Lwów initially came under Soviet occupation. Steinhaus considered escaping to Hungary but ultimately decided to remain in Lwów. The Soviets reorganized the university to give it a more Ukrainian character, but they did appoint Stefan Banach (Steinhaus's student) as the dean of the mathematics department and Steinhaus resumed teaching there. The faculty of the department at the school were also strengthened by several Polish refugees from German-occupied Poland. According to Steinhaus, during the experience of this period, he "acquired an insurmountable physical disgust in regard to all sorts of Soviet administrators, politicians and commissars"[A]
During the interwar period and the time of the Soviet occupation, Steinhaus contributed ten problems to the famous Scottish Book, including the last one, recorded shortly before Lwów was captured by the Nazis in 1941, during Operation Barbarossa.[4]
Steinhaus, because of his Jewish background, spent the Nazi occupation in hiding, first among friends in Lwów, then in the small towns of Osiczyna, near Zamość and Berdechów, near Kraków.[7][8] The Polish anti-Nazi resistance provided him with false documents of a forest ranger who had died sometime earlier, by the name of Grzegorz Krochmalny. Under this name he taught clandestine classes (higher education was forbidden for Poles under the German occupation). Worried about the possibility of imminent death if captured by Germans, Steinhaus, without access to any scholarly material, reconstructed from memory and recorded all the mathematics he knew, in addition to writing other voluminous memoirs, of which only a little part has been published.[8]
Also while in hiding, and cut off from reliable news on the course of the war, Steinhaus devised a statistical means of estimating for himself the German casualties at the front based on sporadic obituaries published in the local press. The method relied on the relative frequency with which the obituaries stated that the soldier who died was someone's son, someone's "second son", someone's "third son" and so on.[8]
According to his student and biographer, Mark Kac, Steinhaus told him that the happiest day of his life were the twenty four hours between the time that the Germans left occupied Poland and the Soviets had not yet arrived ("They had left, and they had not yet come").[8]
After World War II
In the last days of World War II Steinhaus, still in hiding, heard a rumor that University of Lwów was to be transferred to the city of Breslau (Wrocław), which Poland was to acquire as a result of the Potsdam Agreement (Lwów became part of Soviet Ukraine). Although initially he had doubts, he turned down offers for faculty positions in Łódź and Lublin and made his way to the city where he began teaching at University of Wrocław.[7] While there, he revived the idea behind the Scottish Book from Lwów, where prominent and aspiring mathematicians would write down problems of interest along with prizes to be awarded for their solution, by starting the New Scottish Book. It was also most likely Steinhaus who preserved the original Scottish Book from Lwów throughout the war and subsequently sent it to Stanisław Ulam, who translated it into English.[4]
With Steinhaus' help, Wrocław University became renowned for mathematics, much as the University of Lwów had been.[8]
Later, in the 1960s, Steinhaus served as a visiting professor at the University of Notre Dame (1961–62)[4] and the University of Sussex (1966).[9]
Mathematical contributions
See also: Banach–Steinhaus theorem
Steinhaus authored over 170 works.[4] Unlike his student, Stefan Banach, who tended to specialize narrowly in the field of functional analysis, Steinhaus made contributions to a wide range of mathematical sub-disciplines, including geometry, probability theory, functional analysis, theory of trigonometric and Fourier series as well as mathematical logic.[3][4] He also wrote in the area of applied mathematics and enthusiastically collaborated with engineers, geologists, economists, physicians, biologists and, in Kac's words, "even lawyers".[8]
Probably his most notable contribution to functional analysis was the 1927 proof of the Banach–Steinhaus theorem, given along with Stefan Banach, which is now one of the fundamental tools in this branch of mathematics.
His interest in games led him to propose an early formal definition of a strategy, anticipating John von Neumann's more complete treatment of a few years later. Consequently, he is considered an early founder of modern game theory.[6] As a result of his work on infinite games Steinhaus, together with another of his students, Jan Mycielski, proposed the axiom of determinacy.[8]
Steinhaus was also an early contributor to, and co-founder of, probability theory, which at the time was in its infancy and not even considered an actual part of mathematics.[8] He provided the first axiomatic measure-theoretic description of coin-tossing, which was to influence the full axiomatization of probability by the Russian mathematician Andrey Kolmogorov a decade later.[8] Steinhaus was also the first to offer precise definitions of what it means for two events to be "independent", as well as for what it means for a random variable to be "uniformly distributed".[4]
While in hiding during World War II, Steinhaus worked on the fair cake-cutting problem: how to divide a heterogeneous resource among several people with different preferences such that every person believes he received a proportional share. Steinhaus' work has initiated the modern research of the fair cake-cutting problem.[B]
Steinhaus was also the first person to conjecture the ham-sandwich theorem,[8][10] and one of the first to propose the method of k-means clustering.[11]
Legacy
Steinhaus is said to have "discovered" the Polish mathematician Stefan Banach in 1916, after he overheard someone utter the words "Lebesgue integral" while in a Kraków park (Steinhaus referred to Banach as his "greatest mathematical discovery").[12] Together with Banach and the other participant of the park discussion, Otto Nikodym, Steinhaus started the Mathematical Society of Kraków, which later evolved into the Polish Mathematical Society.[4] He was a member of PAU (the Polish Academy of Learning) and PAN (the Polish Academy of Sciences), PTM (the Polish Mathematical Society), the Wrocławskie Towarzystwo Naukowe (Wrocław Scientific Society) as well as many international scientific societies and science academies.[6]
Steinhaus also published one of the first articles in Fundamenta Mathematicae, in 1921.[13] He also co-founded Studia Mathematica along with Stefan Banach (1929),[7] and Zastosowania matematyki (Applications of Mathematics, 1953), Colloquium Mathematicum, and Monografie Matematyczne (Mathematical Monographs).[1]
He received honorary doctorate degrees from Warsaw University (1958), Wrocław Medical Academy (1961), Poznań University (1963) and Wrocław University (1965).[14]
Steinhaus had full command of several foreign languages and was known for his aphorisms, to the point that a booklet of his most famous ones in Polish, French and Latin has been published posthumously.[8]
In 2002, the Polish Academy of Sciences and Wrocław University sponsored "2002, The Year of Hugo Steinhaus", to celebrate his contributions to Polish and world science.[15]
Mathematician Mark Kac, Steinhaus's student, wrote:
"He was one of the architects of the school of mathematics which flowered miraculously in Poland between the two wars and it was he who, perhaps more than any other individual, helped to raise Polish mathematics from the ashes to which it had been reduced by the second World War to the position of new strength and respect which it now occupies. He was a man of great culture and in the best sense of the word a product of Western civilization."[3]
Chief works
• Czym jest, a czym nie jest matematyka (What Mathematics Is, and What It Is Not, 1923).[14]
• Sur le principe de la condensation de la singularités (with Banach, 1927)[3]
• Theorie der Orthogonalreihen (with Stefan Kaczmarz, 1935).[3][16]
• Kalejdoskop matematyczny (Mathematical Snapshots, 1939).[3][14]
• Taksonomia wrocławska (A Wroclaw Taxonomy; with others, 1951).
• Sur la liaison et la division des points d'un ensemble fini (On uniting and separating the points of a finite set, with others, 1951).[17] One of multiple rediscoveries of Borůvka's algorithm.
• Sto zadań (One Hundred Problems In Elementary Mathematics, 1964).[4][18]
• Orzeł czy reszka (Heads or Tails, 1961).[19]
• Słownik racjonalny (A Rational Dictionary, 1980).[20]
Family
His daughter Lidya Steinhaus was married to Jan Kott.
Grave
Steinhaus[21] is buried in Cmentarz Świętej Rodziny in Sępolno, Wrocław, Poland.
On November 1, 2017, a controversy arose as the University of Wrocław took no action to pay for his grave to keep it until 2022 in face of its expiration because it supposedly "had no money". Mayor Rafał Dutkiewicz was also widely criticized for doing nothing. Random people organized a charity and paid for the grave to remain.[22]
See also
• Freiling's axiom of symmetry
• One-seventh area triangle
• Johnson–Trotter algorithm
• Steinhaus conjecture
• Steinhaus polygon notation
• Steinhaus theorem
• Steinhaus longimeter
• Last diminisher
Notes
1. ^
Nabrałem nieprzyzwyciężonej fizycznej wprost odrazy do wszelkich urzędników, polityków i komisarzy sowieckich (Duda, g. 23).
2. ^
The solution to the two person version of the problem is the classic children's rule divide and choose. Steinhaus was the first to generalize the problem definition to three or more people, by inviting the proportional division criterion.
References
1. Foreword to "One hundred problems in elementary mathematics". Courier Dover Publications. 1974. p. 4. ISBN 978-0-486-23875-3.
2. Official webpage of the town of Jasło (2010). "Steinhaus Hugo Dyonizy". Mieszkaniec: Steinhaus Hugo Dyonizy. Jasło. Moje miasto, nasz wspólny dom. Archived from the original on 1 October 2011. Retrieved 16 August 2011.
3. Kac, Mark (1974). "Hugo Steinhaus--A Reminiscence and a Tribute" (PDF). The American Mathematical Monthly. Mathematical Association of America. 81 (6): 572–581. doi:10.2307/2319205. JSTOR 2319205. Archived from the original (PDF) on 2011-09-27.
4. John O'Connor; Edmund F. Robertson (February 2000). "Hugo Dyonizy Steinhaus". The MacTutor History of Mathematics archive. School of Mathematics and Statistics, University of St Andrews, Scotland. Retrieved 16 August 2011.
5. Steven G. Krantz (2002). Mathematical Apocrypha: Stories and Anecdotes of Mathematicians and the Mathematical. Mathematical Association of America. p. 202. ISBN 9780883855393. Steinhaus was an outspoken atheist.
6. Monika Śliwa (May 4, 2010). "Hugo Steinhaus". University of Wrocław. Archived from the original on October 5, 2011.
7. Duda, Roman (2005). "Początki Matematyki w Powojennym Wrocławiu" (PDF). Przegląd Uniwesytetcki. Polskie Towarzystwo Matematyczne. Oddział Wrocławski (September).
8. Kac, Mark (1987). Enigmas of chance: an autobiography. University of California Press. pp. 49–53. ISBN 978-0-520-05986-3.
9. Chełminiak, Wiesław (2002). "Wrocław Europy". Wprost. Retrieved 20 August 2011.
10. Beyer, W. A.; Zardecki, Andrew (2004). "The early history of the ham sandwich theorem". American Mathematical Monthly. 111 (1): 58–61. doi:10.2307/4145019. JSTOR 4145019. ProQuest 203746537.
11. Lindsten, Frederik; Ohlsson, Frederik; Lennard, Ljung (2011). "Just Relax and Come Clustering. A Convexification of k-means Clustering". Technical Report from Automatic Control at Linköpings Universitet. Linköping University: 1.
12. Feferman, Anita Burdman; Feferman, Solomon (2004). Alfred Tarski: life and logic. Cambridge University Press. p. 29. ISBN 978-0-521-80240-6.
13. Kuratowski, Kazimierz; Borsuk, Karol (1978). "One Hundred Volumes of Fundamenta Mathematicae" (PDF). Fundamenta Mathematicae. Polish Academy of Science. 100: 3.
14. "Prof. Hugo Steinhaus". Wrocław University of Technology.
15. Aleksander Weron (January 4, 2002). "2002-Rok Hugona Steinhausa (2002 - Year of Hugo Steinhaus)". Politechnika Wrocławska. Retrieved 26 August 2011.
16. Stefan Kaczmarz; Hugo Steinhaus (1951). Theorie der Orthogonalreihen. Chelsea Pub. Co. Retrieved 2 September 2011.
17. Steinhaus; et al. (1951). "Sur la liaison et la division des points d'un ensemble fini" (PDF). Polish Virtual Library of Science - Mathematical Collection.
18. Steinhaus, Hugo (1974). One hundred problems in elementary mathematics. Courier Dover Publications. ISBN 978-0-486-23875-3.
19. Steinhaus, Hugo (1961), Orzeł czy reszka (in Polish), vol. I, Warszawa : Państwowe Wydaw, OCLC 68678009
20. Steinhaus, Hugo (1980), Słownik racjonalny (in Polish), vol. I, Zakład Narodowy im. Ossolińskich, OCLC 7272718
21. "Hugo Dyonizy Steinhaus". K. Szajowski. Otter Creek Holdings, LLC. May 2010. Retrieved 28 May 2018.
22. "Instytut Matematyczny Uniwersytetu Wrocławskiego on Facebook". Facebook. Archived from the original on 2022-04-27.
Further reading
• Kazimierz Kuratowski, A Half Century of Polish Mathematics: Remembrances and Reflections, Oxford, Pergamon Press, 1980, ISBN 978-0-08-023046-7, pp. 173–79 et passim.
• Hugo Steinhaus, Mathematical Snapshots, second edition, Oxford, 1951, blurb.
External links
• Hugo Steinhaus at the Mathematics Genealogy Project
• Hugo Steinhaus in MathSciNet
• Hugo Steinhaus in Zentralblatt MATH
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| Wikipedia |
Władysław Matwin
Władysław Matwin (17 July 1916 – 21 October 2012) was a Polish politician, journalist and mathematician who was one of the pioneers of computer science in Poland.
Biography
After his parents divorced, he and his mother found themselves in Poznań, where he studied economics. At that time he belonged to the Communist Party of Poland and the Young Communist League of Poland (KZMP).[1] He was secretary of the KZMP District Committee. In January 1935, he was arrested for Communist activities and sentenced to three years in prison. After being released, he went to Czechoslovakia, where he studied chemistry in Brno. He returned to Poland in the spring of 1939. He volunteered to join the army, but was considered a dangerous criminal and was banned from serving in the Polish army.[2]
During World War II he stayed on the territory of the Soviet Union. First, he worked as a miner and later studied at the Metallurgy Institute at night. For a short time he was in the Red Army, from which - due to his origin - he was removed. Later he worked in railway construction. He then stayed in Tbilisi. Later he joined First Polish Army army in Ryazan, where he taught politicsat the officer's school. In 1944 he belonged to the corps of political and educational officers of the 1st Tadeusz Kościuszko Infantry Division. In 1944 he was sent to Tehran, where the Union of Polish Patriots (in which he was active) created an outpost whose main task was to reach the local Polish community through radio broadcasts and newspapers, and above all the soldiers of Władysław Anders.[1]
In 1945, Władysław Matwin was summoned to Moscow, where he became chargé d'affaires at the Polish embassy. In 1946 he returned to Poland and served as an instructor of the Central Committee. In 1947 and 1948 he was the first secretary of the Provincial Committee of the Polish Workers' Party in Wrocław. He also stayed for a year in Davos, Switzerland, where he had his eyes treated (he was in danger of losing his sight as a result of the disease). Together with the PPR, he joined the Polish United Workers' Party, sitting until June 1964 in its Central Committee (he also held the position of the first secretary of the Provincial Committee in Wrocław, which he held until 1949).[3] In the 1950s, he was associated with the Puławy faction.[4]
From 1949 to 1952 he was the chairman of the Main Board of the Union of Polish Youth. From December 1952 to February 1954 he was the first secretary of the Warsaw Committee of the PZPR. From 1954 to March 1956 and again from November 1956 to March 1957 he was editor-in-chief of Trybuna Ludu. From November 1954 to January 1955 he headed the Organizational Department of the Central Committee of the PZPR, and then until November 1963 he was secretary of the Central Committee (until March 1956 responsible for education ). In 1957, he was again sent to Wrocław, where he took the position of the First Secretary of the Provincial Committee of the PZPR, holding the position until his retirement from politics 1963.[2]
In 1963 he began studying mathematics and in 1966 he obtained a diploma in automata theory. The following year, he became the director of the Central Center for Management Staff Improvement, but in 1968 he lost this position for not agreeing to the demand to remove employees of Jewish origin from the institute. He took the job of a senior technologist in Włochy, and in 1970 - at the Institute of Mathematical Machines; in 1973 he became the director of the Department of Electronic Computing Technology. From 1976 to 1991 he worked part-time at the Systems Research Institute of the Polish Academy of Sciences.
Władysław Matwin died in October 2012, being one of the last remaining politicians of the pre-war Polish Communist Party.[1]
References
1. "Zmarł Władysław Matwin - jeden z ostatnich przedwojennych komunistów". dzieje.pl (in Polish). Retrieved 2023-01-18.
2. "Biuletyn Informacji Publicznej Instytutu Pamięci Narodowej". katalog.bip.ipn.gov.pl. Retrieved 2023-01-18.
3. Szumiło, Mirosław. "Elita PZPR w dokumentach dyplomacji andckiej z lat 1959-1964. Komunizm: system – ludzie – dokumentacja 4, 291-328" (PDF).
4. "Sowieckie charakterystyki Władysława Matwina z lat - PDF Free Download". docplayer.pl. Retrieved 2023-01-18.
Authority control
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• ISNI
• VIAF
National
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| Wikipedia |
X-parameters
X-parameters are a generalization of S-parameters and are used for characterizing the amplitudes and relative phase of harmonics generated by nonlinear components under large input power levels. X-parameters are also referred to as the parameters of the Poly-Harmonic Distortion (PHD) nonlinear behavioral model.
Description
X-parameters represent a new category of nonlinear network parameters for high-frequency design (Nonlinear vector network analyzers are sometimes called large signal network analyzers.[1])
X-parameters are applicable to both large-signal and small-signal conditions, for linear and nonlinear components. They are an extension of S-parameters[2] meaning that, in the limit of a small signal, X-parameters reduce to S-parameters.
They help overcome a key challenge in RF engineering, namely that nonlinear impedance differences, harmonic mixing, and nonlinear reflection effects occur when components are cascaded under large signal operating conditions. This means that there is a nonlinear and as such non-trivial relationship between the properties of the individual cascaded components and the composite properties of the resulting cascade. This situation is unlike that at DC, where one can simply add the values of resistors connected in series. X-parameters help solve this cascading problem: if the X-parameters of a set of components are measured individually, the X-parameters (and hence the non-linear transfer function) can be calculated of any cascade made from them. Calculations based on X-parameters are usually performed within a harmonic balance simulator environment.[3]
Development
X-parameters were developed and introduced by Keysight Technologies as functionality included in N5242A Nonlinear Vector Network Analyzer,[4][5] and the W2200 Advanced Design System in 2008. N5242A is a Keysight[6] product that were formerly part of Agilent.[7]
X-parameters are the parameters of the polyharmonic distortion modeling work of Dr. Jan Verspecht[8][9] and Dr. David E. Root.[9]
See also
• Two-port network
Notes
1. Dr. Jan Verspecht (December 2005). "Large-Signal Network Analysis" (PDF). IEEE Microwave Magazine. IEEE. 6 (4): 82–92. doi:10.1109/MMW.2005.1580340. Retrieved May 1, 2009.
2. "EDA Focus: May 2009, Transcript of interview of David E. Root by Microwave Journal Editor David Vye on April 16th, 2009". Microwave Journal. April 16, 2009. Retrieved May 4, 2009.
3. "Keysight NVNA & X-Parameters Simulation in ADS: The new paradigm for nonlinear measurements, modeling, and simulation with ADS (PDF, 1MB) on the X-Parameters MMIC Design Seminar page". Retrieved July 17, 2015.
4. "Agilent Technologies Announces Breakthrough in X-Parameter Nonlinear Model Generation for Components Used in Wireless, Aerospace Defense Industries: X-Parameters Enable Model Generation from Simulation or Measurement, for Fast Development". Keysight.com. December 17, 2008. Retrieved May 6, 2009.
5. "Keysight N5242A PNA-X Series Microwave Network Analyzer, 10 MHz to 26.5 GHz". Retrieved July 17, 2015.
6. "Electronic design, test automation and measurement equipment". Keysight Technologies.{{cite web}}: CS1 maint: url-status (link)
7. "Agilent's Electronic Measurement business is now Keysight Technologies". Agilent Technologies.{{cite web}}: CS1 maint: url-status (link)
8. Dr. Jan Verspecht (October 1996). "Black Box Modelling of Power Transistors in the Frequency Domain" (PDF). Conference paper presented at the INMMC '96, Duisburg, Germany. Retrieved May 6, 2009. (PDF, 85 KB)
9. Dr. Jan Verspecht; Dr. David E. Root (June 2006). "Polyharmonic Distortion Modeling" (PDF). IEEE Microwave Magazine. IEEE. 7 (3): 44–57. doi:10.1109/MMW.2006.1638289. Retrieved May 6, 2009. (PDF, 2.4MB)
External links
• Fundamentally Changing Nonlinear Microwave Design by David Vye Editor, Microwave Journal Vol. 53 No. 3 March 2010 Page 22] (former location)
| Wikipedia |
X-ray transform
In mathematics, the X-ray transform (also called ray transform[1] or John transform) is an integral transform introduced by Fritz John in 1938[2] that is one of the cornerstones of modern integral geometry. It is very closely related to the Radon transform, and coincides with it in two dimensions. In higher dimensions, the X-ray transform of a function is defined by integrating over lines rather than over hyperplanes as in the Radon transform. The X-ray transform derives its name from X-ray tomography (used in CT scans) because the X-ray transform of a function ƒ represents the attenuation data of a tomographic scan through an inhomogeneous medium whose density is represented by the function ƒ. Inversion of the X-ray transform is therefore of practical importance because it allows one to reconstruct an unknown density ƒ from its known attenuation data.
In detail, if ƒ is a compactly supported continuous function on the Euclidean space Rn, then the X-ray transform of ƒ is the function Xƒ defined on the set of all lines in Rn by
$Xf(L)=\int _{L}f=\int _{\mathbf {R} }f(x_{0}+t\theta )dt$
where x0 is an initial point on the line and θ is a unit vector in Rn giving the direction of the line L. The latter integral is not regarded in the oriented sense: it is the integral with respect to the 1-dimensional Lebesgue measure on the Euclidean line L.
The X-ray transform satisfies an ultrahyperbolic wave equation called John's equation.
The Gauss hypergeometric function can be written as an X-ray transform (Gelfand, Gindikin & Graev 2003, 2.1.2).
References
1. Natterer, Frank; Wübbeling, Frank (2001). Mathematical Methods in Image Reconstruction. Philadelphia: SIAM. doi:10.1137/1.9780898718324.fm.
2. Fritz, John (1938). "The ultrahyperbolic differential equation with four independent variables". Duke Mathematical Journal. 4: 300–322. doi:10.1215/S0012-7094-38-00423-5. Retrieved 23 January 2013.
• Berenstein, Carlos A. (2001) [1994], "X-ray transform", Encyclopedia of Mathematics, EMS Press.
• Gelfand, I. M.; Gindikin, S. G.; Graev, M. I. (2003) [2000], Selected topics in integral geometry, Translations of Mathematical Monographs, vol. 220, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-2932-5, MR 2000133
• Helgason, Sigurdur (2008), Geometric analysis on symmetric spaces, Mathematical Surveys and Monographs, vol. 39 (2nd ed.), Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-4530-1, MR 2463854
• Helgason, Sigurdur (1999), The Radon Transform (PDF), Progress in Mathematics (2nd ed.), Boston, M.A.: Birkhauser
| Wikipedia |
ChaCha20-Poly1305
ChaCha20-Poly1305 is an authenticated encryption with additional data (AEAD) algorithm, that combines the ChaCha20 stream cipher with the Poly1305 message authentication code. Its usage in IETF protocols is standardized in RFC 8439.[1] It has fast software performance, and without hardware acceleration, is usually faster than AES-GCM.[2]
History
The two building blocks of the construction, the algorithms Poly1305 and ChaCha20, were both independently designed, in 2005 and 2008, by Daniel J. Bernstein.[3][4]
In 2013–2014, a variant of the original ChaCha20 algorithm (using 32-bit counter and 96-bit nonce) and a variant of the original Poly1305 (authenticating 2 strings) were combined in an IETF draft[5][6] to be used in TLS and DTLS,[7] and chosen by Google, for security and performance reasons, as a newly supported cipher.[8] Shortly after Google's adoption for TLS, ChaCha20, Poly1305 and the combined AEAD mode are added to OpenSSH via [email protected] authenticated encryption cipher[9][10] but kept the original 64-bit counter and 64-bit nonce for the ChaCha20 algorithm.
In 2015, the AEAD algorithm is standardized in RFC 7539[11] and RFC 7905[12] to be used in TLS 1.2 and DTLS 1.2 and in RFC 7634[13] to be used in IPsec. The same year, it was integrated by Cloudflare as an alternative ciphersuite.[14]
In June 2018, the RFC 7539 was updated and replaced by RFC 8439.[1]
Description
The ChaCha20-Poly1305 algorithm as described in RFC 8439[1] takes as input a 256-bit key and a 96-bit nonce to encrypt a plaintext, with a ciphertext expansion of 128-bit (the tag size). In the ChaCha20-Poly1305 construction, ChaCha20 is used in counter mode to derive a key stream that is XORed with the plaintext. The ciphertext and the associated data is then authenticated using a variant of Poly1305 that first encodes the two strings into one.
Variants
XChaCha20-Poly1305 – extended nonce variant
The XChaCha20-Poly1305 construction is an extended 192-bit nonce variant of the ChaCha20-Poly1305 construction, using XChaCha20 instead of ChaCha20. When choosing nonces at random, the XChaCha20-Poly1305 construction allows for better security than the original construction. The draft attempt to standardize the construction expired in July 2020.[15]
Salsa20-Poly1305 and XSalsa20-Poly1305
Salsa20-Poly1305 and XSalsa20-Poly1305 are variants of the ChaCha20-Poly1305 and XChaCha20-Poly1305 algorithms, using Salsa20 and XSalsa20 in place of ChaCha20 and XChaCha20. They are implemented in NaCl[16] and libsodium[17] but not standardized. The variants using ChaCha are preferred in practice as they provide better diffusion per round than Salsa.[3]
Use
ChaCha20-Poly1305 is used in IPsec,[2] SSH,[18] TLS 1.2, DTLS 1.2, TLS 1.3,[12][18] WireGuard,[19] S/MIME 4.0,[20] OTRv4[21] and multiple other protocols. It is used in Software like Borg[22] as standard data encryption. Among others, it is implemented in OpenSSL, and libsodium.
Performance
ChaCha20-Poly1305 usually offers better performance than the more prevalent AES-GCM algorithm on systems where the CPU(s) does not feature the AES-NI instruction set extension.[2] As a result, ChaCha20-Poly1305 is sometimes preferred over AES-GCM due to its similar levels of security and in certain use cases involving mobile devices, which mostly use ARM-based CPUs.
Security
The ChaCha20-Poly1305 construction is proven secure in the standard model and the ideal permutation model, for the single- and multi-user setting.[23] However, similarly to GCM, the security relies on choosing a unique nonce for every message encrypted. Compared to AES-GCM, implementations of ChaCha20-Poly1305 are less vulnerable to timing attacks.
See also
• Authenticated encryption
• Galois/Counter Mode
• Salsa20
• Poly1305
External links
• RFC 8439: ChaCha20 and Poly1305 for IETF Protocols
• RFC 7634: ChaCha20, Poly1305, and Their Use in the Internet Key Exchange Protocol (IKE) and IPsec
• RFC 7905: ChaCha20-Poly1305 Cipher Suites for Transport Layer Security (TLS)
• RFC 8103: Using ChaCha20-Poly1305 Authenticated Encryption in the Cryptographic Message Syntax (CMS)
References
1. Nir, Yoav; Langley, Adam (June 2018). ChaCha20 and Poly1305 for IETF Protocols. doi:10.17487/RFC8439. RFC 8439.
2. Nir, Yoav; Langley, Adam (June 2018). "Performance Measurements of ChaCha20". ChaCha20 and Poly1305 for IETF Protocols. sec. B. doi:10.17487/RFC8439. RFC 8439.
3. Bernstein, D. J. (January 2008). ChaCha, a variant of Salsa20 (PDF). The State of the Art of Stream Ciphers. Vol. 8. pp. 3–5.
4. Bernstein, Daniel J. (2005), "The Poly1305-AES Message-Authentication Code", Fast Software Encryption, Lecture Notes in Computer Science, Berlin, Heidelberg: Springer Berlin Heidelberg, vol. 3557, pp. 32–49, doi:10.1007/11502760_3, ISBN 978-3-540-26541-2
5. Langley, Adam (September 2013). ChaCha20 and Poly1305 based Cipher Suites for TLS. I-D draft-agl-tls-chacha20poly1305-00.
6. Nir, Yoav (27 January 2014). ChaCha20 and Poly1305 for IETF protocols. I-D draft-nir-cfrg-chacha20-poly1305-00.
7. Langley, Adam; Chang, Wan-Teh; Mavrogiannopoulos, Nikos; Strombergson, Joachim; Josefsson, Simon (24 January 2014). The ChaCha Stream Cipher for Transport Layer Security. I-D draft-mavrogiannopoulos-chacha-tls-01.
8. Bursztein, Elie (24 April 2014). "Speeding up and strengthening HTTPS connections for Chrome on Android". Google Online Security Blog. Archived from the original on 2016-09-28. Retrieved 2021-12-27.
9. Miller, Damien. "Super User's BSD Cross Reference: /OpenBSD/usr.bin/ssh/PROTOCOL.chacha20poly1305". bxr.su. Archived from the original on 2013-12-13. Retrieved 2021-12-28.
10. Miller, Damien (29 November 2013). "ChaCha20 and Poly1305 in OpenSSH". Archived from the original on 2013-12-13. Retrieved 2021-12-28.
11. Nir, Yoav; Langley, Adam (May 2015). ChaCha20 and Poly1305 for IETF Protocols. doi:10.17487/RFC7539. RFC 7539.
12. Langley, Adam; Chang, Wan-Teh; Mavrogiannopoulos, Nikos; Strombergson, Joachim; Josefsson, Simon (June 2016). ChaCha20-Poly1305 Cipher Suites for Transport Layer Security (TLS). doi:10.17487/RFC7905. RFC 7905.
13. Nir, Yoav (August 2015). ChaCha20, Poly1305, and Their Use in the Internet Key Exchange Protocol (IKE) and IPsec. doi:10.17487/RFC7634. RFC 7634.
14. "Do the ChaCha: better mobile performance with cryptography". The Cloudflare Blog. 2015-02-23. Retrieved 2021-12-28.
15. Arciszewski, Scott (10 January 2020). XChaCha: eXtended-nonce ChaCha and AEAD_XChaCha20_Poly1305. I-D draft-irtf-cfrg-xchacha.
16. "NaCl: Networking and Cryptography library - Secret-key authenticated encryption". Archived from the original on 2009-06-30.
17. "libsodium - Authenticated encryption". Archived from the original on 2020-08-04.
18. Thomson, Martin; Turner (May 2021). Using TLS to Secure QUIC. doi:10.17487/RFC9001. RFC 9001.
19. Donenfeld, Jason A. "Protocol & Cryptography - WireGuard". www.wireguard.com. Retrieved 2021-12-28.
20. Housley, Russ (February 2017). Using ChaCha20-Poly1305 Authenticated Encryption in the Cryptographic Message Syntax (CMS). doi:10.17487/RFC8103. RFC 8103.
21. OTRv4, OTRv4, 2021-12-25, retrieved 2021-12-28
22. borg rcreate, borgbackup, 2022-08-03, retrieved 2023-01-28
23. Degabriele, Jean Paul; Govinden, Jérôme; Günther, Felix; Paterson, Kenneth G. (2021-11-12), "The Security of ChaCha20-Poly1305 in the Multi-User Setting", Proceedings of the 2021 ACM SIGSAC Conference on Computer and Communications Security, New York, NY, USA: Association for Computing Machinery, pp. 1981–2003, doi:10.1145/3460120.3484814, ISBN 978-1-4503-8454-4, S2CID 244077782, retrieved 2021-12-27
Stream ciphers
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Cryptographic hash functions and message authentication codes
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Mathematics
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| Wikipedia |
XDH assumption
The external Diffie–Hellman (XDH) assumption is a computational hardness assumption used in elliptic curve cryptography. The XDH assumption holds that there exist certain subgroups of elliptic curves which have useful properties for cryptography. Specifically, XDH implies the existence of two distinct groups $\langle {\mathbb {G} }_{1},{\mathbb {G} }_{2}\rangle $ with the following properties:
1. The discrete logarithm problem (DLP), the computational Diffie–Hellman problem (CDH), and the computational co-Diffie–Hellman problem are all intractable in ${\mathbb {G} }_{1}$ and ${\mathbb {G} }_{2}$.
2. There exists an efficiently computable bilinear map (pairing) $e(\cdot ,\cdot ):{\mathbb {G} }_{1}\times {\mathbb {G} }_{2}\rightarrow {\mathbb {G} }_{T}$.
3. The decisional Diffie–Hellman problem (DDH) is intractable in ${\mathbb {G} }_{1}$.
The above formulation is referred to as asymmetric XDH. A stronger version of the assumption (symmetric XDH, or SXDH) holds if DDH is also intractable in ${\mathbb {G} }_{2}$.
The XDH assumption is used in some pairing-based cryptographic protocols. In certain elliptic curve subgroups, the existence of an efficiently-computable bilinear map (pairing) can allow for practical solutions to the DDH problem. These groups, referred to as gap Diffie–Hellman (GDH) groups, facilitate a variety of novel cryptographic protocols, including tri-partite key exchange, identity based encryption, and secret handshakes (to name a few). However, the ease of computing DDH within a GDH group can also be an obstacle when constructing cryptosystems; for example, it is not possible to use DDH-based cryptosystems such as ElGamal within a GDH group. Because the DDH assumption holds within at least one of a pair of XDH groups, these groups can be used to construct pairing-based protocols which allow for ElGamal-style encryption and other novel cryptographic techniques.
In practice, it is believed that the XDH assumption may hold in certain subgroups of MNT elliptic curves. This notion was first proposed by Scott (2002), and later by Boneh, Boyen and Shacham (2002) as a means to improve the efficiency of a signature scheme. The assumption was formally defined by Ballard, Green, de Medeiros and Monrose (2005), and full details of a proposed implementation were advanced in that work. Evidence for the validity of this assumption is the proof by Verheul (2001) and Galbraith and Rotger (2004) of the non-existence of distortion maps in two specific elliptic curve subgroups which possess an efficiently computable pairing. As pairings and distortion maps are currently the only known means to solve the DDH problem in elliptic curve groups, it is believed that the DDH assumption therefore holds in these subgroups, while pairings are still feasible between elements in distinct groups.
References
1. Mike Scott. Authenticated ID-based exchange and remote log-in with simple token and PIN. E-print archive (2002/164), 2002. (pdf file)
2. Dan Boneh, Xavier Boyen, Hovav Shacham. Short Group Signatures. CRYPTO 2004. (pdf file)
3. Lucas Ballard, Matthew Green, Breno de Medeiros, Fabian Monrose. Correlation-Resistant Storage via Keyword-Searchable Encryption. E-print archive (2005/417), 2005. (pdf file)
4. Steven D Galbraith, Victor Rotger. Easy Decision Diffie–Hellman Groups. LMS Journal of Computation and Mathematics, August 2004. ()
5. E.R. Verheul, Evidence that XTR is more secure than supersingular elliptic curve cryptosystems, in B. Pfitzmann (ed.) EUROCRYPT 2001, Springer LNCS 2045 (2001) 195–210.
Computational hardness assumptions
Number theoretic
• Integer factorization
• Phi-hiding
• RSA problem
• Strong RSA
• Quadratic residuosity
• Decisional composite residuosity
• Higher residuosity
Group theoretic
• Discrete logarithm
• Diffie-Hellman
• Decisional Diffie–Hellman
• Computational Diffie–Hellman
Pairings
• External Diffie–Hellman
• Sub-group hiding
• Decision linear
Lattices
• Shortest vector problem (gap)
• Closest vector problem (gap)
• Learning with errors
• Ring learning with errors
• Short integer solution
Non-cryptographic
• Exponential time hypothesis
• Unique games conjecture
• Planted clique conjecture
| Wikipedia |
Fractionally subadditive valuation
A set function is called fractionally subadditive (or XOS) if it is the maximum of several additive set functions. This valuation class was defined, and termed XOS, by Noam Nisan, in the context of combinatorial auctions.[1] The term fractionally subadditive was given by Uriel Feige.[2]
Definition
There is a finite base set of items, $M:=\{1,\ldots ,m\}$.
There is a function $v$ which assigns a number to each subset of $M$.
The function $v$ is called fractionally subadditive (or XOS) if there exists a collection of set functions, $\{a_{1},\ldots ,a_{l}\}$, such that:[3]
• Each $a_{j}$ is additive, i.e., it assigns to each subset $X\subseteq M$, the sum of the values of the items in $X$.
• The function $v$ is the pointwise maximum of the functions $a_{j}$. I.e, for every subset $X\subseteq M$:
$v(X)=\max _{j=1}^{l}a_{j}(X)$
Equivalent Definition
The name fractionally subadditive comes from the following equivalent definition: a set function $v$ is fractionally subadditive if, for any $S\subseteq M$ and any collection $\{\alpha _{i},T_{i}\}_{i=1}^{k}$ with $\alpha _{i}>0$ and $T_{i}\subseteq M$ such that $\sum _{T_{i}\ni j}\alpha _{i}\geq 1$ for all $j\in S$, we have $v(S)\leq \sum _{i=1}^{k}\alpha _{i}v(T_{i})$.
Relation to other utility functions
Every submodular set function is XOS, and every XOS function is a subadditive set function.[1]
See also: Utility functions on indivisible goods.
References
1. Nisan, Noam (2000). "Bidding and allocation in combinatorial auctions". Proceedings of the 2nd ACM conference on Electronic commerce - EC '00. p. 1. doi:10.1145/352871.352872. ISBN 1581132727.
2. Feige, Uriel (2009). "On Maximizing Welfare when Utility Functions Are Subadditive". SIAM Journal on Computing. 39: 122–142. CiteSeerX 10.1.1.86.9904. doi:10.1137/070680977.
3. Christodoulou, George; Kovács, Annamária; Schapira, Michael (2016). "Bayesian Combinatorial Auctions". Journal of the ACM. 63 (2): 1. CiteSeerX 10.1.1.721.5346. doi:10.1145/2835172.
| Wikipedia |
XYZ inequality
In combinatorial mathematics, the XYZ inequality, also called the Fishburn–Shepp inequality, is an inequality for the number of linear extensions of finite partial orders. The inequality was conjectured by Ivan Rival and Bill Sands in 1981. It was proved by Lawrence Shepp in Shepp (1982). An extension was given by Peter Fishburn in Fishburn (1984).
It states that if x, y, and z are incomparable elements of a finite poset, then
$P(x\prec y)P(x\prec z)\leqslant P((x\prec y)\wedge (x\prec z))$,
where P(A) is the probability that a linear order extending the partial order $\prec $ has the property A.
In other words, the probability that $x\prec z$ increases if one adds the condition that $x\prec y$. In the language of conditional probability,
$P(x\prec z)\leqslant P(x\prec z\mid x\prec y).$
The proof uses the Ahlswede–Daykin inequality.
See also
• FKG inequality
References
• Fishburn, Peter C. (1984), "A correlational inequality for linear extensions of a poset", Order, 1 (2): 127–137, doi:10.1007/BF00565648, ISSN 0167-8094, MR 0764320, S2CID 121406218
• "Fishburn-Shepp inequality", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Shepp, L. A. (1982), "The XYZ conjecture and the FKG inequality", The Annals of Probability, Institute of Mathematical Statistics, 10 (3): 824–827, doi:10.1214/aop/1176993791, ISSN 0091-1798, JSTOR 2243391, MR 0659563
| Wikipedia |
Multiplication
Multiplication (often denoted by the cross symbol ×, by the mid-line dot operator ⋅, by juxtaposition, or, on computers, by an asterisk *) is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction, and division. The result of a multiplication operation is called a product.
Arithmetic operations
Addition (+)
$\scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,+\,{\text{term}}\\\scriptstyle {\text{summand}}\,+\,{\text{summand}}\\\scriptstyle {\text{addend}}\,+\,{\text{addend}}\\\scriptstyle {\text{augend}}\,+\,{\text{addend}}\end{matrix}}\right\}\,=\,$ $\scriptstyle {\text{sum}}$
Subtraction (−)
$\scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,-\,{\text{term}}\\\scriptstyle {\text{minuend}}\,-\,{\text{subtrahend}}\end{matrix}}\right\}\,=\,$ $\scriptstyle {\text{difference}}$
Multiplication (×)
$\scriptstyle \left.{\begin{matrix}\scriptstyle {\text{factor}}\,\times \,{\text{factor}}\\\scriptstyle {\text{multiplier}}\,\times \,{\text{multiplicand}}\end{matrix}}\right\}\,=\,$ $\scriptstyle {\text{product}}$
Division (÷)
$\scriptstyle \left.{\begin{matrix}\scriptstyle {\frac {\scriptstyle {\text{dividend}}}{\scriptstyle {\text{divisor}}}}\\[1ex]\scriptstyle {\frac {\scriptstyle {\text{numerator}}}{\scriptstyle {\text{denominator}}}}\end{matrix}}\right\}\,=\,$ $\scriptstyle \left\{{\begin{matrix}\scriptstyle {\text{fraction}}\\\scriptstyle {\text{quotient}}\\\scriptstyle {\text{ratio}}\end{matrix}}\right.$
Exponentiation (^)
$\scriptstyle {\text{base}}^{\text{exponent}}\,=\,$ $\scriptstyle {\text{power}}$
nth root (√)
$\scriptstyle {\sqrt[{\text{degree}}]{\scriptstyle {\text{radicand}}}}\,=\,$ $\scriptstyle {\text{root}}$
Logarithm (log)
$\scriptstyle \log _{\text{base}}({\text{anti-logarithm}})\,=\,$ $\scriptstyle {\text{logarithm}}$
The multiplication of whole numbers may be thought of as repeated addition; that is, the multiplication of two numbers is equivalent to adding as many copies of one of them, the multiplicand, as the quantity of the other one, the multiplier; both numbers can be referred to as factors.
$a\times b=\underbrace {b+\cdots +b} _{a{\text{ times}}}$
For example, 4 multiplied by 3, often written as $3\times 4$ and spoken as "3 times 4", can be calculated by adding 3 copies of 4 together:
$3\times 4=4+4+4=12$
Here, 3 (the multiplier) and 4 (the multiplicand) are the factors, and 12 is the product.
One of the main properties of multiplication is the commutative property, which states in this case that adding 3 copies of 4 gives the same result as adding 4 copies of 3:
$4\times 3=3+3+3+3=12$
Thus the designation of multiplier and multiplicand does not affect the result of the multiplication.[1]
Systematic generalizations of this basic definition define the multiplication of integers (including negative numbers), rational numbers (fractions), and real numbers.
Multiplication can also be visualized as counting objects arranged in a rectangle (for whole numbers) or as finding the area of a rectangle whose sides have some given lengths. The area of a rectangle does not depend on which side is measured first—a consequence of the commutative property.
The product of two measurements (or physical quantities) or is a new type of measurement, usually with a derived unit. For example, multiplying the lengths (in meters or feet) of the two sides of a rectangle gives its area (in square meters or square feet). Such a product is the subject of dimensional analysis.
The inverse operation of multiplication is division. For example, since 4 multiplied by 3 equals 12, 12 divided by 3 equals 4. Indeed, multiplication by 3, followed by division by 3, yields the original number. The division of a number other than 0 by itself equals 1.
Multiplication is also defined for other types of numbers, such as complex numbers, and for more abstract constructs, like matrices. For some of these more abstract constructs, the order in which the operands are multiplied together matters. A listing of the many different kinds of products used in mathematics is given in Product (mathematics).
Notation and terminology
× ⋅
Multiplication signs
In UnicodeU+00D7 × MULTIPLICATION SIGN (×)
U+22C5 ⋅ DOT OPERATOR (⋅)
Different from
Different fromU+00B7 · MIDDLE DOT
U+002E . FULL STOP
Main article: Multiplication sign
In arithmetic, multiplication is often written using the multiplication sign (either × or $\times $) between the terms (that is, in infix notation).[2] For example,
$2\times 3=6$ ("two times three equals six")
$3\times 4=12$
$2\times 3\times 5=6\times 5=30$
$2\times 2\times 2\times 2\times 2=32$
There are other mathematical notations for multiplication:
• To reduce confusion between the multiplication sign × and the common variable x, multiplication is also denoted by dot signs,[3] usually a middle-position dot (rarely period):
$5\cdot 2$ or $5\,.\,3$
The middle dot notation or dot operator, encoded in Unicode as U+22C5 ⋅ DOT OPERATOR, is now standard in the United States and other countries where the period is used as a decimal point. When the dot operator character is not accessible, the interpunct (·) is used. In other countries that use a comma as a decimal mark, either the period or a middle dot is used for multiplication.
Historically, in the United Kingdom and Ireland, the middle dot was sometimes used for the decimal to prevent it from disappearing in the ruled line, and the period/full stop was used for multiplication. However, since the Ministry of Technology ruled to use the period as the decimal point in 1968,[4] and the International System of Units (SI) standard has since been widely adopted, this usage is now found only in the more traditional journals such as The Lancet.[5]
• In algebra, multiplication involving variables is often written as a juxtaposition (e.g., $xy$ for $x$ times $y$ or $5x$ for five times $x$), also called implied multiplication.[6] The notation can also be used for quantities that are surrounded by parentheses (e.g., $5(2)$, $(5)2$ or $(5)(2)$ for five times two). This implicit usage of multiplication can cause ambiguity when the concatenated variables happen to match the name of another variable, when a variable name in front of a parenthesis can be confused with a function name, or in the correct determination of the order of operations.
• In vector multiplication, there is a distinction between the cross and the dot symbols. The cross symbol generally denotes the taking a cross product of two vectors, yielding a vector as its result, while the dot denotes taking the dot product of two vectors, resulting in a scalar.
In computer programming, the asterisk (as in 5*2) is still the most common notation. This is due to the fact that most computers historically were limited to small character sets (such as ASCII and EBCDIC) that lacked a multiplication sign (such as ⋅ or ×), while the asterisk appeared on every keyboard. This usage originated in the FORTRAN programming language.[7]
The numbers to be multiplied are generally called the "factors" (as in factorization). The number to be multiplied is the "multiplicand", and the number by which it is multiplied is the "multiplier". Usually, the multiplier is placed first and the multiplicand is placed second;[1] however sometimes the first factor is the multiplicand and the second the multiplier.[8] Also, as the result of multiplication does not depend on the order of the factors, the distinction between "multiplicand" and "multiplier" is useful only at a very elementary level and in some multiplication algorithms, such as the long multiplication. Therefore, in some sources, the term "multiplicand" is regarded as a synonym for "factor".[9] In algebra, a number that is the multiplier of a variable or expression (e.g., the 3 in $3xy^{2}$) is called a coefficient.
The result of a multiplication is called a product. When one factor is an integer, the product is a multiple of the other or of the product of the others. Thus $2\times \pi $ is a multiple of $\pi $, as is $5133\times 486\times \pi $. A product of integers is a multiple of each factor; for example, 15 is the product of 3 and 5 and is both a multiple of 3 and a multiple of 5.
Definitions
The product of two numbers or the multiplication between two numbers can be defined for common special cases: integers, natural numbers, fractions, real numbers, complex numbers, and quaternions.
Product of two natural numbers
Placing several stones into a rectangular pattern with $r$ rows and $s$ columns gives
$r\cdot s=\sum _{i=1}^{s}r=\underbrace {r+r+\cdots +r} _{s{\text{ times}}}=\sum _{j=1}^{r}s=\underbrace {s+s+\cdots +s} _{r{\text{ times}}}$
stones.
Product of two integers
Integers allow positive and negative numbers. Their product is determined by the product of their positive amounts, combined with the sign derived from the following rule:
${\begin{array}{|c|c c|}\hline \times &-&+\\\hline -&+&-\\+&-&+\\\hline \end{array}}$
(This rule is a necessary consequence of demanding distributivity of multiplication over addition, and is not an additional rule.)
In words, we have:
• A negative number multiplied by a negative number is positive,
• A negative number multiplied by a positive number is negative,
• A positive number multiplied by a negative number is negative,
• A positive number multiplied by a positive number is positive.
Product of two fractions
Two fractions can be multiplied by multiplying their numerators and denominators:
${\frac {z}{n}}\cdot {\frac {z'}{n'}}={\frac {z\cdot z'}{n\cdot n'}}$
Product of two real numbers
There are several equivalent ways for define formally the real numbers; see Construction of the real numbers. The definition of multiplication is a part of all these definitions.
A fundamental aspect of these definitions is that every real number can be approximated to any accuracy by rational numbers. A standard way for expressing this is that every real number is the least upper bound of a set of rational numbers. In particular, every positive real number is the least upper bound of the truncations of its infinite decimal representation; for example, $\pi $ is the least upper bound of $\{3,\;3.1,\;3.14,\;3,141,\ldots \}.$
A fundamental property of real numbers is that rational approximations are compatible with arithmetic operations, and, in particular, with multiplication. This means that, if a and b are positive real numbers such that $a=\sup _{x\in A}x$ and $b=\sup _{y\in b}y,$ then $a\cdot b=\sup _{x\in A,y\in B}x\cdot y.$ In particular, the product of two positive real numbers is the least upper bound of the term-by-term products of the sequences of their decimal representations.
As changing the signs transforms least upper bounds into greatest lower bounds, the simplest way to deal with a multiplication involving one or two negative numbers, is to use the rule of signs described above in § Product of two integers. The construction of the real numbers through Cauchy sequences is often preferred in order to avoid consideration of the four possible sign configurations.
Product of two complex numbers
Two complex numbers can be multiplied by the distributive law and the fact that $i^{2}=-1$, as follows:
${\begin{aligned}(a+b\,i)\cdot (c+d\,i)&=a\cdot c+a\cdot d\,i+b\,i\cdot c+b\cdot d\cdot i^{2}\\&=(a\cdot c-b\cdot d)+(a\cdot d+b\cdot c)\,i\end{aligned}}$
Geometric meaning of complex multiplication can be understood rewriting complex numbers in polar coordinates:
$a+b\,i=r\cdot (\cos(\varphi )+i\sin(\varphi ))=r\cdot e^{i\varphi }$
Furthermore,
$c+d\,i=s\cdot (\cos(\psi )+i\sin(\psi ))=s\cdot e^{i\psi },$
from which one obtains
$(a\cdot c-b\cdot d)+(a\cdot d+b\cdot c)i=r\cdot s\cdot e^{i(\varphi +\psi )}.$
The geometric meaning is that the magnitudes are multiplied and the arguments are added.
Product of two quaternions
The product of two quaternions can be found in the article on quaternions. Note, in this case, that $a\cdot b$ and $b\cdot a$ are in general different.
Computation
Main article: Multiplication algorithm
Many common methods for multiplying numbers using pencil and paper require a multiplication table of memorized or consulted products of small numbers (typically any two numbers from 0 to 9). However, one method, the peasant multiplication algorithm, does not. The example below illustrates "long multiplication" (the "standard algorithm", "grade-school multiplication"):
23958233
× 5830
———————————————
00000000 ( = 23,958,233 × 0)
71874699 ( = 23,958,233 × 30)
191665864 ( = 23,958,233 × 800)
+ 119791165 ( = 23,958,233 × 5,000)
———————————————
139676498390 ( = 139,676,498,390 )
In some countries such as Germany, the above multiplication is depicted similarly but with the original product kept horizontal and computation starting with the first digit of the multiplier:[10]
23958233 · 5830
———————————————
119791165
191665864
71874699
00000000
———————————————
139676498390
Multiplying numbers to more than a couple of decimal places by hand is tedious and error-prone. Common logarithms were invented to simplify such calculations, since adding logarithms is equivalent to multiplying. The slide rule allowed numbers to be quickly multiplied to about three places of accuracy. Beginning in the early 20th century, mechanical calculators, such as the Marchant, automated multiplication of up to 10-digit numbers. Modern electronic computers and calculators have greatly reduced the need for multiplication by hand.
Historical algorithms
Methods of multiplication were documented in the writings of ancient Egyptian, Greek, Indian, and Chinese civilizations.
The Ishango bone, dated to about 18,000 to 20,000 BC, may hint at a knowledge of multiplication in the Upper Paleolithic era in Central Africa, but this is speculative.[11]
Egyptians
Main article: Ancient Egyptian multiplication
The Egyptian method of multiplication of integers and fractions, which is documented in the Rhind Mathematical Papyrus, was by successive additions and doubling. For instance, to find the product of 13 and 21 one had to double 21 three times, obtaining 2 × 21 = 42, 4 × 21 = 2 × 42 = 84, 8 × 21 = 2 × 84 = 168. The full product could then be found by adding the appropriate terms found in the doubling sequence:[12]
13 × 21 = (1 + 4 + 8) × 21 = (1 × 21) + (4 × 21) + (8 × 21) = 21 + 84 + 168 = 273.
Babylonians
The Babylonians used a sexagesimal positional number system, analogous to the modern-day decimal system. Thus, Babylonian multiplication was very similar to modern decimal multiplication. Because of the relative difficulty of remembering 60 × 60 different products, Babylonian mathematicians employed multiplication tables. These tables consisted of a list of the first twenty multiples of a certain principal number n: n, 2n, ..., 20n; followed by the multiples of 10n: 30n 40n, and 50n. Then to compute any sexagesimal product, say 53n, one only needed to add 50n and 3n computed from the table.
Chinese
See also: Chinese multiplication table
In the mathematical text Zhoubi Suanjing, dated prior to 300 BC, and the Nine Chapters on the Mathematical Art, multiplication calculations were written out in words, although the early Chinese mathematicians employed Rod calculus involving place value addition, subtraction, multiplication, and division. The Chinese were already using a decimal multiplication table by the end of the Warring States period.[13]
Modern methods
The modern method of multiplication based on the Hindu–Arabic numeral system was first described by Brahmagupta. Brahmagupta gave rules for addition, subtraction, multiplication, and division. Henry Burchard Fine, then a professor of mathematics at Princeton University, wrote the following:
The Indians are the inventors not only of the positional decimal system itself, but of most of the processes involved in elementary reckoning with the system. Addition and subtraction they performed quite as they are performed nowadays; multiplication they effected in many ways, ours among them, but division they did cumbrously.[14]
These place value decimal arithmetic algorithms were introduced to Arab countries by Al Khwarizmi in the early 9th century and popularized in the Western world by Fibonacci in the 13th century.[15]
Grid method
Grid method multiplication, or the box method, is used in primary schools in England and Wales and in some areas of the United States to help teach an understanding of how multiple digit multiplication works. An example of multiplying 34 by 13 would be to lay the numbers out in a grid as follows:
× 30 4
5 150 20
10 300 40
3 90 12
and then add the entries.
Computer algorithms
Main article: Multiplication algorithm § Fast multiplication algorithms for large inputs
The classical method of multiplying two n-digit numbers requires n2 digit multiplications. Multiplication algorithms have been designed that reduce the computation time considerably when multiplying large numbers. Methods based on the discrete Fourier transform reduce the computational complexity to O(n log n log log n). In 2016, the factor log log n was replaced by a function that increases much slower, though still not constant.[16] In March 2019, David Harvey and Joris van der Hoeven submitted a paper presenting an integer multiplication algorithm with a complexity of $O(n\log n).$[17] The algorithm, also based on the fast Fourier transform, is conjectured to be asymptotically optimal.[18] The algorithm is not practically useful, as it only becomes faster for multiplying extremely large numbers (having more than 2172912 bits).[19]
Products of measurements
Main article: Dimensional analysis
One can only meaningfully add or subtract quantities of the same type, but quantities of different types can be multiplied or divided without problems. For example, four bags with three marbles each can be thought of as:[1]
[4 bags] × [3 marbles per bag] = 12 marbles.
When two measurements are multiplied together, the product is of a type depending on the types of measurements. The general theory is given by dimensional analysis. This analysis is routinely applied in physics, but it also has applications in finance and other applied fields.
A common example in physics is the fact that multiplying speed by time gives distance. For example:
50 kilometers per hour × 3 hours = 150 kilometers.
In this case, the hour units cancel out, leaving the product with only kilometer units.
Other examples of multiplication involving units include:
2.5 meters × 4.5 meters = 11.25 square meters
11 meters/seconds × 9 seconds = 99 meters
4.5 residents per house × 20 houses = 90 residents
Product of a sequence
Capital pi notation
The product of a sequence of factors can be written with the product symbol $\textstyle \prod $, which derives from the capital letter Π (pi) in the Greek alphabet (much like the same way the summation symbol $\textstyle \sum $ is derived from the Greek letter Σ (sigma)).[20][21] The meaning of this notation is given by
$\prod _{i=1}^{4}(i+1)=(1+1)\,(2+1)\,(3+1)\,(4+1),$
which results in
$\prod _{i=1}^{4}(i+1)=120.$
In such a notation, the variable i represents a varying integer, called the multiplication index, that runs from the lower value 1 indicated in the subscript to the upper value 4 given by the superscript. The product is obtained by multiplying together all factors obtained by substituting the multiplication index for an integer between the lower and the upper values (the bounds included) in the expression that follows the product operator.
More generally, the notation is defined as
$\prod _{i=m}^{n}x_{i}=x_{m}\cdot x_{m+1}\cdot x_{m+2}\cdot \,\,\cdots \,\,\cdot x_{n-1}\cdot x_{n},$
where m and n are integers or expressions that evaluate to integers. In the case where m = n, the value of the product is the same as that of the single factor xm; if m > n, the product is an empty product whose value is 1—regardless of the expression for the factors.
Properties of capital pi notation
By definition,
$\prod _{i=1}^{n}x_{i}=x_{1}\cdot x_{2}\cdot \ldots \cdot x_{n}.$
If all factors are identical, a product of n factors is equivalent to exponentiation:
$\prod _{i=1}^{n}x=x\cdot x\cdot \ldots \cdot x=x^{n}.$
Associativity and commutativity of multiplication imply
$\prod _{i=1}^{n}{x_{i}y_{i}}=\left(\prod _{i=1}^{n}x_{i}\right)\left(\prod _{i=1}^{n}y_{i}\right)$ and
$\left(\prod _{i=1}^{n}x_{i}\right)^{a}=\prod _{i=1}^{n}x_{i}^{a}$
if a is a nonnegative integer, or if all $x_{i}$ are positive real numbers, and
$\prod _{i=1}^{n}x^{a_{i}}=x^{\sum _{i=1}^{n}a_{i}}$
if all $a_{i}$ are nonnegative integers, or if x is a positive real number.
Infinite products
Main article: Infinite product
One may also consider products of infinitely many terms; these are called infinite products. Notationally, this consists in replacing n above by the Infinity symbol ∞. The product of such an infinite sequence is defined as the limit of the product of the first n terms, as n grows without bound. That is,
$\prod _{i=m}^{\infty }x_{i}=\lim _{n\to \infty }\prod _{i=m}^{n}x_{i}.$
One can similarly replace m with negative infinity, and define:
$\prod _{i=-\infty }^{\infty }x_{i}=\left(\lim _{m\to -\infty }\prod _{i=m}^{0}x_{i}\right)\cdot \left(\lim _{n\to \infty }\prod _{i=1}^{n}x_{i}\right),$
provided both limits exist.
Exponentiation
Main article: Exponentiation
When multiplication is repeated, the resulting operation is known as exponentiation. For instance, the product of three factors of two (2×2×2) is "two raised to the third power", and is denoted by 23, a two with a superscript three. In this example, the number two is the base, and three is the exponent.[22] In general, the exponent (or superscript) indicates how many times the base appears in the expression, so that the expression
$a^{n}=\underbrace {a\times a\times \cdots \times a} _{n}$
indicates that n copies of the base a are to be multiplied together. This notation can be used whenever multiplication is known to be power associative.
Properties
For real and complex numbers, which includes, for example, natural numbers, integers, and fractions, multiplication has certain properties:
Commutative property
The order in which two numbers are multiplied does not matter:
$x\cdot y=y\cdot x.$[23][24]
Associative property
Expressions solely involving multiplication or addition are invariant with respect to the order of operations:
$(x\cdot y)\cdot z=x\cdot (y\cdot z)$[23][24]
Distributive property
Holds with respect to multiplication over addition. This identity is of prime importance in simplifying algebraic expressions:
$x\cdot (y+z)=x\cdot y+x\cdot z$[23][24]
Identity element
The multiplicative identity is 1; anything multiplied by 1 is itself. This feature of 1 is known as the identity property:
$x\cdot 1=x$[23][24]
Property of 0
Any number multiplied by 0 is 0. This is known as the zero property of multiplication:
$x\cdot 0=0$[23]
Negation
−1 times any number is equal to the additive inverse of that number.
$(-1)\cdot x=(-x)$ where $(-x)+x=0$
–1 times –1 is 1.
$(-1)\cdot (-1)=1$
Inverse element
Every number x, except 0, has a multiplicative inverse, ${\frac {1}{x}}$, such that $x\cdot \left({\frac {1}{x}}\right)=1$.[25]
Order preservation
Multiplication by a positive number preserves the order:
For a > 0, if b > c then ab > ac.
Multiplication by a negative number reverses the order:
For a < 0, if b > c then ab < ac.
The complex numbers do not have an ordering that is compatible with both addition and multiplication.[26]
Other mathematical systems that include a multiplication operation may not have all these properties. For example, multiplication is not, in general, commutative for matrices and quaternions.[23]
Axioms
Main article: Peano axioms
In the book Arithmetices principia, nova methodo exposita, Giuseppe Peano proposed axioms for arithmetic based on his axioms for natural numbers. Peano arithmetic has two axioms for multiplication:
$x\times 0=0$
$x\times S(y)=(x\times y)+x$
Here S(y) represents the successor of y; i.e., the natural number that follows y. The various properties like associativity can be proved from these and the other axioms of Peano arithmetic, including induction. For instance, S(0), denoted by 1, is a multiplicative identity because
$x\times 1=x\times S(0)=(x\times 0)+x=0+x=x.$
The axioms for integers typically define them as equivalence classes of ordered pairs of natural numbers. The model is based on treating (x,y) as equivalent to x − y when x and y are treated as integers. Thus both (0,1) and (1,2) are equivalent to −1. The multiplication axiom for integers defined this way is
$(x_{p},\,x_{m})\times (y_{p},\,y_{m})=(x_{p}\times y_{p}+x_{m}\times y_{m},\;x_{p}\times y_{m}+x_{m}\times y_{p}).$
The rule that −1 × −1 = 1 can then be deduced from
$(0,1)\times (0,1)=(0\times 0+1\times 1,\,0\times 1+1\times 0)=(1,0).$
Multiplication is extended in a similar way to rational numbers and then to real numbers.
Multiplication with set theory
The product of non-negative integers can be defined with set theory using cardinal numbers or the Peano axioms. See below how to extend this to multiplying arbitrary integers, and then arbitrary rational numbers. The product of real numbers is defined in terms of products of rational numbers; see construction of the real numbers.[27]
Multiplication in group theory
There are many sets that, under the operation of multiplication, satisfy the axioms that define group structure. These axioms are closure, associativity, and the inclusion of an identity element and inverses.
A simple example is the set of non-zero rational numbers. Here we have identity 1, as opposed to groups under addition where the identity is typically 0. Note that with the rationals, we must exclude zero because, under multiplication, it does not have an inverse: there is no rational number that can be multiplied by zero to result in 1. In this example, we have an abelian group, but that is not always the case.
To see this, consider the set of invertible square matrices of a given dimension over a given field. Here, it is straightforward to verify closure, associativity, and inclusion of identity (the identity matrix) and inverses. However, matrix multiplication is not commutative, which shows that this group is non-abelian.
Another fact worth noticing is that the integers under multiplication do not form a group—even if we exclude zero. This is easily seen by the nonexistence of an inverse for all elements other than 1 and −1.
Multiplication in group theory is typically notated either by a dot or by juxtaposition (the omission of an operation symbol between elements). So multiplying element a by element b could be notated as a $\cdot $ b or ab. When referring to a group via the indication of the set and operation, the dot is used. For example, our first example could be indicated by $\left(\mathbb {Q} /\{0\},\,\cdot \right)$.[28]
Multiplication of different kinds of numbers
Numbers can count (3 apples), order (the 3rd apple), or measure (3.5 feet high); as the history of mathematics has progressed from counting on our fingers to modelling quantum mechanics, multiplication has been generalized to more complicated and abstract types of numbers, and to things that are not numbers (such as matrices) or do not look much like numbers (such as quaternions).
Integers
$N\times M$ is the sum of N copies of M when N and M are positive whole numbers. This gives the number of things in an array N wide and M high. Generalization to negative numbers can be done by
$N\times (-M)=(-N)\times M=-(N\times M)$ and
$(-N)\times (-M)=N\times M$
The same sign rules apply to rational and real numbers.
Rational numbers
Generalization to fractions ${\frac {A}{B}}\times {\frac {C}{D}}$ is by multiplying the numerators and denominators respectively: ${\frac {A}{B}}\times {\frac {C}{D}}={\frac {(A\times C)}{(B\times D)}}$. This gives the area of a rectangle ${\frac {A}{B}}$ high and ${\frac {C}{D}}$ wide, and is the same as the number of things in an array when the rational numbers happen to be whole numbers.[23]
Real numbers
Real numbers and their products can be defined in terms of sequences of rational numbers.
Complex numbers
Considering complex numbers $z_{1}$ and $z_{2}$ as ordered pairs of real numbers $(a_{1},b_{1})$ and $(a_{2},b_{2})$, the product $z_{1}\times z_{2}$ is $(a_{1}\times a_{2}-b_{1}\times b_{2},a_{1}\times b_{2}+a_{2}\times b_{1})$. This is the same as for reals $a_{1}\times a_{2}$ when the imaginary parts $b_{1}$ and $b_{2}$ are zero.
Equivalently, denoting ${\sqrt {-1}}$ as $i$, we have $z_{1}\times z_{2}=(a_{1}+b_{1}i)(a_{2}+b_{2}i)=(a_{1}\times a_{2})+(a_{1}\times b_{2}i)+(b_{1}\times a_{2}i)+(b_{1}\times b_{2}i^{2})=(a_{1}a_{2}-b_{1}b_{2})+(a_{1}b_{2}+b_{1}a_{2})i.$[23]
Alternatively, in trigonometric form, if $z_{1}=r_{1}(\cos \phi _{1}+i\sin \phi _{1}),z_{2}=r_{2}(\cos \phi _{2}+i\sin \phi _{2})$, then$ z_{1}z_{2}=r_{1}r_{2}(\cos(\phi _{1}+\phi _{2})+i\sin(\phi _{1}+\phi _{2})).$[23]
Further generalizations
See Multiplication in group theory, above, and multiplicative group, which for example includes matrix multiplication. A very general, and abstract, concept of multiplication is as the "multiplicatively denoted" (second) binary operation in a ring. An example of a ring that is not any of the above number systems is a polynomial ring (you can add and multiply polynomials, but polynomials are not numbers in any usual sense.)
Division
Often division, ${\frac {x}{y}}$, is the same as multiplication by an inverse, $x\left({\frac {1}{y}}\right)$. Multiplication for some types of "numbers" may have corresponding division, without inverses; in an integral domain x may have no inverse "${\frac {1}{x}}$" but ${\frac {x}{y}}$ may be defined. In a division ring there are inverses, but ${\frac {x}{y}}$ may be ambiguous in non-commutative rings since $x\left({\frac {1}{y}}\right)$ need not be the same as $\left({\frac {1}{y}}\right)x$.
See also
• Dimensional analysis
• Multiplication algorithm
• Karatsuba algorithm, for large numbers
• Toom–Cook multiplication, for very large numbers
• Schönhage–Strassen algorithm, for huge numbers
• Multiplication table
• Binary multiplier, how computers multiply
• Booth's multiplication algorithm
• Floating-point arithmetic
• Multiply–accumulate operation
• Fused multiply–add
• Wallace tree
• Multiplicative inverse, reciprocal
• Factorial
• Genaille–Lucas rulers
• Lunar arithmetic
• Napier's bones
• Peasant multiplication
• Product (mathematics), for generalizations
• Slide rule
Notes
1. Devlin, Keith (January 2011). "What Exactly is Multiplication?". Mathematical Association of America. Archived from the original on May 27, 2017. Retrieved May 14, 2017. With multiplication you have a multiplicand (written second) multiplied by a multiplier (written first)
2. Khan Academy (2015-08-14), Intro to multiplication | Multiplication and division | Arithmetic | Khan Academy, archived from the original on 2017-03-24, retrieved 2017-03-07
3. Khan Academy (2012-09-06), Why aren't we using the multiplication sign? | Introduction to algebra | Algebra I | Khan Academy, archived from the original on 2017-03-27, retrieved 2017-03-07
4. "Victory on Points". Nature. 218 (5137): 111. 1968. Bibcode:1968Natur.218S.111.. doi:10.1038/218111c0.
5. "The Lancet – Formatting guidelines for electronic submission of manuscripts" (PDF). Retrieved 2017-04-25.
6. Announcing the TI Programmable 88! (PDF). Texas Instruments. 1982. Archived (PDF) from the original on 2017-08-03. Retrieved 2017-08-03.
7. Fuller, William R. (1977). FORTRAN Programming: A Supplement for Calculus Courses. Universitext. Springer. p. 10. doi:10.1007/978-1-4612-9938-7. ISBN 978-0-387-90283-8.
8. Crewton Ramone. "Multiplicand and Multiplier". Crewton Ramone's House of Math. Archived from the original on 26 October 2015. Retrieved 10 November 2015..
9. Chester Litvin (2012). Advance Brain Stimulation by Psychoconduction. Trafford. pp. 2–3, 5–6. ISBN 978-1-4669-0152-0 – via Google Book Search.
10. "Multiplication". www.mathematische-basteleien.de. Retrieved 2022-03-15.
11. Pletser, Vladimir (2012-04-04). "Does the Ishango Bone Indicate Knowledge of the Base 12? An Interpretation of a Prehistoric Discovery, the First Mathematical Tool of Humankind". arXiv:1204.1019 [math.HO].
12. "Peasant Multiplication". www.cut-the-knot.org. Retrieved 2021-12-29.
13. Qiu, Jane (7 January 2014). "Ancient times table hidden in Chinese bamboo strips". Nature. doi:10.1038/nature.2014.14482. S2CID 130132289. Archived from the original on 22 January 2014. Retrieved 22 January 2014.
14. Fine, Henry B. (1907). The Number System of Algebra – Treated Theoretically and Historically (PDF) (2nd ed.). p. 90.
15. Bernhard, Adrienne. "How modern mathematics emerged from a lost Islamic library". www.bbc.com. Retrieved 2022-04-22.
16. Harvey, David; van der Hoeven, Joris; Lecerf, Grégoire (2016). "Even faster integer multiplication". Journal of Complexity. 36: 1–30. arXiv:1407.3360. doi:10.1016/j.jco.2016.03.001. ISSN 0885-064X. S2CID 205861906.
17. David Harvey, Joris Van Der Hoeven (2019). Integer multiplication in time O(n log n) Archived 2019-04-08 at the Wayback Machine
18. Hartnett, Kevin (11 April 2019). "Mathematicians Discover the Perfect Way to Multiply". Quanta Magazine. Retrieved 2020-01-25.
19. Klarreich, Erica. "Multiplication Hits the Speed Limit". cacm.acm.org. Archived from the original on 2020-10-31. Retrieved 2020-01-25.
20. Weisstein, Eric W. "Product". mathworld.wolfram.com. Retrieved 2020-08-16.
21. "Summation and Product Notation". math.illinoisstate.edu. Retrieved 2020-08-16.
22. Weisstein, Eric W. "Exponentiation". mathworld.wolfram.com. Retrieved 2021-12-29.
23. "Multiplication - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2021-12-29.
24. Biggs, Norman L. (2002). Discrete Mathematics. Oxford University Press. p. 25. ISBN 978-0-19-871369-2.
25. Weisstein, Eric W. "Multiplicative Inverse". mathworld.wolfram.com. Retrieved 2022-04-19.
26. Angell, David. "ORDERING COMPLEX NUMBERS... NOT*" (PDF). web.maths.unsw.edu.au. Retrieved 29 December 2021.{{cite web}}: CS1 maint: url-status (link)
27. "10.2: Building the Real Numbers". Mathematics LibreTexts. 2018-04-11. Retrieved 2023-06-23.
28. Burns, Gerald (1977). Introduction to group theory with applications. New York: Academic Press. ISBN 9780121457501.
References
• Boyer, Carl B. (revised by Merzbach, Uta C.) (1991). History of Mathematics. John Wiley and Sons, Inc. ISBN 978-0-471-54397-8.{{cite book}}: CS1 maint: multiple names: authors list (link)
External links
• Multiplication and Arithmetic Operations In Various Number Systems at cut-the-knot
• Modern Chinese Multiplication Techniques on an Abacus
Elementary arithmetic
+
Addition
(+)
−
Subtraction
(−)
×
Multiplication
(× or ·)
÷
Division
(÷ or ∕)
Hyperoperations
Primary
• Successor (0)
• Addition (1)
• Multiplication (2)
• Exponentiation (3)
• Tetration (4)
• Pentation (5)
Inverse for left argument
• Predecessor (0)
• Subtraction (1)
• Division (2)
• Root extraction (3)
• Super-root (4)
Inverse for right argument
• Predecessor (0)
• Subtraction (1)
• Division (2)
• Logarithm (3)
• Super-logarithm (4)
Related articles
• Ackermann function
• Conway chained arrow notation
• Grzegorczyk hierarchy
• Knuth's up-arrow notation
• Steinhaus–Moser notation
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| Wikipedia |
Xavier Fernique
Xavier Fernique (3 May 1934 – 15 March 2020) was a mathematician, noted mostly for his contributions to the theory of stochastic processes. Fernique's theorem, a result on the integrability of Gaussian measures, is named after him.
External links
• Xavier Fernique at the Mathematics Genealogy Project
• Photograph, courtesy of the Mathematical Research Institute of Oberwolfach
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| Wikipedia |
Xavier Tolsa
Xavier Tolsa (born 1966) is a Catalan mathematician, specializing in analysis.
Tolsa is a professor at the Autonomous University of Barcelona and at the Institució Catalana de Recerca i Estudis Avançats (ICREA), the Catalan Institute for Advanced Scientific Studies.
Tolsa does research on harmonic analysis (Calderón-Zygmund theory), complex analysis, geometric measure theory, and potential theory. Specifically, he is known for his research on analytic capacity and removable sets. He solved the problem of A. G. Vitushkin[1][2] about the semi-additivity of analytic capacity. This enabled him to solve an even older problem of Paul Painlevé on the geometric characterization of removable sets. Tolsa succeeded in solving the Painlevé problem by using the concept of so-called curvatures of measures introduced by Mark Melnikov in 1995. Tolsa's proof involves estimates of Cauchy transforms. He has also done research on the so-called David-Semmes problem involving Riesz transforms and rectifiability.[3]
In 2002 he was awarded the Salem Prize.[4] In 2006 in Madrid he was an Invited Speaker at the ICM with talk Analytic capacity, rectifiability, and the Cauchy integral. He received in 2004 the EMS Prize[5] and was an Invited Lecturer at the 2004 ECM with talk Painlevé's problem, analytic capacity and curvature of measures. In 2013 he received the Ferran Sunyer i Balaguer Prize for his monograph Analytic capacity, the Cauchy transform, and non-homogeneous Calderón-Zygmund theory (Birkhäuser Verlag, 2013}.[6] In 2019 he received the Rei Jaume I prize for his contributions to Mathematics.
Selected publications
• Tolsa, Xavier (2000). "Principal Values for the Cauchy Integral and Rectifiability". Proceedings of the American Mathematical Society. 128 (7): 2111–2119. doi:10.1090/S0002-9939-00-05264-3. JSTOR 119706.
• Tolsa, Xavier (2003). "Painlevé's problem and the semiadditivity of analytic capacity". Acta Mathematica. 190: 105–149. doi:10.1007/BF02393237.
• Nazarov, Fedor; Volberg, Alexander; Tolsa, Xavier (2014). "On the uniform rectifiability of AD-regular measures with bounded Riesz transform operator: the case of codimension 1". Acta Mathematica. 213 (2): 237–321. doi:10.1007/s11511-014-0120-7. ISSN 0001-5962.
References
1. Vitushkin, A. G. (1967). "The analytic capacity of sets in problems of approximation theory". Russian Mathematical Surveys. 22 (6): 139–200. Bibcode:1967RuMaS..22..139V. doi:10.1070/RM1967v022n06ABEH003763. S2CID 250869451.
2. Dudziak, James (2011-02-03). Vitushkin's Conjecture for Removable Sets. ISBN 9781441967091.
3. "Xavier Tolsa, ICREA Research Professor". Departament de Matemàtiques Universitat Autonoma de Barcelona.
4. "Premi Salem", Societat Catalana de Matemàtiques Notícies, July 2002, n°17, page 9
5. "Prizes Presented at the European Congress of Mathematicians" (PDF). Notices of the AMS. 51 (9): 1070–1071. October 2004.
6. Tolsa, Xavier (2013-12-16). Analytic Capacity, the Cauchy Transform, and Non-homogeneous Calderón–Zygmund Theory. ISBN 9783319005966.
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| Wikipedia |
x̅ and s chart
In statistical quality control, the ${\bar {x}}$ and s chart is a type of control chart used to monitor variables data when samples are collected at regular intervals from a business or industrial process.[1] This is connected to traditional statistical quality control (SQC) and statistical process control (SPC). However, Woodall[2] noted that "I believe that the use of control charts and other monitoring methods should be referred to as “statistical process monitoring,” not “statistical process control (SPC).”"
${\bar {x}}$ and s chart
Originally proposed byWalter A. Shewhart
Process observations
Rational subgroup size{{{subgroupsize}}}
Measurement typeAverage quality characteristuu7hic uper unit
Quality characteristic typeVariables data
Underlying distributionNormal distribution
Performance
Size of shift to detect≥ 1.5σ
Process variation chart
Center line${\bar {s}}={\frac {\sum _{i=1}^{m}{\sqrt {\frac {\sum _{j=1}^{n}\left(x_{ij}-{\bar {\bar {x}}}\right)^{2}}{n-1}}}}{m}}$
Upper control limit$B_{4}{\bar {S}}$
Lower control limit$B_{3}{\bar {S}}$
Plotted statistic${\bar {s}}_{i}={\sqrt {\frac {\sum _{j=1}^{n}\left(x_{ij}-{\bar {x}}_{i}\right)^{2}}{n-1}}}$
Process mean chart
Center line${\bar {x}}={\frac {\sum _{i=1}^{m}\sum _{j=1}^{n}x_{ij}}{mn}}$
Control limits${\bar {x}}\pm A_{3}{\bar {s}}$
Plotted statistic${\bar {x}}_{i}={\frac {\sum _{j=1}^{n}x_{ij}}{n}}$
Uses
The chart is advantageous in the following situations:[3]
1. The sample size is relatively large (say, n > 10—${\bar {x}}$ and R charts are typically used for smaller sample sizes)
2. The sample size is variable
3. Computers can be used to ease the burden of calculation
The "chart" actually consists of a pair of charts: One to monitor the process standard deviation and another to monitor the process mean, as is done with the ${\bar {x}}$ and R and individuals control charts. The ${\bar {x}}$ and s chart plots the mean value for the quality characteristic across all units in the sample, ${\bar {x}}_{i}$, plus the standard deviation of the quality characteristic across all units in the sample as follows:
$s={\sqrt {\frac {\sum _{i=1}^{n}{\left(x_{i}-{\bar {x}}\right)}^{2}}{n-1}}}$.
Assumptions
The normal distribution is the basis for the charts and requires the following assumptions:
• The quality characteristic to be monitored is adequately modeled by a normally-distributed random variable
• The parameters μ and σ for the random variable are the same for each unit and each unit is independent of its predecessors or successors
• The inspection procedure is same for each sample and is carried out consistently from sample to sample
Control limits
The control limits for this chart type are:[4]
• $B_{3}{\bar {s}}$ (lower) and $B_{4}{\bar {s}}$ (upper) for monitoring the process variability
• ${\bar {x}}\pm A_{3}{\bar {s}}$ for monitoring the process mean
where ${\bar {x}}$ and ${\bar {s}}={\frac {\sum _{i=1}^{m}s_{i}}{m}}$ are the estimates of the long-term process mean and range established during control-chart setup and A3, B3, and B4 are sample size-specific anti-biasing constants. The anti-biasing constants are typically found in the appendices of textbooks on statistical process control. NIST provides guidance on manually calculating these constants "6.3.2. What are Variables Control Charts?".
Validity
As with the ${\bar {x}}$ and R and individuals control charts, the ${\bar {x}}$ chart is only valid if the within-sample variability is constant.[5] Thus, the s chart is examined before the ${\bar {x}}$ chart; if the s chart indicates the sample variability is in statistical control, then the ${\bar {x}}$ chart is examined to determine if the sample mean is also in statistical control. If on the other hand, the sample variability is not in statistical control, then the entire process is judged to be not in statistical control regardless of what the ${\bar {x}}$ chart indicates.
Unequal samples
When samples collected from the process are of unequal sizes (arising from a mistake in collecting them, for example), there are two approaches:
TechniqueDescription
Use variable-width control limits[6] Each observation plots against its own control limits as determined by the sample size-specific values, ni, of A3, B3, and B4
Use control limits based on an average sample size[7] Control limits are fixed at the modal (or most common) sample size-specific value of A3, B3, and B4
Limitations and improvements
Effect of estimation of parameters plays a major role. Also a change in variance affects the performance of ${\bar {X}}$chart while a shift in mean affects the performance of the S chart.
Therefore, several authors recommend using a single chart that can simultaneously monitor ${\bar {X}}$and S.[8] McCracken, Chackrabori and Mukherjee [9] developed one of the most modern and efficient approach for jointly monitoring the Gaussian process parameters, using a set of reference sample in absence of any knowledge of true process parameters.
See also
• ${\bar {x}}$ and R chart
• Shewhart individuals control chart
• Simultaneous monitoring of mean and variance of Gaussian Processes with estimated parameters (when standards are unknown)[9]
References
1. "Shewhart X-bar and R and S Control Charts". NIST/Sematech Engineering Statistics Handbook. National Institute of Standards and Technology. Retrieved 2009-01-13.
2. Woodall, William H. (2016-07-19). "Bridging the Gap between Theory and Practice in Basic Statistical Process Monitoring". Quality Engineering: 00. doi:10.1080/08982112.2016.1210449. ISSN 0898-2112. S2CID 113516285.
3. Montgomery, Douglas (2005). Introduction to Statistical Quality Control. Hoboken, New Jersey: John Wiley & Sons, Inc. p. 222. ISBN 978-0-471-65631-9. OCLC 56729567.
4. Montgomery, Douglas (2005). Introduction to Statistical Quality Control. Hoboken, New Jersey: John Wiley & Sons, Inc. p. 225. ISBN 978-0-471-65631-9. OCLC 56729567.
5. Montgomery, Douglas (2005). Introduction to Statistical Quality Control. Hoboken, New Jersey: John Wiley & Sons, Inc. p. 214. ISBN 978-0-471-65631-9. OCLC 56729567.
6. Montgomery, Douglas (2005). Introduction to Statistical Quality Control. Hoboken, New Jersey: John Wiley & Sons, Inc. p. 227. ISBN 978-0-471-65631-9. OCLC 56729567.
7. Montgomery, Douglas (2005). Introduction to Statistical Quality Control. Hoboken, New Jersey: John Wiley & Sons, Inc. p. 229. ISBN 978-0-471-65631-9. OCLC 56729567.
8. Chen, Gemai; Cheng, Smiley W. (1998). "Max Chart: Combining X-Bar Chart and S Chart". Statistica Sinica. 8 (1): 263–271. ISSN 1017-0405. JSTOR 24306354.
9. McCracken, A. K.; Chakraborti, S.; Mukherjee, A. (October 2013). "Control Charts for Simultaneous Monitoring of Unknown Mean and Variance of Normally Distributed Processes". Journal of Quality Technology. 45 (4): 360–376. doi:10.1080/00224065.2013.11917944. ISSN 0022-4065. S2CID 117307669.
| Wikipedia |
Cross-entropy method
The cross-entropy (CE) method is a Monte Carlo method for importance sampling and optimization. It is applicable to both combinatorial and continuous problems, with either a static or noisy objective.
The method approximates the optimal importance sampling estimator by repeating two phases:[1]
1. Draw a sample from a probability distribution.
2. Minimize the cross-entropy between this distribution and a target distribution to produce a better sample in the next iteration.
Reuven Rubinstein developed the method in the context of rare event simulation, where tiny probabilities must be estimated, for example in network reliability analysis, queueing models, or performance analysis of telecommunication systems. The method has also been applied to the traveling salesman, quadratic assignment, DNA sequence alignment, max-cut and buffer allocation problems.
Estimation via importance sampling
Consider the general problem of estimating the quantity
$\ell =\mathbb {E} _{\mathbf {u} }[H(\mathbf {X} )]=\int H(\mathbf {x} )\,f(\mathbf {x} ;\mathbf {u} )\,{\textrm {d}}\mathbf {x} $ ;\mathbf {u} )\,{\textrm {d}}\mathbf {x} } ,
where $H$ is some performance function and $f(\mathbf {x} ;\mathbf {u} )$ ;\mathbf {u} )} is a member of some parametric family of distributions. Using importance sampling this quantity can be estimated as
${\hat {\ell }}={\frac {1}{N}}\sum _{i=1}^{N}H(\mathbf {X} _{i}){\frac {f(\mathbf {X} _{i};\mathbf {u} )}{g(\mathbf {X} _{i})}}$,
where $\mathbf {X} _{1},\dots ,\mathbf {X} _{N}$ is a random sample from $g\,$. For positive $H$, the theoretically optimal importance sampling density (PDF) is given by
$g^{*}(\mathbf {x} )=H(\mathbf {x} )f(\mathbf {x} ;\mathbf {u} )/\ell $ ;\mathbf {u} )/\ell } .
This, however, depends on the unknown $\ell $. The CE method aims to approximate the optimal PDF by adaptively selecting members of the parametric family that are closest (in the Kullback–Leibler sense) to the optimal PDF $g^{*}$.
Generic CE algorithm
1. Choose initial parameter vector $\mathbf {v} ^{(0)}$; set t = 1.
2. Generate a random sample $\mathbf {X} _{1},\dots ,\mathbf {X} _{N}$ from $f(\cdot ;\mathbf {v} ^{(t-1)})$ ;\mathbf {v} ^{(t-1)})}
3. Solve for $\mathbf {v} ^{(t)}$, where
$\mathbf {v} ^{(t)}=\mathop {\textrm {argmax}} _{\mathbf {v} }{\frac {1}{N}}\sum _{i=1}^{N}H(\mathbf {X} _{i}){\frac {f(\mathbf {X} _{i};\mathbf {u} )}{f(\mathbf {X} _{i};\mathbf {v} ^{(t-1)})}}\log f(\mathbf {X} _{i};\mathbf {v} )$
4. If convergence is reached then stop; otherwise, increase t by 1 and reiterate from step 2.
In several cases, the solution to step 3 can be found analytically. Situations in which this occurs are
• When $f\,$ belongs to the natural exponential family
• When $f\,$ is discrete with finite support
• When $H(\mathbf {X} )=\mathrm {I} _{\{\mathbf {x} \in A\}}$ and $f(\mathbf {X} _{i};\mathbf {u} )=f(\mathbf {X} _{i};\mathbf {v} ^{(t-1)})$, then $\mathbf {v} ^{(t)}$ corresponds to the maximum likelihood estimator based on those $\mathbf {X} _{k}\in A$.
Continuous optimization—example
The same CE algorithm can be used for optimization, rather than estimation. Suppose the problem is to maximize some function $S$, for example, $S(x)={\textrm {e}}^{-(x-2)^{2}}+0.8\,{\textrm {e}}^{-(x+2)^{2}}$. To apply CE, one considers first the associated stochastic problem of estimating $\mathbb {P} _{\boldsymbol {\theta }}(S(X)\geq \gamma )$ for a given level $\gamma \,$, and parametric family $\left\{f(\cdot ;{\boldsymbol {\theta }})\right\}$ ;{\boldsymbol {\theta }})\right\}} , for example the 1-dimensional Gaussian distribution, parameterized by its mean $\mu _{t}\,$ and variance $\sigma _{t}^{2}$ (so ${\boldsymbol {\theta }}=(\mu ,\sigma ^{2})$ here). Hence, for a given $\gamma \,$, the goal is to find ${\boldsymbol {\theta }}$ so that $D_{\mathrm {KL} }({\textrm {I}}_{\{S(x)\geq \gamma \}}\|f_{\boldsymbol {\theta }})$ is minimized. This is done by solving the sample version (stochastic counterpart) of the KL divergence minimization problem, as in step 3 above. It turns out that parameters that minimize the stochastic counterpart for this choice of target distribution and parametric family are the sample mean and sample variance corresponding to the elite samples, which are those samples that have objective function value $\geq \gamma $. The worst of the elite samples is then used as the level parameter for the next iteration. This yields the following randomized algorithm that happens to coincide with the so-called Estimation of Multivariate Normal Algorithm (EMNA), an estimation of distribution algorithm.
Pseudocode
// Initialize parameters
μ := −6
σ2 := 100
t := 0
maxits := 100
N := 100
Ne := 10
// While maxits not exceeded and not converged
while t < maxits and σ2 > ε do
// Obtain N samples from current sampling distribution
X := SampleGaussian(μ, σ2, N)
// Evaluate objective function at sampled points
S := exp(−(X − 2) ^ 2) + 0.8 exp(−(X + 2) ^ 2)
// Sort X by objective function values in descending order
X := sort(X, S)
// Update parameters of sampling distribution
μ := mean(X(1:Ne))
σ2 := var(X(1:Ne))
t := t + 1
// Return mean of final sampling distribution as solution
return μ
Related methods
• Simulated annealing
• Genetic algorithms
• Harmony search
• Estimation of distribution algorithm
• Tabu search
• Natural Evolution Strategy
See also
• Cross entropy
• Kullback–Leibler divergence
• Randomized algorithm
• Importance sampling
Journal papers
• De Boer, P-T., Kroese, D.P, Mannor, S. and Rubinstein, R.Y. (2005). A Tutorial on the Cross-Entropy Method. Annals of Operations Research, 134 (1), 19–67.
• Rubinstein, R.Y. (1997). Optimization of Computer simulation Models with Rare Events, European Journal of Operational Research, 99, 89–112.
Software implementations
• CEoptim R package
• Novacta.Analytics .NET library
References
1. Rubinstein, R.Y. and Kroese, D.P. (2004), The Cross-Entropy Method: A Unified Approach to Combinatorial Optimization, Monte-Carlo Simulation, and Machine Learning, Springer-Verlag, New York ISBN 978-0-387-21240-1.
| Wikipedia |
Minkowski Portal Refinement
The Minkowski Portal Refinement collision detection algorithm is a technique for determining whether two convex shapes overlap.
The algorithm was created by Gary Snethen in 2006 and was first published in Game Programming Gems 7. The algorithm was used in Tomb Raider: Underworld and other games created by Crystal Dynamics and its sister studios within Eidos Interactive.
MPR, like its cousin GJK, relies on shapes that are defined using support mappings. This allows the algorithm to support a limitless variety of shapes that are problematic for other algorithms. Support mappings require only a single mathematical function to represent a point, line segment, disc, cylinder, cone, ellipsoid, football, bullet, frustum or most any other common convex shape. Once a set of basic primitives have been created, they can easily be combined with one another using operations such as sweep, shrink-wrap and affine transformation.
Unlike GJK, MPR does not provide the shortest distance between separated shapes. However, according to its author, MPR is simpler, more numerically robust and handles translational sweeping with very little modification. This makes it well-suited for games and other real-time applications.
External links
• Snethen, Gary (2008) "Complex Collision Made Simple", Game Programming Gems 7, 165–178
• Snethen, Gary (2008) "XenoCollide Homepage"
• Open source implementation: libccd
| Wikipedia |
Xhezair Teliti
Xhezair Teliti (born 17 February 1948, in Kavajë) is a professor of mathematics and has served as chief of the Department of Mathematics at Tirana University since 2008. He was Albania's Minister of Education from 1993-1996.[1][2]
Xhezair Teliti
Minister of Education
In office
6 April 1993 – 11 July 1996
Preceded byYlli Vejsiu
Succeeded byEdmond Lulja
Personal details
Born (1948-02-17) February 17, 1948
Kavajë, Albania
Career
His field of study is Functional Analysis and Theory of Mass and Integration. Teliti is author of many text-books:
1. "Teoria e Funksioneve te Variablit Real, I, II(Theory of Functions of Real Variable)", 1980, Tirana;
2. "Përgjithësimi i Konceptit të Integrali(Generalization of the Concept of Integral)", 1981;
3. "Teoria Konstruktive e Funksioneve(The Constructive Theory of Functions)", P. Pilika, Xh. Teliti – 1984;
4. "Përmbledhje Problemash në Analizën Funksional(Summary of Problems for Functional Analysis)", 1989, Tirana;
5. "Probleme dhe Ushtrime të Analizës Matematike (Problems and Exercises of Mathematical Analysis", 1997, Tirana;
6. "Teoria e Masës dhe e Integrimit(Theory of Mass and Integration)", 1997, Tirana;
7. "Problema në Teorinë e Masës e të Integrimit(Problems for the Theory of Mass and Integration)", 1998, Tirana;
8. "Topologjia e Përgjithshme dhe Analiza Funksionale(General Topology and Funbctional Analysis)", 2002, Tirana;
9. "Elemente Strukturorë dhe Topologjikë në Hapësirat R dhe R (n) (Topological and Structural Elements in R and R (n) spaces", 2008, Tirana.
Prof. Teliti has also written many articles in the Bulletin of Natural Sciences at the University.[3]
References
1. "Historiku i Departamentit të Matematikës".
2. "www.unesco.org/bpi/eng/unescopress/96-40e.htm". Archived from the original on 2014-04-24. Retrieved 2014-04-24.
3. Kllogjeri, Pellumb. "The Albanian Mathematicians by the Flowside of the Mathematicians of the World".
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Xi
Xi may refer to:
Look up xi or Xi in Wiktionary, the free dictionary.
Arts and entertainment
• Xi (alternate reality game), a console-based game
• Xi, Japanese name for the video game Devil Dice
Language
• Xi (letter), a Greek letter
• Xi, a Latin digraph used in British English to write the sound /kʃ/
People
• Xi (surname), any of several Chinese surnames
• Xi Jinping, current General Secretary of the Chinese Communist Party since 2012
Places
• Xi (state), an ancient Chinese state during the Shang and Zhou Dynasties
• Xi County, Henan, China
• Xi County, Shanxi, China
• Xi River, western tributary of the Pearl River in southern China
Other uses
• Xi (business), a Chinese form of business organization
• Xi baryon, a range of baryons with one up or down quark and two heavier quarks
• Xi, a brand name for the 4G LTE mobile telecommunications service operated by NTT DoCoMo in Japan
• Xi (apartment), a brand name for some apartments constructed by GS Construction in Korea.
See also
• XI (disambiguation)
• 11 (disambiguation)
• Kumo Xi, an ancient Mongolic people
• Shuang Xi (双喜, written 囍), a Chinese calligraphic design
• Hsi (disambiguation) — "Xi" and "Hsi" are different transliterations of the same sound in Mandarin Chinese
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Xiahou Yang Suanjing
Xiahou Yang Suanjing (Xiahou Yang's Mathematical Manual) is a mathematical treatise attributed to the fifth century CE Chinese mathematician Xiahou Yang (also known as Hsiahou Yang). However, some historians are of the opinion that Xiahou Yang Suanjing was not written by Xiahou Yang.[1] It is one of the books in The Ten Computational Canons, a collection of mathematical texts assembled by Li Chunfeng and used as the official mathematical for the imperial examinations.
Though little is known about the period of the author, there is some evidence which more or less conclusively establishes the date of the work. These suggest 468 CE as the latest possible date for the work to be written and 425 CE as the earliest date.[1]
Contents
The treatise is divided into three parts and these are spoken of as the higher, the middle and the lower sections.[2] The first chapter contains 19 problems, the second chapter contains 29 problems and the last chapter contains 44 problems. As in all the older Chinese books, no technical rules are given, and the problems are simply followed by the answers, occasionally with brief explanations.[2]
Section 1
In the first section the five operations of addition, subtraction, multiplication, division, and square and cube roots are given. The work on division is subdivided into (1) "ordinary division"; (2) "division by ten, hundred, and so on," especially intended for work in mensuration; (3) "division by simplification" (yo ch'ut). The last problem in the section is as follows:
"There are 1843 k'o, 8 t'ow, 3 ho of coarse rice. A contract requires that this be exchanged for refined rice at the rate of 1 k'o, 4 t'ow for 3 k'o. How much refined rice must be given?" The answer is 860 k'o, 534 ho. The solution is given as follows: "Multiply the given number by 1 k'o, 4 t'ow and divide by 3 k'o and you will obtain the result." (1 k'o = 10 t'ow = 100 ho)
Fractions are also mentioned, special names being given to the four most common ones, as follows:
1/2 is called chung p'an (even part)
1/3 is called shaw p'an (small part)
2/3 is called thai p'an (large part)
1/4 is called joh p'an (weak part)
Section 2
In the second section there are twenty-eight applied problems relating to taxes, commissions, and such questions as concern the division by army officers of loot and food (silk, rice, wine, soy sauce, vinegar, and the like) among their soldiers.[2]
Section 3
The third section contains forty-two problems. The translations of some of these problems are given below.[2]
1. "Now for 1 pound of gold one gets 1200 pieces of silk. How many can you get for 1 ounce?" Answer: For 1 ounce you get exactly 75 pieces. Solution: Take the given number of pieces, have it divided by 16 ounces, and you will obtain the answer. (Chinese pound was divided into 16 ounces.)
2. "Now you have 192 ounces of silk. How many choo have you?" Answer: Four thousand six hundred eight. (It appears that in obtaining the given solution to the problem, pound was divided into 24 choos.)
3. "Now 2000 packages of cash must be carried to the town at the rate of 10 cash per bundle. How much will be given to the mandarin and how much to the carrier?" Answer: 1980 packages and 198 2/101 cash to the mandarin; 19 packages and 801 98/101 to the carrier. Solution: Take the total number as the dividend, and 1 package plus 10 cash as the divisor.
4. "Out of 3485 ounces of silk how many pieces of satin can be made, 5 ounces being required for each piece?" Answer: 697. Solution: Multiply the number of ounces by 2 and go back by one row. Dividing by 5 will also give the answer.
5. "Now they build a wall, high 3 rods, broad 5 feet at the upper part and 15 feet at the lower part; the length 100 rods. For a 2-foot square a man works 1 day. How many days are required?" Answer: 75,000. Solution: Take-half the sum of the upper and lower breadths, have it multiplied by the height and length; the product will be the dividend. As the divisor you will use the square of the given 2 feet.
References
1. J J O'Connor and E F Robertson. "Xiahou Yang". MacTutor History of Mathematics Archive. University of St Andrews, Scotland. Retrieved 5 December 2016.
2. Pere Louis Vanhee (May 1924). "The Arithmetic Classic of Hsia-Hou Yang". The American Mathematical Monthly. 31 (5): 235–237. doi:10.1080/00029890.1924.11986334. JSTOR 2299246.
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Wu-Chung Hsiang
Wu-Chung Hsiang (Chinese: 項武忠; pinyin: Xiàng Wǔzhōng; Wade–Giles: Hsiang Wu-chung; born 12 June 1935 in Zhejiang) is a Chinese-American mathematician, specializing in topology. Hsiang served as chairman of the Department of Mathematics at Princeton University from 1982 to 1985 and was one of the most influential topologists of the second half of the 20th century.[1]
Biography
Hsiang hails from Wenzhou, Zhejiang.[2] He received in 1957 his bachelor's degree from the National Taiwan University and in 1963 his Ph.D. under Norman Steenrod from Princeton University with thesis Obstructions to sectioning fibre bundles.[3] At Yale University he became in 1962 a lecturer, in 1963 an assistant professor, and in 1968 a full professor. At Princeton University he was a full professor from 1972 until retiring in 2006 as professor emeritus and was the department chair from 1982 to 1985.[4] He was a visiting scholar at the Institute for Advanced Study for the academic years 1965–1966, 1971–1972, and 1979–1980. He was a visiting professor at the University of Warwick in 1966, the University of Amsterdam in 1969, the University of Bonn in 1971, the University of California, Berkeley in 1976, and the Mathematical Sciences Research Institute and Stanford University in 1980.
Hsiang has made important contributions to algebraic and differential topology. Works by Hsiang, Julius Shaneson, C. T. C. Wall, Robion Kirby, Laurent Siebenmann and Andrew Casson led in the 1960s to the proof of the annulus theorem (previously known as the annulus conjecture).[5] The annulus theorem is important in the theory of triangulation of manifolds.
With F. Thomas Farrell he worked on a program to prove the Novikov conjecture and the Borel conjecture with methods from geometric topology[6] and gave proofs for special cases. For example, they gave a proof of the integral Novikov conjecture for compact Riemannian manifolds with non-positive sectional curvature.[7] Hsiang also made contributions to the topological study of simply-connected 4-manifolds.[8]
From 1967 to 1969 he was a Sloan Fellow and for the academic year 1975–1976 a Guggenheim Fellow. In 1980 he was elected a member of Academia Sinica. He was an Invited Speaker at the International Congress of Mathematicians in 1970 in Nice, with a talk on Differentiable actions of compact connected Lie groups on $R^{n}$[9] and a Plenary Speaker in 1983 in Warsaw, with a talk on Geometric applications of algebraic K-theory.[10] In 2005 there was a conference at Stanford University in honor of his 70th birthday.[11]
His doctoral students include Ruth Charney, F. Thomas Farrell, Thomas Goodwillie, Michael W. Davis, and Lowell E. Jones.[3]
References
1. "Wu-Chung Hsiang | Dean of the Faculty".
2. "釣運文獻館-項武忠院士訪談". archives.lib.nthu.edu.tw. Archived from the original on 2015-04-02.
3. Wu-Chung Hsiang at the Mathematics Genealogy Project
4. "Eight faculty members transfer to emeritus status". Princeton Weekly Bulletin. Vol. 95, no. 29. Princeton University. 19 June 2006.
5. Hsiang, Wu-Chung and Shaneson, Julius L. (1969). Fake tori, the annulus conjecture, and the conjectures of Kirby. Proceedings of the National Academy of Sciences of the United States of America, 62 (3), 687–691.
6. Hsiang, Wu-Chung: Borel's conjecture, Novikov's conjecture and K-theoretic analogues, in: Algebra, Analysis and Topology, World Scientific 1989
7. Farrell, F. Thomas; Hsiang, Wu-Chung (1981). "On Novikov's conjecture for non-positively curved manifolds, I". Annals of Mathematics. 113 (1): 199–209. doi:10.2307/1971138. JSTOR 1971138.
8. Curtis, Cynthia L.; Freedman, Michael H.; Hsiang, Wu-Chung; Stong, Richard (1996). "A decomposition theorem for h-cobordant smooth simply-connected compact 4-manifolds". Inventiones Mathematicae. 123 (2): 343–348. doi:10.1007/s002220050031. MR 1374205. S2CID 189819783.
9. Hsiang, Wu-Chung. "Differentiable actions of compact connected Lie groups on $R^{n}$." Actes Congr. Int. Mathématiciens (1970): Tome 2, 73–77.
10. Hsiang, Wu-Chung. "Geometric applications of algebraic K-theory." In Proceedings of the International Congress of Mathematicians, vol. 1, p. 2. 1983.
11. Algebraic & Differential Topology: A Conference in Honor of Wu-chung's 70th Birthday, Stanford U., August 6th and 7th, 2005
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Xiangyu Zhou
Xiangyu Zhou (Zhou Xiangyu, 周向宇, born March 1965 in Chenzhou, Hunan Province) is a Chinese mathematician, specializing in several complex variables and complex geometry. He is known for his 1998 proof of the "extended future tube conjecture", which was an unsolved problem for almost forty years.[1]
Education and career
Zhou matriculated in 1981 at Xiangtan University, where he graduated in 1985 with a bachelor's degree in mathematics. From 1985 to 1990 he studied at the Institute of Mathematics of the Chinese Academy of Sciences, where he received an M.Sc. in 1988 and a Ph.D. in 1990. At Beijing's Institute of Mathematics of the Chinese Academy of Sciences he was from 1990 to 1992 an assistant professor and from 1992 to 1998 an associate professor and is since 1998 a full professor. During 1990 to 1992 he was on academic leave as a senior scientific member of the Steklov Mathematical Institute, where he was awarded in 1998 a Russian Doctor of Sciences degree.[2]
In the Chinese Academy of Sciences, Zhou was from 2003 to 2012 the director of the Institute of Mathematics and is since 2008 the director of the Hua Loo-Keng Key Laboratory of Mathematics. From 2008 to 2011 he was the vice-chair of the Chinese Mathematical Society.[2]
Zhou was awarded in 1999 the First Class Prize of the Natural Science Award of the Chinese Academy of Sciences, in 2001 the S.S. Chern Mathematics Prize of the Chinese Mathematical Society, and in 2004 the National Natural Science Award of China of the State Council of China.[2] In 2002 he was an invited speaker at the International Congress of Mathematicians in Beijing.[3] In 2013 he was elected an Academician by the Chinese Academy of Sciences[2] and was a keynote speaker of the Abel Symposium.[4]
Selected publications
• Zhou, Xiangyu (2003). "Some results related to group actions in several complex variables". arXiv:math/0304337.
• Guan, Qiʼan; Zhou, Xiangyu; Zhu, Langfeng (2011). "On the Ohsawa–Takegoshi extension theorem and the twisted Bochner–Kodaira identity". Comptes Rendus Mathematique. 349 (13–14): 797–800. doi:10.1016/j.crma.2011.06.001.
• Zhu, Langfeng; Guan, Qiʼan; Zhou, Xiangyu (2012). "On the Ohsawa–Takegoshi $L^{2}$ extension theorem and the Bochner–Kodaira identity with non-smooth twist factor". Journal de Mathématiques Pures et Appliquées. 97 (6): 579–601. doi:10.1016/j.matpur.2011.09.010.
• Guan, Qiʼan; Zhou, Xiangyu (2012). "Optimal constant problem in the extension theorem". Comptes Rendus Mathematique. 350 (15–16): 753–756. doi:10.1016/j.crma.2012.08.007.
• Guan, Qi'an; Zhou, Xiangyu (2013). "Strong openness conjecture for plurisubharmonic functions". arXiv:1311.3781 [math.CV].
• Guan, Qi'an; Zhou, Xiangyu (2015). "A proof of Demailly's strong openness conjecture". Annals of Mathematics. 182 (2): 605–616. doi:10.4007/annals.2015.182.2.5. JSTOR 24523344.
• Guan, Qi'an; Zhou, Xiangyu (2015). "A solution of an $L^{2}$ extension problem with an optimal estimate and applications". Annals of Mathematics. 181 (3): 1139–1208. doi:10.4007/annals.2015.181.3.6. JSTOR 24523356. S2CID 56205818.
• Guan, Qi'an; Zhou, Xiangyu (2015). "Effectiveness of Demailly's strong openness conjecture and related problems". Inventiones Mathematicae. 202 (2): 635–676. arXiv:1403.7247. Bibcode:2015InMat.202..635G. doi:10.1007/s00222-014-0575-3. S2CID 119317767.
• Guan, Qi'An; Zhou, Xiangyu (2015). "Optimal constant in an $L^{2}$ extension problem and a proof of a conjecture of Ohsawa". Science China Mathematics. 58 (1): 35–59. arXiv:1412.0054. Bibcode:2015ScChA..58...35G. doi:10.1007/s11425-014-4946-4. S2CID 119139395.
• Guan, Qi'an; Zhou, Xiangyu (2015). "Characterization of multiplier ideal sheaves with weights of Lelong number one". Advances in Mathematics. 285: 1688–1705. doi:10.1016/j.aim.2015.08.002. S2CID 118853973.
• Guan, Qi'An; Zhou, Xiangyu (2017). "Strong openness of multiplier ideal sheaves and optimal $L^{2}$ extension". Science China Mathematics. 60 (6): 967–976. arXiv:1703.08387. doi:10.1007/s11425-017-9055-5. S2CID 119150408.
• Zhou, Xiangyu; Zhu, Langfeng (2018). "An optimal $L^{2}$ extension theorem on weakly pseudoconvex Kähler manifolds". Journal of Differential Geometry. 110. doi:10.4310/jdg/1536285628. S2CID 125514454. 2018
• Deng, Fusheng; Wang, Zhiwei; Zhang, Liyou; Zhou, Xiangyu (2018). "New characterizations of plurisubharmonic functions and positivity of direct image sheaves". arXiv:1809.10371 [math.CV].
• Zhou, Xiangyu; Zhu, Langfeng (2019). "Extension of cohomology classes and holomorphic sections defined on subvarieties". arXiv:1909.08822 [math.CV].
• Zhou, Xiangyu; Zhu, Langfeng (2020). "Siu's lemma, optimal $L^{2}$ extension and applications to twisted pluricanonical sheaves". Mathematische Annalen. 377 (1–2): 675–722. doi:10.1007/s00208-018-1783-8. S2CID 126209780.
• Guan, Qi'an; Zhou, Xiangyu (2020). "Restriction formula and subadditivity property related to multiplier ideal sheaves". Journal für die Reine und Angewandte Mathematik. 2020 (769): 1–33. doi:10.1515/crelle-2019-0043. S2CID 213427163.
References
1. Zhou, Xiang-Yu (1998). "The exterded future tube is a domain of holomorphy" (PDF). Mathematical Research Letters. 5 (2): 185–190. doi:10.4310/MRL.1998.v5.n2.a4.
2. "周向宇 (with CV & list of "Representative Research Works")". 中国科学院 (Chinese Academy of Sciences — math.ac.cn).
3. Zhou, Xiangyu (2003). "Some results related to group actions in several complex variables". arXiv:math/0304337.
4. "Abel Symposium 2013" (PDF).
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Mei-Chi Shaw
Mei-Chi Shaw (Chinese: 蕭美琪; pinyin: Xiāo Měiqí; born 1955) is a professor of mathematics at the University of Notre Dame.[1] Her research concerns partial differential equations.
Mei-Chi Shaw
蕭美琪
Born1955
Taipei, Taiwan
Alma materPrinceton University
Awards
• Fellow of the American Mathematical Society
Stefan Bergman Prize (2019)
Scientific career
FieldsComplex analysis
Partial differential equations
Complex geometry
InstitutionsNotre Dame University
ThesisHodge Theory on Domains with Cone-Like or Horn-Like Singularities (1981)
Doctoral advisorJoseph Kohn
Life and career
Shaw was born in Taipei, Taiwan in 1955.[2] She graduated with an undergraduate degree in mathematics from National Taiwan University in 1977. Shaw received her PhD from Princeton University four years later in 1981, working with Joseph Kohn.[3] She then took a postdoctoral position at Purdue University[2] During this time, she married her husband, Hsueh-Chia Chang. In 1983, Shaw took a tenure-track position at Texas A&M University, moving to University of Houston in 1986 and finally relocating to the University of Notre Dame in 1987, first as an associate professor and then as full professor.
Awards and honors
In 2012, Shaw became a fellow of the American Mathematical Society.[4] For 2019 she received the Stefan Bergman Prize.[5]
Selected publications
• Chen, So-Chin; Shaw, Mei-Chi. Partial differential equations in several complex variables. AMS/IP Studies in Advanced Mathematics, 19. American Mathematical Society, Providence, RI; International Press, Boston, MA, 2001. xii+380 pp. ISBN 0-8218-1062-6
• Shaw, Mei-Chi. L2-estimates and existence theorems for the tangential Cauchy-Riemann complex. Invent. Math. 82 (1985), no. 1, 133–150.
• Boas, Harold P.; Shaw, Mei-Chi Sobolev estimates for the Lewy operator on weakly pseudoconvex boundaries. Math. Ann. 274 (1986), no. 2, 221–231.
References
1. "Mei-Chi Shaw". Retrieved Feb 9, 2015.
2. Shaw, Mei-Chi. "A Woman Mathematician's Journey," ICCM Not. 2 (2014), no. 1, 59-74.
3. Mei-Chi Shaw at the Mathematics Genealogy Project
4. List of Fellows of the American Mathematical Society
5. Stefan Bergman Prize 2019
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Yum-Tong Siu
Yum-Tong Siu (Chinese: 蕭蔭堂; born May 6, 1943, in Guangzhou, China) is the William Elwood Byerly Professor of Mathematics at Harvard University.
Yum-Tong Siu
Yum-Tong Siu in 2000
Born (1943-05-06) May 6, 1943
Guangzhou, China
Alma materUniversity of Hong Kong
OccupationMathematician
Scientific career
FieldsComplex analysis
Differential geometry
Algebraic geometry
InstitutionsHarvard University
Doctoral studentsJun-Muk Hwang, Ngaiming Mok
Chinese name
Traditional Chinese蕭蔭堂
Simplified Chinese萧荫堂
Hanyu PinyinXiāo Yìntáng
Yale RomanizationSīu Yamtòhng
JyutpingSiu1 Jam3-tong4
Siu is a prominent figure in the study of functions of several complex variables. His research interests involve the intersection of complex variables, differential geometry, and algebraic geometry. He has resolved various conjectures by applying estimates of the complex Neumann problem and the theory of multiplier ideal sheaves to algebraic geometry.[1][2]
Education and career
Siu obtained his B.A. in mathematics from the University of Hong Kong in 1963, his M.A. from the University of Minnesota, and his Ph.D. from Princeton University in 1966.[3] Siu completed his doctoral dissertation, titled "Coherent Noether-Lasker decomposition of subsheaves and sheaf cohomology", under the supervision of Robert C. Gunning.[4] Before joining Harvard, he taught at Purdue University, the University of Notre Dame, Yale, and Stanford. In 1982 he joined Harvard as a Professor, of Mathematics. He previously served as the Chairman of the Harvard Math Department.[5]
In 2006, Siu published a proof of the finite generation of the pluricanonical ring.[6]
Awards, honors and professional memberships
In 1993, Siu received the Stefan Bergman Prize[7] of the American Mathematical Society. He has holds honorary doctorates from the University of Hong Kong, University of Bochum, Germany, and University of Macau. He is a Corresponding Member of the Goettingen Academy of Sciences (elected 1993); a Foreign member of the Chinese Academy of Sciences (elected 2004); and a member of the American Academy of Arts & Sciences (elected 1998),[8] the National Academy of Sciences (elected 2002),[9] and Academia Sinica, Taiwan (elected 2004).[10][11] He has been an invited speaker at the International Congress of Mathematicians in Helsinki (1978), Warsaw (1983) and Beijing (2002).
Currently, Siu is a member of the Scientific Advisory Board of the Clay Mathematics Institute (since 2003); the Advisory Committee for the Shaw Prize In Mathematical Sciences (since 2010);[12] the Advisory Committee for the Millennium Prize Problems under the sponsorship of the Clay Mathematics Institute; the Scientific Advisory Board for the Institute for Mathematics Sciences, National University of Singapore[13] (since 2009) and the Institute of Advanced Studies, Nanyang Technological University, Singapore (since 2006).[14]
See also
• Göttingen Academy of Sciences
• Siu's semicontinuity theorem
• List of graduates of University of Hong Kong
• Math 55
Notes
1. "MSRI Publications #37: Several Complex Variables". msri.org.
2. "Yum-Tong Siu: Hongkong-Princeton-Harvard, A Path of Several Complex Variables" (PDF). Asia-Pacific Mathematics Newsletter.
3. "Biographical Note of Professor Yum-Tong Siu". Shaw Prize.
4. Siu, Yum-Tong (1966). Coherent Noether-Lasker decomposition of subsheaves and sheaf cohomology.
5. "Harvard Mathematics Department : Home page". harvard.edu.
6. Siu, Yum-Tong (2006). "[math/0610740] A General Non-Vanishing Theorem and an Analytic Proof of the Finite Generation of the Canonical Ring". arXiv:math.AG/0610740.
7. "Stefan Bergman Prize". American Mathematical Society.
8. "American Academy of Arts & Sciences" (PDF). amacad.org.
9. "National Academy of Sciences". nasonline.org.
10. https://db1n.sinica.edu.tw/textdb/academicians/showPrizeList.p%5B%5D
11. "院士會議". sinica.edu.tw.
12. "The Shaw Prize - Top prizes for astronomy, life science and mathematics". shawprize.org.
13. "Institute for Mathematical Sciences (NUS)". nus.edu.sg.
14. "IAS: IAS Panel: International Advisors". ntu.edu.sg. Archived from the original on 2012-03-22. Retrieved 2011-07-02.
External links
• Yum-Tong Siu at the Mathematics Genealogy Project
• In 2003 and 2004, the Asian Journal of Mathematics dedicated several issues to Siu:
• Vol. 7 #4
• Vol. 8 #1 and 2
• english.cas.cn
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Xiaojun Chen
Xiaojun Chen is a Chinese applied mathematician, Chair Professor of Applied Mathematics at Hong Kong Polytechnic University. Her research interests include nonsmooth and nonconvex optimization, complementarity theory, and stochastic equilibrium problems.[1]
Education and career
Chen completed her Ph.D. in 1987 at Xi'an Jiaotong University.[2] At Hong Kong Polytechnic University, she was head of the applied mathematics department from 2013 to 2019. Since 2020 she has directed the University Research Facility in Big Data Analytics, and co-directed the CAS AMSS-PolyU Joint Laboratory of Applied Mathematics[1]
Recognition
Chen was named a SIAM Fellow in the 2021 class of fellows, "for contributions to optimization, stochastic variational inequalities, and nonsmooth analysis".[3] She was named to the 2023 class of Fellows of the American Mathematical Society, "for contributions to mathematical optimization, stochastic variational inequalities, and the analysis of nondifferentiable functions".[4]
References
1. Xiaojun Chen, Hong Kong Polytechnic University, retrieved 2021-04-03
2. Xiaojun Chen at the Mathematics Genealogy Project
3. "SIAM Announces Class of 2021 Fellows", SIAM News, Society for Industrial and Applied Mathematics, 31 March 2021, retrieved 2021-04-03
4. 2023 Class of Fellows, American Mathematical Society, retrieved 2022-11-09
External links
• Xiaojun Chen publications indexed by Google Scholar
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Xiaoying Han
Xiaoying (Maggie) Han is a Chinese mathematician whose research concerns random dynamical systems, stochastic differential equations, and actuarial science. She is Marguerite Scharnagle Endowed Professor in Mathematics at Auburn University.[1]
Education and career
Han graduated from the University of Science and Technology of China in 2001. She completed her Ph.D. in 2007 at the University at Buffalo.[1] Her dissertation, Interlayer Mixing in Thin Film Growth, was supervised by Brian J. Spencer.[2] She joined the Auburn faculty in 2007. She was promoted to full professor in 2017, and given the Marguerite Scharnagle Endowed Professorship in 2018.[1]
In 2020 she was named a Fulbright Scholar, funding her for a research visit to Brazil in 2021.[3]
Books
Han is the co-author of three books:
• Applied Nonautonomous and Random Dynamical Systems (with T. Caraballo, Springer, 2016)[4]
• Attractors Under Discretisation (with P. E. Kloeden, Springer, 2017)[5]
• Random Ordinary Differential Equations and Their Numerical Solution (with P. E. Kloeden, Springer, 2017)[6]
References
1. "Xiaoying (Maggie) Han", Faculty Directory, Auburn University Department of Mathematics and Statistics, retrieved 2019-09-05
2. Xiaoying Han at the Mathematics Genealogy Project
3. Gebhardt, Maria (June 16, 2020), Han Awarded Fulbright Grant to Conduct Research in Brazil in 2021, Auburn College of Sciences and Mathematics
4. Reviews of Applied Nonautonomous and Random Dynamical Systems:
• Schurz, Henri, Mathematical Reviews, MR 3586630{{citation}}: CS1 maint: untitled periodical (link)
• Pötzsche, Christian, zbMATH, Zbl 1370.37001{{citation}}: CS1 maint: untitled periodical (link)
5. Reviews of Attractors Under Discretisation:
• Meijer, Hil G. E., Mathematical Reviews, MR 3699511{{citation}}: CS1 maint: untitled periodical (link)
• Gheorghiu, Calin Ioan, zbMATH, Zbl 1381.65099{{citation}}: CS1 maint: untitled periodical (link)
6. Review of Random Ordinary Differential Equations and Their Numerical Solution:
• Lax, Melvin D., zbMATH, Zbl 1392.60003{{citation}}: CS1 maint: untitled periodical (link)
External links
• Home page
• Xiaoying Han publications indexed by Google Scholar
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Xiaoyu Luo
Xiaoyu Luo FIMechE FRSE (Chinese: 罗小玉,[1] born 1960[2]) is a Chinese and British applied mathematician who studies biomechanics, fluid dynamics, and the interactions of fluid flows with soft biological tissues. She is a professor of applied mathematics at the University of Glasgow.[3]
Education and career
Luo was born in the UK[2] but grew up in Xi'an in a family of artists.[4] After earning bachelor's and master's degrees in theoretical mechanics at Xi'an Jiaotong University, in 1982 and 1985 respectively, she became a lecturer at Xi'an Jiaotong University. There, she studied for a Ph.D. from 1987 until 1990,[4][5] with a visit to the UK through a joint doctoral program with the University of Sheffield. When she earned her Ph.D. at Xi'an Jiaotong University in 1990, she became the first woman to do so.[4]
She moved to the UK in 1992 to become a postdoctoral researcher at the University of Leeds. She worked as a lecturer in engineering at Queen Mary and Westfield College from 1997 to 2000, and in mechanical engineering at the University of Sheffield from 2000 to 2004, before becoming a senior lecturer in mathematics at the University of Glasgow in 2005. She was promoted to professor in 2008,[4][5] the first female professor of applied mathematics at Glasgow.[2]
In 2014 she was named a chair professor at Northwestern Polytechnical University in Xi'an.[2] She has also been a visitor to the International Center for Applied Mechanics at Xi'an Jiaotong University.[1]
Recognition
Luo became a Fellow of the Institution of Mechanical Engineers in 2004[4] and a Fellow of the Royal Society of Edinburgh in 2014.[3][4]
References
1. "People", International Center for Applied Mechanics, Xi'an Jiaotong University, retrieved 2020-09-25
2. Professor Luo Xiaoyu from University of Glasgow was employed as "Chair Professor"of NPU, Northwestern Polytechnical University, 21 December 2014, retrieved 2020-09-25
3. Professor Xiaoyu Luo FRSE, Royal Society of Edinburgh, retrieved 2020-09-23
4. Curriculum vitae, University of Glasgow, retrieved 2020-09-23
5. "Xiaoyu Luo", ORCID, retrieved 2020-09-25
External links
• Home page
• Xiaoyu Luo publications indexed by Google Scholar
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Ding Xieping
Ding Xieping (Chinese: 丁协平; 16 April 1938 – 4 January 2020) was a Chinese mathematician and a professor at Sichuan Normal University. He served as Director of the Institute of Mathematics at the university.
Biography
Ding was born on 16 April 1938 in Zigong, Sichuan, Republic of China.[1] After graduating from Sichuan University in 1961, he taught as an assistant professor at the Department of Mathematics of the former Chengdu University (now Southwestern University of Finance and Economics).[1][2]
In 1964, Ding transferred to Sichuan Normal University, where he taught until his retirement in 2010. He served as Director of the university's Institute of Mathematics.[1] His research focus was nonlinear analysis and applications. Starting in 1979, he published more than 360 research papers, including 160 on Science Citation Index (SCI) journals.[1] In 1999 and 2000, he was China's most prolific authors of papers on SCI journals in the field of mathematics.[2]
He was named a National Outstanding Scientist in 1986 and was awarded a special pension for distinguished scholars by the State Council of the People's Republic of China. He was named a National Outstanding Teacher in 2001.[2]
Ding died on 4 January 2020, aged 81.[1]
References
1. Zhang Cha 张叉 (2020-01-04). "著名数学家丁协平教授". Sichuan Normal University. Archived from the original on 2020-01-14. Retrieved 2020-01-14.
2. Yue, Huairang (2020-01-05). "82岁著名数学家、四川师范大学教授丁协平逝世". The Paper. Archived from the original on 2020-01-14. Retrieved 2020-01-14.
| Wikipedia |
Xin Zhou
Xin Zhou is a mathematician known for his contributions in scattering theory, integrable systems, random matrices and Riemann–Hilbert problems.
He is Professor Emeritus of Mathematics at Duke University. Zhou had obtained M.Sc. from the University of the Chinese Academy of Sciences in 1982 and then got his Ph.D. in 1988 from the University of Rochester.[1] He received the Pólya prize in 1998[2] and was awarded with the Guggenheim Fellowship in 1999.[3] He is most well known for his work with Percy Deift on the steepest descent method for oscillatory Riemann–Hilbert problems.[4]
References
1. "Xin Zhou". Duke University. Retrieved 2019-01-29.
2. "Mathematics People" (PDF). American Mathematics Society. October 1998.
3. "Xin Zhou". Guggenheim Foundation. Retrieved 2019-01-29.
4. Deift, P.; Zhou, X. (1 January 1993). "A Steepest Descent Method for Oscillatory Riemann--Hilbert Problems. Asymptotics for the MKdV Equation". Annals of Mathematics. 137 (2): 295–368. arXiv:math/9201261. doi:10.2307/2946540. JSTOR 2946540. S2CID 12699956.
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Xinyi Yuan
Xinyi Yuan (Chinese: 袁新意; born 1981) is a Chinese mathematician who is currently a professor of mathematics at Peking University working in number theory, arithmetic geometry, and automorphic forms.[1] In particular, his work focuses on arithmetic intersection theory, algebraic dynamics, Diophantine equations and special values of L-functions.
Xinyi Yuan
Yuan in 2017
Born1981 (age 41–42)
Macheng, Hubei, China
Alma materColumbia University
Peking University
Awards
• Clay Research Fellow (2008)
Scientific career
FieldsMathematics
InstitutionsPeking University
University of California, Berkeley
Institute for Advanced Study
Princeton University
Harvard University
ThesisEquidistribution Theory over Algebraic Dynamical Systems (2008)
Doctoral advisorShou-Wu Zhang
Education
Yuan is from Macheng, Huanggang, Hubei province, and graduated from Huanggang Middle School in 2000.[2] That year, he received a gold medal at the International Mathematical Olympiad while representing China.[3] Yuan obtained his A.B. in mathematics from Peking University in 2003 and his Ph.D. in mathematics from the Columbia University in 2008 under the direction of Shou-Wu Zhang.[4] His article "Big Line Bundles over Arithmetic Varieties," published in Inventiones Mathematicae, demonstrates a natural sufficient condition for when the orbit under the absolute Galois group is equidistributed.[5]
Career
He spent time at the Institute for Advanced Study, Princeton University, and Harvard University before joining the Berkeley faculty in 2012.[6]
Yuan was appointed a Clay Research Fellow for a three-year term from 2008 to 2013.[7] Together with a number of other collaborators, Yuan was profiled in Quanta Magazine and Business Insider for, among other things, his research on L-functions.[8][9]
Yuan left UC Berkeley to become a full professor at Peking University in 2020.[10]
Research
Together with Shou-Wu Zhang, Yuan proved the averaged Colmez conjecture which was later shown to imply the André–Oort conjecture for Siegel modular varieties by Jacob Tsimerman.[11][12]
Publications (selected)
• (with Tong Zhang) "Effective Bound of Linear Series on Arithmetic Surfaces", Duke Mathematical Journal 162 (2013), no. 10, 1723–1770.
• "On Volumes of Arithmetic Line Bundles", Compositio Mathematica 145 (2009), 1447–1464.
• "Big Line Bundles over Arithmetic Varieties", Inventiones mathematicae 173 (2008), no. 3, 603–649.
• (with Tong Zhang) "Relative Noether inequality on fibered surfaces", Advances in Mathematics 259 (2014), 89–115.
• (with Shou-Wu Zhang) "The arithmetic Hodge index theorem for adelic line bundles", Mathematische Annalen (2016), 1–49.
• (with Wei Zhang, Shou-Wu Zhang) "The Gross–Kohnen–Zagier theorem over totally real fields", Compositio Mathematica 145 (2009), no. 5, 1147–1162.
• (with Wei Zhang, Shou-Wu Zhang) "The Gross–Zagier formula on Shimura curves", Annals of Mathematics Studies vol. 184, Princeton University Press, 2012.
• (with Wei Zhang, Shou-Wu Zhang) "Triple product L-series and Gross–Kudla–Schoen cycles", preprint.
• Yuan, Xinyi; Zhang, Shou-Wu (2018). "On the averaged Colmez conjecture". Annals of Mathematics. 187 (2): 553–638. arXiv:1507.06903. doi:10.4007/annals.2018.187.2.4. S2CID 118916754.
References
1. "Xinyi Yuan". math.berkeley.edu. Retrieved 2020-11-14.
2. "黄冈中学近14年来未出省状元 发展过程中矛盾凸显". Xinhua News Agency. 6 April 2015. Archived from the original on August 3, 2017. Retrieved 3 August 2017.
3. "Xinyi Yuan – Official IMO Results", International Mathematical Olympiad. Retrieved on 4 December 2016.
4. "Xinyi Yuan CV", UC Berkeley. Retrieved on 3 December 2016.
5. "Big line bundles over arithmetic varieties", Inventiones Mathematicae. Published September 2008. Retrieved on 4 December 2016.
6. "IAS Member – Xinyi Yuan", Institute of Advanced Study. Retrieved on 4 December 2016.
7. "Xinyi Yuan", Clay Mathematics Institute. Retrieved on 3 December 2016.
8. "Math Quartet Joins Forces on Unified Theory", Quanta Magazine. Retrieved on 3 December 2016.
9. "Math Quartet Joins Forces on Unified Theory", Business Insider. Retrieved on 4 December 2016.
10. "Xinyi Yuan | Department of Mathematics at University of California Berkeley".
11. "February 2018". Notices of the American Mathematical Society. 65 (2): 191. 2018. ISSN 1088-9477.
12. Yuan, Xinyi; Zhang, Shou-Wu (2018). "On the averaged Colmez conjecture". Annals of Mathematics. 187 (2): 553–638. arXiv:1507.06903. doi:10.4007/annals.2018.187.2.4. S2CID 118916754.
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| Wikipedia |
Xiuxiong Chen
Xiuxiong Chen (Chinese: 陈秀雄; pinyin: Chén Xiùxióng) is a Chinese-American mathematician whose research concerns differential geometry and differential equations.[1] A professor at Stony Brook University since 2010, he was elected a Fellow of the American Mathematical Society in 2015 and awarded the Oswald Veblen Prize in Geometry in 2019. In 2019, he was awarded the Simons Investigator award.[2]
Biography
Chen was born in Qingtian County, Zhejiang, China.[3] He entered the Department of Mathematics of the University of Science and Technology of China in 1982,[4] and graduated in 1987.[3] He subsequently studied under Peng Jiagui (彭家贵) at the Graduate School of the Chinese Academy of Sciences, where he earned his master's degree. [3]
In 1989 , he moved to the United States to study at the University of Pennsylvania.[3] The last doctoral student of Eugenio Calabi,[3][5] he obtained his Ph.D. in mathematics in 1994, with his dissertation on "Extremal Hermitian Matrices with Curvature Distortion in a Riemann Surface".[5]
Chen was an instructor at McMaster University in Canada from 1994 to 1996 . For the next two years he was a National Science Foundation postdoctoral fellow at Stanford University. He was an assistant professor at Princeton University from 1998 to 2002, before becoming an associate professor at the University of Wisconsin–Madison. He was promoted to full professor in 2005. Since October 2010 he has been a professor at Stony Brook University.[6] In 2006, he founded the Pacific Rim Conference on Complex Geometry at the University of Science and Technology of China.[3]
As of 2019, Chen has advised 17 Ph.D. students, including Song Sun and Bing Wang (王兵).[3][5] He was elected a Fellow of the American Mathematical Society in 2015 "for contributions to differential geometry, particularly the theory of extremal Kahler metrics".[7] He was an invited speaker at the 2002 International Congress of Mathematicians, in Beijing.[8]
Conjecture on Fano manifolds and Veblen Prize
In 2019, Chen was awarded the prestigious Oswald Veblen Prize in Geometry, together with Simon Donaldson and Chen's former student Song Sun, for proving a long-standing conjecture on Fano manifolds, which states "that a Fano manifold admits a Kähler–Einstein metric if and only if it is K-stable". It had been one of the most actively investigated topics in geometry since a loose version of it was first proposed in the 1980s by Shing-Tung Yau after his proof of the Calabi conjecture. More precise versions were subsequently proposed by Gang Tian and Donaldson. The solution by Chen, Donaldson and Sun was published in the Journal of the American Mathematical Society in 2015 as a three-article series, "Kähler–Einstein metrics on Fano manifolds, I, II and III".[9][10]
Major publications
• Chen, Xiuxiong. The space of Kähler metrics. J. Differential Geom. 56 (2000), no. 2, 189–234.
• Chen, X. X.; Tian, G. Geometry of Kähler metrics and foliations by holomorphic discs. Publ. Math. Inst. Hautes Études Sci. No. 107 (2008), 1–107.
• Chen, Xiuxiong; Donaldson, Simon; Sun, Song. Kähler-Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities. J. Amer. Math. Soc. 28 (2015), no. 1, 183–197.
• Chen, Xiuxiong; Donaldson, Simon; Sun, Song. Kähler-Einstein metrics on Fano manifolds. II: Limits with cone angle less than 2π. J. Amer. Math. Soc. 28 (2015), no. 1, 199–234.
• Chen, Xiuxiong; Donaldson, Simon; Sun, Song. Kähler-Einstein metrics on Fano manifolds. III: Limits as cone angle approaches 2π and completion of the main proof. J. Amer. Math. Soc. 28 (2015), no. 1, 235–278.
• Chen, Xiuxiong; Wang, Bing. Space of Ricci flows (II)—Part B: Weak compactness of the flows. J. Differential Geom. 116 (2020), no. 1, 1 - 123.
• Chen, Xiuxiong; Cheng, Jingrui. On the constant scalar curvature Kähler metrics (I)—A priori estimates. J. Amer. Math. Soc. 34 (2021), no. 4, 909–936.
• Chen, Xiuxiong; Cheng, Jingrui. On the constant scalar curvature Kähler metrics (II)—Existence results. J. Amer. Math. Soc. 34 (2021), no. 4, 937–1009.
References
1. "Xiuxiong Chen". Stony Brook University. Retrieved 8 April 2019.
2. Simons Investigators Awardees, The Simons Foundation
3. "陈秀雄─卡拉比高徒". Ta Kung Pao. 15 May 2014. Retrieved 9 April 2019.
4. "陈秀雄孙崧荣获维布伦奖". University of Science and Technology of China Initiative Foundation. 20 November 2018. Retrieved 9 April 2019.
5. "Xiu-Xiong Chen". The Mathematics Genealogy Project. Retrieved 8 April 2019.
6. Chen, Xiuxiong. "Vitae" (PDF). Stony Brook University. Retrieved 8 April 2019.
7. "List of Fellows of the American Mathematical Society". American Mathematical Society. Retrieved 8 April 2019.
8. "Invited Speakers". International Congress of Mathematicians, Beijing 2002. 9 May 2002. Retrieved 24 May 2019.
9. "2019 Oswald Veblen Prize in Geometry to Xiuxiong Chen, Simon Donaldson, and Song Sun". American Mathematical Society. 19 November 2018. Retrieved 9 April 2019.
10. "Stony Brook Faculty Win Prestigious Veblen Prize in Geometry". Stony Brook University. 20 November 2018. Retrieved 8 April 2019.
Recipients of the Oswald Veblen Prize in Geometry
• 1964 Christos Papakyriakopoulos
• 1964 Raoul Bott
• 1966 Stephen Smale
• 1966 Morton Brown and Barry Mazur
• 1971 Robion Kirby
• 1971 Dennis Sullivan
• 1976 William Thurston
• 1976 James Harris Simons
• 1981 Mikhail Gromov
• 1981 Shing-Tung Yau
• 1986 Michael Freedman
• 1991 Andrew Casson and Clifford Taubes
• 1996 Richard S. Hamilton and Gang Tian
• 2001 Jeff Cheeger, Yakov Eliashberg and Michael J. Hopkins
• 2004 David Gabai
• 2007 Peter Kronheimer and Tomasz Mrowka; Peter Ozsváth and Zoltán Szabó
• 2010 Tobias Colding and William Minicozzi; Paul Seidel
• 2013 Ian Agol and Daniel Wise
• 2016 Fernando Codá Marques and André Neves
• 2019 Xiuxiong Chen, Simon Donaldson and Song Sun
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Xu Guangqi
Xu Guangqi or Hsü Kuang-ch'i (April 24, 1562 – November 8, 1633), also known by his baptismal name Paul, was a Chinese agronomist, astronomer, mathematician, politician, and writer during the Ming dynasty.[6] Xu was appointed by the Chinese Emperor in 1629 to be the leader of the ShiXian calendar reform, which he embarked on with the assistance of Jesuits.[7] Xu was a colleague and collaborator of the Italian Jesuits Matteo Ricci and Sabatino de Ursis and assisted their translation of several classic Western texts into Chinese, including part of Euclid's Elements. He was also the author of the Nong Zheng Quan Shu, a treatise on agriculture. He was one of the "Three Pillars of Chinese Catholicism"; the Roman Catholic Church considers him a Servant of God.[8] On April 15, 2011, Vatican spokesman Federico Lombardi announced the beatification of Xu Guangqi.[9][10]
Servant of God
Xu Guangqi
徐光啓
Portrait of Xu Guangqi
Vice Minister of Rites
In office
1629–1630
Preceded byMa Zhiqi
Succeeded byLi Sunchen
Grand Secretary of the Wenyuan Library
In office
1632–1633
Senior Grand SecretaryZhou Tingru
Wen Tiren
Personal details
BornApril 24, 1562
Shanghai County, Songjiang, South Zhili, Ming Empire[1]
DiedNovember 8, 1633(1633-11-08) (aged 71)[2]
Beijing, Shuntian, North Zhili, Ming Empire
Resting placeGuangqi Park, Xujiahui, Xuhui District, Shanghai, China
NationalityChinese
SpouseWu[3]
RelationsCandida Xu (granddaughter)[4]
(Xu Zhun)[4]
(Xu Maheux)[4]
ChildrenXu Ji (徐驥)
Parent(s)Xu Sicheng (徐思誠), father
EducationJinshi Degree (1604)[5]
Occupationscholar-official (Minister of Rites and Grand Secretary), agronomist, astronomer, mathematician, writer
Known forThree Pillars of Chinese Catholicism
Chinese translation of Euclid's Elements
Chongzhen calendar
Complete Treatise on Agriculture
Baptismal namePaul Xu
Xu Guangqi
Traditional Chinese徐光啓
Simplified Chinese徐光启
Transcriptions
Standard Mandarin
Hanyu PinyinXú Guāngqǐ
Wade–GilesHsü2 Kuang1-ch‘i3
IPA[ɕy̌ kwáŋtɕʰì]
Courtesy name
Chinese子先
Transcriptions
Standard Mandarin
Hanyu PinyinZǐxiān
Wade–GilesTzu-hsien
Second alternative Chinese name
Chinese玄扈
Transcriptions
Standard Mandarin
Hanyu PinyinXuánhù
Wade–GilesHsüan-hu
Third alternative Chinese name
Traditional Chinese保祿
Simplified Chinese保禄
Literal meaningPaulus
Transcriptions
Standard Mandarin
Hanyu PinyinBǎolù
Wade–GilesPao-lu
Servant of God
Xu Guangqi
Xu Guangqi with Matteo Ricci (left)
From Athanasius Kircher's China Illustrata, 1667
BornApril 24, 1562
Shanghai, China
DiedNovember 8, 1633
Beijing, China
Name
Xu Guangqi is the pinyin romanization of the Mandarin Chinese pronunciation of Xu's Chinese name. His name is written Hsü Kuang-ch‘i using the Wade–Giles system. His courtesy name was Zixian and his penname was Xuanhu. In the Jesuits' records, it is the last which is used as his Chinese name, in the form "Siù Hsven Hú".[11]
At his conversion, he adopted the baptismal name Paul (Latin: Paulus). In Chinese, its transcription is employed as a kind of courtesy name (i.e., Xu Baolu) and the Jesuits sometimes referred to him as "Siù Pao Lò"[11] or Ciù Paulus.[12] More often, however, they describe him as "Doctor Paul" (Latin: Doctor Paulus;[13][11] Portuguese: Doutor Paulo),[14] "Our Paul" (Latin: noster Paulus), or "Paul Siu"[15] or "Ciu".[12][11][16]
Early life
Xu Guangqi was born in Shanghai in Southern Zhili's Songjiang Prefecture on April 24, 1562,[1] under China's Ming dynasty. At the time, Shanghai was merely a small walled county seat in the old quarter around the present city's Yu Garden. His family, including an older and younger sister, lived in the Taiqing Quarter at the south end of the town. Guangqi's branch of the Xus were not related to those who had passed the imperial examinations and joined Shanghai's local gentry.[17] His father Xu Sicheng (died c. 1607)[18] had been orphaned at age 5 and seen most of his inheritance lost to "Japanese" pirate raids and insolvent friends in the 1550s.[17]
At the time of Guangqi's birth, his father worked twenty mu (1¼ ha)[19] or less south of the city wall.[20] About half of this would have been used to feed the family,[21] with the rest used to supplement his income from small-scale trading.[17] By the time Guangqi was 6, the family had saved enough to send him to a local school, where a later hagiographer records him piously upbraiding his classmates when they spoke of wanting to use their education for wealth or mystical power. Instead, he supposedly advised, "None of these things is worth doing. If you want to talk about the sort of person you want to become, then it should be to establish yourself and to follow the Way. Bring order to the state and the people. Revere the orthodox and expose the heterodox. Don't waste the chance to be someone who matters in this world."[20] From 1569 to 1573, the family sent Guangqi to the school at the Buddhist monastery at Longhua.[1] It is not recorded, but this school was probably a separate secular and fee-based institution for families too poor to hire private tutors for their sons.[20]
His mother died on May 8, 1592, and he undertook the ritual mourning period in her honor.[22] His whereabouts over the next few years are uncertain but he seems to have failed the provincial exam at Beijing in 1594, after the mourning period was over.[22]
Career
In 1596, he moved to Xunzhou (now Guiping) in Guangxi to assist its prefect Zhao Fengyu, a Shanghai native who had passed the juren exams in 1555.[22] The next year, he traveled to Beijing in the spring and passed its provincial exam, becoming a juren.[22] He seems to have stayed there for the imperial exam the next year, but failed to pass. He then returned to Shanghai around April, turning his attention to the study of military and agricultural subjects.[22] The next year he studied under Cheng Jiasui.[22]
He first met Matteo Ricci, the Italian Jesuit, in Nanjing in March or April 1600.[22] He collaborated with Ricci in translating several classic Western texts—most notably the first part of Euclid's Elements—into Chinese, as well as several Chinese Confucian texts into Latin.[7] Ricci's influence led to Xu being baptized as a Roman Catholic in 1603. His descendants remained Catholics or Protestants into the 21st century..
From 1607 until 1610, Xu was forced to retire from public office and returned to his home in Shanghai. It was during this time that he experimented with Western-style irrigation methods.[23] He also experimented with the cultivation of sweet potatoes, cotton, and the nu zhen tree.[23] He was called once more to serve the Chinese bureaucracy, where he rose to a high rank and became known late in his career simply as "The Minister".[7] Yet he continued to experiment and learn of new agricultural practices while he served his office, promoting the use of wet-rice in the Northeast China.[23] From 1613 until 1620 he often visited Tianjin, where he helped organize self-sufficient military settlements (tun tian).[23]
In 1629, memorials by Xu successfully moved the court to permit the Portuguese captain Gonçalo Teixeira-Correa to bring 10 artillery pieces and 4 "excellent bombards" across China to the capital to demonstrate the effectiveness of Western artillery.[13] An earlier demonstration in 1623 had gone disastrously, with an exploding cannon killing one Portuguese artillerist and three Chinese observers, but on this occasion the pieces were accepted and directed to Dengzhou (now Penglai) in Shandong.[24] The Christian convert Ignatius Sun, a protégé of Xu's, was governor there[25] and had also been a strong advocate of modernizing China's military. Together with Captain Teixeira and his translator João Rodrigues, Sun used the pieces to train his troops to oppose the ongoing Manchu invasion. However, Sun's lenient treatment of a 1632 mutiny under Kong Yude and Geng Zhongming permitted them to successfully capture Dengzhou, seize the artillery, and establish an independent power base that eventually joined the Manchus.[25] Xu's memorials for clemency were unsuccessful and Sun was court-martialed and executed.[15]
He held the positions of Minister of Rites (禮部尙書), overseeing government programs related to culture, education, and foreign affairs, and Deputy Senior Grand Secretary (內閣次輔), effectively a deputy premier for the imperial cabinet.
Johann Adam Schall von Bell stayed with Xu during his final illness in 1633 and oversaw the return of his body to his family in Shanghai.[26] There, it was publicly displayed at his villa until 1641, when it was buried "in a time of hardship".[15]
Legacy
Xu Guangqi's tomb remains the centerpiece of Shanghai's Guangqi Park on Nandan Road (南丹路), just south of Xujiahui Cathedral.
The 350th anniversary of his death in 1983 was celebrated very publicly, both with ceremonies in Shanghai and a laudatory article in the Beijing Review. The vocal Communist support for these memorials has been seen as signaling support for Deng Xiaoping's policies of opening up and modernizing China.[27] Most Chinese treatments of his life and legacy, however, focus upon his desire for scientific, technological, and political progress and its effect upon Chinese development, whereas western treatments nearly universally attach great importance to his Christian conversion and identity.[27]
Works
Military science
Xu Guangqi wrote a book on military techniques and strategies entitled Mr Xu's Amateur Observations in response to the criticisms he faced for daring discuss military matters in spite of being a mere scholar.[28] He frequently cited the Xunzi and Guanzi, and made use of rewards and punishments along the lines of the Legalists, at least in relief activities.[29]
Xu Guangqi put forward the concept of a "Rich Country and Strong Army" (富國強兵), which would be adopted by Japan for its modernization in the end of the 19th century, under the name Fukoku Kyohei.
Mathematics
In 1607, Xu and Ricci translated the first parts of Euclid's Elements (a treatise on mathematics, geometry, and logic) into Chinese. Some Chinese scholars credit Xu as having "started China's enlightenment".[7]
Astronomy
After followers of Xu and Ricci publicly predicted a solar eclipse in 1629, Xu was appointed by the Emperor as the leader of an effort to reform the Chinese calendar. The reform, which constituted the first major collaboration between scientists from Europe and from the Far East, was completed after his death and became known as the Chongzhen calendar.[7] It's notable for systematically introducing the concepts and development of European mathematics and astronomy to China for the first time, including extensive translations and references to Euclid's Elements and the works of Nicolaus Copernicus, Johannes Kepler, Galileo Galilei, and Tycho Brahe, whose Tychonic system was used its main theoretical basis.[30][31]
Agriculture
Xu Guangqi wrote the Complete Treatise on Agriculture, an outstanding agricultural treatise that followed in the tradition of those such as Wang Zhen (wrote the Wang Zhen Nong Shu of 1313 AD) and Jia Sixia (wrote the Chi Min Yao Shu of 535 AD).[32] Like Wang Zhen, Xu lived in troubled times, and was devoted as a patriot to aiding the rural farmers of China.[23] His main interests were in irrigation, fertilizers, famine relief, economic crops, and empirical observation with early notions of chemistry.[23] It was an enormous written work, some 700,000 written Chinese characters, making it 7 times as large as the work of both Jia Sixia and Wang Zhen.[33] Although its final draft was unfinished by Xu Guangqi by the time of his death in 1633, the famous Jiangnan scholar Chen Zilong assembled a group of scholars to edit the draft, publishing it in 1639.[33]
The topics covered by his book are as follows:[33]
• The Fundamentals of Agriculture (Nong Ben): quotations from the classics on the importance of encouraging agriculture
• Field System (Tian Zhi): land distribution, field management
• Agricultural Tasks (Nong Shi): clearing land, tilling; also a detailed exposition on settlement schemes
• Water Control (Shui Li): various methods of irrigation, types of irrigation equipment, and the last two chapters devoted to new Western-style irrigation equipment
• Illustrated Treatise on Agricultural Implements (Nong Chi Tu Pu): based largely on Wang Zhen's book of 1313 AD
• Horticulture (Shi Yi): vegetables and fruit
• Sericulture (Can Sang): silk production
• Further Textile Crops (Can Sang Guang Lei): cotton, hemp, etc.
• Silviculture (Chong Chi): forestry preservation
• Animal Husbandry (Mu Yang)
• Culinary Preparations (Zhi Zao)
• Famine Control (Huang Zheng): administrative measures, famine flora
Family
Xu's only son was John Xu[15] (t 徐驥, s 徐骥, Xú Jì),[3] whose daughter was Candida Xu (徐甘第大, Xú Gāndìdà; 1607–1680). A devout Christian, she was recognized as an important patron of Christianity in Jiangnan during the early Qing era. The Jesuit Philippe Couplet, who worked closely with her, composed her biography in Latin. This was published in French translation as A History of the Christian Lady of China, Candide Hiu (Histoire d'une Dame Chrétienne de la Chine, Candide Hiu) in 1688.[34] Her son was Basil Xu, who served as an official under the Qing.[15]
Gallery
• Paul Xu (bottom left) and his granddaughter Candida (bottom right), along with Ricci, Schall, and Verbiest (top row)
• Xu's statue in Xujiahui, on North Caoxi Road
• Xu's tomb in Shanghai's Guangqi Park
• Introduction to Astronomy, translated by Xu Guangqi
• Portuguese translation of "Doctor Paul"'s letter to the king of Portugal
See also
• Xu Guangqi Memorial Hall
• Roman Catholicism in China
• Jesuit China missions
• Three Pillars of Chinese Catholicism
• Shanghainese people
• History of agriculture
• Xavier School in San Juan, Metro Manila, the Philippines
References
Citations
1. Dudink (2001), p. 399
2. Dudink (2001), p. 409.
3. Dudink (2001), p. 400
4. Brockey (2008), p. 140.
5. Brockey (2008), p. 59.
6. Hummel, Arthur W. Sr., ed. (1943). "Hsü Kuang-ch'i" . Eminent Chinese of the Ch'ing Period. United States Government Printing Office.
7. Stone (2007).
8. Roman Catholic Diocese of Shanghai: 徐光启列品案筹备进程
9. "Church to beatify early China convert in Shanghai". UCA News. 2011. Retrieved May 24, 2018.
10. "In a first, Shanghai-born Ming Dynasty bureaucrat and scientist Xu Guangqi to be beatified". Shanghaiist. May 5, 2018.
11. Blue (2001), p. 48
12. Blue (2001), p. 33
13. Blue (2001), p. 44
14. Vasconcelos (2012), p. 163.
15. Blue (2001), p. 45
16. Blue (2001), p. 49
17. Brook (2001), p. 93
18. Blue (2001), p. 35.
19. Wilkinson (2000), p. 243.
20. Brook (2001), p. 94
21. Clunas (1996), p. 40.
22. Dudink (2001), p. 402
23. Needham (1984), p. 65.
24. Chan (1976), p. 1147.
25. Fang (1943).
26. Blue (2001), pp. 42–3.
27. Blue (2001), p. 19
28. Xu Guangqi Memorial Hall permanent exhibit
29. Smith (2009), p. 252.
30. "明版《崇祯历书》原貌再现" (in Chinese). Archived from the original on June 2, 2015. Retrieved May 28, 2010.
31. 徐光启和《崇祯历书》 Archived 2011-05-24 at the Wayback Machine
32. Needham (1984), pp. 64–5.
33. Needham (1984), p. 66.
34. Mungello (1989), pp. 253–254.
Sources
• Blue, Gregory (2001), "Xu Guangqi in the West: Early Jesuit Sources and the Construction of an Identity", Statecraft & Intellectual Renewal in Late Ming China: The Cross-Cultural Synthesis of Xu Guangqi (1562–1633), Sinica Leidensia, Vol. 50, Leiden, South Holland: Brill, pp. 19–71, ISBN 9004120580.
• Brockey, Liam Matthew (2008), Journey to the East: The Jesuit Mission to China, 1579–1724, Cambridge, MA: The Belknap Press of Harvard University Press
• Brook, Timothy (2001), "Xu Guangqi in His Context: The World of the Shanghai Gentry", Statecraft & Intellectual Renewal in Late Ming China: The Cross-Cultural Synthesis of Xu Guangqi (1562–1633), ISBN 9004120580.
• Chan, Albert (1976), "João Rodrígues", Dictionary of Ming Biography, 1368–1644, Vol. II: M–Z, New York, NY: Columbia University Press, pp. 1145–47, ISBN 9780231038331.
• Clunas, Craig (1996), Fruitful Sites: Garden Culture in Ming Dynasty China, Envisioning Asia, London, England: Reaktion Books, ISBN 9780948462887.
• Dudink, Adrianus Cornelis (2001), "Xu Guangqi's Career: An Annotated Chronology", Statecraft & Intellectual Renewal in Late Ming China, ISBN 9004120580.
• Wilkinson, Endymion Porter (2000), Chinese History: A Manual, Cambridge, MA: Harvard University Asia Center for the Harvard–Yenching Institute, ISBN 9780674002494.
• Fang Zhaoying (1943). "Sun Yüan-hua" . In Hummel, Arthur W. Sr. (ed.). Eminent Chinese of the Ch'ing Period. United States Government Printing Office. p. 686.
• Mungello, David E. (1989), Curious Land: Jesuit Accommodation and the Origins of Sinology, Honolulu: University of Hawaii Press, ISBN 0-8248-1219-0.
• Needham, Joseph (1984), Science and Civilisation in China, Vol. VI: Biology and Biological Technology, Part 2: Agriculture, Cambridge, England: Cambridge University Press.
• Smith, Joanna Handlin (2009), The Art of Doing Good: Charity in Late Ming China.
• Stone, Richard (2007), "Scientists Fete China's Supreme Polymath", Science, vol. 318, p. 733, doi:10.1126/science.318.5851.733, ISSN 0036-8075, PMID 17975042, S2CID 162156995.
• Vasconcelos de Saldanha, António (2012), "The Last Imperial Honours: From Tomás Pereira to the Eulogium Europeorum Doctorum in 1711", In the Light and Shadow of an Emperor: Tomás Pereira, SJ (1645–1708), the Kangxi Emperor, and the Jesuit Mission in China, Newcastle, England: Cambridge Scholars Publishing, pp. 144–227, ISBN 9781443838542.
Further reading
• Needham, Joseph (1959). Science and Civilisation in China: Volume 3, Mathematics and the Sciences of the Heavens and the Earth. Cambridge: Cambridge University Press; reprinted: Taipei: Caves Books, 1986. OCLC
• Zhao, Jikai, "Xu Guangqi". Encyclopedia of China (Economics Edition), 1st ed.
• Mei, Rongzhao, "Xue Guangqi". Encyclopedia of China (Mathematics Edition), 1st ed.
• Hummel, Arthur W. Sr., ed. (1943). "Hsü Kuang-ch'i" . Eminent Chinese of the Ch'ing Period. United States Government Printing Office.
External links
Wikimedia Commons has media related to:
Xu Guangqi (category)
• O'Connor, John J.; Robertson, Edmund F., "Xu Guangqi", MacTutor History of Mathematics Archive, University of St Andrews
• Ricci Roundtable at the University of San Francisco
• University of Texas article mentioning him
• Xu Guangqi at the University of Maine Archived August 28, 2008, at the Wayback Machine
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| Wikipedia |
Xuong tree
In graph theory, a Xuong tree is a spanning tree $T$ of a given graph $G$ with the property that, in the remaining graph $G-T$, the number of connected components with an odd number of edges is as small as possible.[1] They are named after Nguyen Huy Xuong, who used them to characterize the cellular embeddings of a given graph having the largest possible genus.[2]
According to Xuong's results, if $T$ is a Xuong tree and the numbers of edges in the components of $G-T$ are $m_{1},m_{2},\dots ,m_{k}$, then the maximum genus of an embedding of $G$ is $\textstyle \sum _{i=1}^{k}\lfloor m_{i}/2\rfloor $.[1][2] Any one of these components, having $m_{i}$ edges, can be partitioned into $\lfloor m_{i}/2\rfloor $ edge-disjoint two-edge paths, with possibly one additional left-over edge.[3] An embedding of maximum genus may be obtained from a planar embedding of the Xuong tree by adding each two-edge path to the embedding in such a way that it increases the genus by one.[1][2]
A Xuong tree, and a maximum-genus embedding derived from it, may be found in any graph in polynomial time, by a transformation to a more general computational problem on matroids, the matroid parity problem for linear matroids.[1][4]
References
1. Beineke, Lowell W.; Wilson, Robin J. (2009), Topics in topological graph theory, Encyclopedia of Mathematics and its Applications, vol. 128, Cambridge University Press, Cambridge, p. 36, doi:10.1017/CBO9781139087223, ISBN 978-0-521-80230-7, MR 2581536
2. Xuong, Nguyen Huy (1979), "How to determine the maximum genus of a graph", Journal of Combinatorial Theory, Series B, 26 (2): 217–225, doi:10.1016/0095-8956(79)90058-3, MR 0532589
3. Sumner, David P. (1974), "Graphs with 1-factors", Proceedings of the American Mathematical Society, American Mathematical Society, 42 (1): 8–12, doi:10.2307/2039666, JSTOR 2039666, MR 0323648
4. Furst, Merrick L.; Gross, Jonathan L.; McGeoch, Lyle A. (1988), "Finding a maximum-genus graph imbedding", Journal of the ACM, 35 (3): 523–534, doi:10.1145/44483.44485, MR 0963159, S2CID 17991210
| Wikipedia |
x̅ and R chart
In statistical process control (SPC), the ${\bar {x}}$ and R chart is a type of scheme, popularly known as control chart, used to monitor the mean and range of a normally distributed variables simultaneously, when samples are collected at regular intervals from a business or industrial process.[1] It is often used to monitor the variables data but the performance of the ${\bar {x}}$ and R chart may suffer when the normality assumption is not valid.
${\bar {x}}$ and R chart
Originally proposed byWalter A. Shewhart
Process observations
Rational subgroup size1 < n ≤ 10
Measurement typeAverage quality characteristic per unit
Quality characteristic typeVariables data
Underlying distributionNormal distribution
Performance
Size of shift to detect≥ 1.5σ
Process variation chart
Center line${\bar {R}}={\frac {\sum _{i=1}^{m}max(x_{ij})-min(x_{ij})}{m}}$
Upper control limit$D_{4}{\bar {R}}$
Lower control limit$D_{3}{\bar {R}}$
Plotted statisticRi = max(xj) - min(xj)
Process mean chart
Center line${\bar {\bar {x}}}={\frac {\sum _{i=1}^{m}\sum _{j=1}^{n}x_{ij}}{mn}}$
Control limits${\bar {\bar {x}}}\pm A_{2}{\bar {R}}$
Plotted statistic${\bar {x}}_{i}={\frac {\sum _{j=1}^{n}x_{j}}{n}}$
Properties
The "chart" actually consists of a pair of charts: One to monitor the process standard deviation (as approximated by the sample moving range) and another to monitor the process mean, as is done with the ${\bar {x}}$ and s and individuals control charts. The ${\bar {x}}$ and R chart plots the mean value for the quality characteristic across all units in the sample, ${\bar {x}}_{i}$, plus the range of the quality characteristic across all units in the sample as follows:
R = xmax - xmin.
The normal distribution is the basis for the charts and requires the following assumptions:
• The quality characteristic to be monitored is adequately modeled by a normally distributed random variable
• The parameters μ and σ for the random variable are the same for each unit and each unit is independent of its predecessors or successors
• The inspection procedure is same for each sample and is carried out consistently from sample to sample
The control limits for this chart type are:[2]
• $D_{3}{\bar {R}}$ (lower) and $D_{4}{\bar {R}}$ (upper) for monitoring the process variability
• ${\bar {\bar {x}}}\pm A_{2}{\bar {R}}$ for monitoring the process mean
where ${\bar {\bar {x}}}$ and ${\bar {R}}$ are the estimates of the long-term process mean and range established during control-chart setup and A2, D3, and D4 are sample size-specific anti-biasing constants. The anti-biasing constants are typically found in the appendices of textbooks on statistical process control.
Usage of the chart
The chart is advantageous in the following situations:[3]
1. The sample size is relatively small (say, n ≤ 10—${\bar {x}}$ and s charts are typically used for larger sample sizes)
2. The sample size is constant
3. Humans must perform the calculations for the chart
As with the ${\bar {x}}$ and s and individuals control charts, the ${\bar {x}}$ chart is only valid if the within-sample variability is constant.[4] Thus, the R chart is examined before the ${\bar {x}}$ chart; if the R chart indicates the sample variability is in statistical control, then the ${\bar {x}}$ chart is examined to determine if the sample mean is also in statistical control. If on the other hand, the sample variability is not in statistical control, then the entire process is judged to be not in statistical control regardless of what the ${\bar {x}}$ chart indicates.
Limitations
For monitoring the mean and variance of a normal distribution, the ${\bar {x}}$ and s chart chart is usually better than the ${\bar {x}}$ and R chart.
See also
• ${\bar {x}}$ and s chart
• Shewhart individuals control chart
• Simultaneous monitoring of mean and variance of Gaussian Processes with estimated parameters (when standards are unknown)[5]
References
1. "Shewhart X-bar and R and S Control Charts". NIST/Sematech Engineering Statistics Handbook]. National Institute of Standards and Technology. Retrieved 2009-01-13.
2. Montgomery, Douglas (2005). Introduction to Statistical Quality Control. Hoboken, New Jersey: John Wiley & Sons, Inc. p. 197. ISBN 978-0-471-65631-9. OCLC 56729567.
3. Montgomery, Douglas (2005). Introduction to Statistical Quality Control. Hoboken, New Jersey: John Wiley & Sons, Inc. p. 222. ISBN 978-0-471-65631-9. OCLC 56729567.
4. Montgomery, Douglas (2005). Introduction to Statistical Quality Control. Hoboken, New Jersey: John Wiley & Sons, Inc. p. 214. ISBN 978-0-471-65631-9. OCLC 56729567.
5. McCracken, A. K.; Chakraborti, S.; Mukherjee, A. (2013-10-01). "Control Charts for Simultaneous Monitoring of Unknown Mean and Variance of Normally Distributed Processes". Journal of Quality Technology. 45 (4): 360–376. doi:10.1080/00224065.2013.11917944. ISSN 0022-4065. S2CID 117307669.
| Wikipedia |
Y-homeomorphism
In mathematics, the y-homeomorphism, or crosscap slide, is a special type of auto-homeomorphism in non-orientable surfaces.
It can be constructed by sliding a Möbius band included on the surface around an essential 1-sided closed curve until the original position; thus it is necessary that the surfaces have genus greater than one. The projective plane ${\mathbb {R} P}^{2}$ has no y-homeomorphism.
See also
• Lickorish-Wallace theorem
References
• J. S. Birman, D. R. J. Chillingworth, On the homeotopy group of a non-orientable surface, Trans. Amer. Math. Soc. 247 (1979), 87-124.
• D. R. J. Chillingworth, A finite set of generators for the homeotopy group of a non-orientable surface, Proc. Camb. Phil. Soc. 65 (1969), 409–430.
• M. Korkmaz, Mapping class group of non-orientable surface, Geometriae Dedicata 89 (2002), 109–133.
• W. B. R. Lickorish, Homeomorphisms of non-orientable two-manifolds, Math. Proc. Camb. Phil. Soc. 59 (1963), 307–317.
| Wikipedia |
y-intercept
In analytic geometry, using the common convention that the horizontal axis represents a variable x and the vertical axis represents a variable y, a y-intercept or vertical intercept is a point where the graph of a function or relation intersects the y-axis of the coordinate system.[1] As such, these points satisfy x = 0.
Using equations
If the curve in question is given as $y=f(x),$ the y-coordinate of the y-intercept is found by calculating $f(0).$ Functions which are undefined at x = 0 have no y-intercept.
If the function is linear and is expressed in slope-intercept form as $f(x)=a+bx$, the constant term $a$ is the y-coordinate of the y-intercept.[2]
Multiple y-intercepts
Some 2-dimensional mathematical relationships such as circles, ellipses, and hyperbolas can have more than one y-intercept. Because functions associate x values to no more than one y value as part of their definition, they can have at most one y-intercept.
x-intercepts
Main article: x-intercept
Analogously, an x-intercept is a point where the graph of a function or relation intersects with the x-axis. As such, these points satisfy y=0. The zeros, or roots, of such a function or relation are the x-coordinates of these x-intercepts.[3]
Unlike y-intercepts, functions of the form y = f(x) may contain multiple x-intercepts. The x-intercepts of functions, if any exist, are often more difficult to locate than the y-intercept, as finding the y intercept involves simply evaluating the function at x=0.
In higher dimensions
The notion may be extended for 3-dimensional space and higher dimensions, as well as for other coordinate axes, possibly with other names. For example, one may speak of the I-intercept of the current–voltage characteristic of, say, a diode. (In electrical engineering, I is the symbol used for electric current.)
See also
• Regression intercept
References
1. Weisstein, Eric W. "y-Intercept". MathWorld--A Wolfram Web Resource. Retrieved 2010-09-22.
2. Stapel, Elizabeth. "x- and y-Intercepts." Purplemath. Available from http://www.purplemath.com/modules/intrcept.htm.
3. Weisstein, Eric W. "Root". MathWorld--A Wolfram Web Resource. Retrieved 2010-09-22.
| Wikipedia |
Quadratic function
In mathematics, a quadratic polynomial is a polynomial of degree two in one or more variables. A quadratic function is the polynomial function defined by a quadratic polynomial. Before the 20th century, the distinction was unclear between a polynomial and its associated polynomial function; so "quadratic polynomial" and "quadratic function" were almost synonymous. This is still the case in many elementary courses, where both terms are often abbreviated as "quadratic".
For the zeros of a quadratic function, see Quadratic equation and Quadratic formula.
For example, a univariate (single-variable) quadratic function has the form[1]
$f(x)=ax^{2}+bx+c,\quad a\neq 0,$
where x is its variable. The graph of a univariate quadratic function is a parabola, a curve that has an axis of symmetry parallel to the y-axis.
If a quadratic function is equated with zero, then the result is a quadratic equation. The solutions of a quadratic equation are the zeros of the corresponding quadratic function.
The bivariate case in terms of variables x and y has the form
$f(x,y)=ax^{2}+bxy+cy^{2}+dx+ey+f,$
with at least one of a, b, c not equal to zero. The zeros of this quadratic function is, in general (that is, if a certain expression of the coefficients is not equal to zero), a conic section (a circle or other ellipse, a parabola, or a hyperbola).
A quadratic function in three variables x, y, and z contains exclusively terms x2, y2, z2, xy, xz, yz, x, y, z, and a constant:
$f(x,y,z)=ax^{2}+by^{2}+cz^{2}+dxy+exz+fyz+gx+hy+iz+j,$
where at least one of the coefficients a, b, c, d, e, f of the second-degree terms is not zero.
A quadratic function can have an arbitrarily large number of variables. The set of its zero form a quadric, which is a surface in the case of three variables and a hypersurface in general case.
Etymology
The adjective quadratic comes from the Latin word quadrātum ("square"). A term raised to the second power like x2 is called a square in algebra because it is the area of a square with side x.
Terminology
Coefficients
The coefficients of a quadric function are often taken to be real or complex numbers, but they may be taken in any ring, in which case the domain and the codomain are this ring (see polynomial evaluation).
Degree
When using the term "quadratic polynomial", authors sometimes mean "having degree exactly 2", and sometimes "having degree at most 2". If the degree is less than 2, this may be called a "degenerate case". Usually the context will establish which of the two is meant.
Sometimes the word "order" is used with the meaning of "degree", e.g. a second-order polynomial. However, where the "degree of a polynomial" refers to the largest degree of a non-zero term of the polynomial, more typically "order" refers to the lowest degree of a non-zero term of a power series.
Variables
A quadratic polynomial may involve a single variable x (the univariate case), or multiple variables such as x, y, and z (the multivariate case).
The one-variable case
Any single-variable quadratic polynomial may be written as
$ax^{2}+bx+c,$
where x is the variable, and a, b, and c represent the coefficients. Such polynomials often arise in a quadratic equation $ax^{2}+bx+c=0.$ The solutions to this equation are called the roots and can be expressed in terms of the coefficients as the quadratic formula. Each quadratic polynomial has an associated quadratic function, whose graph is a parabola.
Bivariate and multivariate cases
Any quadratic polynomial with two variables may be written as
$ax^{2}+by^{2}+cxy+dx+ey+f,$
where x and y are the variables and a, b, c, d, e, f are the coefficients, and one of a, b and c is nonzero. Such polynomials are fundamental to the study of conic sections, as the implicit equation of a conic section is obtained by equating to zero a quadratic polynomial, and the zeros of a quadratic function form a (possibly degenerate) conic section.
Similarly, quadratic polynomials with three or more variables correspond to quadric surfaces or hypersurfaces.
Quadratic polynomials that have only terms of degree two are called quadratic forms.
Forms of a univariate quadratic function
A univariate quadratic function can be expressed in three formats:[2]
• $f(x)=ax^{2}+bx+c$ is called the standard form,
• $f(x)=a(x-r_{1})(x-r_{2})$ is called the factored form, where r1 and r2 are the roots of the quadratic function and the solutions of the corresponding quadratic equation.
• $f(x)=a(x-h)^{2}+k$ is called the vertex form, where h and k are the x and y coordinates of the vertex, respectively.
The coefficient a is the same value in all three forms. To convert the standard form to factored form, one needs only the quadratic formula to determine the two roots r1 and r2. To convert the standard form to vertex form, one needs a process called completing the square. To convert the factored form (or vertex form) to standard form, one needs to multiply, expand and/or distribute the factors.
Graph of the univariate function
Regardless of the format, the graph of a univariate quadratic function $f(x)=ax^{2}+bx+c$ is a parabola (as shown at the right). Equivalently, this is the graph of the bivariate quadratic equation $y=ax^{2}+bx+c$.
• If a > 0, the parabola opens upwards.
• If a < 0, the parabola opens downwards.
The coefficient a controls the degree of curvature of the graph; a larger magnitude of a gives the graph a more closed (sharply curved) appearance.
The coefficients b and a together control the location of the axis of symmetry of the parabola (also the x-coordinate of the vertex and the h parameter in the vertex form) which is at
$x=-{\frac {b}{2a}}.$
The coefficient c controls the height of the parabola; more specifically, it is the height of the parabola where it intercepts the y-axis.
Vertex
The vertex of a parabola is the place where it turns; hence, it is also called the turning point. If the quadratic function is in vertex form, the vertex is (h, k). Using the method of completing the square, one can turn the standard form
$f(x)=ax^{2}+bx+c$
into
${\begin{aligned}f(x)&=ax^{2}+bx+c\\&=a(x-h)^{2}+k\\&=a\left(x-{\frac {-b}{2a}}\right)^{2}+\left(c-{\frac {b^{2}}{4a}}\right),\\\end{aligned}}$
so the vertex, (h, k), of the parabola in standard form is
$\left(-{\frac {b}{2a}},c-{\frac {b^{2}}{4a}}\right).$
If the quadratic function is in factored form
$f(x)=a(x-r_{1})(x-r_{2})$
the average of the two roots, i.e.,
${\frac {r_{1}+r_{2}}{2}}$
is the x-coordinate of the vertex, and hence the vertex (h, k) is
$\left({\frac {r_{1}+r_{2}}{2}},f\left({\frac {r_{1}+r_{2}}{2}}\right)\right).$
The vertex is also the maximum point if a < 0, or the minimum point if a > 0.
The vertical line
$x=h=-{\frac {b}{2a}}$
that passes through the vertex is also the axis of symmetry of the parabola.
Maximum and minimum points
Using calculus, the vertex point, being a maximum or minimum of the function, can be obtained by finding the roots of the derivative:
$f(x)=ax^{2}+bx+c\quad \Rightarrow \quad f'(x)=2ax+b$
x is a root of f '(x) if f '(x) = 0 resulting in
$x=-{\frac {b}{2a}}$
with the corresponding function value
$f(x)=a\left(-{\frac {b}{2a}}\right)^{2}+b\left(-{\frac {b}{2a}}\right)+c=c-{\frac {b^{2}}{4a}},$
so again the vertex point coordinates, (h, k), can be expressed as
$\left(-{\frac {b}{2a}},c-{\frac {b^{2}}{4a}}\right).$
Roots of the univariate function
Further information: Quadratic equation
Exact roots
The roots (or zeros), r1 and r2, of the univariate quadratic function
${\begin{aligned}f(x)&=ax^{2}+bx+c\\&=a(x-r_{1})(x-r_{2}),\\\end{aligned}}$
are the values of x for which f(x) = 0.
When the coefficients a, b, and c, are real or complex, the roots are
$r_{1}={\frac {-b-{\sqrt {b^{2}-4ac}}}{2a}},$
$r_{2}={\frac {-b+{\sqrt {b^{2}-4ac}}}{2a}}.$
Upper bound on the magnitude of the roots
The modulus of the roots of a quadratic $ax^{2}+bx+c$ can be no greater than ${\frac {\max(|a|,|b|,|c|)}{|a|}}\times \phi ,$ where $\phi $ is the golden ratio ${\frac {1+{\sqrt {5}}}{2}}.$[4]
The square root of a univariate quadratic function
The square root of a univariate quadratic function gives rise to one of the four conic sections, almost always either to an ellipse or to a hyperbola.
If $a>0,$ then the equation $y=\pm {\sqrt {ax^{2}+bx+c}}$ describes a hyperbola, as can be seen by squaring both sides. The directions of the axes of the hyperbola are determined by the ordinate of the minimum point of the corresponding parabola $y_{p}=ax^{2}+bx+c.$ If the ordinate is negative, then the hyperbola's major axis (through its vertices) is horizontal, while if the ordinate is positive then the hyperbola's major axis is vertical.
If $a<0,$ then the equation $y=\pm {\sqrt {ax^{2}+bx+c}}$ describes either a circle or other ellipse or nothing at all. If the ordinate of the maximum point of the corresponding parabola $y_{p}=ax^{2}+bx+c$ is positive, then its square root describes an ellipse, but if the ordinate is negative then it describes an empty locus of points.
Iteration
To iterate a function $f(x)=ax^{2}+bx+c$, one applies the function repeatedly, using the output from one iteration as the input to the next.
One cannot always deduce the analytic form of $f^{(n)}(x)$, which means the nth iteration of $f(x)$. (The superscript can be extended to negative numbers, referring to the iteration of the inverse of $f(x)$ if the inverse exists.) But there are some analytically tractable cases.
For example, for the iterative equation
$f(x)=a(x-c)^{2}+c$
one has
$f(x)=a(x-c)^{2}+c=h^{(-1)}(g(h(x))),$
where
$g(x)=ax^{2}$ and $h(x)=x-c.$
So by induction,
$f^{(n)}(x)=h^{(-1)}(g^{(n)}(h(x)))$
can be obtained, where $g^{(n)}(x)$ can be easily computed as
$g^{(n)}(x)=a^{2^{n}-1}x^{2^{n}}.$
Finally, we have
$f^{(n)}(x)=a^{2^{n}-1}(x-c)^{2^{n}}+c$
as the solution.
See Topological conjugacy for more detail about the relationship between f and g. And see Complex quadratic polynomial for the chaotic behavior in the general iteration.
The logistic map
$x_{n+1}=rx_{n}(1-x_{n}),\quad 0\leq x_{0}<1$
with parameter 2<r<4 can be solved in certain cases, one of which is chaotic and one of which is not. In the chaotic case r=4 the solution is
$x_{n}=\sin ^{2}(2^{n}\theta \pi )$
where the initial condition parameter $\theta $ is given by $\theta ={\tfrac {1}{\pi }}\sin ^{-1}(x_{0}^{1/2})$. For rational $\theta $, after a finite number of iterations $x_{n}$ maps into a periodic sequence. But almost all $\theta $ are irrational, and, for irrational $\theta $, $x_{n}$ never repeats itself – it is non-periodic and exhibits sensitive dependence on initial conditions, so it is said to be chaotic.
The solution of the logistic map when r=2 is
$x_{n}={\frac {1}{2}}-{\frac {1}{2}}(1-2x_{0})^{2^{n}}$
for $x_{0}\in [0,1)$. Since $(1-2x_{0})\in (-1,1)$ for any value of $x_{0}$ other than the unstable fixed point 0, the term $(1-2x_{0})^{2^{n}}$ goes to 0 as n goes to infinity, so $x_{n}$ goes to the stable fixed point ${\tfrac {1}{2}}.$
Bivariate (two variable) quadratic function
Further information: Quadric and Quadratic form
A bivariate quadratic function is a second-degree polynomial of the form
$f(x,y)=Ax^{2}+By^{2}+Cx+Dy+Exy+F,$
where A, B, C, D, and E are fixed coefficients and F is the constant term. Such a function describes a quadratic surface. Setting $f(x,y)$ equal to zero describes the intersection of the surface with the plane $z=0,$ which is a locus of points equivalent to a conic section.
Minimum/maximum
If $4AB-E^{2}<0,$ the function has no maximum or minimum; its graph forms a hyperbolic paraboloid.
If $4AB-E^{2}>0,$ the function has a minimum if both A > 0 and B > 0, and a maximum if both A < 0 and B < 0; its graph forms an elliptic paraboloid. In this case the minimum or maximum occurs at $(x_{m},y_{m}),$ where:
$x_{m}=-{\frac {2BC-DE}{4AB-E^{2}}},$
$y_{m}=-{\frac {2AD-CE}{4AB-E^{2}}}.$
If $4AB-E^{2}=0$ and $DE-2CB=2AD-CE\neq 0,$ the function has no maximum or minimum; its graph forms a parabolic cylinder.
If $4AB-E^{2}=0$ and $DE-2CB=2AD-CE=0,$ the function achieves the maximum/minimum at a line—a minimum if A>0 and a maximum if A<0; its graph forms a parabolic cylinder.
See also
• Quadratic form
• Quadratic equation
• Matrix representation of conic sections
• Quadric
• Periodic points of complex quadratic mappings
• List of mathematical functions
References
1. "Quadratic Equation from Wolfram MathWorld". Retrieved January 6, 2013.
2. Hughes-Hallett, Deborah; Connally, Eric; McCallum, William G. (2007), College Algebra, John Wiley & Sons Inc., p. 205, ISBN 9780471271758
3. "Complex Roots Made Visible – Math Fun Facts". Retrieved 1 October 2016.
4. Lord, Nick, "Golden bounds for the roots of quadratic equations", Mathematical Gazette 91, November 2007, 549.
• Algebra 1, Glencoe, ISBN 0-07-825083-8
• Algebra 2, Saxon, ISBN 0-939798-62-X
External links
• Weisstein, Eric W. "Quadratic". MathWorld.
Polynomials and polynomial functions
By degree
• Zero polynomial (degree undefined or −1 or −∞)
• Constant function (0)
• Linear function (1)
• Linear equation
• Quadratic function (2)
• Quadratic equation
• Cubic function (3)
• Cubic equation
• Quartic function (4)
• Quartic equation
• Quintic function (5)
• Sextic equation (6)
• Septic equation (7)
By properties
• Univariate
• Bivariate
• Multivariate
• Monomial
• Binomial
• Trinomial
• Irreducible
• Square-free
• Homogeneous
• Quasi-homogeneous
Tools and algorithms
• Factorization
• Greatest common divisor
• Division
• Horner's method of evaluation
• Resultant
• Discriminant
• Gröbner basis
Authority control: National
• Germany
| Wikipedia |
YBC 7289
YBC 7289 is a Babylonian clay tablet notable for containing an accurate sexagesimal approximation to the square root of 2, the length of the diagonal of a unit square. This number is given to the equivalent of six decimal digits, "the greatest known computational accuracy ... in the ancient world".[1] The tablet is believed to be the work of a student in southern Mesopotamia from some time between 1800 and 1600 BC.
Content
The tablet depicts a square with its two diagonals. One side of the square is labeled with the sexagesimal number 30. The diagonal of the square is labeled with two sexagesimal numbers. The first of these two, 1;24,51,10 represents the number 305470/216000 ≈ 1.414213, a numerical approximation of the square root of two that is off by less than one part in two million. The second of the two numbers is 42;25,35 = 30547/720 ≈ 42.426. This number is the result of multiplying 30 by the given approximation to the square root of two, and approximates the length of the diagonal of a square of side length 30.[2]
Because the Babylonian sexagesimal notation did not indicate which digit had which place value, one alternative interpretation is that the number on the side of the square is 30/60 = 1/2. Under this alternative interpretation, the number on the diagonal is 30547/43200 ≈ 0.70711, a close numerical approximation of $1/{\sqrt {2}}$, the length of the diagonal of a square of side length 1/2, that is also off by less than one part in two million. David Fowler and Eleanor Robson write, "Thus we have a reciprocal pair of numbers with a geometric interpretation…". They point out that, while the importance of reciprocal pairs in Babylonian mathematics makes this interpretation attractive, there are reasons for skepticism.[2]
The reverse side is partly erased, but Robson believes it contains a similar problem concerning the diagonal of a rectangle whose two sides and diagonal are in the ratio 3:4:5.[3]
Interpretation
Although YBC 7289 is frequently depicted (as in the photo) with the square oriented diagonally, the standard Babylonian conventions for drawing squares would have made the sides of the square vertical and horizontal, with the numbered side at the top.[4] The small round shape of the tablet, and the large writing on it, suggests that it was a "hand tablet" of a type typically used for rough work by a student who would hold it in the palm of his hand.[1][2] The student would likely have copied the sexagesimal value of the square root of 2 from another tablet, but an iterative procedure for computing this value can be found in another Babylonian tablet, BM 96957 + VAT 6598.[2]
The mathematical significance of this tablet was first recognized by Otto E. Neugebauer and Abraham Sachs in 1945.[2][5] The tablet "demonstrates the greatest known computational accuracy obtained anywhere in the ancient world", the equivalent of six decimal digits of accuracy.[1] Other Babylonian tablets include the computations of areas of hexagons and heptagons, which involve the approximation of more complicated algebraic numbers such as ${\sqrt {3}}$.[2] The same number ${\sqrt {3}}$ can also be used in the interpretation of certain ancient Egyptian calculations of the dimensions of pyramids. However, the much greater numerical precision of the numbers on YBC 7289 makes it more clear that they are the result of a general procedure for calculating them, rather than merely being an estimate.[6]
The same sexagesimal approximation to ${\sqrt {2}}$, 1;24,51,10, was used much later by Greek mathematician Claudius Ptolemy in his Almagest.[7][8] Ptolemy did not explain where this approximation came from and it may be assumed to have been well known by his time.[7]
Provenance and curation
It is unknown where in Mesopotamia YBC 7289 comes from, but its shape and writing style make it likely that it was created in southern Mesopotamia, sometime between 1800BC and 1600BC.[1][2]
At Yale, the Institute for the Preservation of Cultural Heritage has produced a digital model of the tablet, suitable for 3D printing.[9][10][11] The original tablet is currently kept in the Yale Babylonian Collection at Yale University.[10]
See also
Wikimedia Commons has media related to YBC 7289.
• Plimpton 322
• IM 67118
References
1. Beery, Janet L.; Swetz, Frank J. (July 2012), "The best known old Babylonian tablet?", Convergence, Mathematical Association of America, doi:10.4169/loci003889
2. Fowler, David; Robson, Eleanor (1998), "Square root approximations in old Babylonian mathematics: YBC 7289 in context", Historia Mathematica, 25 (4): 366–378, doi:10.1006/hmat.1998.2209, MR 1662496
3. Robson, Eleanor (2007), "Mesopotamian Mathematics", in Katz, Victor J. (ed.), The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, Princeton University Press, p. 143, ISBN 978-0-691-11485-9
4. Friberg, Jöran (2007), Friberg, Jöran (ed.), A remarkable collection of Babylonian mathematical texts, Sources and Studies in the History of Mathematics and Physical Sciences, Springer, New York, p. 211, doi:10.1007/978-0-387-48977-3, ISBN 978-0-387-34543-7, MR 2333050
5. Neugebauer, O.; Sachs, A. J. (1945), Mathematical Cuneiform Texts, American Oriental Series, American Oriental Society and the American Schools of Oriental Research, New Haven, Conn., p. 43, MR 0016320
6. Rudman, Peter S. (2007), How mathematics happened: the first 50,000 years, Prometheus Books, Amherst, NY, p. 241, ISBN 978-1-59102-477-4, MR 2329364
7. Neugebauer, O. (1975), A History of Ancient Mathematical Astronomy, Part One, Springer-Verlag, New York-Heidelberg, pp. 22–23, ISBN 978-3-642-61910-6, MR 0465672
8. Pedersen, Olaf (2011), Jones, Alexander (ed.), A Survey of the Almagest, Sources and Studies in the History of Mathematics and Physical Sciences, Springer, p. 57, ISBN 978-0-387-84826-6
9. Lynch, Patrick (April 11, 2016), "A 3,800-year journey from classroom to classroom", Yale News, retrieved 2017-10-25
10. A 3D-print of ancient history: one of the most famous mathematical texts from Mesopotamia, Yale Institute for the Preservation of Cultural Heritage, January 16, 2016, retrieved 2017-10-25
11. Kwan, Alistair (April 20, 2019), Mesopotamian tablet YBC 7289, University of Auckland, doi:10.17608/k6.auckland.6114425.v1
| Wikipedia |
Y and H transforms
In mathematics, the Y transforms and H transforms are complementary pairs of integral transforms involving, respectively, the Neumann function (Bessel function of the second kind) Yν of order ν and the Struve function Hν of the same order.
For a given function f(r), the Y-transform of order ν is given by
$F(k)=\int _{0}^{\infty }f(r)Y_{\nu }(kr){\sqrt {kr}}\,dr$
The inverse of above is the H-transform of the same order; for a given function F(k), the H-transform of order ν is given by
$f(r)=\int _{0}^{\infty }F(k)\mathbf {H} _{\nu }(kr){\sqrt {kr}}\,dk$
These transforms are closely related to the Hankel transform, as both involve Bessel functions. In problems of mathematical physics and applied mathematics, the Hankel, Y, H transforms all may appear in problems having axial symmetry. Hankel transforms are however much more commonly seen due to their connection with the 2-dimensional Fourier transform. The Y, H transforms appear in situations with singular behaviour on the axis of symmetry (Rooney).
References
• Bateman Manuscript Project: Tables of Integral Transforms Vol. II. Contains extensive tables of transforms: Chapter IX (Y-transforms) and Chapter XI (H-transforms).
• Rooney, P. G. (1980). "On the Yν and Hν transformations". Canadian Journal of Mathematics. 32 (5): 1021. doi:10.4153/CJM-1980-079-4.
| Wikipedia |
Yael Dowker
Yael Naim Dowker (1919–2016)[1] was an English mathematician, prominent especially due to her work in the fields of measure theory, ergodic theory and topological dynamics.
Yael Dowker
Yael Dowker in New Orleans, 1961.
Born
Yael Naim
1919
Tel Aviv, Mandatory Palestine
Died2016 (aged 96–97)
Academic background
Alma materJohns Hopkins University
Radcliffe College
ThesisThe Ergodic Theorems and Invariant Measure (1948)
Doctoral advisorWitold Hurewicz
Academic work
DisciplineMathematics
InstitutionsInstitute for Advanced Study,
Victoria University of Manchester,
University of London
Biography
Yael Naim (later Dowker) was born in Tel Aviv.[1] She left for the United States to study at Johns Hopkins University in Baltimore, Maryland. In 1941, as a graduate student, she met Clifford Hugh Dowker, a Canadian topologist working as an instructor there. The couple married in 1944. From 1943 to 1946 they worked together at the Radiation Laboratory at Massachusetts Institute of Technology. Clifford also worked as a civilian adviser for the United States Air Force during World War II.[2]
Dowker did her doctorate at Radcliffe College (in Cambridge, Massachusetts) under Witold Hurewicz (a Polish mathematician known for the Hurewicz theorem). She published her thesis Invariant measure and the ergodic theorems in 1947 and received her Ph.D in 1948.[3] In the period between 1948 and 1949, she did post-doctoral work at the Institute for Advanced Study, located in Princeton, New Jersey. A few years after the war, McCarthyism became a common phenomenon in the academic world, with several of the Dowker couple's friends in the mathematical community harassed and one arrested. In 1950, they emigrated to the United Kingdom.[2]
In 1951 Dowker served as a professor at the University of Manchester,[4] and later went on as a professor at the Imperial College London, where she was the first female reader within the department.[1] While there, among the students she advised was Bill Parry, who published his thesis in 1960.[3] She also cooperated on some of her work with the Hungarian mathematician Paul Erdős (Erdős' number of one). She worked with her husband with gifted children who were having difficulties at school for the National association for gifted children.[1][2]
Legacy
The best PhD award at Imperial College London is given in her name each year.[5]
Works
• Invariant measure and the ergodic theorems, Duke Math. J. 14 (1947), 1051–1061
• Finite and $\sigma $-finite measures, Annals of Mathematics, 54 (1951), 595–608
• The mean and transitive points of homeomorphisms, Annals of Mathematics, 58 (1953), 123–133
• On limit sets in dynamical systems, Proc. London Math. Soc. 4 (1954), 168–176 (with Friedlander, F. G.)
• On minimal sets in dynamical systems, Quart. J. Math. Oxford Ser. (2) 7 (1956), 5–16
• Some examples in ergodic theory, Proc. London Math. Soc. 9 (1959), 227–241 (with Erdős, Paul)
References
1. Barrett, Anne (2017). Women At Imperial College; Past, Present And Future. World Scientific. p. 303. ISBN 9781786342645.
2. James, I. M.; Kronheimer, E. H. (31 January 1985). Aspects of Topology: In Memory of Hugh Dowker 1912–1982. Cambridge University Press. pp. 11–12. ISBN 978-0-521-27815-7.
3. "Yael Dowker". Mathematics Genealogy Project. Retrieved 14 October 2015.
4. "Bulletin of the American Mathematical Society". American Mathematical Society. 1951: 100. {{cite journal}}: Cite journal requires |journal= (help)
5. Anne, Barrett (2017-02-24). Women At Imperial College; Past, Present And Future. World Scientific. ISBN 9781786342645.
External links
• "Yael N. Dowker". Institute for Advanced Study. 1948-09-20. Retrieved 2018-02-13.
• "Yael Dowker". The Mathematics Genealogy Project. 2017-04-04. Retrieved 2018-02-13.
Authority control: Academics
• MathSciNet
• Mathematics Genealogy Project
• zbMATH
| Wikipedia |
Yair Minsky
Yair Nathan Minsky (born in 1962) is an Israeli-American mathematician whose research concerns three-dimensional topology, differential geometry, group theory and holomorphic dynamics. He is a professor at Yale University.[1] He is known for having proved Thurston's ending lamination conjecture and as a student of curve complex geometry.
Biography
Minsky obtained his Ph.D. from Princeton University in 1989 under the supervision of William Paul Thurston, with the thesis Harmonic Maps and Hyperbolic Geometry.[2]
His Ph.D. students include Jason Behrstock, Erica Klarreich, Hossein Namazi and Kasra Rafi.[2]
Honors and awards
He received a Sloan Fellowship in 1995.[3][4]
He was a speaker at the ICM (Madrid) 2006.
He was named to the 2021 class of fellows of the American Mathematical Society "for contributions to hyperbolic 3-manifolds, low-dimensional topology, geometric group theory and Teichmuller theory".[5] He was elected to the American Academy of Arts and Sciences in 2023.[6]
Selected invited talks
• Coxeter lectures (Fields Institute) 2006
• Mallat Lectures (Technion) 2008
Selected publications
• with Howard Masur: "Geometry of the complex of curves I: Hyperbolicity", Inventiones mathematicae, 138 (1), 103–149.
• with Howard Masur: "Geometry of the complex of curves II: Hierarchical structure", Geometric and Functional Analysis, 10 (4), 902–974.
• "The classification of Kleinian surface groups, I: Models and bounds", Annals of Mathematics, 171 (2010), 1–107.
• with Jeffrey Brock, and Richard Canary: "The classification of Kleinian surface groups, II: The ending lamination conjecture", Annals of Mathematics, 176 (2012), 1–149.
• with Jason Behrstock: "Dimension and rank for mapping class groups", Annals of Mathematics (2) 167 (2008), no. 3, 1055–1077.
• "The classification of punctured-torus groups", Annals of Mathematics, 149 (1999), 559–626.
• "On rigidity, limit sets, and end invariants of hyperbolic 3-manifolds", Journal of the American Mathematical Society, 7 (3), 539–588.
See also
• Ending lamination theorem
• Curve complex
Quotes
• "When Thurston proposed it, the virtual Haken conjecture seemed like a small question, but it hung on stubbornly, shining a spotlight on how little we knew about the field."[7]
References
1. Minsky's home page at Yale University
2. Yair Nathan Minsky at the Mathematics Genealogy Project
3. Alfred P. Sloan Foundation
4. Stony Brook University
5. 2021 Class of Fellows of the AMS, American Mathematical Society, retrieved 2020-11-02
6. New members, American Academy of Arts and Sciences, 2023, retrieved 2023-04-21
7. Klarreich, Erica (2 October 2012), "Getting Into Shapes: From Hyperbolic Geometry to Cube Complexes and Back", Quanta Magazine
External links
• Minsky's home page at Yale University
• Minsky's profile at Google Scholar
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Yaiza Canzani
Yaiza Canzani García is a Spanish and Uruguayan mathematician known for her work in mathematical analysis, and particularly in spectral geometry and microlocal analysis. She is an associate professor of mathematics at the University of North Carolina at Chapel Hill.[1]
Yaiza Canzani
Born
Spain
Alma materMcGill University
Scientific career
FieldsMathematics
InstitutionsUniversity of North Carolina at Chapel Hill
Doctoral advisorDmitry Jakobson, John Toth
Education and career
Canzani was born in Spain and grew up in Uruguay.[2] She was an undergraduate at the University of the Republic (Uruguay), where she earned a bachelor's degree in mathematics in 2008.[3] She completed a Ph.D. in 2013 at McGill University in Montreal, Canada, with the dissertation Spectral Geometry of Conformally Covariant Operators jointly supervised by Dmitry Jakobson and John Toth.[4]
After postdoctoral study at the Institute for Advanced Study and as a Benjamin Peirce Fellow at Harvard University, she became an assistant professor of mathematics at the University of North Carolina at Chapel Hill in 2016.[3] In 2021 she was promoted to associate professor.[3][1]
Recognition
Canzani is a recipient of a National Science Foundation CAREER Award and a Sloan Research Fellowship.[3][5] She is the 2022 winner of the Sadosky Prize in analysis of the Association for Women in Mathematics.[1][5] The award was given "in recognition of outstanding contributions in spectral geometry and microlocal analysis", citing her "breakthrough results on nodal sets, random waves, Weyl Laws, $L^{p}$-norms, and other problems on eigenfunctions and eigenvalues on Riemannian manifolds".[5]
References
1. Yaiza Canzani – recipient of the 2022 AWM–Sadosky Research Prize in Analysis, UNC Chapel Hill Department of Mathematics, retrieved 2021-09-15
2. "Yaiza Canzani", Calendar 2018, Lathisms, retrieved 2021-09-15
3. Canzani, Yaiza (21 January 2021), Curriculum vitae (PDF), retrieved 2021-09-15
4. Yaiza Canzani at the Mathematics Genealogy Project
5. "2022 Winner: Yaiza Canzani", AWM Sadosky Research Prize in Analysis, Association for Women in Mathematics, retrieved 2021-09-15
External links
• Home page
• Yaiza Canzani publications indexed by Google Scholar
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Yakov Eliashberg
Yakov Matveevich Eliashberg (also Yasha Eliashberg; Russian: Яков Матвеевич Элиашберг; born 11 December 1946) is an American mathematician who was born in Leningrad, USSR.
Yakov Eliashberg
Eliashberg in 1988.
Born (1946-12-11) 11 December 1946
Leningrad, USSR
NationalityAmerican
Alma materSt. Petersburg State University
Known forHomotopy principle
AwardsOswald Veblen Prize in Geometry (2001)
Heinz Hopf Prize (2013)
Crafoord Prize (2016)
Wolf Prize in Mathematics (2020)
Scientific career
FieldsMathematics
InstitutionsStanford University
ThesisSurgery of Singularities of Smooth Mappings (1972)
Doctoral advisorVladimir Rokhlin[1]
Doctoral students
• Eric Katz
• Emmy Murphy
• John Pardon
Websitemathematics.stanford.edu/people/yakov-eliashberg
Education and career
Eliashberg received his PhD, entitled Surgery of Singularities of Smooth Mappings, from Leningrad University in 1972, under the direction of Vladimir Rokhlin.[1]
Due to the growing anti-Semitism in the Soviet Union, from 1972 to 1979 he had to work at the Syktyvkar State University in the isolated Komi Republic. In 1980 Eliashberg returned to Leningrad and applied for a visa, but his request was denied and he became a refusenik until 1987. He was cut off from mathematical life and was prevented to work in academia, but due to a friend's intercession, he managed to secure a job in industry as the head of a computer software group.[2][3][4]
In 1988 Eliashberg managed to move to the United States, and since 1989 he has been Herald L. and Caroline L. Ritch professor of mathematics at Stanford University.[5] Between 2001 and 2002 he was Distinguished Visiting professor at the Institute of Advanced Studies.[6]
Awards
Eliashberg received the "Young Mathematician" Prize from the Leningrad Mathematical Society in 1972.[7] He was an invited speaker at the International Congress of Mathematicians in 1986,[8] 1998[9] and 2006 (plenary lecture).[10] In 1995 he was a recipient of the Guggenheim Fellowship.[11]
In 2001 Eliashberg was awarded the Oswald Veblen Prize in Geometry from the AMS for his work in symplectic and contact topology,[12] in particular for his proof of the symplectic rigidity[13] and the development of 3-dimensional contact topology.[14]
In 2002 Eliashberg was elected to the National Academy of Sciences of the US[15] and in 2012 he became a fellow of the American Mathematical Society.[16] He also was a member of the Selection Committee in mathematical sciences of the Shaw Prize.[17] He received a Doctorat Honoris Causa from the ENS Lyon in 2009[18] and from the University of Uppsala in 2017.[19]
In 2013 Eliashberg shared with Helmut Hofer the Heinz Hopf Prize from the ETH, Zurich, for their pioneering research in symplectic topology.[20] In 2016 Yakov Eliashberg was awarded the Crafoord Prize in Mathematics from the Swedish Academy of Sciences for the development of contact and symplectic topology and groundbreaking discoveries of rigidity and flexibility phenomena.[21]
In 2020 he received the Wolf Prize in Mathematics (jointly with Simon K. Donaldson).[2][22][23] He was elected to the American Academy of Arts and Sciences in 2021.[24]
Research
Eliashberg's research interests are in differential topology, especially in symplectic and contact topology.[4]
In the 80's he developed a combinatorial technique[13] which he used to prove that the group of symplectomorphisms is ${\mathcal {C}}^{0}$-closed in the diffeomorphism group.[25] This fundamental result, proved in a different way also by Gromov[26] is now called the Eliashberg-Gromov theorem, and is one of the first manifestation of symplectic rigidity.
In 1990 he discovered a complete topological characterization of Stein manifolds of complex dimension greater than 2.[27]
Eliashberg classified contact structures into "tight" and "overtwisted" ones.[28] Using this dichotomy, he gave the complete classification of contact structures on the 3-sphere.[14] Together with Thurston, he developed the theory of confoliations, which unifies foliations and contact structures.[29]
Eliashberg worked on various aspects of the h-principle, introduced by Mikhail Gromov, and he wrote in 2002 an introductory book on the subject.[30]
Together with Givental and Hofer, Eliashberg pioneered the foundations of symplectic field theory.[31]
He supervised 41 PhD students as of 2022.[1]
Major publications
• Eliashberg, Y. (1989). "Classification of overtwisted contact structures on 3-manifolds". Inventiones Mathematicae. Springer Science and Business Media LLC. 98 (3): 623–637. Bibcode:1989InMat..98..623E. doi:10.1007/bf01393840. ISSN 0020-9910. S2CID 121666486.
• Eliashberg, Yakov (24 January 1991). "Filling by holomorphic discs and its applications". Geometry of Low-Dimensional Manifolds. Cambridge University Press. pp. 45–68. doi:10.1017/cbo9780511629341.006. ISBN 978-0-521-40001-5.
• Eliashberg, Yakov (1990). "Topological Characterization of Stein Manifolds of Dimension >2". International Journal of Mathematics. World Scientific Pub Co Pte Lt. 01 (1): 29–46. doi:10.1142/s0129167x90000034. ISSN 0129-167X.
• Eliashberg, Yakov; Ogawa, Noboru; Yoshiyasu, Toru (1 June 2021). "Stabilized convex symplectic manifolds are Weinstein". Kyoto Journal of Mathematics. Duke University Press. 61 (2). arXiv:2003.12251. doi:10.1215/21562261-2021-0004. ISSN 2156-2261. S2CID 214693087.
• Eliashberg, Yakov (1992). "Contact 3-manifolds twenty years since J. Martinet's work". Annales de l'Institut Fourier. Cellule MathDoc/CEDRAM. 42 (1–2): 165–192. doi:10.5802/aif.1288. ISSN 0373-0956.
• Eliashberg, Y.; Glvental, A.; Hofer, H. (2000). "Introduction to Symplectic Field Theory". Visions in Mathematics. Basel: Birkhäuser Basel. doi:10.1007/978-3-0346-0425-3_4. ISBN 978-3-0346-0424-6. S2CID 6725644.
• Bourgeois, Frederic; Eliashberg, Yakov; Hofer, Helmut; Wysocki, Kris; Zehnder, Eduard (4 December 2003). "Compactness results in Symplectic Field Theory". Geometry & Topology. Mathematical Sciences Publishers. 7 (2): 799–888. arXiv:math/0308183. doi:10.2140/gt.2003.7.799. ISSN 1364-0380. S2CID 11794561.
Books
• Eliashberg, Yakov M.; Thurston, William P. Confoliations. University Lecture Series, 13. American Mathematical Society, Providence, RI, 1998. x+66 pp. ISBN 0-8218-0776-5
• Eliashberg, Y.; Mishachev, N. Introduction to the h-principle. Graduate Studies in Mathematics, 48. American Mathematical Society, Providence, RI, 2002. xviii+206 pp. ISBN 0-8218-3227-1
• Cieliebak, Kai; Eliashberg, Yakov. From Stein to Weinstein and back. Symplectic geometry of affine complex manifolds. American Mathematical Society Colloquium Publications, 59. American Mathematical Society, Providence, RI, 2012. xii+364 pp. ISBN 978-0-8218-8533-8
References
1. Yakov Eliashberg at the Mathematics Genealogy Project
2. "Yakov Eliashberg". Wolf Foundation. 2020-01-13. Retrieved 2022-08-08.
3. Schulman, Julia; Hsieh, Michael (2021-02-11). "Coffin Problems: Soviet Anti-Semitism Buried Rising Jewish Scientists". Tablet Magazine. Retrieved 2022-08-08.
4. New perspectives and challenges in symplectic field theory (PDF). Miguel Abreu, François Lalonde, Leonid Polterovich. Providence, R.I.: American Mathematical Society. 2009. ISBN 978-0-8218-4356-7. OCLC 370387862.{{cite book}}: CS1 maint: others (link)
5. "Yakov Eliashberg". mathematics.stanford.edu. Retrieved 2022-08-09.
6. "Yakov Eliashberg". www.ias.edu. 2019-12-09. Retrieved 2022-08-08.
7. "SPb. Math. Society: the awards". Saint Petersburg Mathematical Society. Retrieved 2022-08-09.
8. Gleason, Andrew M., ed. (1986). Proceedings of the International Congress of Mathematician 1986 (PDF). Vol. 1. Berkeley: American Mathematical Society. pp. 531–539.
9. Louis, Alfred K.; Schneider, Peter, eds. (1998). Proceedings of the International Congress of Mathematician 1998 (PDF). Vol. 2. Berlin: German Mathematical Society. pp. 327–338.
10. Sanz-Solé, Marta; Soria, Javier; Varona, Juan Luis; Verdera, Joan, eds. (2007). Proceedings of the International Congress of Mathematician 2006 (PDF). Vol. 1. Madrid: European Mathematical Society. pp. 217–246.
11. "Yakov Eliashberg". John Simon Guggenheim Memorial Foundation. Retrieved 2022-08-08.
12. "2001 Veblen Prize" (PDF). Notices of the AMS. 48 (4): 408–410.
13. Eliashberg, Ya M. (1986). "Combinatorial methods in symplectic geometry". Proc. of the International Congress of Mathematicians, 1986. pp. 531–539.
14. Eliashberg, Yakov (1992). "Contact 3-manifolds twenty years since J. Martinet's work". Annales de l'Institut Fourier. 42 (1–2): 165–192. doi:10.5802/aif.1288. ISSN 0373-0956.
15. "Yakov Eliashberg". www.nasonline.org. Retrieved 2022-08-08.
16. List of Fellows of the American Mathematical Society, retrieved 2012-12-02.
17. "Award Presentation Ceremony 2012 | The Shaw Prize". www.shawprize.org. Retrieved 2022-08-08.
18. Mangin, Fabienne (2019). "La remise des insignes de Docteur 'Honoris Causa', une tradition au sein de l'ENS de Lyon" [The presentation of the insignia of doctor "Honoris Causa", a tradition within the ENS of Lyon]. alumni.ens-lyon.fr (in French). Retrieved 2022-08-08.
19. Piehl, Jakob. "Honorary Doctors of the Faculty of Science and Technology - Uppsala University, Sweden". www.uu.se. Retrieved 2022-08-08.
20. "Laureates 2013". math.ethz.ch. Retrieved 2022-08-08.
21. "The Crafoord Prizes in Mathematics and Astronomy 2016".
22. University, Stanford (2020-01-17). "Yakov Eliashberg awarded Wolf Prize in Mathematics". Stanford News. Retrieved 2022-08-08.
23. Kehoe, Elaine (2020-06-01). "Donaldson and Eliashberg Awarded 2020 Wolf Prize". Notices of the American Mathematical Society. 67 (6): 1. doi:10.1090/noti2109. ISSN 0002-9920. S2CID 225820459.
24. "Yakov Eliashberg". American Academy of Arts & Sciences. Retrieved 2022-08-08.
25. Eliashberg, Ya. M. (1987-07-01). "A theorem on the structure of wave fronts and its applications in symplectic topology". Functional Analysis and Its Applications. 21 (3): 227–232. doi:10.1007/BF02577138. ISSN 1573-8485. S2CID 121961311.
26. Gromov, Mikhael (1986). Partial Differential Relations. doi:10.1007/978-3-662-02267-2. ISBN 978-3-642-05720-5.
27. Eliashberg, Yakov (1990-03-01). "Topological characterization of stein manifolds of dimension >2". International Journal of Mathematics. 01 (1): 29–46. doi:10.1142/S0129167X90000034. ISSN 0129-167X.
28. Eliashberg, Y. (1989-10-01). "Classification of overtwisted contact structures on 3-manifolds". Inventiones Mathematicae. 98 (3): 623–637. Bibcode:1989InMat..98..623E. doi:10.1007/BF01393840. ISSN 1432-1297. S2CID 121666486.
29. Eliashberg, Y.; Thurston, William P. (1998). Confoliations. Providence, R.I.: American Mathematical Society. ISBN 0-8218-0776-5. OCLC 37748408.
30. Eliashberg, Y.; Mishachev, N. (2002). Introduction to the h-principle. Providence, Rhode Island. ISBN 0-8218-3227-1. OCLC 49312496.{{cite book}}: CS1 maint: location missing publisher (link)
31. Eliashberg, Y.; Glvental, A.; Hofer, H. (2010), Alon, N.; Bourgain, J.; Connes, A.; Gromov, M. (eds.), "Introduction to Symplectic Field Theory", Visions in Mathematics: GAFA 2000 Special volume, Part II, Basel: Birkhäuser, pp. 560–673, doi:10.1007/978-3-0346-0425-3_4, ISBN 978-3-0346-0425-3, S2CID 6725644, retrieved 2022-08-09
Recipients of the Oswald Veblen Prize in Geometry
• 1964 Christos Papakyriakopoulos
• 1964 Raoul Bott
• 1966 Stephen Smale
• 1966 Morton Brown and Barry Mazur
• 1971 Robion Kirby
• 1971 Dennis Sullivan
• 1976 William Thurston
• 1976 James Harris Simons
• 1981 Mikhail Gromov
• 1981 Shing-Tung Yau
• 1986 Michael Freedman
• 1991 Andrew Casson and Clifford Taubes
• 1996 Richard S. Hamilton and Gang Tian
• 2001 Jeff Cheeger, Yakov Eliashberg and Michael J. Hopkins
• 2004 David Gabai
• 2007 Peter Kronheimer and Tomasz Mrowka; Peter Ozsváth and Zoltán Szabó
• 2010 Tobias Colding and William Minicozzi; Paul Seidel
• 2013 Ian Agol and Daniel Wise
• 2016 Fernando Codá Marques and André Neves
• 2019 Xiuxiong Chen, Simon Donaldson and Song Sun
Laureates of the Wolf Prize in Mathematics
1970s
• Israel Gelfand / Carl L. Siegel (1978)
• Jean Leray / André Weil (1979)
1980s
• Henri Cartan / Andrey Kolmogorov (1980)
• Lars Ahlfors / Oscar Zariski (1981)
• Hassler Whitney / Mark Krein (1982)
• Shiing-Shen Chern / Paul Erdős (1983/84)
• Kunihiko Kodaira / Hans Lewy (1984/85)
• Samuel Eilenberg / Atle Selberg (1986)
• Kiyosi Itô / Peter Lax (1987)
• Friedrich Hirzebruch / Lars Hörmander (1988)
• Alberto Calderón / John Milnor (1989)
1990s
• Ennio de Giorgi / Ilya Piatetski-Shapiro (1990)
• Lennart Carleson / John G. Thompson (1992)
• Mikhail Gromov / Jacques Tits (1993)
• Jürgen Moser (1994/95)
• Robert Langlands / Andrew Wiles (1995/96)
• Joseph Keller / Yakov G. Sinai (1996/97)
• László Lovász / Elias M. Stein (1999)
2000s
• Raoul Bott / Jean-Pierre Serre (2000)
• Vladimir Arnold / Saharon Shelah (2001)
• Mikio Sato / John Tate (2002/03)
• Grigory Margulis / Sergei Novikov (2005)
• Stephen Smale / Hillel Furstenberg (2006/07)
• Pierre Deligne / Phillip A. Griffiths / David B. Mumford (2008)
2010s
• Dennis Sullivan / Shing-Tung Yau (2010)
• Michael Aschbacher / Luis Caffarelli (2012)
• George Mostow / Michael Artin (2013)
• Peter Sarnak (2014)
• James G. Arthur (2015)
• Richard Schoen / Charles Fefferman (2017)
• Alexander Beilinson / Vladimir Drinfeld (2018)
• Jean-François Le Gall / Gregory Lawler (2019)
2020s
• Simon K. Donaldson / Yakov Eliashberg (2020)
• George Lusztig (2022)
• Ingrid Daubechies (2023)
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Yakov Pesin
Yakov Borisovich Pesin (Russian: Яков Борисович Песин) was born in Moscow, Russia (former USSR) on December 12, 1946. Pesin is currently a Distinguished Professor in the Department of Mathematics and the Director of the Anatole Katok Center for Dynamical Systems and Geometry at the Pennsylvania State University (PSU). His primary areas of research are the theory of dynamical systems with an emphasis on smooth ergodic theory, dimension theory in dynamical systems, and Riemannian geometry, as well as mathematical and statistical physics.
Yakov Pesin
Born (1946-12-12) December 12, 1946
Moscow, USSR (Russia)
Alma materMoscow State University
Known forPesin theory
AwardsMember of the European Academy – Academia Europaea
Fellow of the American Mathematical Society
Scientific career
FieldsMathematics – Dynamical Systems
InstitutionsPennsylvania State University
Doctoral advisorDmitry Anosov
Professional life and education
Pesin became interested in mathematics in school but his real involvement began when he entered the boarding school with emphasis on teaching physics and mathematics that was organized by Andrei Kolmogorov. Following his graduation of the school (with honors) in 1965, he successfully passed entry exams to the Department of Mechanics and Mathematics (or "Mech-Mat") of Moscow State University. Pesin graduated from Moscow State University (also with honors) in 1970, receiving a master's degree in mathematics. His master thesis advisor was Yakov Sinai.
Pesin naturally wanted to pursue a PhD in mathematics but faced significant challenges due to the oppressive nature and anti-Semitic policies of the Soviet regime. Thus, he was not permitted to continue his study at the university graduate school and was subsequently assigned to work at a research institute in Moscow (for a more complete historical account of the anti-Semitic sentiment in the Soviet mathematics establishment during this period see the article).[1]
Since Pesin always dreamed to be a "pure" mathematician, under the circumstances, he chose to combine his work at the institute with his after-hours research in mathematics and within a few years after graduation, he made a number of outstanding breakthroughs in the theory of smooth dynamical systems. His research at this time was conducted under the supervision of his PhD advisor, Dmitry Anosov, and also Anatole Katok.
In 1989 Pesin immigrated to the United States with his family. He first worked as a visiting Professor in the Department of Mathematics at the University of Chicago before getting the position of Full Professor at Penn State University. In 2003 Pesin received the title of Distinguished Professor of mathematics.
Yakov Pesin is married to Natasha Pesin who while in Russia worked for several years as a senior editor in the division of mathematics at the "Prosvechenie" ("Education") Publishing House in Moscow. After moving to the US she started a new career as a ceramicist (see her artworks at www.natashapesin.com).
Yakov Pesin also has two daughters, Elena and Irina who reside in the US.
Research accomplishments
Yakov Pesin is famous for several fundamental discoveries in the theory of dynamical systems (relevant references can be found on Pesin's website).
1) In a joint work with Michael Brin "Flows of frames on manifolds of negative curvature" (Russian Math. Surveys, 1973), Pesin laid down the foundations of partial hyperbolicity theory. As an application, they studied ergodic properties of the frame flows on manifolds of negative curvature.[2] In a later work with Yakov Sinai "Gibbs measures for partially hyperbolic attractors" (Ergodic Theory and Dynamical Systems, 1983) Pesin constructed a special class of u-measures for partially hyperbolic systems which are a direct analog in this setting of the famous Sinai-Ruelle-Bowen (SRB) measures.
2) Pesin's greatest contribution to dynamics is creation of non-uniform hyperbolicity theory, which is commonly known as Pesin Theory.[3][4][5] This theory serves as the mathematical foundation for the principal phenomenon known as "deterministic chaos" – the appearance of highly irregular chaotic motions in completely deterministic dynamical systems. Among the highlights of this theory is the formula for the Kolmogorov-Sinai entropy of the system (also known as Pesin entropy formula). His main article on this topic "Characteristic Lyapunov exponents and smooth ergodic theory" (Russian Mathematical Surveys, 1977) has a very high number of citations in mathematical literature and beyond (in physics, biology, etc.).
3) Pesin's later work on non-uniform hyperbolicity includes establishing presence of systems with non-zero Lyapunov exponents on any manifold; a proof of the Eckmann—Ruelle conjecture; the study of the essential coexistence phenomenon of regular and chaotic dynamics; constructions of SRB measures for hyperbolic attractors with singularities, partially hyperbolic and non-uniformly hyperbolic attractors; and effecting thermodynamic formalism for some classes of non-uniformly hyperbolic dynamical systems.
4) Pesin designed a construction (known also as the Caratheodory-Pesin construction) that allows one to introduce and study various dimension-type characteristics of dynamical systems. Among other things his work reveals "dimension nature" of many of the well-known thermodynamics invariants such as metric and topological entropies and topological pressure. It also provides a unified approach to describe various dimension spectra and related multi-fractal formalism (see [6]).
5) Pesin's work in Mathematical Physics includes the study of Coupled Map Lattices associated with infinite chains of hyperbolic systems as well as the ones generated by some diffusion-type PDEs such as FitzHu-Nagumo and Belousov-Zhabotinsky equations.
Teaching
Yakov Pesin holds a tenured faculty position at the Pennsylvania State University, where he has advised numerous PhD students on their thesis. In addition to his regular teaching responsibilities, he designed and taught courses at the special MASS (Mathematics Advanced Study Semester) program on Dynamical Systems[7] and Analytic and Projective Geometry. He has also delivered mini-courses at numerous International Mathematical Schools.
Honors and recognition
In 1986 Yakov Pesin was invited to speak at the International Congress of Mathematicians (ICM) in Berkeley, CA, but Soviet authorities did not allow him to travel to the US. Nevertheless, his talk on "Ergodic properties and dimension-like characteristics of strange attractors that are close to hyperbolic" was published in the proceedings of the ICM in 1987.
In 2012 Yakov Pesin became a Fellow of the American Mathematical Society (in its inaugural class) and in 2019 he was elected a (foreign) member of the European Academy—Academia Europaea. He was elected to the American Academy of Arts and Sciences in 2023.[8]
Yakov Pesin was invited to give many distinguished lectures including Invited Address at SIAM Annual Meeting (Kansas City, 1996), Invited Address, at the AMS Annual Meeting (Ohio State University, 2001), and Bernoulli Lecture at the Centre Interfacultaire Bernoulli, Ecole Polytechnique Federale de Lausanne, Switzerland (2013).
References
1. Anatole Katok, "Moscow dynamics seminars of the nineteen seventies and the early career of Yasha Pesin", Discrete and Continuous Dynamical Systems, v. 22, N1, 2 (2008) 1--22
2. Pesin, Yakov (2004). Lectures on Partial Hyperbolicity and Stable Ergodicity. Zurich Lectures in Advanced Mathematics. EMS. ISBN 3-03719-003-5.
3. Luis Barreira and Yakov Pesin (2013). Introduction to Smooth Ergodic Theory. Graduate Studies in Mathematics, v. 148, AMS. ISBN 978-0-8218-9853-6.
4. Luis Barreira and Yakov Pesin (2007). Nonuniform Hyperbolicity: Dynamics of Systems with Nonzero Lyapunov Exponents. Encyclopedia of Mathematics and Its Applications, 115, Cambridge University Press. ISBN 978-0-521-83258-8.
5. Pesin Theory, Encyclopedia of Mathematics.
6. Pesin, Yakov (1997). Dimension Theory in Dynamical Systems: Contemporary Views and Applications. Chicago Lectures in Mathematics Series, University of Chicago Press. ISBN 0-226-66222-5.
7. Vaughn Climenhaga and Yakov Pesin (2009). Lectures on Fractal Geometry and Dynamical Systems. Student Mathematical Library, v. 52, AMS, Providence, RI. ISBN 978-0-8218-4889-0.
8. "New members". American Academy of Arts and Sciences. 2023. Retrieved 2023-04-21.
External links
• Anatole Katok Center for Dynamical Systems and Geometry at Penn State: http://www.math.psu.edu/dynsys/
• Yakov Pesin website: http://www.math.psu.edu/pesin
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Yakov Sinai
Yakov Grigorevich Sinai (Russian: Я́ков Григо́рьевич Сина́й; born September 21, 1935) is a Russian–American mathematician known for his work on dynamical systems. He contributed to the modern metric theory of dynamical systems and connected the world of deterministic (dynamical) systems with the world of probabilistic (stochastic) systems.[1] He has also worked on mathematical physics and probability theory.[2] His efforts have provided the groundwork for advances in the physical sciences.[1]
Yakov Sinai
Yakov G. Sinai
Born
Yakov Grigorevich Sinai
(1935-09-21) September 21, 1935
Moscow, Russian SFSR, Soviet Union
NationalityRussian / American
Alma materMoscow State University
Known forMeasure-preserving dynamical systems, various works on dynamical systems, mathematical and statistical physics, probability theory, mathematical fluid dynamics
SpouseElena B. Vul
AwardsBoltzmann Medal (1986)
Dannie Heineman Prize (1990)
Dirac Prize (1992)
Wolf Prize (1997)
Nemmers Prize (2002)
Lagrange Prize (2008)
Henri Poincaré Prize (2009)
Foreign Member of the Royal Society (2009)
Leroy P. Steele Prize (2013)
Abel Prize (2014)
Marcel Grossmann Award (2015)
Scientific career
FieldsMathematics
InstitutionsMoscow State University, Landau Institute for Theoretical Physics, Princeton University
Doctoral advisorAndrey Kolmogorov
Doctoral studentsLeonid Bunimovich
Nikolai Chernov
Dmitry Dolgopyat
Svetlana Jitomirskaya
Anatole Katok
Konstantin Khanin
Grigory Margulis
Leonid Polterovich
Marina Ratner
Corinna Ulcigrai
Sinai has won several awards, including the Nemmers Prize, the Wolf Prize in Mathematics and the Abel Prize. He serves as the professor of mathematics at Princeton University since 1993 and holds the position of Senior Researcher at the Landau Institute for Theoretical Physics in Moscow, Russia.
Biography
Yakov Grigorevich Sinai was born into a Russian Jewish academic family on September 21, 1935, in Moscow, Soviet Union (now Russia).[3] His parents, Nadezda Kagan and Gregory Sinai, were both microbiologists. His grandfather, Veniamin Kagan, headed the Department of Differential Geometry at Moscow State University and was a major influence on Sinai's life.[3]
Sinai received his bachelor's and master's degrees from Moscow State University.[2] In 1960, he earned his Ph.D., also from Moscow State; his adviser was Andrey Kolmogorov. Together with Kolmogorov, he showed that even for "unpredictable" dynamic systems, the level of unpredictability of motion can be described mathematically. In their idea, which became known as Kolmogorov–Sinai entropy, a system with zero entropy is entirely predictable, while a system with non-zero entropy has an unpredictability factor directly related to the amount of entropy.[1]
In 1963, Sinai introduced the idea of dynamical billiards, also known as "Sinai Billiards". In this idealized system, a particle bounces around inside a square boundary without loss of energy. Inside the square is a circular wall, of which the particle also bounces off. He then proved that for most initial trajectories of the ball, this system is ergodic, that is, after a long time, the amount of that time the ball will have spent in any given region on the surface of the table is approximately proportional to the area of that region. It was the first time anyone proved a dynamic system was ergodic.[1]
Also in 1963, Sinai announced a proof of the ergodic hypothesis for a gas consisting of n hard spheres confined to a box. The complete proof, however, was never published, and in 1987 Sinai declared that the announcement was premature. The problem remains open to this day.[4]
Other contributions in mathematics and mathematical physics include the rigorous foundations of Kenneth Wilson's renormalization group-method, which led to Wilson's Nobel Prize for Physics in 1982, Gibbs measures in ergodic theory, hyperbolic Markov partitions, proof of the existence of Hamiltonian dynamics for infinite particle systems by the idea of "cluster dynamics", description of the discrete Schrödinger operators by the localization of eigenfunctions, Markov partitions for billiards and Lorenz map (with Bunimovich and Chernov), a rigorous treatment of subdiffusions in dynamics, verification of asymptotic Poisson distribution of energy level gaps for a class of integrable dynamical systems, and his version of the Navier–Stokes equations together with Khanin, Mattingly and Li.
From 1960 to 1971, Sinai was a researcher in the Laboratory of Probabilistic and Statistical Methods at Moscow State University. In 1971 he accepted a position as senior researcher at the Landau Institute for Theoretical Physics in Russia, while continuing to teach at Moscow State. He had to wait until 1981 to become a professor at Moscow State, likely because he had supported the dissident poet, mathematician and human rights activist Alexander Esenin-Volpin in 1968.[5]
Since 1993, Sinai has been a professor of mathematics at Princeton University, while maintaining his position at the Landau Institute. For the 1997–98 academic year, he was the Thomas Jones Professor at Princeton, and in 2005, the Moore Distinguished Scholar at the California Institute of Technology.[3]
In 2002, Sinai won the Nemmers Prize for his "revolutionizing" work on dynamical systems, statistical mechanics, probability theory, and statistical physics.[2] In 2005, the Moscow Mathematical Journal dedicated an issue to Sinai writing "Yakov Sinai is one of the greatest mathematicians of our time ... his exceptional scientific enthusiasm inspire[d] several generations of scientists all over the world."[3]
In 2013, Sinai received the Leroy P. Steele Prize for Lifetime Achievement.[3] In 2014, the Norwegian Academy of Science and Letters awarded him the Abel Prize, for his contributions to dynamical systems, ergodic theory, and mathematical physics.[6] Presenting the award, Jordan Ellenberg said Sinai had solved real world physical problems "with the soul of a mathematician".[1] He praised the tools developed by Sinai which demonstrate how systems that look different may in fact have fundamental similarities. The prize comes with 6 million Norwegian krone,[1] equivalent at the time to $US 1 million or £600,000. He was also inducted into the Norwegian Academy of Science and Letters.[7]
Other awards won by Sinai include the Boltzmann Medal (1986), the Dannie Heineman Prize for Mathematical Physics (1990), the Dirac Prize (1992), the Wolf Prize in Mathematics (1997), the Lagrange Prize (2008) and the Henri Poincaré Prize (2009).[2][3] He is a member of the United States National Academy of Sciences, the Russian Academy of Sciences, and the Hungarian Academy of Sciences.[2] He is an honorary member of the London Mathematical Society (1992) and, in 2012, he became a fellow of the American Mathematical Society.[2][8] Sinai has been selected an honorary member of the American Academy of Arts and Sciences (1983), Brazilian Academy of Sciences (2000), the Academia Europaea, the Polish Academy of Sciences, and the Royal Society of London. He holds honorary degrees from the Budapest University of Technology and Economics, the Hebrew University of Jerusalem, Warwick University, and Warsaw University.[3]
Sinai has authored more than 250 papers and books. Concepts in mathematics named after him include Minlos–Sinai theory of phase separation, Sinai's random walk, Sinai–Ruelle–Bowen measures, and Pirogov–Sinai theory, Bleher–Sinai renormalization theory. Sinai has overseen more than 50 PhD candidates.[3] He has spoken at the International Congress of Mathematicians four times.[2] In 2000, he was a plenary speaker at the First Latin American Congress in Mathematics.[3]
Sinai is married to mathematician and physicist Elena B. Vul. The couple have written several joint papers.[3]
Selected works
• Introduction to Ergodic Theory. Princeton 1976.[9]
• Topics in Ergodic Theory. Princeton 1977, 1994.[10]
• Probability Theory – an Introductory Course. Springer, 1992.[10]
• Theory of probability and Random Processes (with Koralov). 2nd edition, Springer, 2007.[10]
• Theory of Phase Transitions – Rigorous Results. Pergamon, Oxford 1982.[10]
• Ergodic Theory (with Isaac Kornfeld and Sergei Fomin). Springer, Grundlehren der mathematischen Wissenschaften 1982.[10]
• "What is a Billiard?", Notices AMS 2004.[10]
• "Mathematicians and physicists = Cats and Dogs?" in Bulletin of the AMS. 2006, vol. 4.[10]
• "How mathematicians and physicists found each other in the theory of dynamical systems and in statistical mechanics", in Mathematical Events of the Twentieth Century (editors: Bolibruch, Osipov, & Sinai). Springer 2006, p. 399.[10]
References
1. Ball, Philip (March 26, 2014). "Chaos-theory pioneer nabs Abel Prize". Nature. Retrieved March 29, 2014.
2. "2002 Frederic Esser Nemmers Mathematics Prize Recipient". Northwestern University. Retrieved March 30, 2014.
3. "Yakov G. Sinai" (PDF). Abel Prize. Retrieved August 2, 2022.{{cite web}}: CS1 maint: url-status (link)
4. Uffink, Jos (2006). Compendium of the foundations of classical statistical physics (PDF). p. 91.
5. "Sinai biography". www-history.mcs.st-andrews.ac.uk. Retrieved June 28, 2017.
6. "2014: Yakov G. Sinai". www.abelprize.no. Retrieved August 2, 2022.{{cite web}}: CS1 maint: url-status (link)
7. "Gruppe 1: Matematiske fag" (in Norwegian). Norwegian Academy of Science and Letters. Retrieved March 30, 2016.
8. "List of Fellows of the American Mathematical Society". Retrieved July 20, 2013.
9. Chacon, R. V. (1978). "Review: Introduction to ergodic theory, by Ya. G. Sinai" (PDF). Bull. Amer. Math. Soc. 84 (4): 656–660. doi:10.1090/s0002-9904-1978-14515-7.
10. "Yakov Bibliography" (PDF). Princeton University. Retrieved March 30, 2014.
External links
Wikimedia Commons has media related to Yakov Grigorevich Sinai.
• Sinai on scholarpedia
• O'Connor, John J.; Robertson, Edmund F., "Yakov Sinai", MacTutor History of Mathematics Archive, University of St Andrews
• Yakov Sinai at the Mathematics Genealogy Project
• List of publications on the website of the Landau Institute for Theoretical Physics
Laureates of the Wolf Prize in Mathematics
1970s
• Israel Gelfand / Carl L. Siegel (1978)
• Jean Leray / André Weil (1979)
1980s
• Henri Cartan / Andrey Kolmogorov (1980)
• Lars Ahlfors / Oscar Zariski (1981)
• Hassler Whitney / Mark Krein (1982)
• Shiing-Shen Chern / Paul Erdős (1983/84)
• Kunihiko Kodaira / Hans Lewy (1984/85)
• Samuel Eilenberg / Atle Selberg (1986)
• Kiyosi Itô / Peter Lax (1987)
• Friedrich Hirzebruch / Lars Hörmander (1988)
• Alberto Calderón / John Milnor (1989)
1990s
• Ennio de Giorgi / Ilya Piatetski-Shapiro (1990)
• Lennart Carleson / John G. Thompson (1992)
• Mikhail Gromov / Jacques Tits (1993)
• Jürgen Moser (1994/95)
• Robert Langlands / Andrew Wiles (1995/96)
• Joseph Keller / Yakov G. Sinai (1996/97)
• László Lovász / Elias M. Stein (1999)
2000s
• Raoul Bott / Jean-Pierre Serre (2000)
• Vladimir Arnold / Saharon Shelah (2001)
• Mikio Sato / John Tate (2002/03)
• Grigory Margulis / Sergei Novikov (2005)
• Stephen Smale / Hillel Furstenberg (2006/07)
• Pierre Deligne / Phillip A. Griffiths / David B. Mumford (2008)
2010s
• Dennis Sullivan / Shing-Tung Yau (2010)
• Michael Aschbacher / Luis Caffarelli (2012)
• George Mostow / Michael Artin (2013)
• Peter Sarnak (2014)
• James G. Arthur (2015)
• Richard Schoen / Charles Fefferman (2017)
• Alexander Beilinson / Vladimir Drinfeld (2018)
• Jean-François Le Gall / Gregory Lawler (2019)
2020s
• Simon K. Donaldson / Yakov Eliashberg (2020)
• George Lusztig (2022)
• Ingrid Daubechies (2023)
Mathematics portal
Abel Prize laureates
• 2003 Jean-Pierre Serre
• 2004 Michael Atiyah
• Isadore Singer
• 2005 Peter Lax
• 2006 Lennart Carleson
• 2007 S. R. Srinivasa Varadhan
• 2008 John G. Thompson
• Jacques Tits
• 2009 Mikhail Gromov
• 2010 John Tate
• 2011 John Milnor
• 2012 Endre Szemerédi
• 2013 Pierre Deligne
• 2014 Yakov Sinai
• 2015 John Forbes Nash Jr.
• Louis Nirenberg
• 2016 Andrew Wiles
• 2017 Yves Meyer
• 2018 Robert Langlands
• 2019 Karen Uhlenbeck
• 2020 Hillel Furstenberg
• Grigory Margulis
• 2021 László Lovász
• Avi Wigderson
• 2022 Dennis Sullivan
• 2023 Luis Caffarelli
Fellows of the Royal Society elected in 2009
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• Robert Ainsworth
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| Wikipedia |
Yale shooting problem
The Yale shooting problem is a conundrum or scenario in formal situational logic on which early logical solutions to the frame problem fail. The name of this problem comes from a scenario proposed by its inventors, Steve Hanks and Drew McDermott, working at Yale University when they proposed it. In this scenario, Fred (later identified as a turkey) is initially alive and a gun is initially unloaded. Loading the gun, waiting for a moment, and then shooting the gun at Fred is expected to kill Fred. However, if inertia is formalized in logic by minimizing the changes in this situation, then it cannot be uniquely proved that Fred is dead after loading, waiting, and shooting. In one solution, Fred indeed dies; in another (also logically correct) solution, the gun becomes mysteriously unloaded and Fred survives.
Technically, this scenario is described by two fluents (a fluent is a condition that can change truth value over time): $alive$ and $loaded$. Initially, the first condition is true and the second is false. Then, the gun is loaded, some time passes, and the gun is fired. Such problems can be formalized in logic by considering four time points $0$, $1$, $2$, and $3$, and turning every fluent such as $alive$ into a predicate $alive(t)$ depending on time. A direct formalization of the statement of the Yale shooting problem in logic is the following one:
$alive(0)$
$\neg loaded(0)$
$true\rightarrow loaded(1)$
$loaded(2)\rightarrow \neg alive(3)$
The first two formulae represent the initial state. The third formula formalizes the effect of loading the gun at time $0$. The fourth formula formalizes the effect of shooting at Fred at time $2$. This is a simplified formalization in which action names are neglected and the effects of actions are directly specified for the time points in which the actions are executed. See situation calculus for details.
The formulae above, while being direct formalizations of the known facts, do not suffice to correctly characterize the domain. Indeed, $\neg alive(1)$ is consistent with all these formulae, although there is no reason to believe that Fred dies before the gun has been shot. The problem is that the formulae above only include the effects of actions, but do not specify that all fluents not changed by the actions remain the same. In other words, a formula $alive(0)\equiv alive(1)$ must be added to formalize the implicit assumption that loading the gun only changes the value of $loaded$ and not the value of $alive$. The necessity of a large number of formulae stating the obvious fact that conditions do not change unless an action changes them is known as the frame problem.
An early solution to the frame problem was based on minimizing the changes. In other words, the scenario is formalized by the formulae above (that specify only the effects of actions) and by the assumption that the changes in the fluents over time are as minimal as possible. The rationale is that the formulae above enforce all effect of actions to take place, while minimization should restrict the changes to exactly those due to the actions.
In the Yale shooting scenario, one possible evaluation of the fluents in which the changes are minimized is the following one.
$alive(0)$ $alive(1)$ $alive(2)$ $\neg alive(3)$
$\neg loaded(0)$ $loaded(1)$ $loaded(2)$ $loaded(3)$
This is the expected solution. It contains two fluent changes: $loaded$ becomes true at time 1 and $alive$ becomes false at time 3. The following evaluation also satisfies all formulae above.
$alive(0)$ $alive(1)$ $alive(2)$ $alive(3)$
$\neg loaded(0)$ $loaded(1)$ $\neg loaded(2)$ $\neg loaded(3)$
In this evaluation, there are still two changes only: $loaded$ becomes true at time 1 and false at time 2. As a result, this evaluation is considered a valid description of the evolution of the state, although there is no valid reason to explain $loaded$ being false at time 2. The fact that minimization of changes leads to wrong solution is the motivation for the introduction of the Yale shooting problem.
While the Yale shooting problem has been considered a severe obstacle to the use of logic for formalizing dynamical scenarios, solutions to it have been known since the late 1980s. One solution involves the use of predicate completion in the specification of actions: in this solution, the fact that shooting causes Fred to die is formalized by the preconditions: alive and loaded, and the effect is that alive changes value (since alive was true before, this corresponds to alive becoming false). By turning this implication into an if and only if statement, the effects of shooting are correctly formalized. (Predicate completion is more complicated when there is more than one implication involved.)
A solution proposed by Erik Sandewall was to include a new condition of occlusion, which formalizes the “permission to change” for a fluent. The effect of an action that might change a fluent is therefore that the fluent has the new value, and that the occlusion is made (temporarily) true. What is minimized is not the set of changes, but the set of occlusions being true. Another constraint specifying that no fluent changes unless occlusion is true completes this solution.
The Yale shooting scenario is also correctly formalized by the Reiter version of the situation calculus, the fluent calculus, and the action description languages.
In 2005, the 1985 paper in which the Yale shooting scenario was first described received the AAAI Classic Paper award. In spite of being a solved problem, that example is still sometimes mentioned in recent research papers, where it is used as an illustrative example (e.g., for explaining the syntax of a new logic for reasoning about actions), rather than being presented as a problem.
See also
• Circumscription (logic)
• Frame problem
• Situation calculus
References
• M. Gelfond and V. Lifschitz (1993). Representing action and change by logic programs. Journal of Logic Programming, 17:301–322.
• S. Hanks and D. McDermott (1987). Nonmonotonic logic and temporal projection. Artificial Intelligence, 33(3):379–412.
• J. McCarthy (1986). Applications of circumscription to formalizing common-sense knowledge. Artificial Intelligence, 28:89–116.
• T. Mitchell and H. Levesque (2006). The 2005 AAAI Classic Paper awards. "AI Magazine", 26(4):98–99.
• R. Reiter (1991). The frame problem in the situation calculus: a simple solution (sometimes) and a completeness result for goal regression. In Vladimir Lifschitz, editor, Artificial Intelligence and Mathematical Theory of Computation: Papers in Honor of John McCarthy, pages 359–380. Academic Press, New York.
• E. Sandewall (1994). Features and Fluents. Oxford University Press.
| Wikipedia |
Yamabe flow
In differential geometry, the Yamabe flow is an intrinsic geometric flow—a process which deforms the metric of a Riemannian manifold. First introduced by Richard S. Hamilton,[1] Yamabe flow is for noncompact manifolds, and is the negative L2-gradient flow of the (normalized) total scalar curvature, restricted to a given conformal class: it can be interpreted as deforming a Riemannian metric to a conformal metric of constant scalar curvature, when this flow converges.
The Yamabe flow was introduced in response to Richard S. Hamilton's own work on the Ricci flow and Rick Schoen's solution of the Yamabe problem on manifolds of positive conformal Yamabe invariant.
Main results
The fixed points of the Yamabe flow are metrics of constant scalar curvature in the given conformal class. The flow was first studied in the 1980s in unpublished notes of Richard Hamilton. Hamilton conjectured that, for every initial metric, the flow converges to a conformal metric of constant scalar curvature. This was verified by Rugang Ye in the locally conformally flat case.[2] Later, Simon Brendle proved convergence of the flow for all conformal classes and arbitrary initial metrics.[3] The limiting constant-scalar-curvature metic is typically no longer a Yamabe minimizer in this context. While the compact case is settled, the flow on complete, non-compact manifolds is not completely understood, and remains a topic of current research.
Notes
1. Hamilton, Richard S. (1988). "The Ricci flow on surfaces". Mathematics and general relativity (Santa Cruz, CA, 1986). Contemp. Math. Vol. 71. Amer. Math. Soc., Providence, RI. pp. 237–262. doi:10.1090/conm/071/954419. MR 0954419.
2. Ye, Rugang (1994). "Global existence and convergence of Yamabe flow". J. Differential Geom. 39 (1): 35–50. doi:10.4310/jdg/1214454674.
3. Brendle, Simon (2005). "Convergence of the Yamabe flow for arbitrary initial energy". J. Differential Geom. 69 (2): 217–278. doi:10.4310/jdg/1121449107.
| Wikipedia |
Yamabe problem
The Yamabe problem refers to a conjecture in the mathematical field of differential geometry, which was resolved in the 1980s. It is a statement about the scalar curvature of Riemannian manifolds:
Let (M,g) be a closed smooth Riemannian manifold. Then there exists a positive and smooth function f on M such that the Riemannian metric fg has constant scalar curvature.
By computing a formula for how the scalar curvature of fg relates to that of g, this statement can be rephrased in the following form:
Let (M,g) be a closed smooth Riemannian manifold. Then there exists a positive and smooth function φ on M, and a number c, such that
${\frac {4(n-1)}{n-2}}\Delta ^{g}\varphi +R^{g}\varphi +c\varphi ^{(n+2)/(n-2)}=0.$
Here n denotes the dimension of M, Rg denotes the scalar curvature of g, and ∆g denotes the Laplace-Beltrami operator of g.
The mathematician Hidehiko Yamabe, in the paper Yamabe (1960), gave the above statements as theorems and provided a proof; however, Trudinger (1968) discovered an error in his proof. The problem of understanding whether the above statements are true or false became known as the Yamabe problem. The combined work of Yamabe, Trudinger, Thierry Aubin, and Richard Schoen provided an affirmative resolution to the problem in 1984.
It is now regarded as a classic problem in geometric analysis, with the proof requiring new methods in the fields of differential geometry and partial differential equations. A decisive point in Schoen's ultimate resolution of the problem was an application of the positive energy theorem of general relativity, which is a purely differential-geometric mathematical theorem first proved (in a provisional setting) in 1979 by Schoen and Shing-Tung Yau.
There has been more recent work due to Simon Brendle, Marcus Khuri, Fernando Codá Marques, and Schoen, dealing with the collection of all positive and smooth functions f such that, for a given Riemannian manifold (M,g), the metric fg has constant scalar curvature. Additionally, the Yamabe problem as posed in similar settings, such as for complete noncompact Riemannian manifolds, is not yet fully understood.
The Yamabe problem in special cases
Here, we refer to a "solution of the Yamabe problem" on a Riemannian manifold $(M,{\overline {g}})$ as a Riemannian metric g on M for which there is a positive smooth function $\varphi :M\to \mathbb {R} ,$ with $g=\varphi ^{-2}{\overline {g}}.$
On a closed Einstein manifold
Let $(M,{\overline {g}})$ be a smooth Riemannian manifold. Consider a positive smooth function $\varphi :M\to \mathbb {R} ,$ so that $g=\varphi ^{-2}{\overline {g}}$ is an arbitrary element of the smooth conformal class of ${\overline {g}}.$ A standard computation shows
${\overline {R}}_{ij}-{\frac {1}{n}}{\overline {R}}{\overline {g}}_{ij}=R_{ij}-{\frac {1}{n}}Rg_{ij}+{\frac {n-2}{\varphi }}{\Big (}\nabla _{i}\nabla _{j}\varphi +{\frac {1}{n}}g_{ij}\Delta \varphi {\Big )}.$
Taking the g-inner product with $\textstyle \varphi (\operatorname {Ric} -{\frac {1}{n}}Rg)$ results in
$\varphi \left\langle {\overline {\operatorname {Ric} }}-{\frac {1}{n}}{\overline {R}}{\overline {g}},\operatorname {Ric} -{\frac {1}{n}}Rg\right\rangle _{g}=\varphi {\Big |}\operatorname {Ric} -{\frac {1}{n}}Rg{\Big |}_{g}^{2}+(n-2){\Big (}{\big \langle }\operatorname {Ric} ,\operatorname {Hess} \varphi {\big \rangle }_{g}-{\frac {1}{n}}R\Delta \varphi {\Big )}.$
If ${\overline {g}}$ is assumed to be Einstein, then the left-hand side vanishes. If $M$ is assumed to be closed, then one can do an integration by parts, recalling the Bianchi identity $\textstyle \operatorname {div} \operatorname {Ric} ={\frac {1}{2}}\nabla R,$ to see
$\int _{M}\varphi {\Big |}\operatorname {Ric} -{\frac {1}{n}}Rg{\Big |}^{2}\,d\mu _{g}=(n-2){\Big (}{\frac {1}{2}}-{\frac {1}{n}}{\Big )}\int _{M}\langle \nabla R,\nabla \varphi \rangle \,d\mu _{g}.$
If g has constant scalar curvature, then the right-hand side vanishes. The consequent vanishing of the left-hand side proves the following fact, due to Obata (1971):
Every solution to the Yamabe problem on a closed Einstein manifold is Einstein.
Obata then went on to prove that, except in the case of the standard sphere with its usual constant-sectional-curvature metric, the only constant-scalar-curvature metrics in the conformal class of an Einstein metric (on a closed manifold) are constant multiples of the given metric. The proof proceeds by showing that the gradient of the conformal factor is actually a conformal Killing field. If the conformal factor is not constant, following flow lines of this gradient field, starting at a minimum of the conformal factor, then allows one to show that the manifold is conformally related to the cylinder $S^{n-1}\times \mathbb {R} $, and hence has vanishing Weyl curvature.
The non-compact case
A closely related question is the so-called "non-compact Yamabe problem", which asks: Is it true that on every smooth complete Riemannian manifold (M,g) which is not compact, there exists a metric that is conformal to g, has constant scalar curvature and is also complete? The answer is no, due to counterexamples given by Jin (1988). Various additional criteria under which a solution to the Yamabe problem for a non-compact manifold can be shown to exist are known (for example Aviles & McOwen (1988)); however, obtaining a full understanding of when the problem can be solved in the non-compact case remains a topic of research.
See also
• Yamabe flow
• Yamabe invariant
References
Research articles
• Aubin, Thierry (1976), "Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire", J. Math. Pures Appl., 55: 269–296
• Aviles, P.; McOwen, R. C. (1988), "Conformal deformation to constant negative scalar curvature on noncompact Riemannian manifolds", J. Differ. Geom., 27 (2): 225–239, doi:10.4310/jdg/1214441781, MR 0925121
• Jin, Zhi Ren (1988), "A counterexample to the Yamabe problem for complete noncompact manifolds", Partial differential equations (Tianjin, 1986), Lecture Notes in Mathematics, vol. 1306, Berlin: Springer, pp. 93–101, doi:10.1007/BFb0082927, MR 1032773
• Lee, John M.; Parker, Thomas H. (1987), "The Yamabe problem", Bulletin of the American Mathematical Society, 17: 37–81, doi:10.1090/s0273-0979-1987-15514-5.
• Obata, Morio (1971), "The conjectures on conformal transformations of Riemannian manifolds", Journal of Differential Geometry, 6: 247–258, doi:10.4310/jdg/1214430407, MR 0303464
• Schoen, Richard (1984), "Conformal deformation of a Riemannian metric to constant scalar curvature", J. Differ. Geom., 20 (2): 479–495, doi:10.4310/jdg/1214439291
• Trudinger, Neil S. (1968), "Remarks concerning the conformal deformation of Riemannian structures on compact manifolds", Ann. Scuola Norm. Sup. Pisa (3), 22: 265–274, MR 0240748
• Yamabe, Hidehiko (1960), "On a deformation of Riemannian structures on compact manifolds", Osaka Journal of Mathematics, 12: 21–37, ISSN 0030-6126, MR 0125546
Textbooks
• Aubin, Thierry. Some nonlinear problems in Riemannian geometry. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1998. xviii+395 pp. ISBN 3-540-60752-8
• Schoen, R.; Yau, S.-T. Lectures on differential geometry. Lecture notes prepared by Wei Yue Ding, Kung Ching Chang [Gong Qing Zhang], Jia Qing Zhong and Yi Chao Xu. Translated from the Chinese by Ding and S. Y. Cheng. With a preface translated from the Chinese by Kaising Tso. Conference Proceedings and Lecture Notes in Geometry and Topology, I. International Press, Cambridge, MA, 1994. v+235 pp. ISBN 1-57146-012-8
• Struwe, Michael. Variational methods. Applications to nonlinear partial differential equations and Hamiltonian systems. Fourth edition. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 34. Springer-Verlag, Berlin, 2008. xx+302 pp. ISBN 978-3-540-74012-4
| Wikipedia |
Yamamoto's reciprocity law
In mathematics, Yamamoto's reciprocity law is a reciprocity law related to class numbers of quadratic number fields, introduced by Yamamoto (1986).
References
• Williams, Kenneth S. (1991), "On Yamamoto's reciprocity law", Proceedings of the American Mathematical Society, 111 (3): 607–609, doi:10.2307/2048395, ISSN 0002-9939, JSTOR 2048395, MR 1047009
• Yamamoto, Yoshihiko (1986), "Congruences modulo 2i (i=3,4) for the class numbers of quadratic fields", Proceedings of the international conference on class numbers and fundamental units of algebraic number fields (Katata, 1986), Nagoya: Nagoya Univ., pp. 205–215, MR 0891897
| Wikipedia |
Lattice word
In mathematics, a lattice word (or lattice permutation) is a string composed of positive integers, in which every prefix contains at least as many positive integers i as integers i + 1.
A reverse lattice word, or Yamanouchi word, is a string whose reversal is a lattice word.
Examples
For instance, 11122121 is a lattice permutation, so 12122111 is a Yamanouchi word, but 12122111 is not a lattice permutation, since the prefix 12122 contains more 2s than 1s.
See also
• Dyck word
References
• Fulton, William (1997), Young tableaux, London Mathematical Society Student Texts, vol. 35, Cambridge University Press, ISBN 978-0-521-56724-4, MR 1464693
• Macdonald, Ian G. (1995), Symmetric functions and Hall polynomials, Oxford Mathematical Monographs (Second ed.), The Clarendon Press and Oxford University Press, ISBN 0-19-853489-2, MR 1354144
| Wikipedia |
Yan's theorem
In probability theory, Yan's theorem is a separation and existence result. It is of particular interest in financial mathematics where one uses it to prove the Kreps-Yan theorem.
The theorem was published by Jia-An Yan.[1] It was proven for the L1 space and later generalized by Jean-Pascal Ansel to the case $1\leq p<+\infty $.[2]
Yan's theorem
Notation:
${\overline {\Omega }}$ is the closure of a set $\Omega $.
$A-B=\{f-g:f\in A,\;g\in B\}$.
$I_{A}$ is the indicator function of $A$.
$q$ is the conjugate index of $p$.
Statement
Let $(\Omega ,{\mathcal {F}},P)$ be a probability space, $1\leq p<+\infty $ and $B_{+}$ be the space of non-negative and bounded random variables. Further let $K\subseteq L^{p}(\Omega ,{\mathcal {F}},P)$ be a convex subset and $0\in K$.
Then the following three conditions are equivalent:
1. For all $f\in L_{+}^{p}(\Omega ,{\mathcal {F}},P)$ with $f\neq 0$ exists a constant $c>0$, such that $cf\not \in {\overline {K-B_{+}}}$.
2. For all $A\in {\mathcal {F}}$ with $P(A)>0$ exists a constant $c>0$, such that $cI_{A}\not \in {\overline {K-B_{+}}}$.
3. There exists a random variable $Z\in L^{q}$, such that $Z>0$ almost surely and
$\sup \limits _{Y\in K}\mathbb {E} [ZY]<+\infty $.
Literature
• Yan, Jia-An (1980). "Caracterisation d' une Classe d'Ensembles Convexes de $L^{1}$ ou $H^{1}$". Séminaire de probabilités de Strasbourg. 14: 220–222.
• Freddy Delbaen and Walter Schachermayer: The Mathematics of Arbitrage (2005). Springer Finance
References
1. Yan, Jia-An (1980). "Caracterisation d' une Classe d'Ensembles Convexes de $L^{1}$ ou $H^{1}$". Séminaire de probabilités de Strasbourg. 14: 220–222.
2. Ansel, Jean-Pascal; Stricker, Christophe (1990). "Quelques remarques sur un théorème de Yan". Séminaire de Probabilités XXIV, Lect. Notes Math. Springer.
| Wikipedia |
Yan Rachinsky
Yan Zbignevich Rachinsky (Russian: Ян Збигневич Рачинский, also spelt Jan Raczynski,[1][2] born 6 December 1958 in Moscow, USSR) is a Russian human rights activist, programmer and mathematician.
Yan Rachinsky
Ян Рачинский
Born6 December 1958 (age 64)
Moscow
Alma mater
• MSU Faculty of Mechanics and Mathematics
OccupationHistorian, mathematician, programmer
Employer
• Memorial
He has been a human rights activist since the late 1980s when he first became involved in the work and activities of Memorial, a human rights organization examining the crimes of Stalin's regime. When the long-serving chairman of International Memorial, Arseny Roginsky, died in 2018, the board elected Rachinsky as his successor.[3][4]
When Memorial was awarded the Nobel Peace Prize in 2022, Rachinsky received the prize on behalf of Memorial and also gave the Nobel Lecture.[5]
Family
Rachinsky's grandfather Sigismund Raczynski was Polish; his grandmother Rebecca (Rivka) Fyalka (1888-1975) was a prominent member of the Socialist Revolutionary Party. She was sentenced to 13 years "hard labour" (katorga) by a field tribunal after the 1905 Revolution. She began her sentence in 1907 in Buryatia (east Siberia) and was sent into permanent exile in 1910. She escaped and after the February 1917 Revolution was elected to the Soviet of Workers and Soldiers Deputies in Svobodny (Amur Region).[6]
Work with Memorial HRC
In 1990-1995, Raczynski worked extensively with the Memorial Human Rights Centre (HRC), travelling to many hotspots in and around Russia: Karabakh in Azerbaijan; Transnistria in Moldova; and the Prigorodny district of North Ossetia.[7] He was a member of the organisation's team of observers during the first Chechen conflict (1994-1996).[8]
International recognition
In April 2011, Raczynski was awarded the Order of Merit of the Republic of Poland for his research[9][10] on the 1940 massacres in the Smolensk, Tver and Kharkov Regions of the USSR of POWs and others from the occupied territories of eastern Poland.[11][12]
"Victims of Political Terror"
Over the past 15 years, Raczynski has served as director of the project to assemble a single resource from the information scattered between the numerous Books of Remembrance compiled and published in Russia since the early 1990s. By 2016, its fifth edition, an online database entitled "Victims of Political Terror in the USSR", contained the names of about three million victims of the Soviet regime: those who were deported, imprisoned or executed from 1918 onwards.[13] This impressive figure was estimated to represent only a quarter of those who would qualify for rehabilitation under the terms of the October 1991 Law.
A controversy arose in August 2021 when Israeli historian Aron Schneer publicly announced[14] that Nazi collaborators guilty of war crimes had been included in the database as "victims of political terror".[15] In December 2021, Raczynski responded on behalf of Memorial to Vladimir Putin: on 10 December the Russian president publicly named three Latvian polizei, who had already been excluded from the database in September 2021.[16] Raczynski and Memorial suggested that the Russian authorities should express some appreciation for Memorial's work in compiling such an extensive database.
In 2015, formulating the State program for the Commemoration of the Victims of Political Repression, President Putin had talked of creating a unified database of victims. In January 2021 he instructed the FSB, Ministry of Internal Affairs and other relevant bodies to report back on this proposal in early October 2021. By the end of that year, however, nothing more was known of their activities. Memorial, meanwhile, was hampered as before by a lack of access to the archival materials at the disposal of the police and security services.[17]
The threat of closure
In mid-November 2021 lawsuits were brought against International Memorial and Memorial HRC in the RF Supreme Court and the Moscow City Court, respectively.
After a number of hearings, the Moscow courts ruled on two consecutive days, 28–29 December 2021, that both organisations should dissolve. "We never counted on love from the State," commented Raczynski.[18]
There were international protests, and a petition in many languages attracted tens of thousands of signatures worldwide.
As board chairman of International Memorial, Raczynski said that the organisation would appeal against the decision.[19]
When Memorial was awarded the 2022 Nobel Peace Prize, Raczynski told the BBC that he was ordered to turn down the prize by the Russian authorities.[20]
See also
• Chechnya
• Katyn massacre (1940) in the USSR.
• Memorial (society)
• Arseny Roginsky
Bibliography
• Alexander Cherkasov and Oleg Orlov, Russia and Chechnya: a trail of crimes and errors, 1998 Zvenya publishers: Moscow, 9785787000214, 398 pp (in Russian).
References and notes
1. See membership of International Memorial Board, elected 23 November 2018 (in English). Raczynski is No. 19 of 28.
2. He prefers his surname and first name to be written as "Jan Raczynski" in the Latin script, reflecting his Polish ancestry
3. "Jan Raczynski has been elected chairman of Memorial (22 March 2018)". TASS (in Russian). Retrieved 2 January 2022.
4. "Jan Raczynski now heads the board of International Memorial (23 March 2018)". Radio Liberty (in Russian). Retrieved 2 January 2022.
5. "Memorial", The Nobel Foundation, December 2022
6. Rebecca Fyalka, "An autobiographical sketch", early 1950s manuscript.
7. "Jan Raczynski now heads the board of International Memorial (23 March 2018)". Radio Liberty (in Russian). Retrieved 2 January 2022.
8. See Cherkasov and Orlov, Chechnya, 1998, p. 381.
9. "A decree of the Polish President dated 7 April 2011 on the award of an honour". isap.sejm.gov.pl (in Polish). Retrieved 2 January 2022.
10. "Order award ceremony". Embassy of Poland in Russia. 26 June 2013. Retrieved 2 January 2022.
11. Katyn Memorial Complex (Smolensk Region), "Russia's Necropolis" website.
12. Mednoe Memorial Complex (Tver Region) "Russia's Necropolis" website.
13. "Victims of Political Terror in the USSR", 1918-1987.
14. Aron Shneyer, Facebook, 24 August 2021. Retrieved 18 January 2022.
15. "Israeli historian Aron Schneer about Memory and the book Volokolamsk Highway". The recording shows Schneer speaking at a gathering in Moscow attended by Vladimir Putin.
16. "Memorial has responded to Putin's claims", Grani.ru, 13 December 2021.
17. "Memorial has responded to Putin's claims", Grani.ru, 13 December 2021.
18. Rachinsky: "We never counted on love from the State", Meduza news website, 28 December 2021.
19. "What next?" International Memorial, 12 January 2022.
20. "Nobel Peace Prize: Russian laureate 'told to turn down award'". BBC. 10 December 2022.
External links
Wikiquote has quotations related to Yan Rachinsky.
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| Wikipedia |
Yan Soibelman
Iakov (Yan) Soibelman (Russian: Яков Семенович Сойбельман) born 15 April 1956 (Kiev, USSR) is a Russian American mathematician, professor at Kansas State University (Manhattan, USA), member of the Kyiv Mathematical Society (Ukraine), founder of Manhattan Mathematical Olympiad.
Scientific work
Yan Soibelman is a specialist in theory of quantum groups, representation theory and symplectic geometry. He introduced the notion of quantum Weyl group, studied representation theory of the algebras of functions on compact quantum groups, and meromorphic braided monoidal categories. His long term collaboration with Maxim Kontsevich is devoted to various aspects of homological mirror symmetry, a proof of Deligne conjecture about operations on the cohomological Hochschild complex, a direct construction of Calabi-Yau varieties based on SYZ conjecture and non-archimedean geometry, and more recently to Donaldson-Thomas theory. Together with Kontsevich he laid the foundation and developed the theory of motivic Donaldson-Thomas invariants. Kontsevich-Soibelman wall-crossing formula for Donaldson-Thomas invariants (a.k.a BPS invariants) found important applications in physics.[1]They also introduced the notion of Cohomological Hall algebra which has numerous applications in geometric representation theory and quantum physics.[2]
See also
• bio
• Yan Soibelman's papers posted to math archives
• Manhattan Mathematical Olympiad, past years problems
References
1. Cecotti, Sergio; Vafa, Cumrun (2009). "BPS Wall Crossing and Topological Strings". arXiv:0910.2615 [hep-th].
2. "Cohomological Hall Algebras in Mathematics and Physics | Perimeter Institute".
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| Wikipedia |
Yang-Hui He
Yang-Hui He (simplified Chinese: 何杨辉; traditional Chinese: 何楊輝; pinyin: Hé Yáng Huī) is a mathematical physicist, who is a Fellow at the London Institute,[1] which is based at the Royal Institution of Great Britain, as well as lecturer and former Fellow at Merton College, Oxford.[2][3] He holds honorary positions as visiting professor of mathematics at City, University of London,[4] Chang-Jiang Chair professor at Nankai University,[5] and President of STEMM Global scientific society.[6]
Yang-Hui He
Alma materPrinceton University
Cambridge University
Massachusetts Institute of Technology
Scientific career
FieldsMathematical Physics
InstitutionsLondon Institute for Mathematical Sciences
University of Oxford
University of London
Nankai University
University of Pennsylvania
Doctoral advisorAmihay Hanany
Yang works on the interface between quantum field theory, string theory, algebraic geometry and number theory, as well as how AI and machine-learning help with these problems.[7][8] He is one of the pioneers of the field of using AI for pure mathematics.
Yang is author of over 200 scientific publications [9] and is also a keen communicator of science, giving regular public lectures [10] [11] [12] [13][14] including the Royal Institution Friday Evening Discourse,[15] as well as podcasts.[16][17][18] His other outreach activities include acting an advisor to BMUCO [19] and being a fellow of the One Garden.[20]
Education and career
Born in Wuhu, China, Yang attended primary schools in China and Australia,[21] and high schools in Australia and Canada.[22] He received his A.B. in Physics from Princeton University in 1996, with Highest Honours (summa cum laude, Allen Shenstone Prize and Kusaka Memorial Prize), joint with certificates in applied mathematics and in engineering physics. He received his Masters from University of Cambridge in 1997 with Distinction and then obtained his PhD from MIT in 2002 in the Center for Theoretical Physics (NSF Scholarship and MIT Presidential Award) under the supervision of Amihay Hanany.[23]
After postdoctoral work at the University of Pennsylvania, in the group of Burt Ovrut, Yang joined the University of Oxford as FitzJames Fellow and Advanced Fellow of the STFC, UK, working closely with Philip Candelas.[24] He remains a tutor at Merton College, Oxford when taking up his professorships at the University of London and Nankai University, and more recently, when he joined the London Institute.
Works
Yang has authored over 200 journal papers,[25] as well as several books, notably:
• Topology and Physics,[26] co-edited with C. N. Yang and Mo-Lin Ge, with contributions from Sir Michael Atiyah, Edward Witten, Sir Roger Penrose, Robbert Dijkgraaf et al., recommended by Book Authority as one of the 20 most influential books in quantum field theory of all time.[27]
• The Calabi-Yau Landscape: from geometry, to physics, to machine-learning,[28] textbook aimed at early PhD students, introducing mathematics to physicists, physics to mathematicians and machine-learning to both, the first textbook on the AI mathematician.
• Dialogues Between Physics and Mathematics: C. N. Yang at 100,[29] co-edited with Mo-Lin Ge, with contributions from Edward Witten, Sir Roger Penrose, Sir Anthony James Leggett, Alexander Polyakov, Vladimir Drinfeld et al., celebrating the 100th birthday of C. N. Yang.
• Machine Learning in Pure Mathematics and Theoretical Physics,[30] the first of its kind, as a collection of essays on the interactions between AI and pure mathematics/fundamental physics.
References
1. "Yang-Hui He". LIMS - London Institute for Mathematical Sciences. Retrieved 2021-10-05.
2. "Professor Yang-Hui He". Merton College, Oxford. Retrieved 2021-10-05.
3. "Professor Yang-Hui He". University of Oxford Department of Physics. Retrieved 2021-10-05.
4. "Professor Yang-Hui He | City, University of London". www.city.ac.uk. 2020-01-31. Retrieved 2021-10-05.
5. "Professor He, Chair Professor, Nankai". www.nankai.edu.cn.
6. "STEMM Global". stemm.global.
7. Lu, Donna. "AI is helping tackle one of the biggest unsolved problems in maths". New Scientist. Retrieved 2021-10-05.
8. Hutson, Matthew. "Companies make it easier for scientists to use AI". Science Magazine. Retrieved 2021-10-05.
9. "Yang-Hui He Publications". www.scholar.google.com.
10. "ToE".
11. "Universes as Big Data". Youtube.
12. "Muß Es Sein? – Epigraph to a String Quartet". ICMS NEWS. 2014-05-09. Retrieved 2021-10-11.
13. "慕校". nk.umlink.cn. Retrieved 2021-10-11.
14. "In conversation with Sir Roger Penrose". www.rigb.org. Retrieved 2021-10-12.
15. "Geometry and Physics".
16. "Muss es Sein?". Know it all Wall.
17. "The 23 Challenges". onegarden.com.
18. "Paradigm interview". Machine-Learning the universe.
19. "BMUCO, a student NGO for outreach". www.bmuco.org.
20. "OneGarden". onegarden.com.
21. "Yu Hong Primary School". "Cathedral College, Melbourne".
22. "Sydney Boys High". "Jarvis Collegiate".
23. "Yang-Hui He - The Mathematics Genealogy Project". www.genealogy.math.ndsu.nodak.edu. Retrieved 2021-10-05.
24. Ananthaswamy, Anil. "String Theory may Predict the Universe". New Scientist. Retrieved 2021-10-05.
25. "Yang-Hui He Publications". www.scholar.google.com.
26. "Topology and Physics". www.worldscientific.com. 2019.
27. "best QFT books".
28. "The Calabi-Yau Landscape". www.springer.com. 2021. arXiv preprint
29. "Dialogues Between Physics and Mathematics". www.springer.com. 2022.
30. "ML math-physics".
External links
• Yang-Hui He publications indexed by Google Scholar
• "String Math 2020, Day 2: Yang-Hui He". YouTube. Strings 2020. July 30, 2020.
• "The Search for a Theory of Everything - with Yang-Hui He". YouTube. The Royal Institution. December 1, 2022.
• "How Geometry created modern physics - with Yang-Hui He". YouTube. The Royal Institution. February 24, 2023.
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| Wikipedia |
Yang–Mills existence and mass gap
The Yang–Mills existence and mass gap problem is an unsolved problem in mathematical physics and mathematics, and one of the seven Millennium Prize Problems defined by the Clay Mathematics Institute, which has offered a prize of US$1,000,000 for its solution.
Millennium Prize Problems
• Birch and Swinnerton-Dyer conjecture
• Hodge conjecture
• Navier–Stokes existence and smoothness
• P versus NP problem
• Poincaré conjecture (solved)
• Riemann hypothesis
• Yang–Mills existence and mass gap
The problem is phrased as follows:[1]
Yang–Mills Existence and Mass Gap. Prove that for any compact simple gauge group G, a non-trivial quantum Yang–Mills theory exists on $\mathbb {R} ^{4}$ and has a mass gap Δ > 0. Existence includes establishing axiomatic properties at least as strong as those cited in Streater & Wightman (1964), Osterwalder & Schrader (1973) and Osterwalder & Schrader (1975).
In this statement, a quantum Yang–Mills theory is a non-abelian quantum field theory similar to that underlying the Standard Model of particle physics; $\mathbb {R} ^{4}$ is Euclidean 4-space; the mass gap Δ is the mass of the least massive particle predicted by the theory.
Therefore, the winner must prove that:
• Yang–Mills theory exists and satisfies the standard of rigor that characterizes contemporary mathematical physics, in particular constructive quantum field theory,[2][3] and
• The mass of all particles of the force field predicted by the theory are strictly positive.
For example, in the case of G=SU(3)—the strong nuclear interaction—the winner must prove that glueballs have a lower mass bound, and thus cannot be arbitrarily light.
The general problem of determining the presence of a spectral gap in a system is known to be undecidable.[4][5]
Background
[...] one does not yet have a mathematically complete example of a quantum gauge theory in four-dimensional space-time, nor even a precise definition of quantum gauge theory in four dimensions. Will this change in the 21st century? We hope so!
— From the Clay Institute's official problem description by Arthur Jaffe and Edward Witten.
The problem requires the construction of a QFT satisfying the Wightman axioms and showing the existence of a mass gap. Both of these topics are described in sections below.
The Wightman axioms
The Millennium problem requires the proposed Yang–Mills theory to satisfy the Wightman axioms or similarly stringent axioms.[1] There are four axioms:
W0 (assumptions of relativistic quantum mechanics)
Quantum mechanics is described according to von Neumann; in particular, the pure states are given by the rays, i.e. the one-dimensional subspaces, of some separable complex Hilbert space.
The Wightman axioms require that the Poincaré group acts unitarily on the Hilbert space. In other words, they have position dependent operators called quantum fields which form covariant representations of the Poincaré group.
The group of space-time translations is commutative, and so the operators can be simultaneously diagonalised. The generators of these groups give us four self-adjoint operators, $P_{j},j=0,1,2,3$, which transform under the homogeneous group as a four-vector, called the energy-momentum four-vector.
The second part of the zeroth axiom of Wightman is that the representation U(a, A) fulfills the spectral condition—that the simultaneous spectrum of energy-momentum is contained in the forward cone:
$P_{0}\geq 0,\;\;\;\;P_{0}^{2}-P_{j}P_{j}\geq 0.$
The third part of the axiom is that there is a unique state, represented by a ray in the Hilbert space, which is invariant under the action of the Poincaré group. It is called a vacuum.
W1 (assumptions on the domain and continuity of the field)
For each test function f, there exists a set of operators $A_{1}(f),\ldots ,A_{n}(f)$ which, together with their adjoints, are defined on a dense subset of the Hilbert state space, containing the vacuum. The fields A are operator-valued tempered distributions. The Hilbert state space is spanned by the field polynomials acting on the vacuum (cyclicity condition).
W2 (transformation law of the field)
The fields are covariant under the action of Poincaré group, and they transform according to some representation S of the Lorentz group, or SL(2,C) if the spin is not integer:
$U(a,L)^{\dagger }A(x)U(a,L)=S(L)A(L^{-1}(x-a)).$
W3 (local commutativity or microscopic causality)
If the supports of two fields are space-like separated, then the fields either commute or anticommute.
Cyclicity of a vacuum, and uniqueness of a vacuum are sometimes considered separately. Also, there is the property of asymptotic completeness—that the Hilbert state space is spanned by the asymptotic spaces $H^{in}$ and $H^{out}$, appearing in the collision S matrix. The other important property of field theory is the mass gap which is not required by the axioms—that the energy-momentum spectrum has a gap between zero and some positive number.
Mass gap
In quantum field theory, the mass gap is the difference in energy between the vacuum and the next lowest energy state. The energy of the vacuum is zero by definition, and assuming that all energy states can be thought of as particles in plane-waves, the mass gap is the mass of the lightest particle.
For a given real field $\phi (x)$, we can say that the theory has a mass gap if the two-point function has the property
$\langle \phi (0,t)\phi (0,0)\rangle \sim \sum _{n}A_{n}\exp \left(-\Delta _{n}t\right)$
with $\Delta _{0}>0$ being the lowest energy value in the spectrum of the Hamiltonian and thus the mass gap. This quantity, easy to generalize to other fields, is what is generally measured in lattice computations. It was proved in this way that Yang–Mills theory develops a mass gap on a lattice.[6][7]
Importance of Yang–Mills theory
Most known and nontrivial (i.e. interacting) quantum field theories in 4 dimensions are effective field theories with a cutoff scale. Since the beta-function is positive for most models, it appears that most such models have a Landau pole as it is not at all clear whether or not they have nontrivial UV fixed points. This means that if such a QFT is well-defined at all scales, as it has to be to satisfy the axioms of axiomatic quantum field theory, it would have to be trivial (i.e. a free field theory).
Quantum Yang–Mills theory with a non-abelian gauge group and no quarks is an exception, because asymptotic freedom characterizes this theory, meaning that it has a trivial UV fixed point. Hence it is the simplest nontrivial constructive QFT in 4 dimensions. (QCD is a more complicated theory because it involves quarks.)
Quark confinement
Main articles: Quantum chromodynamics, color confinement, and lattice gauge theory
At the level of rigor of theoretical physics, it has been well established that the quantum Yang–Mills theory for a non-abelian Lie group exhibits a property known as confinement; though proper mathematical physics has more demanding requirements on a proof. A consequence of this property is that above the confinement scale, the color charges are connected by chromodynamic flux tubes leading to a linear potential between the charges. Hence free color charge and free gluons cannot exist. In the absence of confinement, we would expect to see massless gluons, but since they are confined, all we would see are color-neutral bound states of gluons, called glueballs. If glueballs exist, they are massive, which is why a mass gap is expected.
References
1. Arthur Jaffe and Edward Witten "Quantum Yang-Mills theory." Official problem description.
2. R. Streater and A. Wightman, PCT, Spin and Statistics and all That, W. A. Benjamin, New York, 1964.
3. K. Osterwalder and R. Schrader, Axioms for Euclidean Green’s functions, Comm. Math. Phys. 31 (1973), 83–112, and Comm. Math. Phys. 42 (1975), 281–305.
4. Michael Wolf, Toby Cubitt, David Perez Garcia Unsolvable problem // In the world of science — 2018, № 12. — p. 46 — 59
5. Davide Castelvecchi. "Paradox at the heart of mathematics makes physics problem unanswerable". Nature.
6. Lucini, Biagio; Teper, Michael; Wenger, Urs (2004). "Glueballs and k-strings in SU(N) gauge theories : calculations with improved operators". Journal of High Energy Physics. 0406 (6): 012. arXiv:hep-lat/0404008. Bibcode:2004JHEP...06..012L. doi:10.1088/1126-6708/2004/06/012. S2CID 14807677..
7. Chen, Y.; Alexandru, A.; Dong, S. J.; Draper, T.; Horvath, I.; Lee, F. X.; Liu, K. F.; Mathur, N.; Morningstar, C.; Peardon, M.; Tamhankar, S.; Young, B. L.; Zhang, J. B. (2006). "Glueball Spectrum and Matrix Elements on Anisotropic Lattices". Physical Review D. 73 (1): 014516. arXiv:hep-lat/0510074. Bibcode:2006PhRvD..73a4516C. doi:10.1103/PhysRevD.73.014516. S2CID 15741174..
Further reading
• Streater, R.; Wightman, A. (1964). PCT, Spin and Statistics and all That. W. A. Benjamin.
• Osterwalder, K.; Schrader, R. (1973). "Axioms for Euclidean Green's functions". Communications in Mathematical Physics. 31 (2): 83–112. Bibcode:1973CMaPh..31...83O. doi:10.1007/BF01645738. S2CID 189829853.
• Osterwalder, K.; Schrader, R. (1975). "Axioms for Euclidean Green's functions II". Communications in Mathematical Physics. 42 (3): 281–305. Bibcode:1975CMaPh..42..281O. doi:10.1007/BF01608978. S2CID 119389461.
• Bogoliubov, N.; Logunov, A.; Oksak; Todorov, I. (1990). General Principles of Quantum Field Theory. Kluver.
• Strocchi, F. (1994). Selected Topics of the General Properties of Quantum Field Theory FF. World Scientific.
• Dynin, A. (2014). "Quantum Yang-Mills-Weyl dynamics in the Schroedinger paradigm". Russian Journal of Mathematical Physics. 21 (2): 169–188. Bibcode:2014RJMP...21..169D. doi:10.1134/S1061920814020046. S2CID 121878861.
• Dynin, A. (2014). "On the Yang-Mills Mass Gap problem". Russian Journal of Mathematical Physics. 21 (3): 326–328. Bibcode:2014RJMP...21..326D. doi:10.1134/S1061920814030042. S2CID 120135592.
• Bushhorn, G.; Wess, J. (2004). Heisenberg Centennial Symposium "Developments in Modern Physics". Springer.
External links
• The Millennium Prize Problems: Yang–Mills and Mass Gap
| Wikipedia |
Yang–Mills theory
In mathematical physics, Yang–Mills theory is a gauge theory based on a special unitary group SU(n), or more generally any compact, reductive Lie algebra. Yang–Mills theory seeks to describe the behavior of elementary particles using these non-abelian Lie groups and is at the core of the unification of the electromagnetic force and weak forces (i.e. U(1) × SU(2)) as well as quantum chromodynamics, the theory of the strong force (based on SU(3)). Thus it forms the basis of our understanding of the Standard Model of particle physics.
Unsolved problem in physics:
Yang–Mills theory in the non-perturbative regime: The equations of Yang–Mills remain unsolved at energy scales relevant for describing atomic nuclei. How does Yang–Mills theory give rise to the physics of nuclei and nuclear constituents?
(more unsolved problems in physics)
Quantum field theory
Feynman diagram
History
Background
• Field theory
• Electromagnetism
• Weak force
• Strong force
• Quantum mechanics
• Special relativity
• General relativity
• Gauge theory
• Yang–Mills theory
Symmetries
• Symmetry in quantum mechanics
• C-symmetry
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Tools
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Equations
• Dirac equation
• Klein–Gordon equation
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Standard Model
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Incomplete theories
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• Tyutin
• Vainshtein
• Veltman
• Virasoro
• Ward
• Weinberg
• Weisskopf
• Wentzel
• Wess
• Wetterich
• Weyl
• Wick
• Wightman
• Wigner
• Wilczek
• Wilson
• Witten
• Yang
• Yukawa
• Zamolodchikov
• Zamolodchikov
• Zee
• Zimmermann
• Zinn-Justin
• Zuber
• Zumino
History and theoretical description
In 1953, in a private correspondence, Wolfgang Pauli formulated a six-dimensional theory of Einstein's field equations of general relativity, extending the five-dimensional theory of Kaluza, Klein, Fock, and others to a higher-dimensional internal space.[1] However, there is no evidence that Pauli developed the Lagrangian of a gauge field or the quantization of it. Because Pauli found that his theory "leads to some rather unphysical shadow particles", he refrained from publishing his results formally.[1] Although Pauli did not publish his six-dimensional theory, he gave two seminar lectures about it in Zürich in November 1953.[1][lower-alpha 1]
Recent research shows that an extended Kaluza–Klein theory is in general not equivalent to Yang–Mills theory, as the former contains additional terms. Chen Ning Yang long considered the idea of non-abelian gauge theories. Only after meeting Robert Mills did he introduce the junior scientist to the idea and lay the key hypothesis that Mills would use to assist in creating a new theory. This eventually became the Yang–Mills theory, as Mills himself discussed:
"During the academic year 1953-1954, Yang was a visitor to Brookhaven National Laboratory ... I was at Brookhaven also...and was assigned to the same office as Yang. Yang, who has demonstrated on a number of occasions his generosity to physicists beginning their careers, told me about his idea of generalizing gauge invariance and we discussed it at some length...I was able to contribute something to the discussions, especially with regard to the quantization procedures, and to a small degree in working out the formalism; however, the key ideas were Yang's."[2]
In early 1954, Yang and Mills extended the concept of gauge theory for abelian groups, e.g. quantum electrodynamics, to non-abelian groups to provide an explanation for strong interactions.[3] Similar work was done independently in January 1954 by Ronald Shaw, a graduate student at the University of Cambridge.[4] However, the theory needed massless particles in order to maintain gauge invariance. Since no such massless particles were known at the time, Shaw and his supervisor Abdus Salam chose not to publish their work,[4] while Pauli criticized Yang's presentation of his work with Mills in February 1954.[5] Shortly after Yang and Mills published their paper in October 1954, Salam encouraged Shaw to publish his work to mark his contribution. Shaw declined, and instead it only forms a chapter of his PhD thesis published in 1956.[6][7] The idea was set aside until 1960, when the concept of particles acquiring mass through symmetry breaking in massless theories was put forward, initially by Jeffrey Goldstone, Yoichiro Nambu, and Giovanni Jona-Lasinio.
This prompted a significant restart of Yang–Mills theory studies that proved successful in the formulation of both electroweak unification and quantum chromodynamics (QCD). The electroweak interaction is described by the gauge group SU(2) × U(1), while QCD is an SU(3) Yang–Mills theory. The massless gauge bosons of the electroweak SU(2) × U(1) mix after spontaneous symmetry breaking to produce the 3 massive weak bosons (
W+
,
W−
, and
Z0
) as well as the still-massless photon field. The dynamics of the photon field and its interactions with matter are, in turn, governed by the U(1) gauge theory of quantum electrodynamics. The Standard Model combines the strong interaction with the unified electroweak interaction (unifying the weak and electromagnetic interaction) through the symmetry group SU(3) × SU(2) × U(1). In the current epoch the strong interaction is not unified with the electroweak interaction, but from the observed running of the coupling constants it is believed they all converge to a single value at very high energies.
Phenomenology at lower energies in quantum chromodynamics is not completely understood due to the difficulties of managing such a theory with a strong coupling. This may be the reason why confinement has not been theoretically proven, though it is a consistent experimental observation. This shows why QCD confinement at low energy is a mathematical problem of great relevance, and why the Yang–Mills existence and mass gap problem is a Millennium Prize Problem.
Mathematical overview
See also: Yang–Mills equations
The dx1⊗σ3 coefficient of a BPST instanton on the (x1,x2)-slice of ℝ4 where σ3 is the third Pauli matrix (top left). The dx2⊗σ3 coefficient (top right). These coefficients determine the restriction of the BPST instanton A with g=2, ρ=1, z=0 to this slice. The corresponding field strength centered around z=0 (bottom left). A visual representation of the field strength of a BPST instanton with center z on the compactification S4 of ℝ4 (bottom right). The BPST instanton is a classical instanton solution to the Yang–Mills equations on ℝ4.
Yang–Mills theories are special examples of gauge theories with a non-abelian symmetry group given by the Lagrangian
$\ {\mathcal {L}}_{\mathrm {gf} }=-{\tfrac {1}{2}}\operatorname {tr} (F^{2})=-{\tfrac {1}{4}}F^{a\mu \nu }F_{\mu \nu }^{a}\ $
with the generators $\ T^{a}\ $ of the Lie algebra, indexed by a, corresponding to the F-quantities (the curvature or field-strength form) satisfying
$\ \operatorname {tr} \left(T^{a}\ T^{b}\right)={\tfrac {1}{2}}\delta ^{ab}\ ,\qquad \left[T^{a},\ T^{b}\right]=i\ f^{abc}\ T^{c}~.$
Here, the f abc are structure constants of the Lie algebra (totally antisymmetric if the generators of the Lie algebra are normalised such that $\ \operatorname {tr} (T^{a}\ T^{b})\ $ is proportional with $\ \delta ^{ab}\ $), the covariant derivative is defined as
$\ D_{\mu }=I\ \partial _{\mu }-i\ g\ T^{a}\ A_{\mu }^{a}\ ,$
I is the identity matrix (matching the size of the generators), $\ A_{\mu }^{a}\ $ is the vector potential, and g is the coupling constant. In four dimensions, the coupling constant g is a pure number and for a SU(n) group one has $\ a,b,c=1\ldots n^{2}-1~.$
The relation
$\ F_{\mu \nu }^{a}=\partial _{\mu }A_{\nu }^{a}-\partial _{\nu }A_{\mu }^{a}+g\ f^{abc}\ A_{\mu }^{b}\ A_{\nu }^{c}\ $
can be derived by the commutator
$\ \left[D_{\mu },D_{\nu }\right]=-i\ g\ T^{a}\ F_{\mu \nu }^{a}~.$
The field has the property of being self-interacting and the equations of motion that one obtains are said to be semilinear, as nonlinearities are both with and without derivatives. This means that one can manage this theory only by perturbation theory with small nonlinearities.
Note that the transition between "upper" ("contravariant") and "lower" ("covariant") vector or tensor components is trivial for a indices (e.g. $\ f^{abc}=f_{abc}\ $), whereas for μ and ν it is nontrivial, corresponding e.g. to the usual Lorentz signature, $\ \eta _{\mu \nu }={\rm {diag}}(+---)~.$
From the given Lagrangian one can derive the equations of motion given by
$\ \partial ^{\mu }F_{\mu \nu }^{a}+g\ f^{abc}\ A^{\mu b}\ F_{\mu \nu }^{c}=0~.$
Putting $\ F_{\mu \nu }=T^{a}F_{\mu \nu }^{a}\ ,$ these can be rewritten as
$\ \left(D^{\mu }F_{\mu \nu }\right)^{a}=0~.$
A Bianchi identity holds
$\ \left(D_{\mu }\ F_{\nu \kappa }\right)^{a}+\left(D_{\kappa }\ F_{\mu \nu }\right)^{a}+\left(D_{\nu }\ F_{\kappa \mu }\right)^{a}=0\ $
which is equivalent to the Jacobi identity
$\ \left[D_{\mu },\left[D_{\nu },D_{\kappa }\right]\right]+\left[D_{\kappa },\left[D_{\mu },D_{\nu }\right]\right]+\left[D_{\nu },\left[D_{\kappa },D_{\mu }\right]\right]=0\ $
since $\ \left[D_{\mu },F_{\nu \kappa }^{a}\right]=D_{\mu }\ F_{\nu \kappa }^{a}~.$ Define the dual strength tensor $\ {\tilde {F}}^{\mu \nu }={\tfrac {1}{2}}\varepsilon ^{\mu \nu \rho \sigma }F_{\rho \sigma }\ ,$ then the Bianchi identity can be rewritten as
$\ D_{\mu }{\tilde {F}}^{\mu \nu }=0~.$
A source $\ J_{\mu }^{a}\ $ enters into the equations of motion as
$\ \partial ^{\mu }F_{\mu \nu }^{a}+g\ f^{abc}\ A^{b\mu }\ F_{\mu \nu }^{c}=-J_{\nu }^{a}~.$
Note that the currents must properly change under gauge group transformations.
We give here some comments about the physical dimensions of the coupling. In D dimensions, the field scales as $\ \left[A\right]=\left[L^{\left({\tfrac {2-D}{2}}\right)}\right]\ $ and so the coupling must scale as $\ \left[g^{2}\right]=\left[L^{\left(D-4\right)}\right]~.$ This implies that Yang–Mills theory is not renormalizable for dimensions greater than four. Furthermore, for D = 4 , the coupling is dimensionless and both the field and the square of the coupling have the same dimensions of the field and the coupling of a massless quartic scalar field theory. So, these theories share the scale invariance at the classical level.
Quantization
A method of quantizing the Yang–Mills theory is by functional methods, i.e. path integrals. One introduces a generating functional for n-point functions as
$\ Z[j]=\int [\mathrm {d} A]\ \exp \left[-{\tfrac {i}{2}}\int \mathrm {d} ^{4}x\ \operatorname {tr} \left(F^{\mu \nu }\ F_{\mu \nu }\right)+i\ \int \mathrm {d} ^{4}x\ j_{\mu }^{a}(x)\ A^{a\mu }(x)\right]\ ,$
but this integral has no meaning as it is because the potential vector can be arbitrarily chosen due to the gauge freedom. This problem was already known for quantum electrodynamics but here becomes more severe due to non-abelian properties of the gauge group. A way out has been given by Ludvig Faddeev and Victor Popov with the introduction of a ghost field (see Faddeev–Popov ghost) that has the property of being unphysical since, although it agrees with Fermi–Dirac statistics, it is a complex scalar field, which violates the spin–statistics theorem. So, we can write the generating functional as
${\begin{aligned}Z[j,{\bar {\varepsilon }},\varepsilon ]&=\int [\mathrm {d} \ A][\mathrm {d} \ {\bar {c}}][\mathrm {d} \ c]\ \exp {\Bigl \{}i\ S_{F}\ \left[\partial A,A\right]+i\ S_{gf}\left[\partial A\right]+i\ S_{g}\left[\partial c,\partial {\bar {c}},c,{\bar {c}},A\right]{\Bigr \}}\\&\exp \left\{i\int \mathrm {d} ^{4}x\ j_{\mu }^{a}(x)A^{a\mu }(x)+i\int \mathrm {d} ^{4}x\ \left[{\bar {c}}^{a}(x)\ \varepsilon ^{a}(x)+{\bar {\varepsilon }}^{a}(x)\ c^{a}(x)\right]\right\}\end{aligned}}$
being
$S_{F}=-{\tfrac {1}{2}}\int \mathrm {d} ^{4}x\ \operatorname {tr} \left(F^{\mu \nu }\ F_{\mu \nu }\right)\ $
for the field,
$S_{gf}=-{\frac {1}{2\xi }}\int \mathrm {d} ^{4}x\ (\partial \cdot A)^{2}\ $
for the gauge fixing and
$\ S_{g}=-\int \mathrm {d} ^{4}x\ \left({\bar {c}}^{a}\ \partial _{\mu }\partial ^{\mu }c^{a}+g\ {\bar {c}}^{a}\ f^{abc}\ \partial _{\mu }\ A^{b\mu }\ c^{c}\right)\ $
for the ghost. This is the expression commonly used to derive Feynman's rules (see Feynman diagram). Here we have ca for the ghost field while ξ fixes the gauge's choice for the quantization. Feynman's rules obtained from this functional are the following
These rules for Feynman diagrams can be obtained when the generating functional given above is rewritten as
${\begin{aligned}Z[j,{\bar {\varepsilon }},\varepsilon ]&=\exp \left(-i\ g\int \mathrm {d} ^{4}x\ {\frac {\delta }{i\ \delta \ {\bar {\varepsilon }}^{a}(x)}}\ f^{abc}\partial _{\mu }\ {\frac {i\ \delta }{\delta \ j_{\mu }^{b}(x)}}\ {\frac {i\ \delta }{\delta \ \varepsilon ^{c}(x)}}\right)\\&\qquad \times \exp \left(-i\ g\int \mathrm {d} ^{4}x\ f^{abc}\partial _{\mu }{\frac {i\ \delta }{\delta \ j_{\nu }^{a}(x)}}{\frac {i\ \delta }{\delta \ j_{\mu }^{b}(x)}}\ {\frac {i\ \delta }{\delta \ j^{c\nu }(x)}}\right)\\&\qquad \qquad \times \exp \left(-i\ {\frac {g^{2}}{4}}\int \mathrm {d} ^{4}x\ f^{abc}\ f^{ars}{\frac {i\ \delta }{\delta \ j_{\mu }^{b}(x)}}\ {\frac {i\ \delta }{\delta \ j_{\nu }^{c}(x)}}\ {\frac {\ i\delta }{\delta \ j^{r\mu }(x)}}{\frac {i\ \delta }{\delta \ j^{s\nu }(x)}}\right)\\&\qquad \qquad \qquad \times Z_{0}[j,{\bar {\varepsilon }},\varepsilon ]\end{aligned}}$
with
$Z_{0}[j,{\bar {\varepsilon }},\varepsilon ]=\exp \left(-\int \mathrm {d} ^{4}x\ \mathrm {d} ^{4}y\ {\bar {\varepsilon }}^{a}(x)\ C^{ab}(x-y)\ \varepsilon ^{b}(y)\right)\exp \left({\tfrac {1}{2}}\int \mathrm {d} ^{4}x\ \mathrm {d} ^{4}y\ j_{\mu }^{a}(x)\ D^{ab\mu \nu }(x-y)\ j_{\nu }^{b}(y)\right)\ $
being the generating functional of the free theory. Expanding in g and computing the functional derivatives, we are able to obtain all the n-point functions with perturbation theory. Using LSZ reduction formula we get from the n-point functions the corresponding process amplitudes, cross sections and decay rates. The theory is renormalizable and corrections are finite at any order of perturbation theory.
For quantum electrodynamics the ghost field decouples because the gauge group is abelian. This can be seen from the coupling between the gauge field and the ghost field that is $\ {\bar {c}}^{a}\ f^{abc}\ \partial _{\mu }A^{b\mu }\ c^{c}~.$ For the abelian case, all the structure constants $\ f^{abc}\ $ are zero and so there is no coupling. In the non-abelian case, the ghost field appears as a useful way to rewrite the quantum field theory without physical consequences on the observables of the theory such as cross sections or decay rates.
One of the most important results obtained for Yang–Mills theory is asymptotic freedom. This result can be obtained by assuming that the coupling constant g is small (so small nonlinearities), as for high energies, and applying perturbation theory. The relevance of this result is due to the fact that a Yang–Mills theory that describes strong interaction and asymptotic freedom permits proper treatment of experimental results coming from deep inelastic scattering.
To obtain the behavior of the Yang–Mills theory at high energies, and so to prove asymptotic freedom, one applies perturbation theory assuming a small coupling. This is verified a posteriori in the ultraviolet limit. In the opposite limit, the infrared limit, the situation is the opposite, as the coupling is too large for perturbation theory to be reliable. Most of the difficulties that research meets is just managing the theory at low energies. That is the interesting case, being inherent to the description of hadronic matter and, more generally, to all the observed bound states of gluons and quarks and their confinement (see hadrons). The most used method to study the theory in this limit is to try to solve it on computers (see lattice gauge theory). In this case, large computational resources are needed to be sure the correct limit of infinite volume (smaller lattice spacing) is obtained. This is the limit the results must be compared with. Smaller spacing and larger coupling are not independent of each other, and larger computational resources are needed for each. As of today, the situation appears somewhat satisfactory for the hadronic spectrum and the computation of the gluon and ghost propagators, but the glueball and hybrids spectra are yet a questioned matter in view of the experimental observation of such exotic states. Indeed, the σ resonance[8][9] is not seen in any of such lattice computations and contrasting interpretations have been put forward. This is a hotly debated issue.
Open problems
Yang–Mills theories met with general acceptance in the physics community after Gerard 't Hooft, in 1972, worked out their renormalization, relying on a formulation of the problem worked out by his advisor Martinus Veltman.[10] Renormalizability is obtained even if the gauge bosons described by this theory are massive, as in the electroweak theory, provided the mass is only an "acquired" one, generated by the Higgs mechanism.
The mathematics of the Yang–Mills theory is a very active field of research, yielding e.g. invariants of differentiable structures on four-dimensional manifolds via work of Simon Donaldson. Furthermore, the field of Yang–Mills theories was included in the Clay Mathematics Institute's list of "Millennium Prize Problems". Here the prize-problem consists, especially, in a proof of the conjecture that the lowest excitations of a pure Yang–Mills theory (i.e. without matter fields) have a finite mass-gap with regard to the vacuum state. Another open problem, connected with this conjecture, is a proof of the confinement property in the presence of additional Fermion particles.
In physics the survey of Yang–Mills theories does not usually start from perturbation analysis or analytical methods, but more recently from systematic application of numerical methods to lattice gauge theories.
Footnotes
1. See Abraham Pais' account of this period as well as L. Susskind's "Superstrings, Physics World on the first non-abelian gauge theory" where Susskind wrote that Yang–Mills was "rediscovered" only because Pauli had chosen not to publish.
See also
• Aharonov–Bohm effect
• Coulomb gauge
• Deformed Hermitian Yang–Mills equations
• Gauge covariant derivative
• Gauge theory (mathematics)
• Hermitian Yang–Mills equations
• Kaluza–Klein theory
• Lattice gauge theory
• Lorenz gauge
• n = 4 supersymmetric Yang–Mills theory
• Propagator
• Quantum gauge theory
• Field theoretical formulation of the standard model
• Symmetry in physics
• Two-dimensional Yang–Mills theory
• Weyl gauge
• Yang–Mills equations
• Yang–Mills existence and mass gap
• Yang–Mills–Higgs equations
References
1. Straumann, N. (2000). "On Pauli's invention of non-abelian Kaluza-Klein Theory in 1953". arXiv:gr-qc/0012054.
2. Gray, Jeremy; Wilson, Robin (2012-12-06). Mathematical Conversations: Selections from the Mathematical Intelligencer. Springer Science & Business Media. p. 63. ISBN 9781461301950 – via Google Books.
3. Yang, C.N.; Mills, R. (1954). "Conservation of isotopic spin and isotopic gauge invariance". Physical Review. 96 (1): 191–195. Bibcode:1954PhRv...96..191Y. doi:10.1103/PhysRev.96.191.
4. Atiyah, M. (2017). "Ronald Shaw 1929-2016 by Michael Atiyah (1954)". Trinity College Annual Record (memorial). 2017: 137–146.
5. "An anecdote by C.N. Yang". Review of the Universe (universe-review.ca).
6. Shaw, Ronald (September 1956). The problem of particle types and other contributions to the theory of elementary particles (Ph.D. thesis). University of Cambridge. ch. 3, pp. 34–46.
7. Fraser, Gordon (2008). Cosmic Anger: Abdus Salam – the first Muslim Nobel scientist. Oxford, UK: Oxford University Press. p. 117. ISBN 978-0199208463.
8. Caprini, I.; Colangelo, G.; Leutwyler, H. (2006). "Mass and width of the lowest resonance in QCD". Physical Review Letters. 96 (13): 132001. arXiv:hep-ph/0512364. Bibcode:2006PhRvL..96m2001C. doi:10.1103/PhysRevLett.96.132001. PMID 16711979. S2CID 42504317.
9. Yndurain, F.J.; Garcia-Martin, R.; Pelaez, J.R. (2007). "Experimental status of the ππ isoscalar S wave at low energy: f0(600) pole and scattering length". Physical Review D. 76 (7): 074034. arXiv:hep-ph/0701025. Bibcode:2007PhRvD..76g4034G. doi:10.1103/PhysRevD.76.074034. S2CID 119434312.
10. 't Hooft, G.; Veltman, M. (1972). "Regularization and renormalization of gauge fields". Nuclear Physics B. 44 (1): 189–213. Bibcode:1972NuPhB..44..189T. doi:10.1016/0550-3213(72)90279-9. hdl:1874/4845.
Further reading
Books
• Frampton, P. (2008). Gauge Field Theories (3rd ed.). Wiley-VCH. ISBN 978-3-527-40835-1.
• Cheng, T.-P.; Li, L.-F. (1983). Gauge Theory of Elementary Particle Physics. Oxford University Press. ISBN 0-19-851961-3.
• 't Hooft, G., ed. (2005). 50 Years of Yang–Mills theory. Singapore: World Scientific. ISBN 981-238-934-2.
Articles
• Svetlichny, George (1999). "Preparation for Gauge Theory". arXiv:math-ph/9902027.
• Gross, D. (1992). "Gauge theory – Past, Present and Future". Retrieved 2015-05-05.
External links
Wikiquote has quotations related to Yang–Mills theory.
• "Yang-Mills field", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• "Yang–Mills theory". DispersiveWiki.{{cite web}}: CS1 maint: url-status (link)
• "The Clay Mathematics Institute" (main page).
• "The Millennium Prize Problems". The Clay Mathematics Institute.
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| Wikipedia |
Yang–Mills equations
In physics and mathematics, and especially differential geometry and gauge theory, the Yang–Mills equations are a system of partial differential equations for a connection on a vector bundle or principal bundle. They arise in physics as the Euler–Lagrange equations of the Yang–Mills action functional. They have also found significant use in mathematics.
This article discusses the Yang–Mills equations from a mathematical perspective. For the physics perspective, see Yang–Mills theory
The dx1⊗σ3 coefficient of a BPST instanton on the (x1,x2)-slice of R4 where σ3 is the third Pauli matrix (top left). The dx2⊗σ3 coefficient (top right). These coefficients determine the restriction of the BPST instanton A with g=2, ρ=1,z=0 to this slice. The corresponding field strength centered around z=0 (bottom left). A visual representation of the field strength of a BPST instanton with center z on the compactification S4 of R4 (bottom right). The BPST instanton is a solution to the anti-self duality equations, and therefore of the Yang–Mills equations, on R4. This solution can be extended by Uhlenbeck's removable singularity theorem to a topologically non-trivial ASD connection on S4.
Solutions of the equations are called Yang–Mills connections or instantons. The moduli space of instantons was used by Simon Donaldson to prove Donaldson's theorem.
Motivation
Physics
In their foundational paper on the topic of gauge theories, Robert Mills and Chen-Ning Yang developed (essentially independent of the mathematical literature) the theory of principal bundles and connections in order to explain the concept of gauge symmetry and gauge invariance as it applies to physical theories.[1] The gauge theories Yang and Mills discovered, now called Yang–Mills theories, generalised the classical work of James Maxwell on Maxwell's equations, which had been phrased in the language of a $\operatorname {U} (1)$ gauge theory by Wolfgang Pauli and others.[2] The novelty of the work of Yang and Mills was to define gauge theories for an arbitrary choice of Lie group $G$, called the structure group (or in physics the gauge group, see Gauge group (mathematics) for more details). This group could be non-Abelian as opposed to the case $G=\operatorname {U} (1)$ corresponding to electromagnetism, and the right framework to discuss such objects is the theory of principal bundles.
The essential points of the work of Yang and Mills are as follows. One assumes that the fundamental description of a physical model is through the use of fields, and derives that under a local gauge transformation (change of local trivialisation of principal bundle), these physical fields must transform in precisely the way that a connection $A$ (in physics, a gauge field) on a principal bundle transforms. The gauge field strength is the curvature $F_{A}$ of the connection, and the energy of the gauge field is given (up to a constant) by the Yang–Mills action functional
$\operatorname {YM} (A)=\int _{X}\|F_{A}\|^{2}\,d\mathrm {vol} _{g}.$
The principle of least action dictates that the correct equations of motion for this physical theory should be given by the Euler–Lagrange equations of this functional, which are the Yang–Mills equations derived below:
$d_{A}\star F_{A}=0.$
Mathematics
In addition to the physical origins of the theory, the Yang–Mills equations are of important geometric interest. There is in general no natural choice of connection on a vector bundle or principal bundle. In the special case where this bundle is the tangent bundle to a Riemannian manifold, there is such a natural choice, the Levi-Civita connection, but in general there is an infinite-dimensional space of possible choices. A Yang–Mills connection gives some kind of natural choice of a connection for a general fibre bundle, as we now describe.
A connection is defined by its local forms $A_{\alpha }\in \Omega ^{1}(U_{\alpha },\operatorname {ad} (P))$ for a trivialising open cover $\{U_{\alpha }\}$ for the bundle $P\to X$. The first attempt at choosing a canonical connection might be to demand that these forms vanish. However, this is not possible unless the trivialisation is flat, in the sense that the transition functions $g_{\alpha \beta }:U_{\alpha }\cap U_{\beta }\to G$ are constant functions. Not every bundle is flat, so this is not possible in general. Instead one might ask that the local connection forms $A_{\alpha }$ are themselves constant. On a principal bundle the correct way to phrase this condition is that the curvature $F_{A}=dA+{\frac {1}{2}}[A,A]$ vanishes. However, by Chern–Weil theory if the curvature $F_{A}$ vanishes (that is to say, $A$ is a flat connection), then the underlying principal bundle must have trivial Chern classes, which is a topological obstruction to the existence of flat connections: not every principal bundle can have a flat connection.
The best one can hope for is then to ask that instead of vanishing curvature, the bundle has curvature as small as possible. The Yang–Mills action functional described above is precisely (the square of) the $L^{2}$-norm of the curvature, and its Euler–Lagrange equations describe the critical points of this functional, either the absolute minima or local minima. That is to say, Yang–Mills connections are precisely those that minimize their curvature. In this sense they are the natural choice of connection on a principal or vector bundle over a manifold from a mathematical point of view.
Definition
Let $X$ be a compact, oriented, Riemannian manifold. The Yang–Mills equations can be phrased for a connection on a vector bundle or principal $G$-bundle over $X$, for some compact Lie group $G$. Here the latter convention is presented. Let $P$ denote a principal $G$-bundle over $X$. Then a connection on $P$ may be specified by a Lie algebra-valued differential form $A$ on the total space of the principal bundle. This connection has a curvature form $F_{A}$, which is a two-form on $X$ with values in the adjoint bundle $\operatorname {ad} (P)$ of $P$. Associated to the connection $A$ is an exterior covariant derivative $d_{A}$, defined on the adjoint bundle. Additionally, since $G$ is compact, its associated compact Lie algebra admits an invariant inner product under the adjoint representation.
Since $X$ is Riemannian, there is an inner product on the cotangent bundle, and combined with the invariant inner product on $\operatorname {ad} (P)$ there is an inner product on the bundle $\operatorname {ad} (P)\otimes \Lambda ^{2}T^{*}X$ of $\operatorname {ad} (P)$-valued two-forms on $X$. Since $X$ is oriented, there is an $L^{2}$-inner product on the sections of this bundle. Namely,
$\langle s,t\rangle _{L^{2}}=\int _{X}\langle s,t\rangle \,dvol_{g}$
where inside the integral the bundle-wise inner product is being used, and $dvol_{g}$ is the Riemannian volume form of $X$. Using this $L^{2}$-inner product, the formal adjoint operator of $d_{A}$ is defined by
$\langle d_{A}s,t\rangle _{L^{2}}=\langle s,d_{A}^{*}t\rangle _{L^{2}}$.
Explicitly this is given by $d_{A}^{*}=\pm \star d_{A}\star $ where $\star $ is the Hodge star operator acting on two-forms.
Assuming the above set up, the Yang–Mills equations are a system of (in general non-linear) partial differential equations given by
$d_{A}^{*}F_{A}=0.$[3]
(1)
Since the Hodge star is an isomorphism, by the explicit formula for $d_{A}^{*}$ the Yang–Mills equations can equivalently be written
$d_{A}\star F_{A}=0.$
(2)
A connection satisfying (1) or (2) is called a Yang–Mills connection.
Every connection automatically satisfies the Bianchi identity $d_{A}F_{A}=0$, so Yang–Mills connections can be seen as a non-linear analogue of harmonic differential forms, which satisfy
$d\omega =d^{*}\omega =0$.
In this sense the search for Yang–Mills connections can be compared to Hodge theory, which seeks a harmonic representative in the de Rham cohomology class of a differential form. The analogy being that a Yang–Mills connection is like a harmonic representative in the set of all possible connections on a principal bundle.
Derivation
The Yang–Mills equations are the Euler–Lagrange equations of the Yang–Mills functional, defined by
$\operatorname {YM} (A)=\int _{X}\|F_{A}\|^{2}\,d\mathrm {vol} _{g}.$
(3)
To derive the equations from the functional, recall that the space ${\mathcal {A}}$ of all connections on $P$ is an affine space modelled on the vector space $\Omega ^{1}(P;{\mathfrak {g}})$. Given a small deformation $A+ta$ of a connection $A$ in this affine space, the curvatures are related by
$F_{A+ta}=F_{A}+td_{A}a+t^{2}a\wedge a.$
To determine the critical points of (3), compute
${\begin{aligned}{\frac {d}{dt}}\left(\operatorname {YM} (A+ta)\right)_{t=0}&={\frac {d}{dt}}\left(\int _{X}\langle F_{A}+t\,d_{A}a+t^{2}a\wedge a,F_{A}+t\,d_{A}a+t^{2}a\wedge a\rangle \,d\mathrm {vol} _{g}\right)_{t=0}\\&={\frac {d}{dt}}\left(\int _{X}\|F_{A}\|^{2}+2t\langle F_{A},d_{A}a\rangle +2t^{2}\langle F_{A},a\wedge a\rangle +t^{4}\|a\wedge a\|^{2}\,d\mathrm {vol} _{g}\right)_{t=0}\\&=2\int _{X}\langle d_{A}^{*}F_{A},a\rangle \,d\mathrm {vol} _{g}.\end{aligned}}$
The connection $A$ is a critical point of the Yang–Mills functional if and only if this vanishes for every $a$, and this occurs precisely when (1) is satisfied.
Moduli space of Yang–Mills connections
The Yang–Mills equations are gauge invariant. Mathematically, a gauge transformation is an automorphism $g$ of the principal bundle $P$, and since the inner product on $\operatorname {ad} (P)$ is invariant, the Yang–Mills functional satisfies
$\operatorname {YM} (g\cdot A)=\int _{X}\|gF_{A}g^{-1}\|^{2}\,d\mathrm {vol} _{g}=\int _{X}\|F_{A}\|^{2}\,d\mathrm {vol} _{g}=\operatorname {YM} (A)$
and so if $A$ satisfies (1), so does $g\cdot A$.
There is a moduli space of Yang–Mills connections modulo gauge transformations. Denote by ${\mathcal {G}}$ the gauge group of automorphisms of $P$. The set ${\mathcal {B}}={\mathcal {A}}/{\mathcal {G}}$ classifies all connections modulo gauge transformations, and the moduli space ${\mathcal {M}}$ of Yang–Mills connections is a subset. In general neither ${\mathcal {B}}$ or ${\mathcal {M}}$ is Hausdorff or a smooth manifold. However, by restricting to irreducible connections, that is, connections $A$ whose holonomy group is given by all of $G$, one does obtain Hausdorff spaces. The space of irreducible connections is denoted ${\mathcal {A}}^{*}$, and so the moduli spaces are denoted ${\mathcal {B}}^{*}$ and ${\mathcal {M}}^{*}$.
Moduli spaces of Yang–Mills connections have been intensively studied in specific circumstances. Michael Atiyah and Raoul Bott studied the Yang–Mills equations for bundles over compact Riemann surfaces.[4] There the moduli space obtains an alternative description as a moduli space of holomorphic vector bundles. This is the Narasimhan–Seshadri theorem, which was proved in this form relating Yang–Mills connections to holomorphic vector bundles by Donaldson.[5] In this setting the moduli space has the structure of a compact Kähler manifold. Moduli of Yang–Mills connections have been most studied when the dimension of the base manifold $X$ is four.[3][6] Here the Yang–Mills equations admit a simplification from a second-order PDE to a first-order PDE, the anti-self-duality equations.
Anti-self-duality equations
When the dimension of the base manifold $X$ is four, a coincidence occurs: the Hodge star operator maps two-forms to two-forms,
$\star :\Omega ^{2}(X)\to \Omega ^{2}(X)$ :\Omega ^{2}(X)\to \Omega ^{2}(X)} .
The Hodge star operator squares to the identity in this case, and so has eigenvalues $1$ and $-1$. In particular, there is a decomposition
$\Omega ^{2}(X)=\Omega _{+}(X)\oplus \Omega _{-}(X)$
into the positive and negative eigenspaces of $\star $, the self-dual and anti-self-dual two-forms. If a connection $A$ on a principal $G$-bundle over a four-manifold $X$ satisfies either $F_{A}={\star F_{A}}$ or $F_{A}=-{\star F_{A}}$, then by (2), the connection is a Yang–Mills connection. These connections are called either self-dual connections or anti-self-dual connections, and the equations the self-duality (SD) equations and the anti-self-duality (ASD) equations.[3] The spaces of self-dual and anti-self-dual connections are denoted by ${\mathcal {A}}^{+}$ and ${\mathcal {A}}^{-}$, and similarly for ${\mathcal {B}}^{\pm }$ and ${\mathcal {M}}^{\pm }$.
The moduli space of ASD connections, or instantons, was most intensively studied by Donaldson in the case where $G=\operatorname {SU} (2)$ and $X$ is simply-connected.[7][8][9] In this setting, the principal $\operatorname {SU} (2)$-bundle is classified by its second Chern class, $c_{2}(P)\in H^{4}(X,\mathbb {Z} )\cong \mathbb {Z} $.[Note 1] For various choices of principal bundle, one obtains moduli spaces with interesting properties. These spaces are Hausdorff, even when allowing reducible connections, and are generically smooth. It was shown by Donaldson that the smooth part is orientable. By the Atiyah–Singer index theorem, one may compute that the dimension of ${\mathcal {M}}_{k}^{-}$, the moduli space of ASD connections when $c_{2}(P)=k$, to be
$\dim {\mathcal {M}}_{k}^{-}=8k-3(1-b_{1}(X)+b_{+}(X))$
where $b_{1}(X)$ is the first Betti number of $X$, and $b_{+}(X)$ is the dimension of the positive-definite subspace of $H_{2}(X,\mathbb {R} )$ with respect to the intersection form on $X$.[3] For example, when $X=S^{4}$ and $k=1$, the intersection form is trivial and the moduli space has dimension $\dim {\mathcal {M}}_{1}^{-}(S^{4})=8-3=5$. This agrees with existence of the BPST instanton, which is the unique ASD instanton on $S^{4}$ up to a 5 parameter family defining its centre in $\mathbb {R} ^{4}$ and its scale. Such instantons on $\mathbb {R} ^{4}$ may be extended across the point at infinity using Uhlenbeck's removable singularity theorem.
Applications
Donaldson's theorem
Main article: Donaldson's theorem
The moduli space of Yang–Mills equations was used by Donaldson to prove Donaldson's theorem about the intersection form of simply-connected four-manifolds. Using analytical results of Clifford Taubes and Karen Uhlenbeck, Donaldson was able to show that in specific circumstances (when the intersection form is definite) the moduli space of ASD instantons on a smooth, compact, oriented, simply-connected four-manifold $X$ gives a cobordism between a copy of the manifold itself, and a disjoint union of copies of the complex projective plane $\mathbb {CP} ^{2}$.[7][10][11][12] The intersection form is a cobordism invariant up to isomorphism, showing that any such smooth manifold has diagonalisable intersection form.
The moduli space of ASD instantons may be used to define further invariants of four-manifolds. Donaldson defined rational numbers associated to a four-manifold arising from pairings of cohomology classes on the moduli space.[9] This work has subsequently been surpassed by Seiberg–Witten invariants.
Dimensional reduction and other moduli spaces
Through the process of dimensional reduction, the Yang–Mills equations may be used to derive other important equations in differential geometry and gauge theory. Dimensional reduction is the process of taking the Yang–Mills equations over a four-manifold, typically $\mathbb {R} ^{4}$, and imposing that the solutions be invariant under a symmetry group. For example:
• By requiring the anti-self-duality equations to be invariant under translations in a single direction of $\mathbb {R} ^{4}$, one obtains the Bogomolny equations which describe magnetic monopoles on $\mathbb {R} ^{3}$.
• By requiring the self-duality equations to be invariant under translation in two directions, one obtains Hitchin's equations first investigated by Hitchin. These equations naturally lead to the study of Higgs bundles and the Hitchin system.
• By requiring the anti-self-duality equations to be invariant in three directions, one obtains the Nahm equations on an interval.
There is a duality between solutions of the dimensionally reduced ASD equations on $\mathbb {R} ^{3}$ and $\mathbb {R} $ called the Nahm transform, after Werner Nahm, who first described how to construct monopoles from Nahm equation data.[13] Hitchin showed the converse, and Donaldson proved that solutions to the Nahm equations could further be linked to moduli spaces of rational maps from the complex projective line to itself.[14][15]
The duality observed for these solutions is theorized to hold for arbitrary dual groups of symmetries of a four-manifold. Indeed there is a similar duality between instantons invariant under dual lattices inside $\mathbb {R} ^{4}$, instantons on dual four-dimensional tori, and the ADHM construction can be thought of as a duality between instantons on $\mathbb {R} ^{4}$ and dual algebraic data over a single point.[3]
Symmetry reductions of the ASD equations also lead to a number of integrable systems, and Ward's conjecture is that in fact all known integrable ODEs and PDEs come from symmetry reduction of ASDYM. For example reductions of SU(2) ASDYM give the sine-Gordon and Korteweg–de Vries equation, of $\mathrm {SL} (3,\mathbb {R} )$ ASDYM gives the Tzitzeica equation, and a particular reduction to $2+1$ dimensions gives the integrable chiral model of Ward.[16] In this sense it is a 'master theory' for integrable systems, allowing many known systems to be recovered by picking appropriate parameters, such as choice of gauge group and symmetry reduction scheme. Other such master theories are four-dimensional Chern–Simons theory and the affine Gaudin model.
Chern–Simons theory
The moduli space of Yang–Mills equations over a compact Riemann surface $\Sigma $ can be viewed as the configuration space of Chern–Simons theory on a cylinder $\Sigma \times [0,1]$. In this case the moduli space admits a geometric quantization, discovered independently by Nigel Hitchin and Axelrod–Della Pietra–Witten.[17][18]
See also
• Connection (vector bundle)
• Connection (principal bundle)
• Donaldson theory
• Hermitian Yang–Mills equations
• Deformed Hermitian Yang–Mills equations
• Yang–Mills–Higgs equations
Notes
1. For a proof of this fact, see the post https://mathoverflow.net/a/265399.
References
1. Yang, C.N. and Mills, R.L., 1954. Conservation of isotopic spin and isotopic gauge invariance. Physical review, 96(1), p.191.
2. Pauli, W., 1941. Relativistic field theories of elementary particles. Reviews of Modern Physics, 13(3), p.203.
3. Donaldson, S. K., & Kronheimer, P. B. (1990). The geometry of four-manifolds. Oxford University Press.
4. Atiyah, M. F., & Bott, R. (1983). The Yang–Mills equations over riemann surfaces. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 308(1505), 523–615.
5. Donaldson, S. K. (1983). A new proof of a theorem of Narasimhan and Seshadri. Journal of Differential Geometry, 18(2), 269–277.
6. Friedman, R., & Morgan, J. W. (1998). Gauge theory and the topology of four-manifolds (Vol. 4). American Mathematical Soc.
7. Donaldson, S. K. (1983). An application of gauge theory to four-dimensional topology. Journal of Differential Geometry, 18(2), 279–315.
8. Donaldson, S. K. (1986). Connections, cohomology and the intersection forms of 4-manifolds. Journal of Differential Geometry, 24(3), 275–341.
9. Donaldson, S. K. (1990). Polynomial invariants for smooth four-manifolds. Topology, 29(3), 257–315.
10. Taubes, C. H. (1982). Self-dual Yang–Mills connections on non-self-dual 4-manifolds. Journal of Differential Geometry, 17(1), 139–170.
11. Uhlenbeck, K. K. (1982). Connections with Lp bounds on curvature. Communications in Mathematical Physics, 83(1), 31–42.
12. Uhlenbeck, K. K. (1982). Removable singularities in Yang–Mills fields. Communications in Mathematical Physics, 83(1), 11–29.
13. Nahm, W. (1983). All self-dual multimonopoles for arbitrary gauge groups. In Structural elements in particle physics and statistical mechanics (pp. 301–310). Springer, Boston, MA.
14. Hitchin, N. J. (1983). On the construction of monopoles. Communications in Mathematical Physics, 89(2), 145–190.
15. Donaldson, S. K. (1984). Nahm's equations and the classification of monopoles. Communications in Mathematical Physics, 96(3), 387–408.
16. Dunajski, Maciej (2010). Solitons, instantons, and twistors. Oxford: Oxford University Press. pp. 151–154. ISBN 9780198570639.
17. Hitchin, N. J. (1990). Flat connections and geometric quantization. Communications in mathematical physics, 131(2), 347–380.
18. Axelrod, S., Della Pietra, S., & Witten, E. (1991). Geometric quantization of Chern Simons gauge theory. representations, 34, 39.
| Wikipedia |
Yang Le
Yang Le (simplified Chinese: 杨乐; traditional Chinese: 楊樂; pinyin: Yáng Lè; born 10 November 1939) is a Chinese mathematician. He is a member of the Chinese Academy of Sciences.[2]
Yang Le
杨乐
Born (1939-11-10) November 10, 1939
Nantong, Jiangsu, China
Alma materNantong Middle School of Jiangsu Province
Peking University
AwardsHuang Qieyuan[1]
Scientific career
FieldsMathematics
InstitutionsChinese Academy of Sciences
Doctoral advisorXiong Qinglai
Biography
Yang was born and raised in Nantong, Jiangsu. His father, Yang Jingyuan (Chinese: 杨敬渊), was a businessman and assistant manager of Nantong Tongming Electric Company.[3] His mother named Zhou Jingjuan (周静娟). He primarily studied at the First Affiliated Primary School of Nantong Normal College and secondary studied at Nantong Middle School of Jiangsu Province. He was accepted to Peking University in 1956 and graduated in 1962. After college, he studied mathematics under Xiong Qinglai at Chinese Academy of Sciences, and started working there as a research scientist after the graduate program.[3] He was elected a fellow of the Chinese Academy of Sciences in 1980.
Personal life
Yang married Huang Qieyuan (Chinese: 黄且圆), who is Huang Wanli's daughter.
References
1. 熊庆来文革凄惨 华罗庚悼恩师：痛莫痛于不敢啼. Sohu (in Chinese). 2014-02-17.
2. 中国科学院院士、数学家杨乐：以“诚恒”立身. GMW (in Chinese). 2014-07-28.
3. Li Shuya (2012-01-30). 走近院士：数学家杨乐：万事不离其“数”. Chinapictorial.com.cn (in Chinese).
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Yang Hui
Yang Hui (simplified Chinese: 杨辉; traditional Chinese: 楊輝; pinyin: Yáng Huī, ca. 1238–1298), courtesy name Qianguang (謙光), was a Chinese mathematician and writer during the Song dynasty. Originally, from Qiantang (modern Hangzhou, Zhejiang), Yang worked on magic squares, magic circles and the binomial theorem, and is best known for his contribution of presenting Yang Hui's Triangle. This triangle was the same as Pascal's Triangle, discovered by Yang's predecessor Jia Xian. Yang was also a contemporary to the other famous mathematician Qin Jiushao.
Written work
The earliest extant Chinese illustration of 'Pascal's triangle' is from Yang's book Xiangjie Jiuzhang Suanfa (詳解九章算法)[1] of 1261 AD, in which Yang acknowledged that his method of finding square roots and cubic roots using "Yang Hui's Triangle" was invented by mathematician Jia Xian[2] who expounded it around 1100 AD, about 500 years before Pascal. In his book (now lost) known as Rújī Shìsuǒ (如積釋鎖) or Piling-up Powers and Unlocking Coefficients, which is known through his contemporary mathematician Liu Ruxie (劉汝諧).[3] Jia described the method used as 'li cheng shi suo' (the tabulation system for unlocking binomial coefficients).[3] It appeared again in a publication of Zhu Shijie's book Jade Mirror of the Four Unknowns (四元玉鑒) of 1303 AD.[4]
Around 1275 AD, Yang finally had two published mathematical books, which were known as the Xugu Zhaiqi Suanfa (續古摘奇算法) and the Suanfa Tongbian Benmo (算法通變本末, summarily called Yang Hui suanfa 楊輝算法).[5] In the former book, Yang wrote of arrangement of natural numbers around concentric and non concentric circles, known as magic circles and vertical-horizontal diagrams of complex combinatorial arrangements known as magic squares, providing rules for their construction.[6] In his writing, he harshly criticized the earlier works of Li Chunfeng and Liu Yi (劉益), the latter of whom were both content with using methods without working out their theoretical origins or principle.[5] Displaying a somewhat modern attitude and approach to mathematics, Yang once said:
The men of old changed the name of their methods from problem to problem, so that as no specific explanation was given, there is no way of telling their theoretical origin or basis.[5]
In his written work, Yang provided theoretical proof for the proposition that the complements of the parallelograms which are about the diameter of any given parallelogram are equal to one another.[5] This was the same idea expressed in the Greek mathematician Euclid's (fl. 300 BC) forty-third proposition of his first book, only Yang used the case of a rectangle and gnomon.[5] There were also a number of other geometrical problems and theoretical mathematical propositions posed by Yang that were strikingly similar to the Euclidean system.[7] However, the first books of Euclid to be translated into Chinese was by the cooperative effort of the Italian Jesuit Matteo Ricci and the Ming official Xu Guangqi in the early 17th century.[8]
Yang's writing represents the first in which quadratic equations with negative coefficients of 'x' appear, although he attributes this to the earlier Liu Yi.[9] Yang was also well known for his ability to manipulate decimal fractions. When he wished to multiply the figures in a rectangular field with a breadth of 24 paces 3 4⁄10 ft. and length of 36 paces 2 8⁄10, Yang expressed them in decimal parts of the pace, as 24.68 X 36.56 = 902.3008.[10]
See also
• History of mathematics
• List of mathematicians
• Chinese mathematics
Notes
1. Fragments of this book was retained in the Yongle Encyclopedia vol 16344, in British Museum Library
2. Needham, Volume 3, 134-137.
3. Needham, Volume 3, 137.
4. Needham, Volume 3, 134-135.
5. Needham, Volume 3, 104.
6. Needham, Volume 3, 59-60.
7. Needham, Volume 3, 105.
8. Needham, Volume 3, 106.
9. Needham, Volume 3, 46.
10. Needham, Volume 3, 45.
References
• Needham, Joseph (1986). Science and Civilization in China: Volume 3, Mathematics and the Sciences of the Heavens and the Earth. Taipei: Caves Books, Ltd.
• Li, Jimin, "Yang Hui". Encyclopedia of China (Mathematics Edition), 1st ed.
External links
• Yang Hui at MacTutor
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Yang–Mills–Higgs equations
In mathematics, the Yang–Mills–Higgs equations are a set of non-linear partial differential equations for a Yang–Mills field, given by a connection, and a Higgs field, given by a section of a vector bundle (specifically, the adjoint bundle). These equations are
${\begin{aligned}D_{A}*F_{A}+[\Phi ,D_{A}\Phi ]&=0,\\D_{A}*D_{A}\Phi &=0\end{aligned}}$
with a boundary condition
$\lim _{|x|\rightarrow \infty }|\Phi |(x)=1$
where
A is a connection on a vector bundle,
DA is the exterior covariant derivative,
FA is the curvature of that connection,
Φ is a section of that vector bundle,
∗ is the Hodge star, and
[·,·] is the natural, graded bracket.
These equations are named after Chen Ning Yang, Robert Mills, and Peter Higgs. They are very closely related to the Ginzburg–Landau equations, when these are expressed in a general geometric setting.
M.V. Goganov and L.V. Kapitanskii have shown that the Cauchy problem for hyperbolic Yang–Mills–Higgs equations in Hamiltonian gauge on 4-dimensional Minkowski space have a unique global solution with no restrictions at the spatial infinity. Furthermore, the solution has the finite propagation speed property.
Lagrangian
The equations arise as the equations of motion of the Lagrangian density
Yang-Mills-Higgs Lagrangian density
${\mathcal {L}}=-{\frac {1}{4}}\left\langle F_{\mu \nu },F^{\mu \nu }\right\rangle +{\frac {1}{2}}\left\langle D_{\mu }\phi ,D^{\mu }\phi \right\rangle -V(\phi )$
where $\langle \cdot ,\cdot \rangle $ is an invariant symmetric bilinear form on the adjoint bundle. This is sometimes written as ${\text{tr}}$ due to the fact that such a form can arise from the trace on ${\mathfrak {g}}$ under some representation; in particular here we are concerned with the adjoint representation, and the trace on this representation is the Killing form.
For the particular form of the Yang-Mills-Higgs equations given above, the potential $V(\phi )$ is vanishing. Another common choice is $V(\phi )={\frac {1}{2}}m^{2}\langle \phi ,\phi \rangle $, corresponding to a massive Higgs field.
This theory is a particular case of scalar chromodynamics where the Higgs field $\phi $ is valued in the adjoint representation as opposed to a general representation.
See also
• Yang–Mills equations
• Scalar chromodynamics
References
• M.V. Goganov and L.V. Kapitansii, "Global solvability of the initial problem for Yang-Mills-Higgs equations", Zapiski LOMI 147,18–48, (1985); J. Sov. Math, 37, 802–822 (1987).
Quantum field theories
Theories
• Algebraic QFT
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Models
Regular
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• Soler
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Low dimensional
• 2D Yang–Mills
• Bullough–Dodd
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Conformal
• 2D free massless scalar
• Liouville
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Supersymmetric
• Wess–Zumino
• N = 1 super Yang–Mills
• Seiberg–Witten
• Super QCD
Superconformal
• 6D (2,0)
• ABJM
• N = 4 super Yang–Mills
Supergravity
• Higher dimensional
• N = 8
• Pure 4D N = 1
Topological
• BF
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Particle theory
• Chiral
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Related
• Casimir effect
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See also: Template:Quantum mechanics topics
| Wikipedia |
Murakami–Yano formula
In geometry, the Murakami–Yano formula, introduced by Murakami & Yano (2005), is a formula for the volume of a hyperbolic or spherical tetrahedron given in terms of its dihedral angles.
References
• Murakami, Jun; Yano, Masakazu (2005), "On the volume of a hyperbolic and spherical tetrahedron", Communications in Analysis and Geometry, 13 (2): 379–400, ISSN 1019-8385, MR 2154824, archived from the original (PDF) on 2012-04-10, retrieved 2012-02-10 url=http://www.f.waseda.jp/murakami/papers/tetrahedronrev4.pdf
| Wikipedia |
YanYan Li
YanYan Li (also stylized as Yanyan Li, Yan-yan Li, and Yan Yan Li) is a Professor of mathematics at Rutgers University, specializing in elliptic partial differential equations. He received his Ph.D. at New York University in 1988, under the direction of Louis Nirenberg. He joined Rutgers University in 1990.
Li was an invited lecturer at the International Congress of Mathematicians in 2002, and is a Fellow of the American Mathematical Society. He has been an ISI Highly Cited Researcher.[1] He is a member of the editorial board of Advances in Mathematics, among several other academic journals.[2]
Major publications
• Yan Yan Li and Itai Shafrir. Blow-up analysis for solutions of −Δu = Veu in dimension two. Indiana Univ. Math. J. 43 (1994), no. 4, 1255–1270. doi:10.1512/iumj.1994.43.43054
• Yan Yan Li. Prescribing scalar curvature on Sn and related problems. I. J. Differential Equations 120 (1995), no. 2, 319–410. doi:10.1006/jdeq.1995.1115
• Yan Yan Li. Prescribing scalar curvature on Sn and related problems. II. Existence and compactness. Comm. Pure Appl. Math. 49 (1996), no. 6, 541–597. doi:10.1002/(SICI)1097-0312(199606)49:6<541::AID-CPA1>3.0.CO;2-A
References
1. Web of Science
2. Advances in Mathematics - Editorial Board. Accessed October 8, 2020.
External links
• YanYan Li's home page at Rutgers
• Mathematics Genealogy page
• Google Scholar page
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Yao's test
In cryptography and the theory of computation, Yao's test is a test defined by Andrew Chi-Chih Yao in 1982,[1] against pseudo-random sequences. A sequence of words passes Yao's test if an attacker with reasonable computational power cannot distinguish it from a sequence generated uniformly at random.
Formal statement
Boolean circuits
Let $P$ be a polynomial, and $S=\{S_{k}\}_{k}$ be a collection of sets $S_{k}$ of $P(k)$-bit long sequences, and for each $k$, let $\mu _{k}$ be a probability distribution on $S_{k}$, and $P_{C}$ be a polynomial. A predicting collection $C=\{C_{k}\}$ is a collection of boolean circuits of size less than $P_{C}(k)$. Let $p_{k,S}^{C}$ be the probability that on input $s$, a string randomly selected in $S_{k}$ with probability $\mu (s)$, $C_{k}(s)=1$, i.e.
$p_{k,S}^{C}={\mathcal {P}}[C_{k}(s)=1|s\in S_{k}{\text{ with probability }}\mu _{k}(s)]$
Moreover, let $p_{k,U}^{C}$ be the probability that $C_{k}(s)=1$ on input $s$ a $P(k)$-bit long sequence selected uniformly at random in $\{0,1\}^{P(k)}$. We say that $S$ passes Yao's test if for all predicting collection $C$, for all but finitely many $k$, for all polynomial $Q$ :
$|p_{k,S}^{C}-p_{k,U}^{C}|<{\frac {1}{Q(k)}}$
Probabilistic formulation
As in the case of the next-bit test, the predicting collection used in the above definition can be replaced by a probabilistic Turing machine, working in polynomial time. This also yields a strictly stronger definition of Yao's test (see Adleman's theorem). Indeed, One could decide undecidable properties of the pseudo-random sequence with the non-uniform circuits described above, whereas BPP machines can always be simulated by exponential-time deterministic Turing machines.
References
1. Andrew Chi-Chih Yao. Theory and applications of trapdoor functions. In Proceedings of the 23rd IEEE Symposium on Foundations of Computer Science, 1982.
| Wikipedia |
Yao's principle
In computational complexity theory, Yao's principle (also called Yao's minimax principle or Yao's lemma) is a way to prove lower bounds on the worst-case performance of randomized algorithms, by comparing them to deterministic (non-random) algorithms. It states that, for any randomized algorithm, there exists a probability distribution on inputs to the algorithm, so that the expected cost of the randomized algorithm on its worst-case input is at least as large as the cost of the best deterministic algorithm on a random input from this distribution. Thus, to establish a lower bound on the performance of randomized algorithms, it suffices to find an appropriate distribution of difficult inputs, and to prove that no deterministic algorithm can perform well against that distribution. This principle is named after Andrew Yao, who first proposed it.
Yao's principle may be interpreted in game theoretic terms, via a two-player zero-sum game in which one player, Alice, selects a deterministic algorithm, the other player, Bob, selects an input, and the payoff is the cost of the selected algorithm on the selected input. Any randomized algorithm R may be interpreted as a randomized choice among deterministic algorithms, and thus as a mixed strategy for Alice. Similarly, a non-random algorithm may be thought of as a pure strategy for Alice. By von Neumann's minimax theorem, Bob has a randomized strategy that performs at least as well against R as it does against the best pure strategy Alice might have chosen. The worst-case input against Alice's strategy has cost at least as large as Bob's randomly chosen input paired against Alice's strategy, which in turn has cost at least as large as Bob's randomly chosen input paired against any pure strategy.
Statement
The formulation below states the principle for Las Vegas randomized algorithms, i.e., distributions over deterministic algorithms that are correct on every input but have varying costs. It is straightforward to adapt the principle to Monte Carlo algorithms, i.e., distributions over deterministic algorithms that have bounded costs but can be incorrect on some inputs.
Consider a problem over the inputs ${\mathcal {X}}$, and let ${\mathcal {A}}$ be the set of all possible deterministic algorithms that correctly solve the problem. For any algorithm $a\in {\mathcal {A}}$ and input $x\in {\mathcal {X}}$, let $c(a,x)\geq 0$ be the cost of algorithm $a$ run on input $x$.
Let $p$ be a probability distribution over the algorithms ${\mathcal {A}}$, and let $A$ denote a random algorithm chosen according to $p$. Let $q$ be a probability distribution over the inputs ${\mathcal {X}}$, and let $X$ denote a random input chosen according to $q$. Then,
${\underset {x\in {\mathcal {X}}}{\max }}\ \mathbf {E} [c(A,x)]\geq {\underset {a\in {\mathcal {A}}}{\min }}\ \mathbf {E} [c(a,X)].$
That is, the worst-case expected cost of the randomized algorithm is at least the expected cost of the best deterministic algorithm against input distribution $q$.
Proof
Let $C={\underset {x\in {\mathcal {X}}}{\max }}\ \mathbf {E} [c(A,x)]$ and $D={\underset {a\in {\mathcal {A}}}{\min }}\ \mathbf {E} [c(a,X)]$. We have
${\begin{aligned}C&=\sum _{x}q_{x}C\\&\geq \sum _{x}q_{x}\mathbf {E} [c(A,x)]\\&=\sum _{x}q_{x}\sum _{a}p_{a}c(a,x)\\&=\sum _{a}p_{a}\sum _{x}q_{x}c(a,x)\\&=\sum _{a}p_{a}\mathbf {E} [c(a,X)]\\&\geq \sum _{a}p_{a}D=D.\end{aligned}}$
As mentioned above, this theorem can also be seen as a very special case of the Minimax theorem.
See also
• Randomized algorithms as zero-sum games
References
• Borodin, Allan; El-Yaniv, Ran (2005), "8.3 Yao's principle: A technique for obtaining lower bounds", Online Computation and Competitive Analysis, Cambridge University Press, pp. 115–120, ISBN 9780521619462
• Yao, Andrew (1977), "Probabilistic computations: Toward a unified measure of complexity", Proceedings of the 18th IEEE Symposium on Foundations of Computer Science (FOCS), pp. 222–227, doi:10.1109/SFCS.1977.24
External links
• Fortnow, Lance (October 16, 2006), "Favorite theorems: Yao principle", Computational Complexity
| Wikipedia |
Yao graph
In computational geometry, the Yao graph, named after Andrew Yao, is a kind of geometric spanner, a weighted undirected graph connecting a set of geometric points with the property that, for every pair of points in the graph, their shortest path has a length that is within a constant factor of their Euclidean distance.
The basic idea underlying the two-dimensional Yao graph is to surround each of the given points by equally spaced rays, partitioning the plane into sectors with equal angles, and to connect each point to its nearest neighbor in each of these sectors.[1] Associated with a Yao graph is an integer parameter k ≥ 6 which is the number of rays and sectors described above; larger values of k produce closer approximations to the Euclidean distance.[2] The stretch factor is at most $1/(\cos \theta -\sin \theta )$, where $\theta $ is the angle of the sectors.[3] The same idea can be extended to point sets in more than two dimensions, but the number of sectors required grows exponentially with the dimension.
Andrew Yao used these graphs to construct high-dimensional Euclidean minimum spanning trees.[3]
Software for drawing Yao graphs
• Cone-based Spanners in Computational Geometry Algorithms Library (CGAL)
See also
• Theta graph
• Semi-Yao graph
References
1. "Overlay Networks for Wireless Systems" (PDF).
2. "Simple Topologies" (PDF).
3. Yao, A. C. (1982), "On constructing minimum spanning trees in k-dimensional space and related problems", SIAM Journal on Computing, 11 (4): 721–736, CiteSeerX 10.1.1.626.3161, doi:10.1137/0211059.
| Wikipedia |
Yasuo Akizuki
Yasuo Akizuki (23 August 1902 – 11 July 1984) was a Japanese mathematician.[1] He was a professor at Kyoto University.[2] Alongside Wolfgang Krull, Oscar Zariski, and Masayoshi Nagata, he is famous for his early work in commutative algebra. In particular, he is most well known in helping to demonstrate Akizuki–Hopkins–Levitzki theorem.
Yasuo Akizuki
Born(1902-08-23)August 23, 1902
Wakayama, Japan
DiedJuly 11, 1984(1984-07-11) (aged 81)
NationalityJapanese
Citizenship Japan
Alma materKyoto University
Scientific career
Fieldsalgebraic geometry
InstitutionsKyoto University
Gunma University
Doctoral advisorMasazo Sono
Doctoral studentsSatoshi Suzuki
Life
Yasuo Akizuki was born on 23 August 1902 in Wakayama. In 1926, Akizuki graduated Faculty of Mathematics, Department of Science, Kyoto Imperial University.
He was inaugurated as a professor of Kyoto University in 1948.
See also
• Jacob Levitzki
References
1. Atiyah, M.; Bauer, F.L.; Cartan, H.; Chern, S.-S.; Hirzebruch, F.; Conway, J.H.; Eckmann, B.; Faddeev, L.D.; Remmert, Reinhold; Grauert, H.; Hironaka, H.; Hormander, L.; John, F.; Koecher, M.; Narasimhan, R.; Reid, C.; Serre, J-P.; Sloane, N.J.A.; Tits, J.; Weil, A.; Zagier, D. (December 6, 2012). Miscellanea Mathematica. Springer Berlin Heidelberg. p. 164. ISBN 9783642767098 – via Google Books.
2. Shigekawa, Ichiro; Elworthy, K David; Kusuoka, S (May 5, 1997). New Trends In Stochastic Analysis: Proceedings Of The Tanaguchi International Symposium. World Scientific Publishing Company. p. 7. ISBN 9789814547123 – via Google Books.
| Wikipedia |
Yasutaka Ihara
Yasutaka Ihara (伊原 康隆, Ihara Yasutaka; born 1938, Tokyo Prefecture) is a Japanese mathematician and professor emeritus at the Research Institute for Mathematical Sciences. His work in number theory includes Ihara's lemma and the Ihara zeta function.
Yasutaka Ihara
伊原 康隆
EducationUniversity of Tokyo
Scientific career
Doctoral advisorShokichi Iyanaga
Kenkichi Iwasawa
Doctoral studentsKazuya Kato
Career
Ihara received his PhD at the University of Tokyo in 1967 with thesis Hecke polynomials as congruence zeta functions in elliptic modular case.[1]
From 1965 to 1966, Ihara worked at the Institute for Advanced Study. He was a professor at the University of Tokyo and then at the Research Institute for Mathematical Science (RIMS) of the University of Kyōto. In 2002 he retired from RIMS as professor emeritus and then became a professor at Chūō University.
In 1970, he was an invited speaker (with lecture Non abelian class fields over function fields in special cases) at the International Congress of Mathematicians (ICM) in Nice. In 1990, Ihara gave a plenary lecture Braids, Galois groups and some arithmetic functions at the ICM in Kyōto.
His doctoral students include Kazuya Katō.[1]
Research
Ihara has worked on geometric and number theoretic applications of Galois theory. In the 1960s, he introduced the eponymous Ihara zeta function.[2] In graph theory the Ihara zeta function has an interpretation, which was conjectured by Jean-Pierre Serre and proved by Toshikazu Sunada in 1985. Sunada also proved that a regular graph is a Ramanujan graph if and only if its Ihara zeta function satisfies an analogue of the Riemann hypothesis.[3]
Selected works
• On Congruence Monodromy Problems, Mathematical Society of Japan Memoirs, World Scientific 2009 (based on lectures in 1968/1969)
• with Michael Fried (ed.): Arithmetic fundamental groups and noncommutative Algebra, American Mathematical Society, Proc. Symposium Pure Math. vol.70, 2002
• as editor: Galois representations and arithmetic algebraic geometry, North Holland 1987
• with Kenneth Ribet, Jean-Pierre Serre (eds.): Galois Groups over Q, Springer 1989 (Proceedings of a Workshop 1987)
References
1. Yasutaka Ihara at the Mathematics Genealogy Project
2. Ihara: On discrete subgroups of the two by two projective linear group over p-adic fields. J. Math. Soc. Jpn., vol. 18, 1966, pp. 219–235
3. Terras, Audrey (1999). "A survey of discrete trace formulas". In Hejhal, Dennis A.; Friedman, Joel; Gutzwiller, Martin C.; et al. (eds.). Emerging Applications of Number Theory. IMA Vol. Math. Appl. Vol. 109. Springer. pp. 643–681. ISBN 0-387-98824-6. Zbl 0982.11031. See p.678
External links
• Yasutaka Ihara's homepage at RIMS
• The Ihara Zeta Function and the Riemann Zeta Function by Mollie Stein, Amelia Wallace
Authority control
International
• ISNI
• VIAF
National
• France
• BnF data
• Germany
• Israel
• United States
• Japan
• Netherlands
• Poland
Academics
• MathSciNet
• Mathematics Genealogy Project
• zbMATH
People
• Deutsche Biographie
Other
• IdRef
| Wikipedia |
Yates's correction for continuity
In statistics, Yates's correction for continuity (or Yates's chi-squared test) is used in certain situations when testing for independence in a contingency table. It aims at correcting the error introduced by assuming that the discrete probabilities of frequencies in the table can be approximated by a continuous distribution (chi-squared). In some cases, Yates's correction may adjust too far, and so its current use is limited.
Correction for approximation error
Using the chi-squared distribution to interpret Pearson's chi-squared statistic requires one to assume that the discrete probability of observed binomial frequencies in the table can be approximated by the continuous chi-squared distribution. This assumption is not quite correct, and introduces some error.
To reduce the error in approximation, Frank Yates, an English statistician, suggested a correction for continuity that adjusts the formula for Pearson's chi-squared test by subtracting 0.5 from the difference between each observed value and its expected value in a 2 × 2 contingency table.[1] This reduces the chi-squared value obtained and thus increases its p-value.
The effect of Yates's correction is to prevent overestimation of statistical significance for small data. This formula is chiefly used when at least one cell of the table has an expected count smaller than 5. Unfortunately, Yates's correction may tend to overcorrect. This can result in an overly conservative result that fails to reject the null hypothesis when it should (a type II error). So it is suggested that Yates's correction is unnecessary even with quite low sample sizes,[2] such as:
$\sum _{i=1}^{N}O_{i}=20\,$
The following is Yates's corrected version of Pearson's chi-squared statistics:
$\chi _{\text{Yates}}^{2}=\sum _{i=1}^{N}{(|O_{i}-E_{i}|-0.5)^{2} \over E_{i}}$
where:
Oi = an observed frequency
Ei = an expected (theoretical) frequency, asserted by the null hypothesis
N = number of distinct events
2 × 2 table
As a short-cut, for a 2 × 2 table with the following entries:
SF
A aba+b
B cdc+d
a+cb+dN
$\chi _{\text{Yates}}^{2}={\frac {N(|ad-bc|-N/2)^{2}}{(a+b)(c+d)(a+c)(b+d)}}.$
In some cases, this is better.
$\chi _{\text{Yates}}^{2}={\frac {N(\max(0,|ad-bc|-N/2))^{2}}{N_{S}N_{F}N_{A}N_{B}}}.$
See also
• Continuity correction
• Wilson score interval with continuity correction
References
1. Yates, F (1934). "Contingency table involving small numbers and the χ2 test". Supplement to the Journal of the Royal Statistical Society 1(2): 217–235. JSTOR 2983604
2. Sokal RR, Rohlf F.J. (1981). Biometry: The Principles and Practice of Statistics in Biological Research. Oxford: W.H. Freeman, ISBN 0-7167-1254-7.
| Wikipedia |
Yau's conjecture
In differential geometry, Yau's conjecture from 1982, is a mathematical conjecture which states that a closed Riemannian 3-manifold has an infinite number of smooth closed immersed minimal surfaces. It is named after Shing-Tung Yau. It was the first problem in the minimal submanifolds section in Yau's list of open problems.
Not to be confused with Yau's conjecture on the first eigenvalue.
The conjecture has recently been claimed by Kei Irie, Fernando Codá Marques and André Neves in the generic case,[1][2] and by Antoine Song in full generality.[3]
References
1. Irie, Kei; Marques, Fernando Codá; Neves, André (2017). "Density of minimal hypersurfaces for generic metrics". arXiv:1710.10752 [math.DG].
2. Carlos Matheus (November 5, 2017). "Yau's conjecture of abundance of minimal hypersurfaces is generically true (in low dimensions)".
3. Song, Antoine (2018). "Existence of infinitely many minimal hypersurfaces in closed manifolds". arXiv:1806.08816 [math.DG].
Further reading
• Yau, S. T. (1982). Seminar on Differential Geometry. Annals of Mathematics Studies. Vol. 102. Princeton University Press. pp. 669–706. ISBN 0-691-08268-5. (Problem 88)
| Wikipedia |
Yau's conjecture on the first eigenvalue
In mathematics, Yau's conjecture on the first eigenvalue is, as of 2018, an unsolved conjecture proposed by Shing-Tung Yau in 1982. It asks:
Is it true that the first eigenvalue for the Laplace–Beltrami operator on an embedded minimal hypersurface of $S^{n+1}$ is $n$?
If true, it will imply that the area of embedded minimal hypersurfaces in $S^{3}$ will have an upper bound depending only on the genus.
Some possible reformulations are as follows:
• The first eigenvalue of every closed embedded minimal hypersurface $M^{n}$ in the unit sphere $S^{n+1}$(1) is $n$
• The first eigenvalue of an embedded compact minimal hypersurface $M^{n}$ of the standard (n + 1)-sphere with sectional curvature 1 is $n$
• If $S^{n+1}$ is the unit (n + 1)-sphere with its standard round metric, then the first Laplacian eigenvalue on a closed embedded minimal hypersurface ${\sum }^{n}\subset S^{n+1}$ is $n$
The Yau's conjecture is verified for several special cases, but still open in general.
Shiing-Shen Chern conjectured that a closed, minimally immersed hypersurface in $S^{n+1}$(1), whose second fundamental form has constant length, is isoparametric. If true, it would have established the Yau's conjecture for the minimal hypersurface whose second fundamental form has constant length.
A possible generalization of the Yau's conjecture:
Let $M^{d}$ be a closed minimal submanifold in the unit sphere $S^{N+1}$(1) with dimension $d$ of $M^{d}$ satisfying $d\geq {\frac {2}{3}}n+1$. Is it true that the first eigenvalue of $M^{d}$ is $d$?
Further reading
• Yau, S. T. (1982). Seminar on Differential Geometry. Annals of Mathematics Studies. Vol. 102. Princeton University Press. pp. 669–706. ISBN 0-691-08268-5. (Problem 100)
• Ge, J.; Tang, Z. (2012). "Chern Conjecture and Isoparametric Hypersurfaces". Differential Geometry: Under the influence of S.S. Chern. Beijing: Higher Education Press. ISBN 978-1-57146-249-7.
• Tang, Z.; Yan, W. (2013). "Isoparametric Foliation and Yau Conjecture on the First Eigenvalue". Journal of Differential Geometry. 94 (3): 521–540. arXiv:1201.0666. doi:10.4310/jdg/1370979337.
| Wikipedia |
K-stability
In mathematics, and especially differential and algebraic geometry, K-stability is an algebro-geometric stability condition, for complex manifolds and complex algebraic varieties. The notion of K-stability was first introduced by Gang Tian[1] and reformulated more algebraically later by Simon Donaldson.[2] The definition was inspired by a comparison to geometric invariant theory (GIT) stability. In the special case of Fano varieties, K-stability precisely characterises the existence of Kähler–Einstein metrics. More generally, on any compact complex manifold, K-stability is conjectured to be equivalent to the existence of constant scalar curvature Kähler metrics (cscK metrics).
This article covers the general theory of K-stability of projective varieties. The specific case of Fano varieties is covered in K-stability of Fano varieties.
History
In 1954, Eugenio Calabi formulated a conjecture about the existence of Kähler metrics on compact Kähler manifolds, now known as the Calabi conjecture.[3] One formulation of the conjecture is that a compact Kähler manifold $X$ admits a unique Kähler–Einstein metric in the class $c_{1}(X)$. In the particular case where $c_{1}(X)=0$, such a Kähler–Einstein metric would be Ricci flat, making the manifold a Calabi–Yau manifold. The Calabi conjecture was resolved in the case where $c_{1}(X)<0$ by Thierry Aubin and Shing-Tung Yau, and when $c_{1}(X)=0$ by Yau.[4][5][6] In the case where $c_{1}(X)>0$, that is when $X$ is a Fano manifold, a Kähler–Einstein metric does not always exist. Namely, it was known by work of Yozo Matsushima and André Lichnerowicz that a Kähler manifold with $c_{1}(X)>0$ can only admit a Kähler–Einstein metric if the Lie algebra $H^{0}(X,TX)$ is reductive.[7][8] However, it can be easily shown that the blow up of the complex projective plane at one point, ${\text{Bl}}_{p}\mathbb {CP} ^{2}$ is Fano, but does not have reductive Lie algebra. Thus not all Fano manifolds can admit Kähler–Einstein metrics.
After the resolution of the Calabi conjecture for $c_{1}(X)\leq 0$ attention turned to the loosely related problem of finding canonical metrics on vector bundles over complex manifolds. In 1983, Donaldson produced a new proof of the Narasimhan–Seshadri theorem.[9] As proved by Donaldson, the theorem states that a holomorphic vector bundle over a compact Riemann surface is stable if and only if it corresponds to an irreducible unitary Yang–Mills connection. That is, a unitary connection which is a critical point of the Yang–Mills functional
$\operatorname {YM} (\nabla )=\int _{X}\|F_{\nabla }\|^{2}\,d\operatorname {vol} .$
On a Riemann surface such a connection is projectively flat, and its holonomy gives rise to a projective unitary representation of the fundamental group of the Riemann surface, thus recovering the original statement of the theorem by M. S. Narasimhan and C. S. Seshadri.[10] During the 1980s this theorem was generalised through the work of Donaldson, Karen Uhlenbeck and Yau, and Jun Li and Yau to the Kobayashi–Hitchin correspondence, which relates stable holomorphic vector bundles to Hermitian–Einstein connections over arbitrary compact complex manifolds.[11][12][13] A key observation in the setting of holomorphic vector bundles is that once a holomorphic structure is fixed, any choice of Hermitian metric gives rise to a unitary connection, the Chern connection. Thus one can either search for a Hermitian–Einstein connection, or its corresponding Hermitian–Einstein metric.
Inspired by the resolution of the existence problem for canonical metrics on vector bundles, in 1993 Yau was motivated to conjecture the existence of a Kähler–Einstein metric on a Fano manifold should be equivalent to some form of algebro-geometric stability condition on the variety itself, just as the existence of a Hermitian–Einstein metric on a holomorphic vector bundle is equivalent to its stability. Yau suggested this stability condition should be an analogue of slope stability of vector bundles.[14]
In 1997, Tian suggested such a stability condition, which he called K-stability after the K-energy functional introduced by Toshiki Mabuchi.[1][15] The K originally stood for kinetic due to the similarity of the K-energy functional with the kinetic energy, and for the German kanonisch for the canonical bundle. Tian's definition was analytic in nature, and specific to the case of Fano manifolds. Several years later Donaldson introduced an algebraic condition described in this article called K-stability, which makes sense on any polarised variety, and is equivalent to Tian's analytic definition in the case of the polarised variety $(X,-K_{X})$ where $X$ is Fano.[2]
Definition
In this section we work over the complex numbers $\mathbb {C} $, but the essential points of the definition apply over any field. A polarised variety is a pair $(X,L)$ where $X$ is a complex algebraic variety and $L$ is an ample line bundle on $X$. Such a polarised variety comes equipped with an embedding into projective space using the Proj construction,
$X\cong \operatorname {Proj} \bigoplus _{r\geq 0}H^{0}\left(X,L^{kr}\right)\hookrightarrow \mathbb {P} \left(H^{0}\left(X,L^{k}\right)^{*}\right)$
where $k$ is any positive integer large enough that $L^{k}$ is very ample, and so every polarised variety is projective. Changing the choice of ample line bundle $L$ on $X$ results in a new embedding of $X$ into a possibly different projective space. Therefore a polarised variety can be thought of as a projective variety together with a fixed embedding into some projective space $\mathbb {CP} ^{N}$.
Hilbert–Mumford criterion
K-stability is defined by analogy with the Hilbert–Mumford criterion from finite-dimensional geometric invariant theory. This theory describes the stability of points on polarised varieties, whereas K-stability concerns the stability of the polarised variety itself.
The Hilbert–Mumford criterion shows that to test the stability of a point $x$ in a projective algebraic variety $X\subset \mathbb {CP} ^{N}$ under the action of a reductive algebraic group $G\subset \operatorname {GL} (N+1,\mathbb {C} )$, it is enough to consider the one parameter subgroups (1-PS) of $G$. To proceed, one takes a 1-PS of $G$, say $\lambda :\mathbb {C} ^{*}\hookrightarrow G$ :\mathbb {C} ^{*}\hookrightarrow G} , and looks at the limiting point
$x_{0}=\lim _{t\to 0}\lambda (t)\cdot x.$
This is a fixed point of the action of the 1-PS $\lambda $, and so the line over $x$ in the affine space $\mathbb {C} ^{N+1}$ is preserved by the action of $\lambda $. An action of the multiplicative group $\mathbb {C} ^{*}$ on a one dimensional vector space comes with a weight, an integer we label $\mu (x,\lambda )$, with the property that
$\lambda (t)\cdot {\tilde {x}}=t^{\mu (x,\lambda )}{\tilde {x}}$
for any ${\tilde {x}}$ in the fibre over $x_{0}$. The Hilbert-Mumford criterion says:
• The point $x$ is semistable if $\mu (x,\lambda )\leq 0$ for all 1-PS $\lambda <G$.
• The point $x$ is stable if $\mu (x,\lambda )<0$ for all 1-PS $\lambda <G$.
• The point $x$ is unstable if $\mu (x,\lambda )>0$ for any 1-PS $\lambda <G$.
If one wishes to define a notion of stability for varieties, the Hilbert-Mumford criterion therefore suggests it is enough to consider one parameter deformations of the variety. This leads to the notion of a test configuration.
Test Configurations
A test configuration for a polarised variety $(X,L)$ is a pair $({\mathcal {X}},{\mathcal {L}})$ where ${\mathcal {X}}$ is a scheme with a flat morphism $\pi :{\mathcal {X}}\to \mathbb {C} $ :{\mathcal {X}}\to \mathbb {C} } and ${\mathcal {L}}$ is a relatively ample line bundle for the morphism $\pi $, such that:
1. For every $t\in \mathbb {C} $, the Hilbert polynomial of the fibre $({\mathcal {X}}_{t},{\mathcal {L}}_{t})$ is equal to the Hilbert polynomial ${\mathcal {P}}(k)$ of $(X,L)$. This is a consequence of the flatness of $\pi $.
2. There is an action of $\mathbb {C} ^{*}$ on the family $({\mathcal {X}},{\mathcal {L}})$ covering the standard action of $\mathbb {C} ^{*}$ on $\mathbb {C} $.
3. For any (and hence every) $t\in \mathbb {C} ^{*}$, $({\mathcal {X}}_{t},{\mathcal {L}}_{t})\cong (X,L)$ as polarised varieties. In particular away from $0\in \mathbb {C} $, the family is trivial: $({\mathcal {X}}_{t\neq 0},{\mathcal {L}}_{t\neq 0})\cong (X\times \mathbb {C} ^{*},\operatorname {pr} _{1}^{*}L)$ where $\operatorname {pr} _{1}:X\times \mathbb {C} ^{*}\to X$ is projection onto the first factor.
We say that a test configuration $({\mathcal {X}},{\mathcal {L}})$ is a product configuration if ${\mathcal {X}}\cong X\times \mathbb {C} $, and a trivial configuration if the $\mathbb {C} ^{*}$ action on ${\mathcal {X}}\cong X\times \mathbb {C} $ is trivial on the first factor.
Donaldson–Futaki Invariant
To define a notion of stability analogous to the Hilbert–Mumford criterion, one needs a concept of weight $\mu ({\mathcal {X}},{\mathcal {L}})$ on the fibre over $0$ of a test configuration $({\mathcal {X}},{\mathcal {L}})\to \mathbb {C} $ for a polarised variety $(X,L)$. By definition this family comes equipped with an action of $\mathbb {C} ^{*}$ covering the action on the base, and so the fibre of the test configuration over $0\in \mathbb {C} $ is fixed. That is, we have an action of $\mathbb {C} ^{*}$ on the central fibre $({\mathcal {X}}_{0},{\mathcal {L}}_{0})$. In general this central fibre is not smooth, or even a variety. There are several ways to define the weight on the central fiber. The first definition was given by using Ding-Tian's version of generalized Futaki invariant.[1] This definition is differential geometric and is directly related to the existence problems in Kähler geometry. Algebraic definitions were given by using Donaldson-Futaki invariants and CM-weights defined by intersection formula.
By definition an action of $\mathbb {C} ^{*}$ on a polarised scheme comes with an action of $\mathbb {C} ^{*}$ on the ample line bundle ${\mathcal {L}}_{0}$, and therefore induces an action on the vector spaces $H^{0}({\mathcal {X}}_{0},{\mathcal {L}}_{0}^{k})$ for all integers $k\geq 0$. An action of $\mathbb {C} ^{*}$ on a complex vector space $V$ induces a direct sum decomposition $V=V_{1}\oplus \cdots \oplus V_{n}$ into weight spaces, where each $V_{i}$ is a one dimensional subspace of $V$, and the action of $\mathbb {C} ^{*}$ when restricted to $V_{i}$ has a weight $w_{i}$. Define the total weight of the action to be the integer $w=w_{1}+\cdots +w_{n}$. This is the same as the weight of the induced action of $\mathbb {C} ^{*}$ on the one dimensional vector space $ \bigwedge ^{n}V$ where $n=\dim V$.
Define the weight function of the test configuration $({\mathcal {X}},{\mathcal {L}})$ to be the function $w(k)$ where $w(k)$ is the total weight of the $\mathbb {C} ^{*}$ action on the vector space $H^{0}({\mathcal {X}}_{0},{\mathcal {L}}_{0}^{k})$ for each non-negative integer $k\geq 0$. Whilst the function $w(k)$ is not a polynomial in general, it becomes a polynomial of degree $n+1$ for all $k>k_{0}\gg 0$ for some fixed integer $k_{0}$, where $n=\dim X$. This can be seen using an equivariant Riemann-Roch theorem. Recall that the Hilbert polynomial ${\mathcal {P}}(k)$ satisfies the equality ${\mathcal {P}}(k)=\dim H^{0}(X,L^{k})=\dim H^{0}({\mathcal {X}}_{0},{\mathcal {L}}_{0}^{k})$ for all $k>k_{1}\gg 0$ for some fixed integer $k_{1}$, and is a polynomial of degree $n$. For such $k\gg 0$, let us write
${\mathcal {P}}(k)=a_{0}k^{n}+a_{1}k^{n-1}+O(k^{n-2}),\quad w(k)=b_{0}k^{n+1}+b_{1}k^{n}+O(k^{n-1}).$
The Donaldson-Futaki invariant of the test configuration $({\mathcal {X}},{\mathcal {L}})$ is the rational number
$\operatorname {DF} ({\mathcal {X}},{\mathcal {L}})={\frac {b_{0}a_{1}-b_{1}a_{0}}{a_{0}^{2}}}.$
In particular $\operatorname {DF} ({\mathcal {X}},{\mathcal {L}})=-f_{1}$ where $f_{1}$ is the first order term in the expansion
${\frac {w(k)}{k{\mathcal {P}}(k)}}=f_{0}+f_{1}k^{-1}+O(k^{-2}).$
The Donaldson-Futaki invariant does not change if $L$ is replaced by a positive power $L^{r}$, and so in the literature K-stability is often discussed using $\mathbb {Q} $-line bundles.
It is possible to describe the Donaldson-Futaki invariant in terms of intersection theory, and this was the approach taken by Tian in defining the CM-weight.[1] Any test configuration $({\mathcal {X}},{\mathcal {L}})$ admits a natural compactification $({\bar {\mathcal {X}}},{\bar {\mathcal {L}}})$ over $\mathbb {P} ^{1}$ (e.g.,see [16][17]), then the CM-weight is defined by
$CM({\mathcal {X}},{\mathcal {L}})={\frac {1}{2(n+1)\cdot L^{n}}}\left(\mu \cdot n{({\bar {\mathcal {L}}})}^{n+1}+(n+1){K}_{{\bar {\mathcal {X}}}/{\mathbb {P} }^{1}}\cdot {({\bar {\mathcal {L}}})}^{n}\right)$
where $\mu =-{\frac {L^{n-1}\cdot K_{X}}{L^{n}}}$. This definition by intersection formula is now often used in algebraic geometry.
It is known that $\operatorname {DF} ({\mathcal {X}},{\mathcal {L}})$ coincides with $\operatorname {CM} ({\mathcal {X}},{\mathcal {L}})$, so we can take the weight $\mu ({\mathcal {X}},{\mathcal {L}})$ to be either $\operatorname {DF} ({\mathcal {X}},{\mathcal {L}})$ or $\operatorname {CM} ({\mathcal {X}},{\mathcal {L}})$. The weight $\mu ({\mathcal {X}},{\mathcal {L}})$ can be also expressed in terms of the Chow form and hyperdiscriminant.[18] In the case of Fano manifolds, there is an interpretation of the weight in terms of new $\beta $-invariant on valuations found by Chi Li[19] and Kento Fujita.[20]
K-stability
In order to define K-stability, we need to first exclude certain test configurations. Initially it was presumed one should just ignore trivial test configurations as defined above, whose Donaldson-Futaki invariant always vanishes, but it was observed by Li and Xu that more care is needed in the definition.[21][22] One elegant way of defining K-stability is given by Székelyhidi using the norm of a test configuration, which we first describe.[23]
For a test configuration $({\mathcal {X}},{\mathcal {L}})$, define the norm as follows. Let $A_{k}$ be the infinitesimal generator of the $\mathbb {C} ^{*}$ action on the vector space $H^{0}(X,L^{k})$. Then $\operatorname {Tr} (A_{k})=w(k)$. Similarly to the polynomials $w(k)$ and ${\mathcal {P}}(k)$, the function $\operatorname {Tr} (A_{k}^{2})$ is a polynomial for large enough integers $k$, in this case of degree $n+2$. Let us write its expansion as
$\operatorname {Tr} (A_{k}^{2})=c_{0}k^{n+2}+O(k^{n+1}).$
The norm of a test configuration is defined by the expression
$\|({\mathcal {X}},{\mathcal {L}})\|^{2}=c_{0}-{\frac {b_{0}^{2}}{a_{0}}}.$
According to the analogy with the Hilbert-Mumford criterion, once one has a notion of deformation (test configuration) and weight on the central fibre (Donaldson-Futaki invariant), one can define a stability condition, called K-stability.
Let $(X,L)$ be a polarised algebraic variety. We say that $(X,L)$ is:
• K-semistable if $\operatorname {\mu } ({\mathcal {X}},{\mathcal {L}})\geq 0$ for all test configurations $({\mathcal {X}},{\mathcal {L}})$ for $(X,L)$.
• K-stable if $\operatorname {\mu } ({\mathcal {X}},{\mathcal {L}})\geq 0$ for all test configurations $({\mathcal {X}},{\mathcal {L}})$ for $(X,L)$, and additionally $\operatorname {\mu } ({\mathcal {X}},{\mathcal {L}})>0$ whenever $\|({\mathcal {X}},{\mathcal {L}})\|>0$.
• K-polystable if $(X,L)$ is K-semistable, and additionally whenever $\operatorname {\mu } ({\mathcal {X}},{\mathcal {L}})=0$, the test configuration $({\mathcal {X}},{\mathcal {L}})$ is a product configuration.
• K-unstable if it is not K-semistable.
Yau–Tian–Donaldson Conjecture
See also: K-stability of Fano varieties § Existence of Kähler–Einstein metrics
K-stability was originally introduced as an algebro-geometric condition which should characterise the existence of a Kähler–Einstein metric on a Fano manifold. This came to be known as the Yau–Tian–Donaldson conjecture (for Fano manifolds). The conjecture was resolved in the 2010s in works of Xiuxiong Chen, Simon Donaldson, and Song Sun,[24][25][26][27][28][29] The strategy is based on a continuity method with respect to the cone angle of a Kähler–Einstein metric with cone singularities along a fixed anticanonical divisor, as well as an in-depth use of the Cheeger–Colding–Tian theory of Gromov–Hausdorff limits of Kähler manifolds with Ricci bounds.
Theorem (Yau–Tian–Donaldson conjecture for Kähler–Einstein metrics): A Fano Manifold $X$ admits a Kähler–Einstein metric in the class of $c_{1}(X)$ if and only if the pair $(X,-K_{X})$ is K-polystable.
Chen, Donaldson, and Sun have alleged that Tian's claim to equal priority for the proof is incorrect, and they have accused him of academic misconduct.[lower-alpha 1] Tian has disputed their claims.[lower-alpha 2] Chen, Donaldson, and Sun were recognized by the American Mathematical Society's prestigious 2019 Veblen Prize as having had resolved the conjecture.[30] The Breakthrough Prize has recognized Donaldson with the Breakthrough Prize in Mathematics and Sun with the New Horizons Breakthrough Prize, in part based upon their work with Chen on the conjecture.[31][32]
More recently, a proof based on the "classical" continuity method was provided by Ved Datar and Gabor Székelyhidi,[33][34] followed by a proof by Chen, Sun, and Bing Wang using the Kähler–Ricci flow.[35] Robert Berman, Sébastien Boucksom, and Mattias Jonsson also provided a proof from the variational approach.[36]
Extension to constant scalar curvature Kähler metrics
It is expected that the Yau–Tian–Donaldson conjecture should apply more generally to cscK metrics over arbitrary smooth polarised varieties. In fact, the Yau–Tian–Donaldson conjecture refers to this more general setting, with the case of Fano manifolds being a special case, which was conjectured earlier by Yau and Tian. Donaldson built on the conjecture of Yau and Tian from the Fano case after his definition of K-stability for arbitrary polarised varieties was introduced.[2]
Yau–Tian–Donaldson conjecture for constant scalar curvature metrics: A smooth polarised variety $(X,L)$ admits a constant scalar curvature Kähler metric in the class of $c_{1}(L)$ if and only if the pair $(X,L)$ is K-polystable.
As discussed, the Yau–Tian–Donaldson conjecture has been resolved in the Fano setting. It was proven by Donaldson in 2009 that the Yau–Tian–Donaldson conjecture holds for toric varieties of complex dimension 2.[37][38][39] For arbitrary polarised varieties it was proven by Stoppa, also using work of Arezzo and Pacard, that the existence of a cscK metric implies K-polystability.[40][41] This is in some sense the easy direction of the conjecture, as it assumes the existence of a solution to a difficult partial differential equation, and arrives at the comparatively easy algebraic result. The significant challenge is to prove the reverse direction, that a purely algebraic condition implies the existence of a solution to a PDE.
Examples
Smooth Curves
See also: Stable curve
It has been known since the original work of Pierre Deligne and David Mumford that smooth algebraic curves are asymptotically stable in the sense of geometric invariant theory, and in particular that they are K-stable.[42] In this setting, the Yau–Tian–Donaldson conjecture is equivalent to the uniformization theorem. Namely, every smooth curve admits a Kähler–Einstein metric of constant scalar curvature either $+1$ in the case of the projective line $\mathbb {CP} ^{1}$, $0$ in the case of elliptic curves, or $-1$ in the case of compact Riemann surfaces of genus $g>1$.
Fano varieties
Main article: K-stability of Fano varieties
The setting where $L=-K_{X}$ is ample so that $X$ is a Fano manifold is of particular importance, and in that setting many tools are known to verify the K-stability of Fano varieties. For example using purely algebraic techniques it can be proven that all Fermat hypersurfaces
$F_{n,d}=\{z\in \mathbb {CP} ^{n+1}\mid z_{0}^{d}+\cdots z_{n+1}^{d}=0\}\subset \mathbb {CP} ^{n+1}$
are K-stable Fano varieties for $3\leq d\leq n+1$.[43][44][45]
Toric Varieties
K-stability was originally introduced by Donaldson in the context of toric varieties.[2] In the toric setting many of the complicated definitions of K-stability simplify to be given by data on the moment polytope $P$ of the polarised toric variety $(X_{P},L_{P})$. First it is known that to test K-stability, it is enough to consider toric test configurations, where the total space of the test configuration is also a toric variety. Any such toric test configuration can be elegantly described by a convex function on the moment polytope, and Donaldson originally defined K-stability for such convex functions. If a toric test configuration $({\mathcal {X}},{\mathcal {L}})$ for $(X_{P},L_{P})$ is given by a convex function $f$ on $P$, then the Donaldson-Futaki invariant can be written as
$\operatorname {DF} ({\mathcal {X}},{\mathcal {L}})={\frac {1}{2}}{\mathcal {L}}(f)={\frac {1}{2}}\left(\int _{\partial P}f\,d\sigma -a\int _{P}f\,d\mu \right),$
where $d\mu $ is the Lebesgue measure on $P$, $d\sigma $ is the canonical measure on the boundary of $P$ arising from its description as a moment polytope (if an edge of $P$ is given by a linear inequality $h(x)\leq a$ for some affine linear functional h on $\mathbb {R} ^{n}$ with integer coefficients, then $d\mu =\pm dh\wedge d\sigma $), and $a=\operatorname {Vol} (\partial P,d\sigma )/\operatorname {Vol} (P,d\mu )$. Additionally the norm of the test configuration can be given by
$\left\|({\mathcal {X}},{\mathcal {L}})\right\|=\left\|f-{\bar {f}}\right\|_{L^{2}},$
where ${\bar {f}}$ is the average of $f$ on $P$ with respect to $d\mu $.
It was shown by Donaldson that for toric surfaces, it suffices to test convex functions of a particularly simple form. We say a convex function on $P$ is piecewise-linear if it can be written as a maximum $f=\max(h_{1},\dots ,h_{n})$ for some affine linear functionals $h_{1},\dots ,h_{n}$. Notice that by the definition of the constant $a$, the Donaldson-Futaki invariant ${\mathcal {L}}(f)$ is invariant under the addition of an affine linear functional, so we may always take one of the $h_{i}$ to be the constant function $0$. We say a convex function is simple piecewise-linear if it is a maximum of two functions, and so is given by $f=\max(0,h)$ for some affine linear function $h$, and simple rational piecewise-linear if $h$ has rational cofficients. Donaldson showed that for toric surfaces it is enough to test K-stability only on simple rational piecewise-linear functions. Such a result is powerful in so far as it is possible to readily compute the Donaldson-Futaki invariants of such simple test configurations, and therefore computationally determine when a given toric surface is K-stable.
An example of a K-unstable manifold is given by the toric surface $\mathbb {F} _{1}=\operatorname {Bl} _{0}\mathbb {CP} ^{2}$, the first Hirzebruch surface, which is the blow up of the complex projective plane at a point, with respect to the polarisation given by $ L={\frac {1}{2}}(\pi ^{*}{\mathcal {O}}(2)-E)$, where $\pi :\mathbb {F} _{1}\to \mathbb {CP} ^{2}$ :\mathbb {F} _{1}\to \mathbb {CP} ^{2}} is the blow up and $E$ the exceptional divisor.
The measure $d\sigma $ on the horizontal and vertical boundary faces of the polytope are just $dx$ and $dy$. On the diagonal face $x+y=2$ the measure is given by $(dx-dy)/2$. Consider the convex function $f(x,y)=x+y$ on this polytope. Then
$\int _{P}f\,d\mu ={\frac {11}{6}},\qquad \int _{\partial P}f\,d\sigma =6,$
and
$\operatorname {Vol} (P,d\mu )={\frac {3}{2}},\qquad \operatorname {Vol} (\partial P,d\sigma )=5,$
Thus
${\mathcal {L}}(f)=6-{\frac {55}{9}}=-{\frac {1}{9}}<0,$
and so the first Hirzebruch surface $\mathbb {F} _{1}$ is K-unstable.
Alternative Notions
Hilbert and Chow Stability
K-stability arises from an analogy with the Hilbert-Mumford criterion for finite-dimensional geometric invariant theory. It is possible to use geometric invariant theory directly to obtain other notions of stability for varieties that are closely related to K-stability.
Take a polarised variety $(X,L)$ with Hilbert polynomial ${\mathcal {P}}$, and fix an $r>0$ such that $L^{r}$ is very ample with vanishing higher cohomology. The pair $(X,L^{r})$ can then be identified with a point in the Hilbert scheme of subschemes of $\mathbb {P} ^{{\mathcal {P}}(r)-1}$ with Hilbert polynomial ${\mathcal {P}}'(K)={\mathcal {P}}(Kr)$.
This Hilbert scheme can be embedded into projective space as a subscheme of a Grassmannian (which is projective via the Plücker embedding). The general linear group $\operatorname {GL} ({\mathcal {P}}(r),\mathbb {C} )$ acts on this Hilbert scheme, and two points in the Hilbert scheme are equivalent if and only if the corresponding polarised varieties are isomorphic. Thus one can use geometric invariant theory for this group action to give a notion of stability. This construction depends on a choice of $r>0$, so one says a polarised variety is asymptotically Hilbert stable if it is stable with respect to this embedding for all $r>r_{0}\gg 0$ sufficiently large, for some fixed $r_{0}$.
There is another projective embedding of the Hilbert scheme called the Chow embedding, which provides a different linearisation of the Hilbert scheme and therefore a different stability condition. One can similarly therefore define asymptotic Chow stability. Explicitly the Chow weight for a fixed $r>0$ can be computed as
$\operatorname {Chow} _{r}({\mathcal {X}},{\mathcal {L}})={\frac {rb_{0}}{a_{0}}}-{\frac {w(r)}{{\mathcal {P}}(r)}}$
for $r$ sufficiently large.[46] Unlike the Donaldson-Futaki invariant, the Chow weight changes if the line bundle $L$ is replaced by some power $L^{k}$. However, from the expression
$\operatorname {Chow} _{rk}({\mathcal {X}},{\mathcal {L^{k}}})={\frac {krb_{0}}{a_{0}}}-{\frac {w(kr)}{{\mathcal {P}}(kr)}}$
one observes that
$\operatorname {DF} ({\mathcal {X}},{\mathcal {L}})=\lim _{k\to \infty }\operatorname {Chow} _{rk}({\mathcal {X}},{\mathcal {L^{k}}}),$
and so K-stability is in some sense the limit of Chow stability as the dimension of the projective space $X$ is embedded in approaches infinity.
One may similarly define asymptotic Chow semistability and asymptotic Hilbert semistability, and the various notions of stability are related as follows:
Asymptotically Chow stable $\implies $ Asymptotically Hilbert stable $\implies $ Asymptotically Hilbert semistable $\implies $ Asymptotically Chow semistable $\implies $ K-semistable
It is however not know whether K-stability implies asymptotic Chow stability.[47]
Slope K-Stability
It was originally predicted by Yau that the correct notion of stability for varieties should be analogous to slope stability for vector bundles. Julius Ross and Richard Thomas developed a theory of slope stability for varieties, known as slope K-stability. It was shown by Ross and Thomas that any test configuration is essentially obtained by blowing up the variety $X\times \mathbb {C} $ along a sequence of $\mathbb {C} ^{*}$ invariant ideals, supported on the central fibre.[47] This result is essentially due to David Mumford.[48] Explicitly, every test configuration is dominated by a blow up of $X\times \mathbb {C} $ along an ideal of the form
$I=I_{0}+tI_{1}+t^{2}I_{2}+\cdots +t^{r-1}I_{r-1}+(t^{r})\subset {\mathcal {O}}_{X}\otimes \mathbb {C} [t],$
where $t$ is the coordinate on $\mathbb {C} $. By taking the support of the ideals this corresponds to blowing up along a flag of subschemes
$Z_{r-1}\subset \cdots \subset Z_{2}\subset Z_{1}\subset Z_{0}\subset X$
inside the copy $X\times \{0\}$ of $X$. One obtains this decomposition essentially by taking the weight space decomposition of the invariant ideal $I$ under the $\mathbb {C} ^{*}$ action.
In the special case where this flag of subschemes is of length one, the Donaldson-Futaki invariant can be easily computed and one arrives at slope K-stability. Given a subscheme $Z\subset X$ defined by an ideal sheaf $I_{Z}$, the test configuration is given by
${\mathcal {X}}=\operatorname {Bl} _{Z\times \{0\}}(X\times \mathbb {C} ),$
which is the deformation to the normal cone of the embedding $Z\hookrightarrow X$.
If the variety $X$ has Hilbert polynomial ${\mathcal {P}}(k)=a_{0}k^{n}+a_{1}k^{n-1}+O(k^{n-2})$, define the slope of $X$ to be
$\mu (X)={\frac {a_{1}}{a_{0}}}.$
To define the slope of the subscheme $Z$, consider the Hilbert-Samuel polynomial of the subscheme $Z$,
$\chi (L^{r}\otimes I_{Z}^{xr})=a_{0}(x)r^{n}+a_{1}(x)r^{n-1}+O(r^{n-2}),$
for $r\gg 0$ and $x$ a rational number such that $xr\in \mathbb {N} $. The coefficients $a_{i}(x)$ are polynomials in $x$ of degree $n-i$, and the K-slope of $I_{Z}$ with respect to $c$ is defined by
$\mu _{c}(I_{Z})={\frac {\int _{0}^{c}{\big (}a_{1}(x)+{\frac {a_{0}'(x)}{2}}{\big )}\,dx}{\int _{0}^{c}a_{0}(x)\,dx}}.$
This definition makes sense for any choice of real number $c\in (0,\epsilon (Z)]$ where $\epsilon (Z)$ is the Seshadri constant of $Z$. Notice that taking $Z=\emptyset $ we recover the slope of $X$. The pair $(X,L)$ is slope K-semistable if for all proper subschemes $Z\subset X$, $\mu _{c}(I_{Z})\leq \mu (X)$ for all $c\in (0,\epsilon (Z)]$ (one can also define slope K-stability and slope K-polystability by requiring this inequality to be strict, with some extra technical conditions).
It was shown by Ross and Thomas that K-semistability implies slope K-semistability.[49] However, unlike in the case of vector bundles, it is not the case that slope K-stability implies K-stability. In the case of vector bundles it is enough to consider only single subsheaves, but for varieties it is necessary to consider flags of length greater than one also. Despite this, slope K-stability can still be used to identify K-unstable varieties, and therefore by the results of Stoppa, give obstructions to the existence of cscK metrics. For example, Ross and Thomas use slope K-stability to show that the projectivisation of an unstable vector bundle over a K-stable base is K-unstable, and so does not admit a cscK metric. This is a converse to results of Hong, which show that the projectivisation of a stable bundle over a base admitting a cscK metric, also admits a cscK metric, and is therefore K-stable.[50]
Filtration K-Stability
Work of Apostolov–Calderbank–Gauduchon–Tønnesen-Friedman shows the existence of a manifold which does not admit any extremal metric, but does not appear to be destabilised by any test configuration.[51] This suggests that the definition of K-stability as given here may not be precise enough to imply the Yau–Tian–Donaldson conjecture in general. However, this example is destabilised by a limit of test configurations. This was made precise by Székelyhidi, who introduced filtration K-stability.[46][23] A filtration here is a filtration of the coordinate ring
$R=\bigoplus _{k\geq 0}H^{0}(X,L^{k})$
of the polarised variety $(X,L)$. The filtrations considered must be compatible with the grading on the coordinate ring in the following sense: A filtation $\chi $ of $R$ is a chain of finite-dimensional subspaces
$\mathbb {C} =F_{0}R\subset F_{1}R\subset F_{2}R\subset \dots \subset R$
such that the following conditions hold:
1. The filtration is multiplicative. That is, $(F_{i}R)(F_{j}R)\subset F_{i+j}R$ for all $i,j\geq 0$.
2. The filtration is compatible with the grading on $R$ coming from the graded pieces $R_{k}=H^{0}(X,L^{k})$. That is, if $f\in F_{i}R$, then each homogenous piece of $f$ is in $F_{i}R$.
3. The filtration exhausts $R$. That is, we have $\bigcup _{i\geq 0}F_{i}R=R$.
Given a filtration $\chi $, its Rees algebra is defined by
$\operatorname {Rees} (\chi )=\bigoplus _{i\geq 0}(F_{i}R)t^{i}\subset R[t].$
We say that a filtration is finitely generated if its Rees algebra is finitely generated. It was proven by David Witt Nyström that a filtration is finitely generated if and only if it arises from a test configuration, and by Székelyhidi that any filtration is a limit of finitely generated filtrations.[52] Combining these results Székelyhidi observed that the example of Apostolov-Calderbank-Gauduchon-Tønnesen-Friedman would not violate the Yau–Tian–Donaldson conjecture if K-stability was replaced by filtration K-stability. This suggests that the definition of K-stability may need to be edited to account for these limiting examples.
See also
• Kähler manifold
• Kähler–Einstein metric
• K-stability of Fano varieties
• Geometric invariant theory
• Calabi conjecture
• Kobayashi–Hitchin correspondence
• Stable curve
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43. Tian, Gang (1987). "On Kähler-Einstein metrics on certain Kähler manifolds with C1 (M)> 0". Inventiones Mathematicae. 89 (2): 225–246. Bibcode:1987InMat..89..225T. doi:10.1007/BF01389077. S2CID 122352133.
44. Zhuang, Ziquan (2021). "Optimal destabilizing centers and equivariant K-stability". Inventiones Mathematicae. 226 (1): 195–223. arXiv:2004.09413. Bibcode:2021InMat.226..195Z. doi:10.1007/s00222-021-01046-0. S2CID 215827850.
45. Tian, Gang (2000). Canonical Metrics in Kähler Geometry. Notes taken by Meike Akveld. Lectures in Mathematics. ETH Zürich, Birkhäuser Verlag, Basel. doi:10.1007/978-3-0348-8389-4. ISBN 978-3-7643-6194-5. S2CID 120250582.
46. Székelyhidi, Gábor (2015). "Filtrations and test-configurations. With an appendix by Sebastien Boucksom". Mathematische Annalen. 362 (1–2): 451–484. arXiv:1111.4986. doi:10.1007/s00208-014-1126-3. S2CID 253716855.
47. Ross, Julius; Thomas, Richard (2006). "A study of the Hilbert-Mumford criterion for the stability of projective varieties". Journal of Algebraic Geometry. 16 (2): 201–255. doi:10.1090/S1056-3911-06-00461-9. MR 2274514. S2CID 15621023.
48. Mumford, David (1977). "Stability of Projective Varieties". 22 (2). Enseignement Math: 39–110. doi:10.5169/seals-48919. {{cite journal}}: Cite journal requires |journal= (help)
49. Ross, Julius; Thomas, Richard (2006). "An obstruction to the existence of constant scalar curvature Kähler metrics". Journal of Differential Geometry. 72 (3): 429–466. doi:10.4310/jdg/1143593746. MR 2219940. S2CID 15411889.
50. Hong, Ying-Ji (1999). "Constant Hermitian scalar curvature equations on ruled manifolds". Journal of Differential Geometry. 53 (3): 465–516. doi:10.4310/jdg/1214425636.
51. Apostolov, Vestislav; Calderbank, David M.J.; Gauduchon, Paul; Tønnesen-Friedman, Christina W. (2008). "Hamiltonian 2-forms in Kähler geometry, III extremal metrics and stability". Inventiones Mathematicae. 173 (3): 547–601. arXiv:math/0511118. Bibcode:2008InMat.173..547A. doi:10.1007/s00222-008-0126-x. S2CID 17821805.
52. Witt Nyström, David (2012). "Test configurations and Okounkov bodies". Compositio Mathematica. 148 (6): 1736–1756. doi:10.1112/S0010437X12000358.
Notes
1. Xiuxiong Chen, Simon Donaldson, Song Sun. "On some recent developments in Kähler geometry."
2. Gang Tian. "Response to CDS" and "More comments on CDS."
| Wikipedia |
Ya'ish ibn Ibrahim al-Umawi
Abū ʿAbdallāh Yaʿīsh ibn Ibrāhīm ibn Yūsuf ibn Simāk al-Andalusī al-Umawī (Arabic: يعيش بن إبراهيم بن يوسف بن سماك الأموي الأندلسي) (1400? in Al-Andalus – 1489 in Damascus, Syria) was a 15th-century Spanish-Arab[1] mathematician.
Works
• Marasim al-intisab fi'ilm al-hisab ("On arithmetical rules and procedures"), first date written in 1373 and hence the birth date above is controversial.
• Raf'al-ishkal fi ma'rifat al-ashkal (a work on mensuration).
References
1. Gibb, Hamilton Alexander Rosskeen; Kramers, Johannes Hendrik; Lewis, Bernard; Pellat, Charles; Schacht, Joseph (1970). The Encyclopaedia of Islam. Brill.
Bibliography
• Saidan, A. S. (1970). "Al-Umawī, Abū 'Abdallāh Ya'īsh Ibn ibrāHīm Ibn Yūsuf Ibn Simāk Al-Andalusī". Dictionary of Scientific Biography. New York: Charles Scribner's Sons. ISBN 0-684-10114-9.
• Ahmad Salim Saidan (ed.), Yaish ibn Ibrahim al-Umawi, On arithmetical rules and procedures (Aleppo, 1981).
External links
• O'Connor, John J.; Robertson, Edmund F., "Ya'ish ibn Ibrahim al-Umawi", MacTutor History of Mathematics Archive, University of St Andrews
Mathematics in the medieval Islamic world
Mathematicians
9th century
• 'Abd al-Hamīd ibn Turk
• Sanad ibn Ali
• al-Jawharī
• Al-Ḥajjāj ibn Yūsuf
• Al-Kindi
• Qusta ibn Luqa
• Al-Mahani
• al-Dinawari
• Banū Mūsā
• Hunayn ibn Ishaq
• Al-Khwarizmi
• Yusuf al-Khuri
• Ishaq ibn Hunayn
• Na'im ibn Musa
• Thābit ibn Qurra
• al-Marwazi
• Abu Said Gorgani
10th century
• Abu al-Wafa
• al-Khazin
• Al-Qabisi
• Abu Kamil
• Ahmad ibn Yusuf
• Aṣ-Ṣaidanānī
• Sinān ibn al-Fatḥ
• al-Khojandi
• Al-Nayrizi
• Al-Saghani
• Brethren of Purity
• Ibn Sahl
• Ibn Yunus
• al-Uqlidisi
• Al-Battani
• Sinan ibn Thabit
• Ibrahim ibn Sinan
• Al-Isfahani
• Nazif ibn Yumn
• al-Qūhī
• Abu al-Jud
• Al-Sijzi
• Al-Karaji
• al-Majriti
• al-Jabali
11th century
• Abu Nasr Mansur
• Alhazen
• Kushyar Gilani
• Al-Biruni
• Ibn al-Samh
• Abu Mansur al-Baghdadi
• Avicenna
• al-Jayyānī
• al-Nasawī
• al-Zarqālī
• ibn Hud
• Al-Isfizari
• Omar Khayyam
• Muhammad al-Baghdadi
12th century
• Jabir ibn Aflah
• Al-Kharaqī
• Al-Khazini
• Al-Samawal al-Maghribi
• al-Hassar
• Sharaf al-Din al-Tusi
• Ibn al-Yasamin
13th century
• Ibn al‐Ha'im al‐Ishbili
• Ahmad al-Buni
• Ibn Munim
• Alam al-Din al-Hanafi
• Ibn Adlan
• al-Urdi
• Nasir al-Din al-Tusi
• al-Abhari
• Muhyi al-Din al-Maghribi
• al-Hasan al-Marrakushi
• Qutb al-Din al-Shirazi
• Shams al-Din al-Samarqandi
• Ibn al-Banna'
• Kamāl al-Dīn al-Fārisī
14th century
• Nizam al-Din al-Nisapuri
• Ibn al-Shatir
• Ibn al-Durayhim
• Al-Khalili
• al-Umawi
15th century
• Ibn al-Majdi
• al-Rūmī
• al-Kāshī
• Ulugh Beg
• Ali Qushji
• al-Wafa'i
• al-Qalaṣādī
• Sibt al-Maridini
• Ibn Ghazi al-Miknasi
16th century
• Al-Birjandi
• Muhammad Baqir Yazdi
• Taqi ad-Din
• Ibn Hamza al-Maghribi
• Ahmad Ibn al-Qadi
Mathematical
works
• The Compendious Book on Calculation by Completion and Balancing
• De Gradibus
• Principles of Hindu Reckoning
• Book of Optics
• The Book of Healing
• Almanac
• Book on the Measurement of Plane and Spherical Figures
• Encyclopedia of the Brethren of Purity
• Toledan Tables
• Tabula Rogeriana
• Zij
Concepts
• Alhazen's problem
• Islamic geometric patterns
Centers
• Al-Azhar University
• Al-Mustansiriya University
• House of Knowledge
• House of Wisdom
• Constantinople observatory of Taqi ad-Din
• Madrasa
• Maragheh observatory
• University of al-Qarawiyyin
Influences
• Babylonian mathematics
• Greek mathematics
• Indian mathematics
Influenced
• Byzantine mathematics
• European mathematics
• Indian mathematics
Related
• Hindu–Arabic numeral system
• Arabic numerals (Eastern Arabic numerals, Western Arabic numerals)
• Trigonometric functions
• History of trigonometry
• History of algebra
Authority control
International
• VIAF
National
• Germany
Academics
• MathSciNet
• zbMATH
Other
• IdRef
| Wikipedia |
Yaʿqūb ibn Ṭāriq
Yaʿqūb ibn Ṭāriq (يعقوب بن طارق; referred to by some sources as Yaʿqūb; [1] died c. 796) was an 8th-century Persian astronomer and mathematician who lived in Baghdad.
Yaʿqūb ibn Ṭāriq
يعقوب بن طارق
Bornfl. 8th century
Diedc. 796
Academic work
EraIslamic Golden Age
Main interestsastronomer and mathematician
Career
Yaʿqūb ibn Ṭāriq was active in Baghdad as an astronomer during the rule of the second Abbasid caliph, al-Manṣūr (r. 754–775).[2][3] He seems not to have been aware of Ptolemaic astronomy,[2] and used a Zoroastrian calendar, which consisted of 12 months of 30 days each, with any remaining days being added after the eighth month, Ābān.[2]
Yaʿqūb ibn Ṭāriq's treatise Tarkīb al‐aflāk dealt with cosmography (the placement and sizes of the heavenly bodies).[4] The estimations of their sizes and distances in Tarkīb al‐aflāk were tabulated in the 11th century by the polymath al-Bīrūnī, in his work on India. According to al-Bīrūnī, Yaʿqūb ibn Ṭāriq gave the radius of the Earth as 1,050 farsakhs, the diameter of the Moon and Mercury as 5,000 farsakhs (4.8 Earth radii), and the diameter of the other heavenly bodies (Venus, the Sun, Mars, Jupiter, and Saturn) as 20,000 farsakhs (19.0 Earth radii).[5] He wrote that each of the planets had six associated spheres, that the Sun possessed two spheres, and the Moon three. He also spoke of planetary epicycles and speeds.[6] His values for the longitudes and apogees of celestial objects originated from a Persian set of astronomical table, the Zīǧ aš-šāh, although he used methods originating from the work of Indian astronomers to calculate the lunar phases.[1]
The Christian astrologer Ibn Hibintā mentioned Yaʿqūb, noting that he used the positions of the Sun and the stars to determine the latitude of places.[7]
Works
Works ascribed to Yaʿqūb ibn Ṭāriq include:[4]
• Zīj maḥlūl fī al‐Sindhind li‐daraja daraja ("Astronomical Tables in the 'Sindhind' Resolved for each Degree");
• Tarkīb al‐aflāk ("Arrangement of the Orbs"). Part of this work, the earliest surviving description of the celestial sky by an Islamic astronomer, is preserved by Ibn Hibintā.[8]
• Kitāb al‐ʿilal ("Rationales");
• Taqṭīʿ kardajāt al‐jayb ("Distribution of the Kardajas of the Sine");
• Mā irtafaʿa min qaws niṣf al‐nahār ("Elevation along the Arc of the Meridian").
An astrological work, Al‐maqālāt ("The Chapters"), is ascribed to Yaʿqūb ibn Ṭāriq by an unreliable source.[4]
Yaʿqūb ibn Ṭāriq's zij, written in around 770, was based on a Sanskrit work,[4] thought to be similar to the Brāhmasphuṭasiddhānta. It was brought to the court of al-Mansūr, the third caliph of the Fatimid Caliphate, from Sindh,[9] reportedly by the Sindhi astronomer Kankah.[10]
References
1. Sezgin 2021, p. 125.
2. Sezgin 2021, p. 124.
3. Hawting, G.R. "al-Manṣūr". Britannica Online. Retrieved 8 May 2023.
4. Plofker 2007, pp. 1250–1251.
5. Pingree 1976, pp. 106–107.
6. Sezgin 2021, p. 126.
7. Sezgin 2021, p. 127.
8. Sezgin 2021, pp. 126–127.
9. Pingree 1976, p. 97.
10. Kennedy 1956, p. 12.
Sources
• Kennedy, Edward Stewart (1956). "A Survey of Islamic Astronomical Tables". Transactions of the American Philosophical Society. New Series. Philadelphia, Pennsylvania: American Philosophical Society. 46 (2): 123–177. doi:10.2307/1005726. hdl:2027/mdp.39076006359272. ISSN 0065-9746. JSTOR 1005726.
• Pingree, David (1976). "Yaʿqūb ibn Ṭāriq". In Gillispie, Charles Coulston (ed.). Dictionary of Scientific Biography. Vol. 14. New York: Charles Scribner's Sons. ISBN 978-0-684-16962-0.
• Plofker, Kim (2007). "Yaʿqūb ibn Ṭāriq". In Thomas Hockey; et al. (eds.). The Biographical Encyclopedia of Astronomers. New York: Springer. ISBN 978-0-387-31022-0. (PDF version)
• Sezgin, Fuat (2021). "III. Arab Astronomers". Sezgin Online. doi:10.1163/2667-3975_SEZO_COM_603. Retrieved 8 May 2023.
Further reading
• Hogendijk, Jan P. (1988). "New Light on the Lunar Crescent Visibility Table of Yaʿqūb ibn Ṭāriq". Journal of Near Eastern Studies. 47 (2): 95–104. doi:10.1086/373260. JSTOR 544381. S2CID 162371303.
• Kennedy, E. S. (1968). "The Lunar Visibility Theory of Yaʿqūb Ibn Ṭāriq". Journal of Near Eastern Studies. 27 (2): 126–132. doi:10.1086/371945. JSTOR 543759. S2CID 162369374.
• Pingree, David (1968). "The Fragments of the Works of Yaʿqūb Ibn Ṭāriq". Journal of Near Eastern Studies. 27 (2): 97–125. doi:10.1086/371944. JSTOR 543758. S2CID 68584137.
• Steinschneider, Moritz (1870). "Zur Geschichte der Uebersetzungen aus dem Indischen in's Arabische und ihres Einflusses aus die arabische Literatur". Zeitschrift der Deutschen Morgenländischen Gesellschaft. 24: 332.
• Suter, Heinrich (1900). Die Mathematiker und Astronomen der Araber und ihre Werke (in German). Leipzig: Teubner. p. 4. OCLC 230703086.
Astronomy in the medieval Islamic world
Astronomers
• by century
8th
• Ahmad Nahavandi
• Al-Fadl ibn Naubakht
• Muḥammad ibn Ibrāhīm al-Fazārī
• Ibrāhīm al-Fazārī
• Mashallah ibn Athari
• Yaʿqūb ibn Ṭāriq
9th
• Abu Ali al-Khayyat
• Abu Ma'shar al-Balkhi
• Abu Said Gorgani
• Al-Farghani
• Al-Kindi
• Al-Mahani
• Abu Hanifa Dinawari
• Al-Ḥajjāj ibn Yūsuf
• Al-Marwazi
• Ali ibn Isa al-Asturlabi
• Banū Mūsā brothers
• Iranshahri
• Khalid ibn Abd al‐Malik al‐Marwarrudhi
• Al-Khwarizmi
• Sahl ibn Bishr
• Thābit ibn Qurra
• Yahya ibn Abi Mansur
10th
• al-Sufi
• Ibn
• Al-Adami
• al-Khojandi
• al-Khazin
• al-Qūhī
• Abu al-Wafa
• Ahmad ibn Yusuf
• al-Battani
• Al-Qabisi
• Ibn al-A'lam
• Al-Nayrizi
• Al-Saghani
• Aṣ-Ṣaidanānī
• Ibn Yunus
• Ibrahim ibn Sinan
• Ma Yize
• al-Sijzi
• Al-ʻIjliyyah
• Nastulus
• Abolfadl Harawi
• Haseb-i Tabari
• al-Majriti
• Abu al-Hasan al-Ahwazi
11th
• Abu Nasr Mansur
• al-Biruni
• Ali ibn Ridwan
• Al-Zarqālī
• Ibn al-Samh
• Alhazen
• Avicenna
• Ibn al-Saffar
• Kushyar Gilani
• Said al-Andalusi
• Ibrahim ibn Said al-Sahli
• Ibn Mu'adh al-Jayyani
• Al-Isfizari
• Ali ibn Khalaf
12th
• Al-Bitruji
• Avempace
• Ibn Tufail
• Al-Kharaqī
• Al-Khazini
• Al-Samawal al-Maghribi
• Abu al-Salt
• Averroes
• Ibn al-Kammad
• Jabir ibn Aflah
• Omar Khayyam
• Sharaf al-Din al-Tusi
13th
• Ibn al-Banna' al-Marrakushi
• Ibn al‐Ha'im al‐Ishbili
• Jamal ad-Din
• Alam al-Din al-Hanafi
• Najm al‐Din al‐Misri
• Muhyi al-Din al-Maghribi
• Nasir al-Din al-Tusi
• Qutb al-Din al-Shirazi
• Shams al-Din al-Samarqandi
• Zakariya al-Qazwini
• al-Urdi
• al-Abhari
• Muhammad ibn Abi Bakr al‐Farisi
• Abu Ali al-Hasan al-Marrakushi
• Ibn Ishaq al-Tunisi
• Ibn al‐Raqqam
• Al-Ashraf Umar II
• Fakhr al-Din al-Akhlati
14th
• Ibn al-Shatir
• Al-Khalili
• Ibn Shuayb
• al-Battiwi
• Abū al‐ʿUqūl
• Al-Wabkanawi
• Nizam al-Din al-Nisapuri
• al-Jadiri
• Sadr al-Shari'a al-Asghar
• Fathullah Shirazi
15th
• Ali Kuşçu
• Abd al‐Wajid
• Jamshīd al-Kāshī
• Kadızade Rumi
• Ulugh Beg
• Sibt al-Maridini
• Ibn al-Majdi
• al-Wafa'i
• al-Kubunani
• 'Abd al-'Aziz al-Wafa'i
16th
• Al-Birjandi
• al-Khafri
• Baha' al-din al-'Amili
• Piri Reis
• Takiyüddin
17th
• Yang Guangxian
• Ehmedê Xanî
• Al Achsasi al Mouakket
• Muhammad al-Rudani
Topics
Works
• Arabic star names
• Islamic calendar
• Aja'ib al-Makhluqat
• Encyclopedia of the Brethren of Purity
• Tabula Rogeriana
• The Book of Healing
• The Remaining Signs of Past Centuries
Zij
• Alfonsine tables
• Huihui Lifa
• Book of Fixed Stars
• Toledan Tables
• Zij-i Ilkhani
• Zij-i Sultani
• Sullam al-sama'
Instruments
• Alidade
• Analog computer
• Aperture
• Armillary sphere
• Astrolabe
• Astronomical clock
• Celestial globe
• Compass
• Compass rose
• Dioptra
• Equatorial ring
• Equatorium
• Globe
• Graph paper
• Magnifying glass
• Mural instrument
• Navigational astrolabe
• Nebula
• Octant
• Planisphere
• Quadrant
• Sextant
• Shadow square
• Sundial
• Schema for horizontal sundials
• Triquetrum
Concepts
• Almucantar
• Apogee
• Astrology
• Astrophysics
• Axial tilt
• Azimuth
• Celestial mechanics
• Celestial spheres
• Circular orbit
• Deferent and epicycle
• Earth's rotation
• Eccentricity
• Ecliptic
• Elliptic orbit
• Equant
• Galaxy
• Geocentrism
• Gravitational energy
• Gravity
• Heliocentrism
• Inertia
• Islamic cosmology
• Moonlight
• Multiverse
• Muwaqqit
• Obliquity
• Parallax
• Precession
• Qibla
• Salah times
• Specific gravity
• Spherical Earth
• Sublunary sphere
• Sunlight
• Supernova
• Temporal finitism
• Trepidation
• Triangulation
• Tusi couple
• Universe
Institutions
• Al-Azhar University
• House of Knowledge
• House of Wisdom
• University of al-Qarawiyyin
• Observatories
• Constantinople (Taqi al-Din)
• Maragheh
• Samarkand (Ulugh Beg)
Influences
• Babylonian astronomy
• Egyptian astronomy
• Hellenistic astronomy
• Indian astronomy
Influenced
• Byzantine science
• Chinese astronomy
• Medieval European science
• Indian astronomy
Mathematics in the medieval Islamic world
Mathematicians
9th century
• 'Abd al-Hamīd ibn Turk
• Sanad ibn Ali
• al-Jawharī
• Al-Ḥajjāj ibn Yūsuf
• Al-Kindi
• Qusta ibn Luqa
• Al-Mahani
• al-Dinawari
• Banū Mūsā
• Hunayn ibn Ishaq
• Al-Khwarizmi
• Yusuf al-Khuri
• Ishaq ibn Hunayn
• Na'im ibn Musa
• Thābit ibn Qurra
• al-Marwazi
• Abu Said Gorgani
10th century
• Abu al-Wafa
• al-Khazin
• Al-Qabisi
• Abu Kamil
• Ahmad ibn Yusuf
• Aṣ-Ṣaidanānī
• Sinān ibn al-Fatḥ
• al-Khojandi
• Al-Nayrizi
• Al-Saghani
• Brethren of Purity
• Ibn Sahl
• Ibn Yunus
• al-Uqlidisi
• Al-Battani
• Sinan ibn Thabit
• Ibrahim ibn Sinan
• Al-Isfahani
• Nazif ibn Yumn
• al-Qūhī
• Abu al-Jud
• Al-Sijzi
• Al-Karaji
• al-Majriti
• al-Jabali
11th century
• Abu Nasr Mansur
• Alhazen
• Kushyar Gilani
• Al-Biruni
• Ibn al-Samh
• Abu Mansur al-Baghdadi
• Avicenna
• al-Jayyānī
• al-Nasawī
• al-Zarqālī
• ibn Hud
• Al-Isfizari
• Omar Khayyam
• Muhammad al-Baghdadi
12th century
• Jabir ibn Aflah
• Al-Kharaqī
• Al-Khazini
• Al-Samawal al-Maghribi
• al-Hassar
• Sharaf al-Din al-Tusi
• Ibn al-Yasamin
13th century
• Ibn al‐Ha'im al‐Ishbili
• Ahmad al-Buni
• Ibn Munim
• Alam al-Din al-Hanafi
• Ibn Adlan
• al-Urdi
• Nasir al-Din al-Tusi
• al-Abhari
• Muhyi al-Din al-Maghribi
• al-Hasan al-Marrakushi
• Qutb al-Din al-Shirazi
• Shams al-Din al-Samarqandi
• Ibn al-Banna'
• Kamāl al-Dīn al-Fārisī
14th century
• Nizam al-Din al-Nisapuri
• Ibn al-Shatir
• Ibn al-Durayhim
• Al-Khalili
• al-Umawi
15th century
• Ibn al-Majdi
• al-Rūmī
• al-Kāshī
• Ulugh Beg
• Ali Qushji
• al-Wafa'i
• al-Qalaṣādī
• Sibt al-Maridini
• Ibn Ghazi al-Miknasi
16th century
• Al-Birjandi
• Muhammad Baqir Yazdi
• Taqi ad-Din
• Ibn Hamza al-Maghribi
• Ahmad Ibn al-Qadi
Mathematical
works
• The Compendious Book on Calculation by Completion and Balancing
• De Gradibus
• Principles of Hindu Reckoning
• Book of Optics
• The Book of Healing
• Almanac
• Book on the Measurement of Plane and Spherical Figures
• Encyclopedia of the Brethren of Purity
• Toledan Tables
• Tabula Rogeriana
• Zij
Concepts
• Alhazen's problem
• Islamic geometric patterns
Centers
• Al-Azhar University
• Al-Mustansiriya University
• House of Knowledge
• House of Wisdom
• Constantinople observatory of Taqi ad-Din
• Madrasa
• Maragheh observatory
• University of al-Qarawiyyin
Influences
• Babylonian mathematics
• Greek mathematics
• Indian mathematics
Influenced
• Byzantine mathematics
• European mathematics
• Indian mathematics
Related
• Hindu–Arabic numeral system
• Arabic numerals (Eastern Arabic numerals, Western Arabic numerals)
• Trigonometric functions
• History of trigonometry
• History of algebra
Authority control
International
• VIAF
National
• Germany
Other
• İslâm Ansiklopedisi
| Wikipedia |
Yehoshua Bar-Hillel
Yehoshua Bar-Hillel (Hebrew: יהושע בר-הלל; 8 September 1915 – 25 September 1975) was an Israeli philosopher, mathematician, and linguist. He was a pioneer in the fields of machine translation and formal linguistics.
Yehoshua Bar-Hillel
Born(1915-09-08)September 8, 1915
Vienna, Austria-Hungary
DiedSeptember 25, 1975(1975-09-25) (aged 60)
Jerusalem, Israel
EducationHebrew University of Jerusalem
Known forBar-Hillel lemma
ChildrenMaya, Mira
Scientific career
Fieldsphilosophy, mathematics, linguistics
InstitutionsMIT
Hebrew University of Jerusalem[1]
ThesisTheory of syntactic categories[1] (1945)
Doctoral advisorAbraham Fraenkel
Other academic advisorsRudolf Carnap
InfluencedAsa Kasher
Avishai Margalit
Biography
Born Oscar Westreich in Vienna, Austria-Hungary, he was raised in Berlin. In 1933 he emigrated to Palestine with the Bnei Akiva youth movement, and briefly joined the kibbutz Tirat Zvi before settling in Jerusalem and marrying Shulamith.
During World War II, he served in the Jewish Brigade of the British Army. He fought with the Haganah during the 1948 Arab–Israeli War, losing an eye.
Bar-Hillel received his PhD in Philosophy from the Hebrew University where he also studied mathematics under Abraham Fraenkel, with whom he eventually coauthored Foundations of Set Theory (1958, 1973).
Bar-Hillel was a major disciple of Rudolf Carnap, whose Logical Syntax of Language much influenced him. He began a correspondence with Carnap in the 1940s, which led to a 1950 post-doctorate under Carnap at the University of Chicago, and to his collaborating on Carnap's 1952 An Outline of the Theory of Semantic Information.
Bar-Hillel then took up a position at MIT, where he was the first academic to work full-time in the field of machine translation. Bar-Hillel organised the first International Conference on Machine Translation in 1952. Later he expressed doubts that general-purpose fully automatic high-quality machine translation would ever be feasible. He was also a pioneer in the field of information retrieval.
In 1953, Bar-Hillel joined the philosophy department at the Hebrew University, where he taught until his death at age 60. His teachings and writings strongly influenced an entire generation of Israeli philosophers and linguists, including Asa Kasher and Avishai Margalit. In 1953, he founded a pioneering algebraic-computational linguistic group, and in 1961 he contributed to the proof of the pumping lemma for context-free languages (sometimes called the Bar-Hillel lemma). Bar-Hillel helped found the Hebrew University's department of Philosophy of Science. From 1966 to 1969 Bar-Hillel presided over the Division for Logic, Methodology and Philosophy of Science of the International Union of History and Philosophy of Science.
Bar-Hillel's daughter Maya Bar-Hillel is a cognitive psychologist at the Hebrew University, known for her collaborations with Amos Tversky and for her role in critiquing Bible code study. His other daughter, Mira Bar-Hillel, is a freelance journalist who has worked for the London Evening Standard. His granddaughter, Gili Bar-Hillel, is the Hebrew translator of the Harry Potter series of books.
Related terms
• Categorial grammar
• Indexical expression
Notes
1. ^ Melby, Alan. The Possibility of Language (Amsterdam:Benjamins, 1995, 27-41)
2. ^ Appendix III of 'The present status of automatic translation of languages', Advances in Computers, vol.1 (1960), p.158-163. Reprinted in Y.Bar-Hillel: Language and information (Reading, Mass.: Addison-Wesley, 1964), p.174-179.
Select bibliography
• Yehoshua Bar-Hillel (1953). "Some Linguistic Problems Connected With Machine Translation" (PDF). Philosophy of Science. 20 (3): 217–225. doi:10.1086/287266. S2CID 62000879.
• Yehoshua Bar-Hillel (Jul 1954). "Indexical Expressions". Mind. 63 (251): 359–379.
• Yehoshua Bar-Hillel (Jan 1963). Four lectures on Algebraic Linguistics and Machine Translation (PDF) (ASTIA Report). Hebrew University, Jerusalem. Archived from the original (PDF) on May 13, 2014.
• Yehoshua Bar-Hillel (1964). Language and Information: Selected Essays on Their Theory and Application. Reading, MA: Addison-Wesley.
• Yehoshua Bar-Hillel, ed. (1965). Logic, Methodology and Philosophy of Science: Proceedings of the 1964 International Congress. Amsterdam: North-Holland.
• Yehoshua Bar-Hillel (1970). Aspects of Language: Essays in Philosophy of Language, Linguistic Philosophy, and Methodology of Linguistics. Amsterdam: North-Holland.
• Yehoshua Bar-Hillel, ed. (Aug 1970). Mathematical Logic and Foundations of Set Theory: Colloquium Proceedings, Jerusalem, 1968. Study in Logic and Foundation of Mathematics. Amsterdam: North-Holland Pub. Co. ISBN 9780720422559.
• Yehoshua Bar-Hillel, ed. (1971). Pragmatics of Natural Languages: Proceedings of the 1970 International Working Symposium. Synthese Library. Vol. 41. Dordrecht: D. Reidel Publishing Co. doi:10.1007/978-94-010-1713-8. ISBN 978-90-277-0599-0.
• Abraham Halevi Fraenkel; Yehoshua Bar-Hillel; Azriel Levy (1973). Foundations of Set Theory. Studies in Logic and the Foundations of Mathematics. Vol. 67 (2nd ed.). Elsevier. ISBN 9780080887050.
References
1. Bar-Hillel, Yehoshua, Encyclopedia of Linguistics, Philip Strazny (ed), New York, Fitzroy Dearborn, 2005, vol.1, pp. 124-126
External links
• "Yehoshua Bar-Hillel: A Philosopher's Contribution to Machine Translation"
• "Bar-Hillel and Machine Translation: Then and Now. Archived 2010-06-12 at the Wayback Machine"
• Bar-Hillel Colloquium.
• Translation Trouble: Time Magazine article from 1954.
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| Wikipedia |
Japanese mathematics
Japanese mathematics (和算, wasan) denotes a distinct kind of mathematics which was developed in Japan during the Edo period (1603–1867). The term wasan, from wa ("Japanese") and san ("calculation"), was coined in the 1870s[1] and employed to distinguish native Japanese mathematical theory from Western mathematics (洋算 yōsan).[2]
In the history of mathematics, the development of wasan falls outside the Western realm. At the beginning of the Meiji period (1868–1912), Japan and its people opened themselves to the West. Japanese scholars adopted Western mathematical technique, and this led to a decline of interest in the ideas used in wasan.
History
The Japanese mathematical schema evolved during a period when Japan's people were isolated from European influences, but instead borrowed from ancient mathematical texts written in China, including those from the Yuan dynasty and earlier. The Japanese mathematicians Yoshida Shichibei Kōyū, Imamura Chishō, and Takahara Kisshu are among the earliest known Japanese mathematicians. They came to be known to their contemporaries as "the Three Arithmeticians".[3][4]
Yoshida was the author of the oldest extant Japanese mathematical text, the 1627 work called Jinkōki. The work dealt with the subject of soroban arithmetic, including square and cube root operations.[5] Yoshida's book significantly inspired a new generation of mathematicians, and redefined the Japanese perception of educational enlightenment, which was defined in the Seventeen Article Constitution as "the product of earnest meditation".[6]
Seki Takakazu founded enri (円理: circle principles), a mathematical system with the same purpose as calculus at a similar time to calculus's development in Europe. However Seki's investigations did not proceed from the same foundations as those used in Newton's studies in Europe.[7]
Mathematicians like Takebe Katahiro played and important role in developing Enri (" circle principle"), a crude analog to the Western calculus.[8] He obtained power series expansion of $(\arcsin(x))^{2}$ in 1722, 15 years earlier than Euler. He used Richardson extrapolation in 1695, about 200 years earlier than Richardson.[9] He also computed 41 digits of π, based on polygon approximation and Richardson extrapolation.[10]
Select mathematicians
The following list encompasses mathematicians whose work was derived from wasan.
• Yoshida Mitsuyoshi (1598–1672)
• Seki Takakazu (1642–1708)
• Takebe Kenkō (1664–1739)
• Matsunaga Ryohitsu (fl. 1718-1749)[11]
• Kurushima Kinai (d. 1757)
• Arima Raido (1714–1783)[12]
• Fujita Sadasuke (1734-1807)[13]
• Ajima Naonobu (1739–1783)
• Aida Yasuaki (1747–1817)
• Sakabe Kōhan (1759–1824)
• Fujita Kagen (1765–1821)[13]
• Hasegawa Ken (c. 1783-1838)[12]
• Wada Nei (1787–1840)
• Shiraishi Chochu (1796–1862)[14]
• Koide Shuke (1797–1865)[12]
• Omura Isshu (1824–1871)[12]
See also
• Japanese theorem for cyclic polygons
• Japanese theorem for cyclic quadrilaterals
• Sangaku, the custom of presenting mathematical problems, carved in wood tablets, to the public in Shinto shrines
• Soroban, a Japanese abacus
• Category:Japanese mathematicians
Notes
1. Selin, Helaine. (1997). Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures, p. 641. , p. 641, at Google Books
2. Smith, David et al. (1914). A History of Japanese Mathematics, p. 1 n2., p. 1, at Google Books
3. Smith, p. 35. , p. 35, at Google Books
4. Campbell, Douglas et al. (1984). Mathematics: People, Problems, Results, p. 48.
5. Restivo, Sal P. (1984). Mathematics in Society and History, p. 56., p. 56, at Google Books
6. Strayer, Robert (2000). Ways of the World: A Brief Global History with Sources. Bedford/St. Martins. p. 7. ISBN 9780312489168. OCLC 708036979.
7. Smith, pp. 91–127., p. 91, at Google Books
8. Mathematical Society of Japan, Takebe Prize
9. Osada, Naoki (Aug 26, 2011). "収束の加速法の歴史 : 17世紀ヨーロッパと日本の加速法 (数学史の研究)" (PDF). Study of the History of Mathematics RIMS Kôkyûroku (in Japanese). 1787: 100–102 – via Kyoto University.
10. Ogawa, Tsugane (May 13, 1997). "円理の萌芽 : 建部賢弘の円周率計算 : (数学史の研究)" (PDF). Study of the History of Mathematics RIMS Kôkyûroku (in Japanese). 1019: 80–88 – via Kyoto University.
11. Smith, pp. 104, 158, 180., p. 104, at Google Books
12. List of Japanese mathematicians -- Clark University, Dept. of Mathematics and Computer Science
13. Fukagawa, Hidetoshi et al. (2008). Sacred Mathematics: Japanese Temple Geometry, p. 24.
14. Smith, p. 233., p. 233, at Google Books
References
• Campbell, Douglas M. and John C. Iggins. (1984). Mathematics: People, Problems, Results. Belmont, California: Warsworth International. ISBN 9780534032005; ISBN 9780534032012; ISBN 9780534028794; OCLC 300429874
• Endō Toshisada (1896). History of mathematics in Japan (日本數學史, Dai Nihon sūgakush). Tōkyō: _____. OCLC 122770600
• Fukagawa, Hidetoshi, and Dan Pedoe. (1989). Japanese temple geometry problems = Sangaku. Winnipeg: Charles Babbage. ISBN 9780919611214; OCLC 474564475
• __________ and Dan Pedoe. (1991) How to resolve Japanese temple geometry problems? (日本の幾何ー何題解けますか?, Nihon no kika nan dai tokemasu ka) Tōkyō. ISBN 9784627015302; OCLC 47500620
• __________ and Tony Rothman. (2008). Sacred Mathematics: Japanese Temple Geometry. Princeton: Princeton University Press. ISBN 069112745X; OCLC 181142099
• Horiuchi, Annick. (1994). Les Mathematiques Japonaises a L'Epoque d'Edo (1600–1868): Une Etude des Travaux de Seki Takakazu (?-1708) et de Takebe Katahiro (1664–1739). Paris: Librairie Philosophique J. Vrin. ISBN 9782711612130; OCLC 318334322
• __________. (1998). "Les mathématiques peuvent-elles n'être que pur divertissement? Une analyse des tablettes votives de mathématiques à l'époque d'Edo." Extrême-Orient, Extrême-Occident, volume 20, pp. 135–156.
• Kobayashi, Tatsuhiko. (2002) "What kind of mathematics and terminology was transmitted into 18th-century Japan from China?", Historia Scientiarum, Vol.12, No.1.
• Kobayashi, Tatsuhiko. Trigonometry and Its Acceptance in the 18th-19th Centuries Japan.
• Ogawa, Tsukane. "A Review of the History of Japanese Mathematics". Revue d'histoire des mathématiques 7, fascicule 1 (2001), 137-155.
• Restivo, Sal P. (1992). Mathematics in Society and History: Sociological Inquiries. Dordrecht: Kluwer Academic Publishers. ISBN 9780792317654; OCLC 25709270
• Selin, Helaine. (1997). Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures. Dordrecht: Kluwer/Springer. ISBN 9780792340669; OCLC 186451909
• David Eugene Smith and Yoshio Mikami. (1914). A History of Japanese Mathematics. Chicago: Open Court Publishing. OCLC 1515528; see online, multi-formatted, full-text book at archive.org
External links
• Japan Academy, Collection of native Japanese mathematics
• JapanMath, Math program focused on Math Fact Fluency and Japanese originated logic games
• Sangaku
• Sansu Math, translated Tokyo Shoseki Japanese math curriculum
• Kümmerle, Harald. Bibliography on traditional mathematics in Japan (wasan)
Authority control: National
• Japan
| Wikipedia |
Boolean algebras canonically defined
Boolean algebra is a mathematically rich branch of abstract algebra. Stanford Encyclopaedia of Philosophy defines Boolean algebra as 'the algebra of two-valued logic with only sentential connectives, or equivalently of algebras of sets under union and complementation.'[1] Just as group theory deals with groups, and linear algebra with vector spaces, Boolean algebras are models of the equational theory of the two values 0 and 1 (whose interpretation need not be numerical). Common to Boolean algebras, groups, and vector spaces is the notion of an algebraic structure, a set closed under some operations satisfying certain equations.[2]
Boolean algebras are models of the equational theory of two values; this definition is equivalent to the lattice and ring definitions.
Just as there are basic examples of groups, such as the group $\mathbb {Z} $ of integers and the symmetric group Sn of permutations of n objects, there are also basic examples of Boolean algebras such as the following.
• The algebra of binary digits or bits 0 and 1 under the logical operations including disjunction, conjunction, and negation. Applications include the propositional calculus and the theory of digital circuits.
• The algebra of sets under the set operations including union, intersection, and complement. Applications are far-reaching because set theory is the standard foundations of mathematics.
Boolean algebra thus permits applying the methods of abstract algebra to mathematical logic and digital logic.
Unlike groups of finite order, which exhibit complexity and diversity and whose first-order theory is decidable only in special cases, all finite Boolean algebras share the same theorems and have a decidable first-order theory. Instead, the intricacies of Boolean algebra are divided between the structure of infinite algebras and the algorithmic complexity of their syntactic structure.
Definition
Boolean algebra treats the equational theory of the maximal two-element finitary algebra, called the Boolean prototype, and the models of that theory, called Boolean algebras.[3] These terms are defined as follows.
An algebra is a family of operations on a set, called the underlying set of the algebra. We take the underlying set of the Boolean prototype to be {0,1}.
An algebra is finitary when each of its operations takes only finitely many arguments. For the prototype each argument of an operation is either 0 or 1, as is the result of the operation. The maximal such algebra consists of all finitary operations on {0,1}.
The number of arguments taken by each operation is called the arity of the operation. An operation on {0,1} of arity n, or n-ary operation, can be applied to any of 2n possible values for its n arguments. For each choice of arguments, the operation may return 0 or 1, whence there are 22n n-ary operations.
The prototype therefore has two operations taking no arguments, called zeroary or nullary operations, namely zero and one. It has four unary operations, two of which are constant operations, another is the identity, and the most commonly used one, called negation, returns the opposite of its argument: 1 if 0, 0 if 1. It has sixteen binary operations; again two of these are constant, another returns its first argument, yet another returns its second, one is called conjunction and returns 1 if both arguments are 1 and otherwise 0, another is called disjunction and returns 0 if both arguments are 0 and otherwise 1, and so on. The number of (n+1)-ary operations in the prototype is the square of the number of n-ary operations, so there are 162 = 256 ternary operations, 2562 = 65,536 quaternary operations, and so on.
A family is indexed by an index set. In the case of a family of operations forming an algebra, the indices are called operation symbols, constituting the language of that algebra. The operation indexed by each symbol is called the denotation or interpretation of that symbol. Each operation symbol specifies the arity of its interpretation, whence all possible interpretations of a symbol have the same arity. In general it is possible for an algebra to interpret distinct symbols with the same operation, but this is not the case for the prototype, whose symbols are in one-one correspondence with its operations. The prototype therefore has 22n n-ary operation symbols, called the Boolean operation symbols and forming the language of Boolean algebra. Only a few operations have conventional symbols, such as ¬ for negation, ∧ for conjunction, and ∨ for disjunction.[4] It is convenient to consider the i-th n-ary symbol to be nfi as done below in the section on truth tables.
An equational theory in a given language consists of equations between terms built up from variables using symbols of that language. Typical equations in the language of Boolean algebra are x∧y = y∧x, x∧x = x, x∧¬x = y∧¬y, and x∧y = x.
An algebra satisfies an equation when the equation holds for all possible values of its variables in that algebra when the operation symbols are interpreted as specified by that algebra. The laws of Boolean algebra are the equations in the language of Boolean algebra satisfied by the prototype. The first three of the above examples are Boolean laws, but not the fourth since 1∧0 ≠ 1.
The equational theory of an algebra is the set of all equations satisfied by the algebra. The laws of Boolean algebra therefore constitute the equational theory of the Boolean prototype.
A model of a theory is an algebra interpreting the operation symbols in the language of the theory and satisfying the equations of the theory.
A Boolean algebra is any model of the laws of Boolean algebra.
That is, a Boolean algebra is a set and a family of operations thereon interpreting the Boolean operation symbols and satisfying the same laws as the Boolean prototype.[5]
If we define a homologue of an algebra to be a model of the equational theory of that algebra, then a Boolean algebra can be defined as any homologue of the prototype.
Example 1. The Boolean prototype is a Boolean algebra, since trivially it satisfies its own laws. It is thus the prototypical Boolean algebra. We did not call it that initially in order to avoid any appearance of circularity in the definition.
Basis
The operations need not be all explicitly stated. A basis is any set from which the remaining operations can be obtained by composition. A "Boolean algebra" may be defined from any of several different bases. Three bases for Boolean algebra are in common use, the lattice basis, the ring basis, and the Sheffer stroke or NAND basis. These bases impart respectively a logical, an arithmetical, and a parsimonious character to the subject.
• The lattice basis originated in the 19th century with the work of Boole, Peirce, and others seeking an algebraic formalization of logical thought processes.
• The ring basis emerged in the 20th century with the work of Zhegalkin and Stone and became the basis of choice for algebraists coming to the subject from a background in abstract algebra. Most treatments of Boolean algebra assume the lattice basis, a notable exception being Halmos[1963] whose linear algebra background evidently endeared the ring basis to him.[6]
• Since all finitary operations on {0,1} can be defined in terms of the Sheffer stroke NAND (or its dual NOR), the resulting economical basis has become the basis of choice for analyzing digital circuits, in particular gate arrays in digital electronics.
The common elements of the lattice and ring bases are the constants 0 and 1, and an associative commutative binary operation, called meet x∧y in the lattice basis, and multiplication xy in the ring basis. The distinction is only terminological. The lattice basis has the further operations of join, x∨y, and complement, ¬x. The ring basis has instead the arithmetic operation x⊕y of addition (the symbol ⊕ is used in preference to + because the latter is sometimes given the Boolean reading of join).
To be a basis is to yield all other operations by composition, whence any two bases must be intertranslatable. The lattice basis translates x∨y to the ring basis as x⊕y⊕xy, and ¬x as x⊕1. Conversely the ring basis translates x⊕y to the lattice basis as (x∨y)∧¬(x∧y).
Both of these bases allow Boolean algebras to be defined via a subset of the equational properties of the Boolean operations. For the lattice basis, it suffices to define a Boolean algebra as a distributive lattice satisfying x∧¬x = 0 and x∨¬x = 1, called a complemented distributive lattice. The ring basis turns a Boolean algebra into a Boolean ring, namely a ring satisfying x2 = x.
Emil Post gave a necessary and sufficient condition for a set of operations to be a basis for the nonzeroary Boolean operations. A nontrivial property is one shared by some but not all operations making up a basis. Post listed five nontrivial properties of operations, identifiable with the five Post's classes, each preserved by composition, and showed that a set of operations formed a basis if, for each property, the set contained an operation lacking that property. (The converse of Post's theorem, extending "if" to "if and only if," is the easy observation that a property from among these five holding of every operation in a candidate basis will also hold of every operation formed by composition from that candidate, whence by nontriviality of that property the candidate will fail to be a basis.) Post's five properties are:
• monotone, no 0-1 input transition can cause a 1-0 output transition;
• affine, representable with Zhegalkin polynomials that lack bilinear or higher terms, e.g. x⊕y⊕1 but not xy;
• self-dual, so that complementing all inputs complements the output, as with x, or the median operator xy⊕yz⊕zx, or their negations;
• strict (mapping the all-zeros input to zero);
• costrict (mapping all-ones to one).
The NAND (dually NOR) operation lacks all these, thus forming a basis by itself.
Truth tables
The finitary operations on {0,1} may be exhibited as truth tables, thinking of 0 and 1 as the truth values false and true.[7] They can be laid out in a uniform and application-independent way that allows us to name, or at least number, them individually.[8] These names provide a convenient shorthand for the Boolean operations. The names of the n-ary operations are binary numbers of 2n bits. There being 22n such operations, one cannot ask for a more succinct nomenclature. Note that each finitary operation can be called a switching function.
This layout and associated naming of operations is illustrated here in full for arities from 0 to 2.
Truth tables for the Boolean operations of arity up to 2
Constants
${}^{0}\!f_{0}$ ${}^{0}\!f_{1}$
01
Unary operations
$x_{0}$ ${}^{1}\!f_{0}$ ${}^{1}\!f_{1}$ ${}^{1}\!f_{2}$ ${}^{1}\!f_{3}$
0 0101
1 0011
Binary operations
$x_{0}$ $x_{1}$ ${}^{2}\!f_{0}$ ${}^{2}\!f_{1}$ ${}^{2}\!f_{2}$ ${}^{2}\!f_{3}$ ${}^{2}\!f_{4}$ ${}^{2}\!f_{5}$ ${}^{2}\!f_{6}$ ${}^{2}\!f_{7}$ ${}^{2}\!f_{8}$ ${}^{2}\!f_{9}$ ${}^{2}\!f_{10}$ ${}^{2}\!f_{11}$ ${}^{2}\!f_{12}$ ${}^{2}\!f_{13}$ ${}^{2}\!f_{14}$ ${}^{2}\!f_{15}$
0 0 0101010101010101
1 0 0011001100110011
0 1 0000111100001111
1 1 0000000011111111
These tables continue at higher arities, with 2n rows at arity n, each row giving a valuation or binding of the n variables x0,...xn−1 and each column headed nfi giving the value nfi(x0,...,xn−1) of the i-th n-ary operation at that valuation. The operations include the variables, for example 1f2 is x0 while 2f10 is x0 (as two copies of its unary counterpart) and 2f12 is x1 (with no unary counterpart). Negation or complement ¬x0 appears as 1f1 and again as 2f5, along with 2f3 (¬x1, which did not appear at arity 1), disjunction or union x0∨x1 as 2f14, conjunction or intersection x0∧x1 as 2f8, implication x0→x1 as 2f13, exclusive-or symmetric difference x0⊕x1 as 2f6, set difference x0−x1 as 2f2, and so on.
As a minor detail important more for its form than its content, the operations of an algebra are traditionally organized as a list. Although we are here indexing the operations of a Boolean algebra by the finitary operations on {0,1}, the truth-table presentation above serendipitously orders the operations first by arity and second by the layout of the tables for each arity. This permits organizing the set of all Boolean operations in the traditional list format. The list order for the operations of a given arity is determined by the following two rules.
(i) The i-th row in the left half of the table is the binary representation of i with its least significant or 0-th bit on the left ("little-endian" order, originally proposed by Alan Turing, so it would not be unreasonable to call it Turing order).
(ii) The j-th column in the right half of the table is the binary representation of j, again in little-endian order. In effect the subscript of the operation is the truth table of that operation. By analogy with Gödel numbering of computable functions one might call this numbering of the Boolean operations the Boole numbering.
When programming in C or Java, bitwise disjunction is denoted x|y, conjunction x&y, and negation ~x. A program can therefore represent for example the operation x∧(y∨z) in these languages as x&(y|z), having previously set x = 0xaa, y = 0xcc, and z = 0xf0 (the "0x" indicates that the following constant is to be read in hexadecimal or base 16), either by assignment to variables or defined as macros. These one-byte (eight-bit) constants correspond to the columns for the input variables in the extension of the above tables to three variables. This technique is almost universally used in raster graphics hardware to provide a flexible variety of ways of combining and masking images, the typical operations being ternary and acting simultaneously on source, destination, and mask bits.
Examples
Bit vectors
Example 2. All bit vectors of a given length form a Boolean algebra "pointwise", meaning that any n-ary Boolean operation can be applied to n bit vectors one bit position at a time. For example, the ternary OR of three bit vectors each of length 4 is the bit vector of length 4 formed by oring the three bits in each of the four bit positions, thus 0100∨1000∨1001 = 1101. Another example is the truth tables above for the n-ary operations, whose columns are all the bit vectors of length 2n and which therefore can be combined pointwise whence the n-ary operations form a Boolean algebra.[9] This works equally well for bit vectors of finite and infinite length, the only rule being that the bit positions all be indexed by the same set in order that "corresponding position" be well defined.
The atoms of such an algebra are the bit vectors containing exactly one 1. In general the atoms of a Boolean algebra are those elements x such that x∧y has only two possible values, x or 0.
Power set algebra
Example 3. The power set algebra, the set 2W of all subsets of a given set W.[10] This is just Example 2 in disguise, with W serving to index the bit positions. Any subset X of W can be viewed as the bit vector having 1's in just those bit positions indexed by elements of X. Thus the all-zero vector is the empty subset of W while the all-ones vector is W itself, these being the constants 0 and 1 respectively of the power set algebra. The counterpart of disjunction x∨y is union X∪Y, while that of conjunction x∧y is intersection X∩Y. Negation ¬x becomes ~X, complement relative to W. There is also set difference X\Y = X∩~Y, symmetric difference (X\Y)∪(Y\X), ternary union X∪Y∪Z, and so on. The atoms here are the singletons, those subsets with exactly one element.
Examples 2 and 3 are special cases of a general construct of algebra called direct product, applicable not just to Boolean algebras but all kinds of algebra including groups, rings, etc. The direct product of any family Bi of Boolean algebras where i ranges over some index set I (not necessarily finite or even countable) is a Boolean algebra consisting of all I-tuples (...xi,...) whose i-th element is taken from Bi. The operations of a direct product are the corresponding operations of the constituent algebras acting within their respective coordinates; in particular operation nfj of the product operates on n I-tuples by applying operation nfj of Bi to the n elements in the i-th coordinate of the n tuples, for all i in I.
When all the algebras being multiplied together in this way are the same algebra A we call the direct product a direct power of A. The Boolean algebra of all 32-bit bit vectors is the two-element Boolean algebra raised to the 32nd power, or power set algebra of a 32-element set, denoted 232. The Boolean algebra of all sets of integers is 2Z. All Boolean algebras we have exhibited thus far have been direct powers of the two-element Boolean algebra, justifying the name "power set algebra".
Representation theorems
It can be shown that every finite Boolean algebra is isomorphic to some power set algebra.[11] Hence the cardinality (number of elements) of a finite Boolean algebra is a power of 2, namely one of 1,2,4,8,...,2n,... This is called a representation theorem as it gives insight into the nature of finite Boolean algebras by giving a representation of them as power set algebras.
This representation theorem does not extend to infinite Boolean algebras: although every power set algebra is a Boolean algebra, not every Boolean algebra need be isomorphic to a power set algebra. In particular, whereas there can be no countably infinite power set algebras (the smallest infinite power set algebra is the power set algebra 2N of sets of natural numbers, shown by Cantor to be uncountable), there exist various countably infinite Boolean algebras.
To go beyond power set algebras we need another construct. A subalgebra of an algebra A is any subset of A closed under the operations of A. Every subalgebra of a Boolean algebra A must still satisfy the equations holding of A, since any violation would constitute a violation for A itself. Hence every subalgebra of a Boolean algebra is a Boolean algebra.[12]
A subalgebra of a power set algebra is called a field of sets; equivalently a field of sets is a set of subsets of some set W including the empty set and W and closed under finite union and complement with respect to W (and hence also under finite intersection). Birkhoff's [1935] representation theorem for Boolean algebras states that every Boolean algebra is isomorphic to a field of sets. Now Birkhoff's HSP theorem for varieties can be stated as, every class of models of the equational theory of a class C of algebras is the Homomorphic image of a Subalgebra of a direct Product of algebras of C. Normally all three of H, S, and P are needed; what the first of these two Birkhoff theorems shows is that for the special case of the variety of Boolean algebras Homomorphism can be replaced by Isomorphism. Birkhoff's HSP theorem for varieties in general therefore becomes Birkhoff's ISP theorem for the variety of Boolean algebras.
Other examples
It is convenient when talking about a set X of natural numbers to view it as a sequence x0,x1,x2,... of bits, with xi = 1 if and only if i ∈ X. This viewpoint will make it easier to talk about subalgebras of the power set algebra 2N, which this viewpoint makes the Boolean algebra of all sequences of bits.[13] It also fits well with the columns of a truth table: when a column is read from top to bottom it constitutes a sequence of bits, but at the same time it can be viewed as the set of those valuations (assignments to variables in the left half of the table) at which the function represented by that column evaluates to 1.
Example 4. Ultimately constant sequences. Any Boolean combination of ultimately constant sequences is ultimately constant; hence these form a Boolean algebra. We can identify these with the integers by viewing the ultimately-zero sequences as nonnegative binary numerals (bit 0 of the sequence being the low-order bit) and the ultimately-one sequences as negative binary numerals (think two's complement arithmetic with the all-ones sequence being −1). This makes the integers a Boolean algebra, with union being bit-wise OR and complement being −x−1. There are only countably many integers, so this infinite Boolean algebra is countable. The atoms are the powers of two, namely 1,2,4,.... Another way of describing this algebra is as the set of all finite and cofinite sets of natural numbers, with the ultimately all-ones sequences corresponding to the cofinite sets, those sets omitting only finitely many natural numbers.
Example 5. Periodic sequence. A sequence is called periodic when there exists some number n > 0, called a witness to periodicity, such that xi = xi+n for all i ≥ 0. The period of a periodic sequence is its least witness. Negation leaves period unchanged, while the disjunction of two periodic sequences is periodic, with period at most the least common multiple of the periods of the two arguments (the period can be as small as 1, as happens with the union of any sequence and its complement). Hence the periodic sequences form a Boolean algebra.
Example 5 resembles Example 4 in being countable, but differs in being atomless. The latter is because the conjunction of any nonzero periodic sequence x with a sequence of coprime period (greater than 1) is neither 0 nor x. It can be shown that all countably infinite atomless Boolean algebras are isomorphic, that is, up to isomorphism there is only one such algebra.
Example 6. Periodic sequence with period a power of two. This is a proper subalgebra of Example 5 (a proper subalgebra equals the intersection of itself with its algebra). These can be understood as the finitary operations, with the first period of such a sequence giving the truth table of the operation it represents. For example, the truth table of x0 in the table of binary operations, namely 2f10, has period 2 (and so can be recognized as using only the first variable) even though 12 of the binary operations have period 4. When the period is 2n the operation only depends on the first n variables, the sense in which the operation is finitary. This example is also a countably infinite atomless Boolean algebra. Hence Example 5 is isomorphic to a proper subalgebra of itself! Example 6, and hence Example 5, constitutes the free Boolean algebra on countably many generators, meaning the Boolean algebra of all finitary operations on a countably infinite set of generators or variables.
Example 7. Ultimately periodic sequences, sequences that become periodic after an initial finite bout of lawlessness. They constitute a proper extension of Example 5 (meaning that Example 5 is a proper subalgebra of Example 7) and also of Example 4, since constant sequences are periodic with period one. Sequences may vary as to when they settle down, but any finite set of sequences will all eventually settle down no later than their slowest-to-settle member, whence ultimately periodic sequences are closed under all Boolean operations and so form a Boolean algebra. This example has the same atoms and coatoms as Example 4, whence it is not atomless and therefore not isomorphic to Example 5/6. However it contains an infinite atomless subalgebra, namely Example 5, and so is not isomorphic to Example 4, every subalgebra of which must be a Boolean algebra of finite sets and their complements and therefore atomic. This example is isomorphic to the direct product of Examples 4 and 5, furnishing another description of it.
Example 8. The direct product of a Periodic Sequence (Example 5) with any finite but nontrivial Boolean algebra. (The trivial one-element Boolean algebra is the unique finite atomless Boolean algebra.) This resembles Example 7 in having both atoms and an atomless subalgebra, but differs in having only finitely many atoms. Example 8 is in fact an infinite family of examples, one for each possible finite number of atoms.
These examples by no means exhaust the possible Boolean algebras, even the countable ones. Indeed, there are uncountably many nonisomorphic countable Boolean algebras, which Jussi Ketonen [1978] classified completely in terms of invariants representable by certain hereditarily countable sets.
Boolean algebras of Boolean operations
The n-ary Boolean operations themselves constitute a power set algebra 2W, namely when W is taken to be the set of 2n valuations of the n inputs. In terms of the naming system of operations nfi where i in binary is a column of a truth table, the columns can be combined with Boolean operations of any arity to produce other columns present in the table. That is, we can apply any Boolean operation of arity m to m Boolean operations of arity n to yield a Boolean operation of arity n, for any m and n.
The practical significance of this convention for both software and hardware is that n-ary Boolean operations can be represented as words of the appropriate length. For example, each of the 256 ternary Boolean operations can be represented as an unsigned byte. The available logical operations such as AND and OR can then be used to form new operations. If we take x, y, and z (dispensing with subscripted variables for now) to be 10101010, 11001100, and 11110000 respectively (170, 204, and 240 in decimal, 0xaa, 0xcc, and 0xf0 in hexadecimal), their pairwise conjunctions are x∧y = 10001000, y∧z = 11000000, and z∧x = 10100000, while their pairwise disjunctions are x∨y = 11101110, y∨z = 11111100, and z∨x = 11111010. The disjunction of the three conjunctions is 11101000, which also happens to be the conjunction of three disjunctions. We have thus calculated, with a dozen or so logical operations on bytes, that the two ternary operations
$(x\land y)\lor (y\land z)\lor (z\land x)$
and
$(x\lor y)\land (y\lor z)\land (z\lor x)$
are actually the same operation. That is, we have proved the equational identity
$(x\land y)\lor (y\land z)\lor (z\land x)=(x\lor y)\land (y\lor z)\land (z\lor x)$,
for the two-element Boolean algebra. By the definition of "Boolean algebra" this identity must therefore hold in every Boolean algebra.
This ternary operation incidentally formed the basis for Grau's [1947] ternary Boolean algebras, which he axiomatized in terms of this operation and negation. The operation is symmetric, meaning that its value is independent of any of the 3! = 6 permutations of its arguments. The two halves of its truth table 11101000 are the truth tables for ∨, 1110, and ∧, 1000, so the operation can be phrased as if z then x∨y else x∧y. Since it is symmetric it can equally well be phrased as either of if x then y∨z else y∧z, or if y then z∨x else z∧x. Viewed as a labeling of the 8-vertex 3-cube, the upper half is labeled 1 and the lower half 0; for this reason it has been called the median operator, with the evident generalization to any odd number of variables (odd in order to avoid the tie when exactly half the variables are 0).
Axiomatizing Boolean algebras
The technique we just used to prove an identity of Boolean algebra can be generalized to all identities in a systematic way that can be taken as a sound and complete axiomatization of, or axiomatic system for, the equational laws of Boolean logic. The customary formulation of an axiom system consists of a set of axioms that "prime the pump" with some initial identities, along with a set of inference rules for inferring the remaining identities from the axioms and previously proved identities. In principle it is desirable to have finitely many axioms; however as a practical matter it is not necessary since it is just as effective to have a finite axiom schema having infinitely many instances each of which when used in a proof can readily be verified to be a legal instance, the approach we follow here.
Boolean identities are assertions of the form s = t where s and t are n-ary terms, by which we shall mean here terms whose variables are limited to x0 through xn-1. An n-ary term is either an atom or an application. An application mfi(t0,...,tm-1) is a pair consisting of an m-ary operation mfi and a list or m-tuple (t0,...,tm-1) of m n-ary terms called operands.
Associated with every term is a natural number called its height. Atoms are of zero height, while applications are of height one plus the height of their highest operand.
Now what is an atom? Conventionally an atom is either a constant (0 or 1) or a variable xi where 0 ≤ i < n. For the proof technique here it is convenient to define atoms instead to be n-ary operations nfi, which although treated here as atoms nevertheless mean the same as ordinary terms of the exact form nfi(x0,...,xn-1) (exact in that the variables must listed in the order shown without repetition or omission). This is not a restriction because atoms of this form include all the ordinary atoms, namely the constants 0 and 1, which arise here as the n-ary operations nf0 and nf−1 for each n (abbreviating 22n−1 to −1), and the variables x0,...,xn-1 as can be seen from the truth tables where x0 appears as both the unary operation 1f2 and the binary operation 2f10 while x1 appears as 2f12.
The following axiom schema and three inference rules axiomatize the Boolean algebra of n-ary terms.
A1. mfi(nfj0,...,nfjm-1) = nfioĵ where (ioĵ)v = iĵv, with ĵ being j transpose, defined by (ĵv)u = (ju)v.
R1. With no premises infer t = t.
R2. From s = u and t = u infer s = t where s, t, and u are n-ary terms.
R3. From s0 = t0 , ... , sm-1 = tm-1 infer mfi(s0,...,sm-1) = mfi(t0,...,tm-1), where all terms si, ti are n-ary.
The meaning of the side condition on A1 is that ioĵ is that 2n-bit number whose v-th bit is the ĵv-th bit of i, where the ranges of each quantity are u: m, v: 2n, ju: 22n, and ĵv: 2m. (So j is an m-tuple of 2n-bit numbers while ĵ as the transpose of j is a 2n-tuple of m-bit numbers. Both j and ĵ therefore contain m2n bits.)
A1 is an axiom schema rather than an axiom by virtue of containing metavariables, namely m, i, n, and j0 through jm-1. The actual axioms of the axiomatization are obtained by setting the metavariables to specific values. For example, if we take m = n = i = j0 = 1, we can compute the two bits of ioĵ from i1 = 0 and i0 = 1, so ioĵ = 2 (or 10 when written as a two-bit number). The resulting instance, namely 1f1(1f1) = 1f2, expresses the familiar axiom ¬¬x = x of double negation. Rule R3 then allows us to infer ¬¬¬x = ¬x by taking s0 to be 1f1(1f1) or ¬¬x0, t0 to be 1f2 or x0, and mfi to be 1f1 or ¬.
For each m and n there are only finitely many axioms instantiating A1, namely 22m × (22n)m. Each instance is specified by 2m+m2n bits.
We treat R1 as an inference rule, even though it is like an axiom in having no premises, because it is a domain-independent rule along with R2 and R3 common to all equational axiomatizations, whether of groups, rings, or any other variety. The only entity specific to Boolean algebras is axiom schema A1. In this way when talking about different equational theories we can push the rules to one side as being independent of the particular theories, and confine attention to the axioms as the only part of the axiom system characterizing the particular equational theory at hand.
This axiomatization is complete, meaning that every Boolean law s = t is provable in this system. One first shows by induction on the height of s that every Boolean law for which t is atomic is provable, using R1 for the base case (since distinct atoms are never equal) and A1 and R3 for the induction step (s an application). This proof strategy amounts to a recursive procedure for evaluating s to yield an atom. Then to prove s = t in the general case when t may be an application, use the fact that if s = t is an identity then s and t must evaluate to the same atom, call it u. So first prove s = u and t = u as above, that is, evaluate s and t using A1, R1, and R3, and then invoke R2 to infer s = t.
In A1, if we view the number nm as the function type m→n, and mn as the application m(n), we can reinterpret the numbers i, j, ĵ, and ioĵ as functions of type i: (m→2)→2, j: m→((n→2)→2), ĵ: (n→2)→(m→2), and ioĵ: (n→2)→2. The definition (ioĵ)v = iĵv in A1 then translates to (ioĵ)(v) = i(ĵ(v)), that is, ioĵ is defined to be composition of i and ĵ understood as functions. So the content of A1 amounts to defining term application to be essentially composition, modulo the need to transpose the m-tuple j to make the types match up suitably for composition. This composition is the one in Lawvere's previously mentioned category of power sets and their functions. In this way we have translated the commuting diagrams of that category, as the equational theory of Boolean algebras, into the equational consequences of A1 as the logical representation of that particular composition law.
Underlying lattice structure
Underlying every Boolean algebra B is a partially ordered set or poset (B,≤). The partial order relation is defined by x ≤ y just when x = x∧y, or equivalently when y = x∨y. Given a set X of elements of a Boolean algebra, an upper bound on X is an element y such that for every element x of X, x ≤ y, while a lower bound on X is an element y such that for every element x of X, y ≤ x.
A sup of X is a least upper bound on X, namely an upper bound on X that is less or equal to every upper bound on X. Dually an (inf) of X is a greatest lower bound on X. The sup of x and y always exists in the underlying poset of a Boolean algebra, being x∨y, and likewise their inf exists, namely x∧y. The empty sup is 0 (the bottom element) and the empty inf is 1 (top). It follows that every finite set has both a sup and an inf. Infinite subsets of a Boolean algebra may or may not have a sup and/or an inf; in a power set algebra they always do.
Any poset (B,≤) such that every pair x,y of elements has both a sup and an inf is called a lattice. We write x∨y for the sup and x∧y for the inf. The underlying poset of a Boolean algebra always forms a lattice. The lattice is said to be distributive when x∧(y∨z) = (x∧y)∨(x∧z), or equivalently when x∨(y∧z) = (x∨y)∧(x∨z), since either law implies the other in a lattice. These are laws of Boolean algebra whence the underlying poset of a Boolean algebra forms a distributive lattice.
Given a lattice with a bottom element 0 and a top element 1, a pair x,y of elements is called complementary when x∧y = 0 and x∨y = 1, and we then say that y is a complement of x and vice versa. Any element x of a distributive lattice with top and bottom can have at most one complement. When every element of a lattice has a complement the lattice is called complemented. It follows that in a complemented distributive lattice, the complement of an element always exists and is unique, making complement a unary operation. Furthermore, every complemented distributive lattice forms a Boolean algebra, and conversely every Boolean algebra forms a complemented distributive lattice. This provides an alternative definition of a Boolean algebra, namely as any complemented distributive lattice. Each of these three properties can be axiomatized with finitely many equations, whence these equations taken together constitute a finite axiomatization of the equational theory of Boolean algebras.
In a class of algebras defined as all the models of a set of equations, it is usually the case that some algebras of the class satisfy more equations than just those needed to qualify them for the class. The class of Boolean algebras is unusual in that, with a single exception, every Boolean algebra satisfies exactly the Boolean identities and no more. The exception is the one-element Boolean algebra, which necessarily satisfies every equation, even x = y, and is therefore sometimes referred to as the inconsistent Boolean algebra.
Boolean homomorphisms
A Boolean homomorphism is a function h: A→B between Boolean algebras A,B such that for every Boolean operation mfi:
$h(^{m}\!f_{i}(x_{0},...,x_{m-1}))={}^{m}\!f_{i}(h(x_{0},...,x_{m-1}))$
The category Bool of Boolean algebras has as objects all Boolean algebras and as morphisms the Boolean homomorphisms between them.
There exists a unique homomorphism from the two-element Boolean algebra 2 to every Boolean algebra, since homomorphisms must preserve the two constants and those are the only elements of 2. A Boolean algebra with this property is called an initial Boolean algebra. It can be shown that any two initial Boolean algebras are isomorphic, so up to isomorphism 2 is the initial Boolean algebra.
In the other direction, there may exist many homomorphisms from a Boolean algebra B to 2. Any such homomorphism partitions B into those elements mapped to 1 and those to 0. The subset of B consisting of the former is called an ultrafilter of B. When B is finite its ultrafilters pair up with its atoms; one atom is mapped to 1 and the rest to 0. Each ultrafilter of B thus consists of an atom of B and all the elements above it; hence exactly half the elements of B are in the ultrafilter, and there as many ultrafilters as atoms.
For infinite Boolean algebras the notion of ultrafilter becomes considerably more delicate. The elements greater than or equal to an atom always form an ultrafilter, but so do many other sets; for example, in the Boolean algebra of finite and cofinite sets of integers, the cofinite sets form an ultrafilter even though none of them are atoms. Likewise, the powerset of the integers has among its ultrafilters the set of all subsets containing a given integer; there are countably many of these "standard" ultrafilters, which may be identified with the integers themselves, but there are uncountably many more "nonstandard" ultrafilters. These form the basis for nonstandard analysis, providing representations for such classically inconsistent objects as infinitesimals and delta functions.
Infinitary extensions
Recall the definition of sup and inf from the section above on the underlying partial order of a Boolean algebra. A complete Boolean algebra is one every subset of which has both a sup and an inf, even the infinite subsets. Gaifman [1964] and Hales [1964] independently showed that infinite free complete Boolean algebras do not exist. This suggests that a logic with set-sized-infinitary operations may have class-many terms—just as a logic with finitary operations may have infinitely many terms.
There is however another approach to introducing infinitary Boolean operations: simply drop "finitary" from the definition of Boolean algebra. A model of the equational theory of the algebra of all operations on {0,1} of arity up to the cardinality of the model is called a complete atomic Boolean algebra, or CABA. (In place of this awkward restriction on arity we could allow any arity, leading to a different awkwardness, that the signature would then be larger than any set, that is, a proper class. One benefit of the latter approach is that it simplifies the definition of homomorphism between CABAs of different cardinality.) Such an algebra can be defined equivalently as a complete Boolean algebra that is atomic, meaning that every element is a sup of some set of atoms. Free CABAs exist for all cardinalities of a set V of generators, namely the power set algebra 22V, this being the obvious generalization of the finite free Boolean algebras. This neatly rescues infinitary Boolean logic from the fate the Gaifman–Hales result seemed to consign it to.
The nonexistence of free complete Boolean algebras can be traced to failure to extend the equations of Boolean logic suitably to all laws that should hold for infinitary conjunction and disjunction, in particular the neglect of distributivity in the definition of complete Boolean algebra. A complete Boolean algebra is called completely distributive when arbitrary conjunctions distribute over arbitrary disjunctions and vice versa. A Boolean algebra is a CABA if and only if it is complete and completely distributive, giving a third definition of CABA. A fourth definition is as any Boolean algebra isomorphic to a power set algebra.
A complete homomorphism is one that preserves all sups that exist, not just the finite sups, and likewise for infs. The category CABA of all CABAs and their complete homomorphisms is dual to the category of sets and their functions, meaning that it is equivalent to the opposite of that category (the category resulting from reversing all morphisms). Things are not so simple for the category Bool of Boolean algebras and their homomorphisms, which Marshall Stone showed in effect (though he lacked both the language and the conceptual framework to make the duality explicit) to be dual to the category of totally disconnected compact Hausdorff spaces, subsequently called Stone spaces.
Another infinitary class intermediate between Boolean algebras and complete Boolean algebras is the notion of a sigma-algebra. This is defined analogously to complete Boolean algebras, but with sups and infs limited to countable arity. That is, a sigma-algebra is a Boolean algebra with all countable sups and infs. Because the sups and infs are of bounded cardinality, unlike the situation with complete Boolean algebras, the Gaifman-Hales result does not apply and free sigma-algebras do exist. Unlike the situation with CABAs however, the free countably generated sigma algebra is not a power set algebra.
Other definitions of Boolean algebra
We have already encountered several definitions of Boolean algebra, as a model of the equational theory of the two-element algebra, as a complemented distributive lattice, as a Boolean ring, and as a product-preserving functor from a certain category (Lawvere). Two more definitions worth mentioning are:.
Stone (1936)
A Boolean algebra is the set of all clopen sets of a topological space. It is no limitation to require the space to be a totally disconnected compact Hausdorff space, or Stone space, that is, every Boolean algebra arises in this way, up to isomorphism. Moreover, if the two Boolean algebras formed as the clopen sets of two Stone spaces are isomorphic, so are the Stone spaces themselves, which is not the case for arbitrary topological spaces. This is just the reverse direction of the duality mentioned earlier from Boolean algebras to Stone spaces. This definition is fleshed out by the next definition.
Johnstone (1982)
A Boolean algebra is a filtered colimit of finite Boolean algebras.
(The circularity in this definition can be removed by replacing "finite Boolean algebra" by "finite power set" equipped with the Boolean operations standardly interpreted for power sets.)
To put this in perspective, infinite sets arise as filtered colimits of finite sets, infinite CABAs as filtered limits of finite power set algebras, and infinite Stone spaces as filtered limits of finite sets. Thus if one starts with the finite sets and asks how these generalize to infinite objects, there are two ways: "adding" them gives ordinary or inductive sets while "multiplying" them gives Stone spaces or profinite sets. The same choice exists for finite power set algebras as the duals of finite sets: addition yields Boolean algebras as inductive objects while multiplication yields CABAs or power set algebras as profinite objects.
A characteristic distinguishing feature is that the underlying topology of objects so constructed, when defined so as to be Hausdorff, is discrete for inductive objects and compact for profinite objects. The topology of finite Hausdorff spaces is always both discrete and compact, whereas for infinite spaces "discrete"' and "compact" are mutually exclusive. Thus when generalizing finite algebras (of any kind, not just Boolean) to infinite ones, "discrete" and "compact" part company, and one must choose which one to retain. The general rule, for both finite and infinite algebras, is that finitary algebras are discrete, whereas their duals are compact and feature infinitary operations. Between these two extremes, there are many intermediate infinite Boolean algebras whose topology is neither discrete nor compact.
See also
• Boolean domain
• Boolean function
• Boolean-valued function
• Boolean-valued model
• Cartesian closed category
• Closed monoidal category
• Complete Boolean algebra
• Elementary topos
• Field of sets
• Filter (mathematics)
• Finitary boolean function
• Free Boolean algebra
• Functional completeness
• Ideal (order theory)
• Lattice (order)
• Lindenbaum–Tarski algebra
• List of Boolean algebra topics
• Monoidal category
• Propositional calculus
• Robbins algebra
• Truth table
• Ultrafilter
• Universal algebra
References
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| Wikipedia |
Yetter–Drinfeld category
In mathematics a Yetter–Drinfeld category is a special type of braided monoidal category. It consists of modules over a Hopf algebra which satisfy some additional axioms.
Definition
Let H be a Hopf algebra over a field k. Let $\Delta $ denote the coproduct and S the antipode of H. Let V be a vector space over k. Then V is called a (left left) Yetter–Drinfeld module over H if
• $(V,{\boldsymbol {.}})$ is a left H-module, where ${\boldsymbol {.}}:H\otimes V\to V$ denotes the left action of H on V,
• $(V,\delta \;)$ is a left H-comodule, where $\delta :V\to H\otimes V$ denotes the left coaction of H on V,
• the maps ${\boldsymbol {.}}$ and $\delta $ satisfy the compatibility condition
$\delta (h{\boldsymbol {.}}v)=h_{(1)}v_{(-1)}S(h_{(3)})\otimes h_{(2)}{\boldsymbol {.}}v_{(0)}$ for all $h\in H,v\in V$,
where, using Sweedler notation, $(\Delta \otimes \mathrm {id} )\Delta (h)=h_{(1)}\otimes h_{(2)}\otimes h_{(3)}\in H\otimes H\otimes H$ denotes the twofold coproduct of $h\in H$, and $\delta (v)=v_{(-1)}\otimes v_{(0)}$.
Examples
• Any left H-module over a cocommutative Hopf algebra H is a Yetter–Drinfeld module with the trivial left coaction $\delta (v)=1\otimes v$.
• The trivial module $V=k\{v\}$ with $h{\boldsymbol {.}}v=\epsilon (h)v$, $\delta (v)=1\otimes v$, is a Yetter–Drinfeld module for all Hopf algebras H.
• If H is the group algebra kG of an abelian group G, then Yetter–Drinfeld modules over H are precisely the G-graded G-modules. This means that
$V=\bigoplus _{g\in G}V_{g}$,
where each $V_{g}$ is a G-submodule of V.
• More generally, if the group G is not abelian, then Yetter–Drinfeld modules over H=kG are G-modules with a G-gradation
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): V=\bigoplus _{{g\in G}}V_{g} , such that $g.V_{h}\subset V_{ghg^{-1}}$.
• Over the base field $k=\mathbb {C} \;$ all finite-dimensional, irreducible/simple Yetter–Drinfeld modules over a (nonabelian) group H=kG are uniquely given[1] through a conjugacy class $[g]\subset G\;$ together with $\chi ,X\;$ (character of) an irreducible group representation of the centralizer $Cent(g)\;$ of some representing $g\in [g]$:
$V={\mathcal {O}}_{[g]}^{\chi }={\mathcal {O}}_{[g]}^{X}\qquad V=\bigoplus _{h\in [g]}V_{h}=\bigoplus _{h\in [g]}X$
• As G-module take ${\mathcal {O}}_{[g]}^{\chi }$ to be the induced module of $\chi ,X\;$:
$Ind_{Cent(g)}^{G}(\chi )=kG\otimes _{kCent(g)}X$
(this can be proven easily not to depend on the choice of g)
• To define the G-graduation (comodule) assign any element $t\otimes v\in kG\otimes _{kCent(g)}X=V$ to the graduation layer:
$t\otimes v\in V_{tgt^{-1}}$
• It is very custom to directly construct $V\;$ as direct sum of X´s and write down the G-action by choice of a specific set of representatives $t_{i}\;$ for the $Cent(g)\;$-cosets. From this approach, one often writes
$h\otimes v\subset [g]\times X\;\;\leftrightarrow \;\;t_{i}\otimes v\in kG\otimes _{kCent(g)}X\qquad {\text{with uniquely}}\;\;h=t_{i}gt_{i}^{-1}$
(this notation emphasizes the graduation$h\otimes v\in V_{h}$, rather than the module structure)
Braiding
Let H be a Hopf algebra with invertible antipode S, and let V, W be Yetter–Drinfeld modules over H. Then the map $c_{V,W}:V\otimes W\to W\otimes V$,
$c(v\otimes w):=v_{(-1)}{\boldsymbol {.}}w\otimes v_{(0)},$
is invertible with inverse
$c_{V,W}^{-1}(w\otimes v):=v_{(0)}\otimes S^{-1}(v_{(-1)}){\boldsymbol {.}}w.$
Further, for any three Yetter–Drinfeld modules U, V, W the map c satisfies the braid relation
$(c_{V,W}\otimes \mathrm {id} _{U})(\mathrm {id} _{V}\otimes c_{U,W})(c_{U,V}\otimes \mathrm {id} _{W})=(\mathrm {id} _{W}\otimes c_{U,V})(c_{U,W}\otimes \mathrm {id} _{V})(\mathrm {id} _{U}\otimes c_{V,W}):U\otimes V\otimes W\to W\otimes V\otimes U.$
A monoidal category ${\mathcal {C}}$ consisting of Yetter–Drinfeld modules over a Hopf algebra H with bijective antipode is called a Yetter–Drinfeld category. It is a braided monoidal category with the braiding c above. The category of Yetter–Drinfeld modules over a Hopf algebra H with bijective antipode is denoted by ${}_{H}^{H}{\mathcal {YD}}$.
References
1. Andruskiewitsch, N.; Grana, M. (1999). "Braided Hopf algebras over non abelian groups". Bol. Acad. Ciencias (Cordoba). 63: 658–691. arXiv:math/9802074. CiteSeerX 10.1.1.237.5330.
• Montgomery, Susan (1993). Hopf algebras and their actions on rings. Regional Conference Series in Mathematics. Vol. 82. Providence, RI: American Mathematical Society. ISBN 0-8218-0738-2. Zbl 0793.16029.
| Wikipedia |
Yevgenii Vasilevich Zolotov
Evgeniy Vasiljevich Zolotov (29 April 1922 – 26 July 1990) was a Soviet mathematician and a member of the Soviet Academy of Sciences (1987).
Evgeniy Zolotov
Born(1922-04-29)29 April 1922
Tula, USSR
Died26 July 1990(1990-07-26) (aged 68)
Khabarovsk, USSR
OccupationMathematician
AwardsOrder of Friendship (1988)
Academic background
Alma materMoscow State University
Thesis (1962)
Biography
Zolotov was born in Tula (USSR) in 1922. He was educated in MSU from 1939–1942 up to conscripted to the F. E. Dzerzhinsky Artillery Academy, which was completed in 1944 with the engineering degree.[1]
After getting his degree he served in the Research Institute of the Academy of Flak Forces, which was relocated from Moscow into Yevpatoria. In 1962 he defended his thesis for his doctor’s dissertation for the first time in the institute. With his work Zolotov could contribute to a great extent to create and develop the anti-rocket aircraft defence of the country. In 1968 he was demobilized as Colonel Engineer.[2]
Zolotov did his scientific work at the Technical University in Kalinin (1968–1970). He established here the Department of "Automatic System of Management" and headed it along the University career.[3]
In 1970 Zolotov was invited to the Far-East Scientific Centre of the Soviet Academy of Sciences to create and develop the physical, mathematical and technical profile of the scientific research institutions in that region. Also in this year he was elected a corresponding member of the Soviet Academy of Sciences. Between 1970 and 1972 he directed the Division for Applied Mathematics at Khabarovsk Research Institute of the Far-East Scientific Centre of the Soviet Academy of Sciences. From 1972 to 1980, Zolotov was Vice-president of the Presidium of the Far-East Scientific Centre of the Soviet Academy of Sciences.
In 1981, Zolotov became the director of the Computing Centre of the Far-East Scientific Centre of the Soviet Academy of Sciences, which was established by him in Khabarovsk. At the same time he was elected a member of the Presidium of the Far-East Scientific Centre of the Soviet Academy of Sciences and also President of the Scientific Council for Physics-Mathematics and Technology of the Far-East Scientific Centre of the Soviet Academy of Sciences.[4] On this position he invited to the Far-East Scientific Centre of the Soviet Academy of Sciences academician Veniamin Myasnikov who has stand the Director of IACP.
In Khabarovsk, Zolotov gathered a special research team around himself from researchers of Eastern medicines, such as Physicians, Biophysicists, system analysts and programmers. From 1986, after he had created a medical laboratory (it was managed by V. A. Jonicevski, Professor of traditional Chinese Medicine, Candidate of Medicine), the scientists of the Computing Centre started to do research in the area of socio-cultural, medical-ecological and historical-geographical processes in the Far-East.
In 1987 Zolotov was elected a Member of the Soviet Academy of Sciences.
He died in 1990 in Moscow and was buried in Tver at Dmitrovo-Cherkassk cemetery.[5]
Main Research Fields
• Computer simulation
• System effectiveness estimation
• Applied Methods of Stochastic process
Awards
He was awarded with 'Peoples Friendship' Order.
Interesting Life facts
• While he was leading the 2nd Research Institute of the Soviet Flak Forces, where the designing office was working, too, Zolotov, together with A. SZ. Popovic, first rank captain and an enthusiastic group, was building a special ship with his own hands. On the Day of the Navy the ship was rushing along the Volga river as fast as an express train.
• Evgeniy Zolotov became interested in paranormal abilities and had been experimenting with telepathy. He even authored a draft of a university textbook titled Telepathy".[6]
Family
• Wife - Gogol Varvara Dmitrijevna
• Daughter - Olga Evegniyevna Zolotova
• Son - Boris Evegniyevich Zolotov, Ph.D.
Memory
Annual Physics-mathematician young scientists Seminar in Russian Far-East is dedicated to E.V.Zolotov.[7]
References
1. "Золотов Е.В. - Общая информация" [E.V. Zolotov - General information]. www.ras.ru (in Russian).
2. "News : Ministry of Defence of the Russian Federation".
3. "Кафедра Автоматизации технологических процессов" [Department of Automation of Technological Processes] (in Russian). Tver State Technical University (TSTU). Retrieved 5 January 2021 – via www.tstu.tver.ru.
4. "Academician Evgeni Zolotov".
5. "ЗОЛОТОВ Евгений Васильевич (1922-90)" [ZOLOTOV Evgeny Vasilievich (1922-90)] (in Russian). Retrieved 5 January 2021.
6. "Евгений васильевич золотов скачать книгу телепатия" [Evgeny vasilyevich zolotov download the book telepathy] (in Russian). Archived from the original on 21 May 2016.
7. "XXXVIII Дальневосточная Математическая Школа-Семинар имени академика Е.В. Золотова, 2014" [XXXVIII Far Eastern Mathematical School-Seminar named after academician E.V. Zolotova, 2014] (in Russian). Archived from the original on 24 September 2015.
Links
• Evgeniy Vasiljevich Zolotov, Academician, on official website of the Far-East Division of the Russian Academy of Sciences
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| Wikipedia |
Yff center of congruence
In geometry, the Yff center of congruence is a special point associated with a triangle. This special point is a triangle center and Peter Yff initiated the study of this triangle center in 1987.[1]
Isoscelizer
An isoscelizer of an angle A in a triangle ABC is a line through points P1 and Q1, where P1 lies on AB and Q1 on AC, such that the triangle AP1Q1 is an isosceles triangle. An isoscelizer of angle A is a line perpendicular to the bisector of angle A. Isoscelizers were invented by Peter Yff in 1963.[2]
Yff central triangle
Let ABC be any triangle. Let P1Q1 be an isoscelizer of angle A, P2Q2 be an isoscelizer of angle B, and P3Q3 be an isoscelizer of angle C. Let A'B'C' be the triangle formed by the three isoscelizers. The four triangles A'P2Q3, Q1B'P3, P1Q2C', and A'B'C' are always similar.
There is a unique set of three isoscelizers P1Q1, P2Q2, P3Q3 such that the four triangles A'P2Q3, Q1B'P3, P1Q2C', and A'B'C' are congruent. In this special case the triangle A'B'C' formed by the three isoscelizers is called the Yff central triangle of triangle ABC.[3]
The circumcircle of the Yff central triangle is called the Yff central circle of the triangle.
Yff center of congruence
Let ABC be any triangle. Let P1Q1, P2Q2, P3Q3 be the isoscelizers of the angles A, B, C such that the triangle A'B'C' formed by them is the Yff central triangle of triangle ABC. The three isoscelizers P1Q1, P2Q2, P3Q3 are continuously parallel-shifted such that the three triangles A'P2Q3, Q1B'P3, P1Q2C' are always congruent to each other until the triangle A'B'C' formed by the intersections of the isoscelizers reduces to a point. The point to which the triangle A'B'C' reduces to is called the Yff center of congruence of triangle ABC.
Properties
• The trilinear coordinates of the Yff center of congruence are ( sec( A/2 ) : sec ( B/2 ), sec ( C/2 ).[1]
• Any triangle ABC is the triangle formed by the lines which are externally tangent to the three excircles of the Yff central triangle of triangle ABC.
• Let I be the incenter of triangle ABC. Let D be the point on side BC such that ∠BID = ∠DIC, E a point on side CA such that ∠CIE = ∠EIA, and F a point on side AB such that ∠AIF = ∠FIB. Then the lines AD. BE, and CF are concurrent at the Yff center of congruence. This fact gives a geometrical construction for locating the Yff center of congruence.[4]
• A computer assisted search of the properties of the Yff central triangle has generated several interesting results relating to properties of the Yff central triangle.[5]
Generalization
The geometrical construction for locating the Yff center of congruence has an interesting generalization. The generalisation begins with an arbitrary point P in the plane of a triangle ABC. Then points D, E, F are taken on the sides BC, CA, AB such that ∠BPD = ∠DPC, ∠CPE = ∠EPA, and ∠APF = ∠FPB. The generalization asserts that the lines AD, BE, CF are concurrent.[4]
See also
• Congruent isoscelizers point
• Central triangle
References
1. Kimberling, Clark. "Yff Center of Congruence". Retrieved 30 May 2012.
2. Weisstein, Eric W. "Isoscelizer". MathWorld--A Wolfram Web Resource. Retrieved 30 May 2012.
3. Weisstein, Eric W. "Yff central triangle". MathWorld--A Wolfram Web Resource. Retrieved 30 May 2012.
4. Kimberling, Clark. "X(174) = Yff Center of Congruence". Retrieved 2 June 2012.
5. Dekov, Deko (2007). "Yff Center of Congruence". Journal of Computer-Generated Euclidean Geometry. 37: 1–5. Retrieved 30 May 2012.
| Wikipedia |
Yiannis N. Moschovakis
Yiannis Nicholas Moschovakis (Greek: Γιάννης Μοσχοβάκης; born January 18, 1938) is a set theorist, descriptive set theorist, and recursion (computability) theorist, at UCLA.
Yiannis N. Moschovakis
Born
Yiannis Nicholas Moschovakis
(1938-01-18) January 18, 1938
Athens, Greece
Alma materUniversity of Wisconsin–Madison
Known forEffective descriptive set theory
Scientific career
FieldsMathematics
InstitutionsUCLA
Doctoral advisorStephen Kleene
Doctoral studentsAlexander S. Kechris
Phokion G. Kolaitis
His book Descriptive Set Theory (North-Holland) is the primary reference for the subject. He is especially associated with the development of the effective, or lightface, version of descriptive set theory, and he is known for the Moschovakis coding lemma that is named after him.
Biography
Moschovakis earned his Ph.D. from the University of Wisconsin–Madison in 1963 under the direction of Stephen Kleene, with a dissertation entitled Recursive Analysis. In 2015, he was elected as a fellow of the American Mathematical Society "for contributions to mathematical logic, especially set theory and computability theory, and for exposition".[1]
For many years, he has split his time between UCLA and the University of Athens (he retired from the latter in July 2005).
Moschovakis is married to Joan Moschovakis, with whom he gave the 2014 Lindström Lectures at the University of Gothenburg.[2]
Publications
• Elementary induction on abstract structures. North-Holland. 1974.[3] 2nd edn. Dover. 2008. ISBN 9780486152011.
• Descriptive set theory. North-Holland. 1980. ISBN 9780444853059.[4] 2nd edn. 2005. Second edition available online
• Notes on set theory. North-Holland. 1994. ISBN 9783540941804. 2nd edn. 2005.
References
1. 2016 Class of the Fellows of the AMS, American Mathematical Society, retrieved 2015-11-16.
2. "The Lindström Lectures - Department of Philosophy, Linguistics and Theory of Science, University of Gothenburg, Sweden". flov.gu.se. Archived from the original on 2013-11-11.
3. Barwise, K. Jon (1975). "Review: Elementary induction on abstract structures, by Y. Moschovakis". Bull. Amer. Math. Soc. 81 (6): 1031–1035. doi:10.1090/s0002-9904-1975-13893-6.
4. Jech, Thomas (1981). "Review: Descriptive set theory, by Y. Moschovakis". Bull. Amer. Math. Soc. (N.S.). 5 (3): 339–349. doi:10.1090/s0273-0979-1981-14952-1.
External links
• Home page
• Yiannis N. Moschovakis at the Mathematics Genealogy Project
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| Wikipedia |
Yifeng Liu
Yifeng Liu (born July 19, 1985 in Shanghai, China) is a Chinese professor of mathematics at Zhejiang University specializing in number theory, automorphic forms and arithmetic geometry.[1]
Yifeng Liu
Born (1985-07-19) July 19, 1985
Shanghai, China
NationalityChinese
Alma materColumbia University
Peking University
Awards
• Sloan Research Fellowship (2017)
• SASTRA Ramanujan Prize (2018)
Scientific career
FieldsMathematics
Institutions
• Yale University
• Northwestern University
• Massachusetts Institute of Technology
• Zhejiang University
ThesisArithmetic inner product formula for unitary groups (2012)
Doctoral advisorShou-Wu Zhang
Career
Liu received his BS Degree from Peking University in 2007 and PhD degree from Columbia University, New York, in 2012 under the direction of Shou-Wu Zhang. He was a C.L.E. Moore Instructor at MIT from 2012 to 2015 and an assistant professor at Northwestern University from 2015 to 2018 before being appointed an associate professor at Yale University.[2][3] Liu returned to China in 2021 to join Zheijiang University became a full professor of mathematics.[1]
Liu has made important contributions to arithmetic geometry and number theory. His contributions span a wide spectrum of topics such as arithmetic theta lifts and derivatives of L-functions, the Gan–Gross–Prasad conjecture and its arithmetic counterpart, the Beilinson–Bloch–Kato conjecture, the geometric Langlands program, the p-adic Waldspurger theorem, and the study of étale cohomology on Artin stacks.[2]
Awards
He received a Sloan Research Fellowship in 2017.[2]
He was awarded the 2018 SASTRA Ramanujan Prize for his contributions to the field of mathematics. He shared the prize with Jack Thorne.[4][3]
References
1. "百度安全验证".
2. "Yifeng Liu and Jack Thorne to receive 2018 SASTRA Ramanujan Prize" (PDF). DPMMS News. Department of Pure Mathematics and Mathematical Statistics, Cambridge University. Retrieved 3 February 2019.
3. "Yale, Cambridge profs. get SASTRA-Ramanujan Award". The Hindu. December 22, 2018. Retrieved 3 February 2019.
4. Maeve Forti (25 October 2018). "Yifeng Liu wins prestigious award in mathematics". YaleNews. Yale University. Retrieved 3 February 2019.
External links
• Yifeng Liu's personal homepage
Recipients of SASTRA Ramanujan Prize
• Manjul Bhargava (2005)
• Kannan Soundararajan (2005)
• Terence Tao (2006)
• Ben Green (2007)
• Akshay Venkatesh (2008)
• Kathrin Bringmann (2009)
• Wei Zhang (2010)
• Roman Holowinsky (2011)
• Zhiwei Yun (2012)
• Peter Scholze (2013)
• James Maynard (2014)
• Jacob Tsimerman (2015)
• Kaisa Matomäki (2016)
• Maksym Radziwill (2016)
• Maryna Viazovska (2017)
• Yifeng Liu (2018)
• Jack Thorne (2018)
• Adam Harper (2019)
• Shai Evra (2020)
• Will Sawin (2021)
• Yunqing Tang (2022)
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| Wikipedia |
Yigu yanduan
Yigu yanduan (益古演段 Old Mathematics in Expanded Sections) is a 13th-century mathematical work by Yuan dynasty mathematician Li Zhi.
Overview
Yigu yanduan was based on Northern Song mathematician Jiang Zhou's (蒋周) Yigu Ji (益古集 Collection of Old Mathematics) which is not extant. However, from fragments quoted in Yang Hui's work The Complete Algorithms of Acreage (田亩比类算法大全), this lost mathematical treatise Yigu Ji was about solving area problems with geometry.
Li Zhi used the examples of Yigu Ji to introduce the art of Tian yuan shu to newcomers to this field. Although Li Zhi's previous monograph Ceyuan haijing also used Tian yuan shu, it is harder to understand than Yigu yanduan.
Yigu yanduan was later collected into Siku Quanshu.
Yigu yanduan consists of three volumes with 64 problems solved using Tian yuan sh] in parallel with the geometrical method. Li Zhi intended to introduce students to the art of Tian yuan shu through ancient geometry. Yigu yanduan together with Ceyuan haijing are considered major contributions to Tian yuan shu by Li Zhi. These two works are also considered as the earliest extant documents on Tian yuans shu.
All the 64 problems followed more or less the same format, starting with a question (问), followed by an answer (答曰), a diagram, then an algorithm (术), in which Li Zhi explained step by step how to set up algebra equation with Tian yuan shu, then followed by geometrical interpretation (Tiao duan shu). The order of arrangement of Tian yuan shu equation in Yigu yanduan is the reverse of that in Ceyuan haijing, i.e., here with the constant term at top, followed by first order tian yuan, second order tian yuan, third order tian yuan etc. This later arrangement conformed with contemporary convention of algebra equation( for instance, Qin Jiushao's Mathematical Treatise in Nine Sections), and later became a norm.
Yigu yanduan was first introduced to the English readers by the British Protestant Christian missionary to China, Alexander Wylie who wrote:
Yi koo yen t'wan...written in 1282 consists of 64 geometrical problem, illustrated the principle of Plane Measurement, Evolution and other rules, the whole being developed by means of T'een yuen.[1]
In 1913 Van Hée translated all 64 problems in Yigu yanduan into French.[2]
Volume I
Problem 1 to 22, all about the mathematics of a circle embedded in a square.
Example: problem 8
There is a square field, with a circular pool in the middle, given that the land is 13.75 mu, and the sum of the circumferences of the square field and the circular pool equals to 300 steps, what is the circumferences of the square and circle respective ?
Anwwer: The circumference of the square is 240 steps, the circumference of the circle is 60 steps.
Method: set up tian yuan one (celestial element 1) as the diameter of the circle, x
TAI
multiply it by 3 to get the circumference of the circle 3x (pi ~~3)
TAI
subtract this from the sum of circumferences to obtain the circumference of the square $300-3x$
TAI
The square of it equals to 16 times the area of the square $(300-3x)*(300-3x)=90000-1800x+9x^{2}$
TAI
Again set up tian yuan 1 as the diameter of circle, square it up and multiplied by 12 to get 16 times the area of circle as
TAI
subtract from 16 time square area we have 16 times area of land
TAI
put it at right hand side and put 16 times 13.75 mu = 16 * 13.75 *240 =52800 steps at left, after cancellation, we get $-3x^{2}-1800x+37200=0:$
TAI
Solve this equation to get diameter of circle = 20 steps, circumference of circle = 60 steps
Volume II
Problem 23 to 42, 20 problems in all solving geometry of rectangle embedded in circle with tian yuan shu
Example, problem 35
Suppose we have a circular field with a rectangular water pool in the center, and the distance of a corner to the circumference is 17.5 steps, and the sum of length and width of the pool is 85 steps, what is the diameter of the circle, the length and width of the pool ?
Answer: The diameter of the circle is one hundred steps, the length of pool is 60 steps, and the width 25 steps. Method: Let tian yuan one as the diagonal of rectangle, then the diameter of circle is tian yuan one plus 17.5*2
$x+35$
multiply the square of diameter with $\pi \approx 3$ equals to four times the area of the circle:
$3(x+35)^{2}=3x^{2}+210x+3675$
subtracting four times the area of land to obtain:
four times the area of pool = $3x^{2}+210x+3675-4x6000$= $3x^{2}+210x-20325$
now
The square of the sum of length and width of the pool =85*85 =7225 which is four times the pool area plus the square of the difference of its length and width ($(L-W)^{2}$)
Further double the pool area plus $(L-W)^{2}$ equals to $L^{2}+W^{2}$ = the square of the diagonal of the pool thus
( four time pool area + the square of its dimension difference) - (twice the pool area + square if its dimension difference) equals $7225-x^{2}$ = twice the pool area
so four times the area of pool = $2(7225-x^{2})$
equate this with the four times pool area obtained above
$2(7225-x^{2})$ =$3x^{2}+210x-20325$
we get a quadratic equation $5x^{2}+210x-34775$=0 Solve this equation to get
• diagonal of pool =65 steps
• diameter of circle =65 +2*17.5 =100 steps
• Length - width =35 steps
• Length + width =85 steps
• Length =60 steps
• Width =25 steps
Volume III
Problem 42 to 64, altogether 22 questions about the mathematics of more complex diagrams
Q: fifty-fourth. There is a square field, with a rectangular water pool lying on its diagonal. The area outside the pool is one thousand one hundred fifty paces. Given that from the corners of the field to the straight sides of the pool are fourteen paces and nineteen paces. What is the area of the square field, what is the length and width of the pool?
Answer: The area of the square field is 40 square paces, the length of the pool is thirty five paces, and the width is twenty five paces.
Let the width of the pool be Tianyuan 1.
TAI
Add the width of the pool to twice the distance from field corner to short long side of pool equals to the length of diagonal of the field x+38
TAI
Square it to obtain the area of square with the length of the pool diagonal as its sides
$x^{2}+76x+1444$
TAI
The length of pool minus the width of pool multiplied by 2 = 2 (19-14) = 10
Pool length = pool width +10: x+10
TAI
Pool area = pool with times pool length : x(x+10) =$x^{2}+10x$
TAI
Area of pool times 乘 1.96 ( the square root of 2) =1.4
one has $1.96x^{2}+19.6x$
tai
Area of diagonal square subtract area of pool multiplied 1.96 equals to area of land times 1.96:
$x^{2}+76x+1444$ - $1.96x^{2}+19.6x=$：$-0.96x^{2}+56.4x+1444$
TAI
Occupied plot times 1.96 =1150 * 1.96 =2254=$-0.96x^{2}+56.4x+1444$
hence =$-0.96x^{2}+56.4x-810$:
TAI
Solve this equation and we obtain
width of pool 25 paces therefore pool length =pool width +10 =35 paces length of pool =45 paces
References
1. Alexander Wylie, Notes on Chinese Literature, p117, Shanghai 1902, reprinted by kessinger Publishing
2. van Hée Li Yeh, Mathématicien Chinois du XIIIe siècle, TP,1913,14,537
Further reading
• Yoshio Mikami The Development of Mathematics in China and Japan, p. 81
• Annotated Yigu yanduan by Qing dynasty mathematician Li Rui.
| Wikipedia |
Yingying Fan
Yingying Fan is a Chinese-American statistician and Centennial Chair in Business Administration and Professor in Data Sciences and Operations Department of the Marshall School of Business at the University of Southern California.[1] She also holds joint appointments at the USC Dana and David Dornsife College of Letters, Arts and Sciences, and Keck Medicine of USC. Her contributions to statistics and data science were recognized by the Royal Statistical Society Guy Medal in Bronze in 2017[2] and the Institute of Mathematical Statistics Medallion Lecture in 2023.[3] She was elected Fellow of American Statistical Association in 2019[4] and Fellow of Institute of Mathematical Statistics "for seminal contributions to high-dimensional inference, variable selection, classification, networks, and nonparametric methodology, particularly in the field of financial econometrics, and for conscientious professional service" in 2020.[5]
References
1. "Yingying Fan | USC Marshall". www.marshall.usc.edu.
2. "RSS announces honours for 2017 | StatsLife". www.statslife.org.uk.
3. "Institute of Mathematical Statistics | Honored Special Awards & Lecturers Recipient List".
4. https://www.amstat.org/asa/files/pdfs/2019-ASAFellowAnnouncement.pdf
5. "Institute of Mathematical Statistics | Congratulations to the 2020 IMS Fellows!".
Guy Medallists
Gold Medallists
• Charles Booth (1892)
• Robert Giffen (1894)
• Jervoise Athelstane Baines (1900)
• Francis Ysidro Edgeworth (1907)
• Patrick G. Craigie (1908)
• G. Udny Yule (1911)
• T. H. C. Stevenson (1920)
• A. William Flux (1930)
• A. L. Bowley (1935)
• Major Greenwood (1945)
• R. A. Fisher (1946)
• A. Bradford Hill (1953)
• E. S. Pearson (1955)
• Frank Yates (1960)
• Harold Jeffreys (1962)
• Jerzy Neyman (1966)
• M. G. Kendall (1968)
• M. S. Bartlett (1969)
• Harald Cramér (1972)
• David Cox (1973)
• G. A. Barnard (1975)
• Roy Allen (1978)
• D. G. Kendall (1981)
• Henry Daniels (1984)
• Bernard Benjamin (1986)
• Robin Plackett (1987)
• Peter Armitage (1990)
• George E. P. Box (1993)
• Peter Whittle (1996)
• Michael Healy (1999)
• Dennis Lindley (2002)
• John Nelder (2005)
• James Durbin (2008)
• C. R. Rao (2011)
• John Kingman (2013)
• Bradley Efron (2014)
• Adrian Smith (2016)
• Stephen Buckland (2019)
• David Spiegelhalter (2020)
• Nancy Reid (2022)
Silver Medallists
• John Glover (1893)
• Augustus Sauerbeck (1894)
• A. L. Bowley (1895)
• F. J. Atkinson (1897)
• C. S. Loch (1899)
• Richard Crawford (1900)
• Thomas A. Welton (1901)
• R. H. Hooker (1902)
• Yves Guyot (1903)
• D. A. Thomas (1904)
• R. H. Rew (1905)
• W. H. Shaw (1906)
• N. A. Humphreys (1907)
• Edward Brabrook (1909)
• G. H. Wood (1910)
• R. Dudfield (1913)
• S. Rowson (1914)
• S. J. Chapman (1915)
• J. S. Nicholson (1918)
• J. C. Stamp (1919)
• A. William Flux (1921)
• H. W. Macrosty (1927)
• Ethel Newbold (1928)
• H. E. Soper (1930)
• J. H. Jones (1934)
• Ernest Charles Snow (1935)
• R. G. Hawtrey (1936)
• E. C. Ramsbottom (1938)
• L. Isserlis (1939)
• H. Leak (1940)
• M. G. Kendall (1945)
• Harry Campion (1950)
• F. A. A. Menzler (1951)
• M. S. Bartlett (1952)
• J. O. Irwin (1953)
• L. H. C. Tippett (1954)
• D. G. Kendall (1955)
• Henry Daniels (1957)
• G. A. Barnard (1958)
• E. C. Fieller (1960)
• D. R. Cox (1961)
• P. V. Sukhatme (1962)
• George E. P. Box (1964)
• C. R. Rao (1965)
• Peter Whittle (1966)
• Dennis Lindley (1968)
• Robin Plackett (1973)
• James Durbin (1976)
• John Nelder (1977)
• Peter Armitage (1978)
• Michael Healy (1979)
• M. Stone (1980)
• John Kingman (1981)
• Henry Wynn (1982)
• Julian Besag (1983)
• J. C. Gittins (1984)
• A. Bissell (1985)
• W. Pridmore (1985)
• Richard Peto (1986)
• John Copas (1987)
• John Aitchison (1988)
• F. P. Kelly (1989)
• David Clayton (1990)
• R. L. Smith (1991)
• Robert Nicholas Curnow (1992)
• A. F. M. Smith (1993)
• David Spiegelhalter (1994)
• B. W. Silverman (1995)
• Steffen Lauritzen (1996)
• Peter Diggle (1997)
• Harvey Goldstein (1998)
• Peter Green (1999)
• Walter Gilks (2000)
• Philip Dawid (2001)
• David Hand (2002)
• Kanti Mardia (2003)
• Peter Donnelly (2004)
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• Michael Titterington (2006)
• Howell Tong (2007)
• Gareth Roberts (2008)
• Sylvia Richardson (2009)
• Iain M. Johnstone (2010)
• P. G. Hall (2011)
• David Firth (2012)
• Brian Ripley (2013)
• Jianqing Fan (2014)
• Anthony Davison (2015)
• Nancy Reid (2016)
• Neil Shephard (2017)
• Peter Bühlmann (2018)
• Susan Murphy (2019)
• Arnaud Doucet (2020)
• Håvard Rue (2021)
• Paul Fearnhead (2022)
Bronze Medallists
• William Gemmell Cochran (1936)
• R. F. George (1938)
• W. J. Jennett (1949)
• Peter Armitage (1962)
• James Durbin (1966)
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• Robin Plackett (1968)
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• Peter Green (1987)
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• S. M. Gore (1989)
• Valerie Isham (1990)
• M. G. Kenward (1991)
• C. Jennison (1992)
• Jonathan Tawn (1993)
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• J. N. S. Matthews (1996)
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• D. Firth (1998)
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• Peter Lynn (2003)
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• Steve Brooks (2005)
• Matthew Stephens (2006)
• Paul Fearnhead (2007)
• Fiona Steele (2008)
• Chris Holmes (2009)
• Omiros Papaspiliopoulos (2010)
• Nicolai Meinshausen (2011)
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• Piotr Fryzlewicz (2013)
• Ming Yuan (2014)
• Jinchi Lv (2015)
• Yingying Fan (2017)
• Peng Ding (2018)
• Jonas Peters (2019)
• Rachel McCrea (2020)
• Pierre E. Jacob (2021)
• Rajan Shah (2022)
| Wikipedia |
Yinyu Ye
Yinyu Ye (Chinese: 叶荫宇; pinyin: Yè Yīnyǔ; born 1948) is a Chinese American theoretical computer scientist working on mathematical optimization. He is a specialist in interior point methods, especially in convex minimization and linear programming. He is a professor of Management Science and Engineering and Kwoh-Ting Li Chair Professor of Engineering at Stanford University. He also holds a courtesy appointment in the Department of Electrical Engineering. Ye also is a co-founder of minMax Optimization Inc.
Education
Yinyu Ye was born in 1948 in Wuhan, Hubei, China. He attended Huazhong University of Science and Technology and graduated with a B.S. in Systems and Control in 1982. He received a Ph.D in Engineering Economic Systems from Stanford University in 1988, under the supervision of George B. Dantzig.
Research publications
Ye wrote Interior-Point Algorithms: Theory and Analysis. He joined David Luenberger for the third edition of Luenberger's Linear and Nonlinear Programming.[1]
In recent years, Ye has developed computational methods and theory using semidefinite programming for practical problems like the localization of network sensors. In computational economics, Ye has also established new complexity results for problems concerning the computation of an economic equilibrium.[2]
Awards
Ye was a 2009 co-recipient of the John von Neumann Theory Prize.[2] He was elected to the 2006 class of Fellows of the Institute for Operations Research and the Management Sciences.[3]
Positions
Before joining Stanford University, Ye was a Henry B. Tippie Research Professor at the University of Iowa. Ye is a co-founder of minMax Optimization, a technology company based in Palo Alto and Shanghai focused on creating optimization tools for geospatial and financial problems.
References
1. Luenberger, David G.; Ye, Yinyu (2008). Linear and nonlinear programming. International Series in Operations Research & Management Science. Vol. 116 (Third ed.). New York: Springer. pp. xiv+546. ISBN 978-0-387-74502-2. MR 2423726.
2. "John von Neumann Theory Prize / INFORMS Prizes & Awards / Recognize Excellence / IOL Home - INFORMS.org". www.informs.org. Archived from the original on 2010-03-12.
3. Fellows: Alphabetical List, Institute for Operations Research and the Management Sciences, retrieved 2019-10-09
• Luenberger, David G.; Ye, Yinyu (2008). Linear and nonlinear programming. International Series in Operations Research & Management Science. Vol. 116 (Third ed.). New York: Springer. pp. xiv+546. ISBN 978-0-387-74502-2. MR 2423726.
John von Neumann Theory Prize
1975–1999
• George Dantzig (1975)
• Richard Bellman (1976)
• Felix Pollaczek (1977)
• John F. Nash / Carlton E. Lemke (1978)
• David Blackwell (1979)
• David Gale / Harold W. Kuhn / Albert W. Tucker (1980)
• Lloyd Shapley (1981)
• Abraham Charnes / William W. Cooper / Richard J. Duffin (1982)
• Herbert Scarf (1983)
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• Richard E. Barlow / Frank Proschan (1991)
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• Robert Herman (1993)
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• Egon Balas (1995)
• Peter C. Fishburn (1996)
• Peter Whittle (1997)
• Fred W. Glover (1998)
• R. Tyrrell Rockafellar (1999)
2000–present
• Ellis L. Johnson / Manfred W. Padberg (2000)
• Ward Whitt (2001)
• Donald L. Iglehart / Cyrus Derman (2002)
• Arkadi Nemirovski / Michael J. Todd (2003)
• J. Michael Harrison (2004)
• Robert Aumann (2005)
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• Frank Kelly (2008)
• Yurii Nesterov / Yinyu Ye (2009)
• Søren Asmussen / Peter W. Glynn (2010)
• Gérard Cornuéjols (2011)
• George Nemhauser / Laurence Wolsey (2012)
• Michel Balinski (2013)
• Nimrod Megiddo (2014)
• Vašek Chvátal / Jean Bernard Lasserre (2015)
• Martin I. Reiman / Ruth J. Williams (2016)
• Donald Goldfarb / Jorge Nocedal (2017)
• Dimitri Bertsekas / John Tsitsiklis (2018)
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-yllion
-yllion (pronounced /aɪljən/)[1] is a proposal from Donald Knuth for the terminology and symbols of an alternate decimal superbase system. In it, he adapts the familiar English terms for large numbers to provide a systematic set of names for much larger numbers. In addition to providing an extended range, -yllion also dodges the long and short scale ambiguity of -illion.
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List of numeral systems
Knuth's digit grouping is exponential instead of linear; each division doubles the number of digits handled, whereas the familiar system only adds three or six more. His system is basically the same as one of the ancient and now-unused Chinese numeral systems, in which units stand for 104, 108, 1016, 1032, ..., 102n, and so on (with an exception that the -yllion proposal does not use a word for thousand which the original Chinese numeral system has). Today the corresponding Chinese characters are used for 104, 108, 1012, 1016, and so on.
Details and examples
Look up -yllion in Wiktionary, the free dictionary.
In Knuth's -yllion proposal:
• 1 to 999 have their usual names.
• 1000 to 9999 are divided before the 2nd-last digit and named "foo hundred bar." (e.g. 1234 is "twelve hundred thirty-four"; 7623 is "seventy-six hundred twenty-three")
• 104 to 108 − 1 are divided before the 4th-last digit and named "foo myriad bar". Knuth also introduces at this level a grouping symbol (comma) for the numeral. So 382,1902 is "three hundred eighty-two myriad nineteen hundred two."
• 108 to 1016 − 1 are divided before the 8th-last digit and named "foo myllion bar", and a semicolon separates the digits. So 1,0002;0003,0004 is "one myriad two myllion, three myriad four."
• 1016 to 1032 − 1 are divided before the 16th-last digit and named "foo byllion bar", and a colon separates the digits. So 12:0003,0004;0506,7089 is "twelve byllion, three myriad four myllion, five hundred six myriad seventy hundred eighty-nine."
• etc.
Each new number name is the square of the previous one — therefore, each new name covers twice as many digits. Knuth continues borrowing the traditional names changing "illion" to "yllion" on each one. Abstractly, then, "one n-yllion" is $10^{2^{n+2}}$. "One trigintyllion" ($10^{2^{32}}$) would have 232 + 1, or 42;9496,7297, or nearly forty-three myllion (4300 million) digits (by contrast, a conventional "trigintillion" has merely 94 digits — not even a hundred, let alone a thousand million, and still 7 digits short of a googol). Better yet, "one centyllion" ($10^{2^{102}}$) would have 2102 + 1, or 507,0602;4009,1291:7605,9868;1282,1505, or about 1/20 of a tryllion digits, whereas a conventional "centillion" has only 304 digits.
The corresponding Chinese "long scale" numerals are given, with the traditional form listed before the simplified form. Same numerals are used in the Chinese "short scale" (new number name every power of 10 after 1000 (or 103+n)), "myriad scale" (new number name every 104n), and "mid scale" (new number name every 108n). Today these numerals are still in use, but are used in their "myriad scale" values, which is also used in Japanese and in Korean. For a more extensive table, see Myriad system.
ValueNameNotation Standard English name (short scale)Chinese ("long scale")Pīnyīn (Mandarin)Jyutping (Cantonese)Pe̍h-ōe-jī (Hokkien)
100 One 1 One 一 yī jat1 it/chit
101 Ten 10 Ten 十 shí sap6 si̍p/cha̍p
102 One hundred 100 One hundred 百 bǎi baak3 pah
103 Ten hundred 1000 One thousand 千 qiān cin1 chhian
104 One myriad 1,0000 Ten thousand 萬, 万 wàn maan6 bān
105 Ten myriad 10,0000 One hundred thousand 十萬, 十万 shíwàn sap6 maan6 si̍p/cha̍p bān
106 One hundred myriad 100,0000 One million 百萬, 百万 bǎiwàn baak3 maan6 pah bān
107 Ten hundred myriad 1000,0000 Ten million 千萬, 千万 qiānwàn cin1 maan6 chhian bān
108 One myllion 1;0000,0000 One hundred million 億, 亿 yì jik1 ek
109 Ten myllion 10;0000,0000 One billion 十億, 十亿 shíyì sap6 jik1 si̍p/cha̍p ek
1012 One myriad myllion 1,0000;0000,0000 One trillion 萬億, 万亿 wànyì maan6 jik1 bān ek
1016 One byllion 1:0000,0000;0000,0000 Ten quadrillion 兆 zhào siu6 tiāu
1024 One myllion byllion 1;0000,0000:0000,0000;0000,0000 One septillion 億兆, 亿兆 yìzhào jik1 siu6 ek tiāu
1032 One tryllion 1'0000,0000;0000,0000:0000,0000;0000,0000 One hundred nonillion 京 jīng ging1 kiaⁿ
1064 One quadryllion Ten vigintillion 垓 gāi goi1 kai
10128 One quintyllion One hundred unquadragintillion 秭 zǐ zi2 chi
10256 One sextyllion Ten quattuoroctogintillion 穰 ráng joeng4 liōng
10512 One septyllion One hundred novensexagintacentillion 溝, 沟 gōu kau1 kau
101024 One octyllion Ten quadragintatrecentillion 澗, 涧 jiàn gaan3 kán
102048 One nonyllion One hundred unoctogintasescentillion 正 zhēng zing3 chiàⁿ
104096 One decyllion Ten milliquattuorsexagintatrecentillion 載, 载 zài zoi3 chài
Latin- prefix
In order to construct names of the form n-yllion for large values of n, Knuth appends the prefix "latin-" to the name of n without spaces and uses that as the prefix for n. For example, the number "latintwohundredyllion" corresponds to n = 200, and hence to the number $10^{2^{202}}$.
Negative powers
To refer to small quantities with this system, the suffix -th is used.
For instance, $10^{-4}$is a myriadth. $10^{-16777216}$ is a vigintyllionth.
See also
• Nicolas Chuquet – Mathematician
• Jacques Pelletier du Mans – Humanist, Poet, Mathematician
• Knuth's up-arrow notation – Method of notation of very large integers
• The Sand Reckoner – Work by Archimedes
References
1. "Large Numbers (Page 2) at MROB".
• Donald E. Knuth. Supernatural Numbers in The Mathematical Gardener (edited by David A. Klarner). Wadsworth, Belmont, CA, 1981. 310—325.
• Robert P. Munafo. The Knuth -yllion Notation ( Archived 2012-02-13 at the Wayback Machine 2012-02-25), 1996–2012.
Donald Knuth
Publications
• The Art of Computer Programming
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Other
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| Wikipedia |
Jean-Christophe Yoccoz
Jean-Christophe Yoccoz (29 May 1957 – 3 September 2016) was a French mathematician. He was awarded a Fields Medal in 1994, for his work on dynamical systems.[1][2] Yoccoz died on 3 September 2016 at the age of 59.
Jean-Christophe Yoccoz
Jean-Christophe Yoccoz in 2005
Born(1957-05-29)29 May 1957
Paris, France
Died3 September 2016(2016-09-03) (aged 59)
Paris, France
EducationLycée Louis-le-Grand
Alma materÉcole normale supérieure
École Polytechnique
Known forDynamical systems
Yoccoz puzzle
AwardsSalem Prize (1988)
Fields Medal (1994)
Scientific career
FieldsMathematics
InstitutionsCentre de mathématiques Laurent-Schwartz
Paris-Sud 11 University
Collège de France
Doctoral advisorMichael Herman
Doctoral studentsSylvain Crovisier
Ricardo Pérez-Marco
Websitewww.college-de-france.fr/site/jean-christophe-yoccoz/Hommage-a-Jean-Christophe-Yoccoz.htm
Wikimedia Commons has media related to Jean-Christophe Yoccoz.
Biography
Yoccoz attended the Lycée Louis-le-Grand,[3] during which time he was a silver medalist at the 1973 International Mathematical Olympiad and a gold medalist in 1974.[4][5] He entered the École Normale Supérieure in 1975, and completed an agrégation in mathematics in 1977.[6]
After completing military service in Brazil, he completed his PhD under Michael Herman in 1985 at Centre de mathématiques Laurent-Schwartz, which is a research unit jointly operated by the French National Center for Scientific Research (CNRS) and École Polytechnique.[6][7][8]
He took up a position at the University of Paris-Sud in 1987, and became a professor at the Collège de France in 1997, where he remained until his death.[2] He was a member of Bourbaki.[9]
Yoccoz won the Salem Prize in 1988. He was an invited speaker at the International Congress of Mathematicians in 1990 at Kyoto,[10] and was awarded the Fields Medal at the International Congress of Mathematicians in 1994 in Zürich.[11][10] He joined the French Academy of Sciences and Brazilian Academy of Sciences in 1994, became a chevalier in the French Legion of Honor in 1995, and was awarded the Grand Cross of the Brazilian National Order of Scientific Merit in 1998.[5]
Mathematical work
Yoccoz's worked on the theory of dynamical systems, his contributions include advances to KAM theory, and the introduction of the method of Yoccoz puzzles, a combinatorial technique which proved useful to the study of Julia sets.[5]
Notable publications
• Yoccoz, J.-C. Conjugaison différentiable des difféomorphismes du cercle dont le nombre de rotation vérifie une condition diophantienne. Ann. Sci. École Norm. Sup. (4) 17 (1984), no. 3, 333–359. doi:10.24033/asens.1475
• Yoccoz, Jean-Christophe. Théorème de Siegel, nombres de Bruno et polynômes quadratiques. Petits diviseurs en dimension 1. Astérisque No. 231 (1995), 3–88. MR1367353
References
1. O'Connor, John J.; Robertson, Edmund F., "Jean-Christophe Yoccoz", MacTutor History of Mathematics Archive, University of St Andrews
2. "Le mathématicien Jean-Christophe Yoccoz est mort". 5 September 2016. Retrieved 6 September 2016.
3. Administrator. "Des mathématiciens". Retrieved 6 September 2016.
4. Jean-Christophe Yoccoz's results at International Mathematical Olympiad
5. "Jean-Christophe Yoccoz † › Heidelberg Laureate Forum". Archived from the original on 11 September 2016. Retrieved 6 September 2016.
6. "Yoccoz biography". Retrieved 6 September 2016.
7. "Ciência se despede de matemático francês Jean-Christophe Yoccoz". Archived from the original on 19 September 2016. Retrieved 6 September 2016.
8. Jean-Christophe Yoccoz at the Mathematics Genealogy Project
9. Mashaal, Maurice (2006), Bourbaki: a secret society of mathematicians, American Mathematical Society, p. 19, ISBN 978-0-8218-3967-6
10. "International Mathematical Union (IMU)". Archived from the original on 8 November 2017. Retrieved 6 September 2016.
11. "Fields Medals and Nevanlinna Prize 1994". Archived from the original on 29 July 2016. Retrieved 6 September 2016.
Fields Medalists
• 1936 Ahlfors
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| Wikipedia |
Small retrosnub icosicosidodecahedron
In geometry, the small retrosnub icosicosidodecahedron (also known as a retrosnub disicosidodecahedron, small inverted retrosnub icosicosidodecahedron, or retroholosnub icosahedron) is a nonconvex uniform polyhedron, indexed as U72. It has 112 faces (100 triangles and 12 pentagrams), 180 edges, and 60 vertices.[1] It is given a Schläfli symbol sr{⁵/₃,³/₂}.
Small retrosnub icosicosidodecahedron
TypeUniform star polyhedron
ElementsF = 112, E = 180
V = 60 (χ = −8)
Faces by sides(40+60){3}+12{5/2}
Coxeter diagram
Wythoff symbol| 3/2 3/2 5/2
Symmetry groupIh, [5,3], *532
Index referencesU72, C91, W118
Dual polyhedronSmall hexagrammic hexecontahedron
Vertex figure
(35.5/3)/2
Bowers acronymSirsid
The 40 non-snub triangular faces form 20 coplanar pairs, forming star hexagons that are not quite regular. Unlike most snub polyhedra, it has reflection symmetries.
George Olshevsky nicknamed it the yog-sothoth (after the Cthulhu Mythos deity).[2][3]
Convex hull
Its convex hull is a nonuniform truncated dodecahedron.
Truncated dodecahedron
Convex hull
Small retrosnub icosicosidodecahedron
Cartesian coordinates
Cartesian coordinates for the vertices of a small retrosnub icosicosidodecahedron are all the even permutations of
(±(1-ϕ−α), 0, ±(3−ϕα))
(±(ϕ-1−α), ±2, ±(2ϕ-1−ϕα))
(±(ϕ+1−α), ±2(ϕ-1), ±(1−ϕα))
where ϕ = (1+√5)/2 is the golden ratio and α = √3ϕ−2.
See also
• List of uniform polyhedra
• Small snub icosicosidodecahedron
References
1. Maeder, Roman. "72: small retrosnub icosicosidodecahedron". MathConsult.{{cite web}}: CS1 maint: url-status (link)
2. Birrell, Robert J. (May 1992). The Yog-sothoth: analysis and construction of the small inverted retrosnub icosicosidodecahedron (M.S.). California State University.
3. Bowers, Jonathan (2000). "Uniform Polychora" (PDF). In Reza Sarhagi (ed.). Bridges 2000. Bridges Conference. pp. 239–246.
External links
• Weisstein, Eric W. "Small retrosnub icosicosidodecahedron". MathWorld.
• Klitzing, Richard. "3D star small retrosnub icosicosidodecahedron".
| Wikipedia |
Yoichi Miyaoka
Yoichi Miyaoka (宮岡 洋一, Miyaoka Yōichi) is a mathematician who works in algebraic geometry and who proved (independently of Shing-Tung Yau's work) the Bogomolov–Miyaoka–Yau inequality in an Inventiones Mathematicae paper.[1]
In 1984, Miyaoka extended the Bogomolov–Miyaoka–Yau inequality to surfaces with quotient singularities, and in 2008 to orbifold surfaces. Doing so, he obtains sharp bound on the number of quotient singularities on surfaces of general type. Moreover, the inequality for orbifold surfaces gives explicit values for the coefficients of the so-called Lang-Vojta conjecture relating the degree of a curve on a surface with its geometric genus.
References
1. Miyaoka, Yoichi (1977-12-01). "On the Chern numbers of surfaces of general type". Inventiones Mathematicae. 42 (1): 225–237. Bibcode:1977InMat..42..225M. doi:10.1007/BF01389789. ISSN 1432-1297. S2CID 120699065.
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Yom Tov Lipman Lipkin
Yom Tov Lipman Lipkin (Hebrew: יום טוב ליפמן ליפקין, Russian: Липман Израилевич Липкин; 1840 – February 21 [O.S. February 9] 1876) was a Lithuanian Jewish mathematician and inventor. He was the youngest son of Rabbi Yisroel Salanter, the father of the Musar movement.
Lipkin is best known for the Peaucellier–Lipkin linkage which was partly named after him.[1][2] The device is also known as the "Lipkin parallelogram".[3] Lipkin discovered the linkage independent from Peaucellier in 1871.[1] A model of Lipkin's invention was exhibited at the exposition at Vienna in 1873, and was later secured from the inventor by the Museum of the Institute of Engineers of Ways of Communication, St. Petersburg.
Biography
Lipkin was born in Salantai, department of Kovno, in 1846. He became interested in science and mathematics since childhood. Not knowing any non-Jewish languages, he had to derive his information from Hebrew books alone. He later learned German and French and went to study at University of Königsberg at the age of 17. He received a Ph.D. degree at Jena University with a thesis titled "Ueber die Räumlichen Strophoiden." He then moved to St. Petersburg, to work at University of St. Petersburg and continue his studies under Pafnuty Chebyshev. Soon afterwards he died in 1876 from smallpox.
Lipkin broke from traditional Jewish life, but kept interests in Jewish affairs and published in Ha-Tsefirah newspaper.
References
1. Mathematical tutorial of the Peaucellier–Lipkin linkage
2. How to draw a straight line by Daina Taimina
3. Lipkin, in 1906 Jewish Encyclopedia.
• Simona-Mariana Cretu, Gigi-Dragos Ciocioi-Troaca, Emil Soarece, and Eugen Marian Paun, Mechanical Models for Anti-Rhomb Linkage, in Explorations in the History of Machines and Mechanisms, Springer, 2012, pp. 421–430.
• Alan T. Levenson, Roger C. Klein, An Introduction to Modern Jewish Thinkers: From Spinoza to Soloveitchik, Rowman & Littlefield Publishers, 2006.
• I. Etkes, Rabbi Israel Salanter and the Mussar Movement: Seeking the Torah of Truth, Magness Press, Hebrew University of Jerusalem, 1982.
This article incorporates text from a publication now in the public domain: Singer, Isidore; et al., eds. (1901–1906). The Jewish Encyclopedia. New York: Funk & Wagnalls. {{cite encyclopedia}}: Missing or empty |title= (help)
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Yomtov Garti
Yomtov Bonjour Garti (2 September 1915 – 21 February 2011) was a Turkish mathematician and a teacher of mathematics, physics and cosmography in Istanbul, Turkey.
Life
Yomtov Garti was born in Kadıköy, at the Asian part of Istanbul (Ottoman Empire, later to become Turkey). His father Maer Garti was a veterinarian doctor, who died in the typhus epidemics during the World War I, while serving in the Ottoman Army. Yomtov Garti was educated at the French high school Lycée Saint-Joseph, Istanbul . After graduating from the Department of Mathematics and Physics of Istanbul University, he was approached by the famous mathematician Richard Edler von Mises, who was then located in Istanbul[1] and proposed him a PhD position. Garti’s PhD was on statistics and probability theory tutored by Richard Edler von Mises and William Prager.[2] Yomtov Garti received his PhD in 1939, as the first PhD student of von Mises and third PhD of Turkey. He later published his findings in an article (Garti, 1940),[3] which is a generalization of initial distributions to n dimensions published in[4](cited in[5]). Garti served as an assistant for a summer to Harry Dember, a professor in the Institute of Applied Physics in Istanbul. After receiving his doctoral degree in 1939, he started to teach at famous high schools in Istanbul. In 1954, he presented an article to Richard von Mises in a book published in his honor.[6] Yomtov Garti continued teaching until age 92. He died, at the age of 96 in 2011. All main newspapers of Turkey announced the loss of “The teacher of teachers”. At his funeral, the several generations of students overfilled the Hemdat Israel Synagogue at Kadıköy, Istanbul.
Career
Yomtov Garti became a renowned mathematics teacher in Istanbul. He thought mathematics and physics to thousands of students over six decades, in several famous schools, namely Galatasary High School (Galatasaray Lisesi), Haydarpaşa High School, Lycee Saint-Joseph Istanbul, and Notre Dame de Sion Lisesi. He also thought in Musevi Lisesi and Boğaziçi Üniversity. Yomtov Garti was rewarded the title of “Chevalier” of “Ordre des Palmes Academiques” by the French Government in recognition of his educational work in French schools in Turkey.
Publications
• Garti, Y., 1940. Les lois de probabilité pour les fonctions statistiques (cas de collectifs à plusieurs dimensions). Revue Mathématique de l’Union Interbalkanique 3, 21–39.
• Garti Y, Consoli T. 1954. Sur la densite de probabilite du produit de variables aleatoires de Pearson du type III. In: Studies in mathematics and mechanics presented to Richard von Mises. Academic Press, New York. pp 301–309.
References
1. Eden and Irzik, 2012. German mathematicians in exile in Turkey: Richard von Mises, William Prager, Hilda Geiringer, and their impact on Turkish mathematics. Historia Mathematica 2012, 29 (4): 432-459. https://doi.org/10.1016/j.hm.2012.07.002
2. "Yomtov Garti - The Mathematics Genealogy Project". www.genealogy.math.ndsu.nodak.edu. Genealogy of Mathematicians. Retrieved 29 May 2020.
3. Garti, 1940, Les lois de probabilité pour les fonctions statistiques (cas de collectifs à plusieurs dimensions). Revue Mathématique de l’Union Interbalkanique 3, 21–39.
4. Mises 1936 http://www.numdam.org/item?id=AIHP_1936__6_3-4_185_0
5. Mises, 1964. Mathematical theory of probability and statistics. Eited by Hilda Geiringer. Academic Press New York and London. Citing Y. Garti in chapter C.5. page 651.
6. Garti Y, Consoli T. 1954. Sur la densite de probabilite du produit de variables aleatoires de Pearson du type III. In: Studies in mathematics and mechanics presented to Richard von Mises. Academic Press, New York. pp 301-309
• Eden A, Irzik G. 2012. German mathematicians in exile in Turkey: Richard von Mises, William Prager, Hilda Geiringer, and their impact on Turkish mathematics. Historia Mathematica 2012, 29 (4): 432-459. https://doi.org/10.1016/j.hm.2012.07.002
• Garti, Y., 1940. Les lois de probabilité pour les fonctions statistiques (cas de collectifs à plusieurs dimensions). Revue Mathématique de l’Union Interbalkanique 3, 21–39.
• Garti Y, Consoli T. 1954. Sur la densite de probabilite du produit de variables aleatoires de Pearson du type III. In: Studies in mathematics and mechanics presented to Richard von Mises. Academic Press, New York. pp 301–309.
• Mises R. Les lois de probabilité pour les fonctions statistiques. Ann. Inst. Henri Poincare. 6, 185–212, 1936. http://www.numdam.org/item?id=AIHP_1936__6_3-4_185_0
• Mises R. 1964. Mathematical theory of probability and statistics. Eited by Hilda Geiringer. Academic Press New York and London. Citing Y. Garti in chapter C.5. page 651.
External links
• Misés, R. de (1936). "Les lois de probabilité pour les fonctions statistiques". Annales de l'institut Henri Poincaré. pp. 185–212. Retrieved 29 May 2020.
• http://www.matematikdunyasi.org/arsiv/PDF/11_01_59_60_garti.pdf
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| Wikipedia |
Yoneda lemma
In mathematics, the Yoneda lemma is arguably the most important result in category theory.[1] It is an abstract result on functors of the type morphisms into a fixed object. It is a vast generalisation of Cayley's theorem from group theory (viewing a group as a miniature category with just one object and only isomorphisms). It allows the embedding of any locally small category into a category of functors (contravariant set-valued functors) defined on that category. It also clarifies how the embedded category, of representable functors and their natural transformations, relates to the other objects in the larger functor category. It is an important tool that underlies several modern developments in algebraic geometry and representation theory. It is named after Nobuo Yoneda.
Generalities
The Yoneda lemma suggests that instead of studying the locally small category ${\mathcal {C}}$, one should study the category of all functors of ${\mathcal {C}}$ into $\mathbf {Set} $ (the category of sets with functions as morphisms). $\mathbf {Set} $ is a category we think we understand well, and a functor of ${\mathcal {C}}$ into $\mathbf {Set} $ can be seen as a "representation" of ${\mathcal {C}}$ in terms of known structures. The original category ${\mathcal {C}}$ is contained in this functor category, but new objects appear in the functor category, which were absent and "hidden" in ${\mathcal {C}}$. Treating these new objects just like the old ones often unifies and simplifies the theory.
This approach is akin to (and in fact generalizes) the common method of studying a ring by investigating the modules over that ring. The ring takes the place of the category ${\mathcal {C}}$, and the category of modules over the ring is a category of functors defined on ${\mathcal {C}}$.
Formal statement
Yoneda's lemma concerns functors from a fixed category ${\mathcal {C}}$ to the category of sets, $\mathbf {Set} $. If ${\mathcal {C}}$ is a locally small category (i.e. the hom-sets are actual sets and not proper classes), then each object $A$ of ${\mathcal {C}}$ gives rise to a natural functor to $\mathbf {Set} $ called a hom-functor. This functor is denoted:
$h_{A}=\mathrm {Hom} (A,-)$.
The (covariant) hom-functor $h_{A}$ sends $X$ to the set of morphisms $\mathrm {Hom} (A,X)$ and sends a morphism $f\colon X\to Y$ (where $X$ and $Y$ are objects in ${\mathcal {C}}$) to the morphism $f\circ -$ (composition with $f$ on the left) that sends a morphism $g$ in $\mathrm {Hom} (A,X)$ to the morphism $f\circ g$ in $\mathrm {Hom} (A,Y)$. That is,
$h_{A}(f)=\mathrm {Hom} (A,f),{\text{ or}}$
$h_{A}(f)(g)=f\circ g$
Yoneda's lemma says that:
Lemma (Yoneda) — Let $F$ be a functor from a locally small category ${\mathcal {C}}$ to $\mathbf {Set} $. Then for each object $A$ of ${\mathcal {C}}$, the natural transformations $\mathrm {Nat} (h_{A},F)\equiv \mathrm {Hom} (\mathrm {Hom} (A,-),F)$ from $h_{A}$ to $F$ are in one-to-one correspondence with the elements of $F(A)$. That is,
$\mathrm {Nat} (h_{A},F)\cong F(A).$
Moreover, this isomorphism is natural in $A$ and $F$ when both sides are regarded as functors from ${\mathcal {C}}\times \mathbf {Set} ^{\mathcal {C}}$ to $\mathbf {Set} $.
Here the notation $\mathbf {Set} ^{\mathcal {C}}$ denotes the category of functors from ${\mathcal {C}}$ to $\mathbf {Set} $.
Given a natural transformation $\Phi $ from $h_{A}$ to $F$, the corresponding element of $F(A)$ is $u=\Phi _{A}(\mathrm {id} _{A})$;[lower-alpha 1] and given an element $u$ of $F(A)$, the corresponding natural transformation is given by $\Phi _{X}(f)=F(f)(u)$ which assigns to a morphism $f\colon A\to X$ a value of $F(X)$.
Contravariant version
There is a contravariant version of Yoneda's lemma, which concerns contravariant functors from ${\mathcal {C}}$ to $\mathbf {Set} $. This version involves the contravariant hom-functor
$h^{A}=\mathrm {Hom} (-,A),$
which sends $X$ to the hom-set $\mathrm {Hom} (X,A)$. Given an arbitrary contravariant functor $G$ from ${\mathcal {C}}$ to $\mathbf {Set} $, Yoneda's lemma asserts that
$\mathrm {Nat} (h^{A},G)\cong G(A).$
Naming conventions
The use of $h_{A}$ for the covariant hom-functor and $h^{A}$ for the contravariant hom-functor is not completely standard. Many texts and articles either use the opposite convention or completely unrelated symbols for these two functors. However, most modern algebraic geometry texts starting with Alexander Grothendieck's foundational EGA use the convention in this article.[lower-alpha 2]
The mnemonic "falling into something" can be helpful in remembering that $h_{A}$ is the covariant hom-functor. When the letter $A$ is falling (i.e. a subscript), $h_{A}$ assigns to an object $X$ the morphisms from $A$ into $X$.
Proof
Since $\Phi $ is a natural transformation, we have the following commutative diagram:
This diagram shows that the natural transformation $\Phi $ is completely determined by $\Phi _{A}(\mathrm {id} _{A})=u$ since for each morphism $f\colon A\to X$ one has
$\Phi _{X}(f)=(Ff)u.$
Moreover, any element $u\in F(A)$ defines a natural transformation in this way. The proof in the contravariant case is completely analogous.
The Yoneda embedding
An important special case of Yoneda's lemma is when the functor $F$ from ${\mathcal {C}}$ to $\mathbf {Set} $ is another hom-functor $h_{B}$. In this case, the covariant version of Yoneda's lemma states that
$\mathrm {Nat} (h_{A},h_{B})\cong \mathrm {Hom} (B,A).$
That is, natural transformations between hom-functors are in one-to-one correspondence with morphisms (in the reverse direction) between the associated objects. Given a morphism $f\colon B\to A$ the associated natural transformation is denoted $\mathrm {Hom} (f,-)$.
Mapping each object $A$ in ${\mathcal {C}}$ to its associated hom-functor $h_{A}=\mathrm {Hom} (A,-)$ and each morphism $f\colon B\to A$ to the corresponding natural transformation $\mathrm {Hom} (f,-)$ determines a contravariant functor $h_{\bullet }$ from ${\mathcal {C}}$ to $\mathbf {Set} ^{\mathcal {C}}$, the functor category of all (covariant) functors from ${\mathcal {C}}$ to $\mathbf {Set} $. One can interpret $h_{\bullet }$ as a covariant functor:
$h_{\bullet }\colon {\mathcal {C}}^{\text{op}}\to \mathbf {Set} ^{\mathcal {C}}.$
The meaning of Yoneda's lemma in this setting is that the functor $h_{\bullet }$ is fully faithful, and therefore gives an embedding of ${\mathcal {C}}^{\mathrm {op} }$ in the category of functors to $\mathbf {Set} $. The collection of all functors $\{h_{A}|A\in C\}$ is a subcategory of $\mathbf {Set} ^{\mathcal {C}}$. Therefore, Yoneda embedding implies that the category ${\mathcal {C}}^{\mathrm {op} }$ is isomorphic to the category $\{h_{A}|A\in C\}$.
The contravariant version of Yoneda's lemma states that
$\mathrm {Nat} (h^{A},h^{B})\cong \mathrm {Hom} (A,B).$
Therefore, $h^{\bullet }$ gives rise to a covariant functor from ${\mathcal {C}}$ to the category of contravariant functors to $\mathbf {Set} $:
$h^{\bullet }\colon {\mathcal {C}}\to \mathbf {Set} ^{{\mathcal {C}}^{\mathrm {op} }}.$
Yoneda's lemma then states that any locally small category ${\mathcal {C}}$ can be embedded in the category of contravariant functors from ${\mathcal {C}}$ to $\mathbf {Set} $ via $h^{\bullet }$. This is called the Yoneda embedding.
The Yoneda embedding is sometimes denoted by よ, the Hiragana kana Yo.[2]
Representable functor
The Yoneda embedding essentially states that for every (locally small) category, objects in that category can be represented by presheaves, in a full and faithful manner. That is,
$\mathrm {Nat} (h^{A},P)\cong P(A)$
for a presheaf P. Many common categories are, in fact, categories of pre-sheaves, and on closer inspection, prove to be categories of sheaves, and as such examples are commonly topological in nature, they can be seen to be topoi in general. The Yoneda lemma provides a point of leverage by which the topological structure of a category can be studied and understood.
In terms of (co)end calculus
Main article: End (category theory)
Given two categories $\mathbf {C} $ and $\mathbf {D} $ with two functors $F,G:\mathbf {C} \to \mathbf {D} $, natural transformations between them can be written as the following end.[3]
$\mathrm {Nat} (F,G)=\int _{c\in \mathbf {C} }\mathrm {Hom} _{\mathbf {D} }(Fc,Gc)$
For any functors $K\colon \mathbf {C} ^{op}\to \mathbf {Sets} $ and $H\colon \mathbf {C} \to \mathbf {Sets} $ the following formulas are all formulations of the Yoneda lemma.[4]
$K\cong \int ^{c\in \mathbf {C} }Kc\times \mathrm {Hom} _{\mathbf {C} }(-,c),\qquad K\cong \int _{c\in \mathbf {C} }(Kc)^{\mathrm {Hom} _{\mathbf {C} }(c,-)},$
$H\cong \int ^{c\in \mathbf {C} }Hc\times \mathrm {Hom} _{\mathbf {C} }(c,-),\qquad H\cong \int _{c\in \mathbf {C} }(Hc)^{\mathrm {Hom} _{\mathbf {C} }(-,c)}.$
Preadditive categories, rings and modules
A preadditive category is a category where the morphism sets form abelian groups and the composition of morphisms is bilinear; examples are categories of abelian groups or modules. In a preadditive category, there is both a "multiplication" and an "addition" of morphisms, which is why preadditive categories are viewed as generalizations of rings. Rings are preadditive categories with one object.
The Yoneda lemma remains true for preadditive categories if we choose as our extension the category of additive contravariant functors from the original category into the category of abelian groups; these are functors which are compatible with the addition of morphisms and should be thought of as forming a module category over the original category. The Yoneda lemma then yields the natural procedure to enlarge a preadditive category so that the enlarged version remains preadditive — in fact, the enlarged version is an abelian category, a much more powerful condition. In the case of a ring $R$, the extended category is the category of all right modules over $R$, and the statement of the Yoneda lemma reduces to the well-known isomorphism
$M\cong \mathrm {Hom} _{R}(R,M)$ for all right modules $M$ over $R$.
Relationship to Cayley's theorem
As stated above, the Yoneda lemma may be considered as a vast generalization of Cayley's theorem from group theory. To see this, let ${\mathcal {C}}$ be a category with a single object $*$ such that every morphism is an isomorphism (i.e. a groupoid with one object). Then $G=\mathrm {Hom} _{\mathcal {C}}(*,*)$ forms a group under the operation of composition, and any group can be realized as a category in this way.
In this context, a covariant functor ${\mathcal {C}}\to \mathbf {Set} $ consists of a set $X$ and a group homomorphism $G\to \mathrm {Perm} (X)$, where $\mathrm {Perm} (X)$ is the group of permutations of $X$; in other words, $X$ is a G-set. A natural transformation between such functors is the same thing as an equivariant map between $G$-sets: a set function $\alpha \colon X\to Y$ with the property that $\alpha (g\cdot x)=g\cdot \alpha (x)$ for all $g$ in $G$ and $x$ in $X$. (On the left side of this equation, the $\cdot $ denotes the action of $G$ on $X$, and on the right side the action on $Y$.)
Now the covariant hom-functor $\mathrm {Hom} _{\mathcal {C}}(*,-)$ corresponds to the action of $G$ on itself by left-multiplication (the contravariant version corresponds to right-multiplication). The Yoneda lemma with $F=\mathrm {Hom} _{\mathcal {C}}(*,-)$ states that
$\mathrm {Nat} (\mathrm {Hom} _{\mathcal {C}}(*,-),\mathrm {Hom} _{\mathcal {C}}(*,-))\cong \mathrm {Hom} _{\mathcal {C}}(*,*)$,
that is, the equivariant maps from this $G$-set to itself are in bijection with $G$. But it is easy to see that (1) these maps form a group under composition, which is a subgroup of $\mathrm {Perm} (G)$, and (2) the function which gives the bijection is a group homomorphism. (Going in the reverse direction, it associates to every $g$ in $G$ the equivariant map of right-multiplication by $g$.) Thus $G$ is isomorphic to a subgroup of $\mathrm {Perm} (G)$, which is the statement of Cayley's theorem.
History
Yoshiki Kinoshita stated in 1996 that the term "Yoneda lemma" was coined by Saunders Mac Lane following an interview he had with Yoneda in the Gare du Nord station.[5][6]
See also
• Representation theorem
Notes
1. Recall that $\Phi _{A}:\mathrm {Hom} (A,A)\to F(A)$ so the last expression is well-defined and sends a morphism from $A$ to $A$, to an element in $F(A)$.
2. A notable exception to modern algebraic geometry texts following the conventions of this article is Commutative algebra with a view toward algebraic geometry / David Eisenbud (1995), which uses $h_{A}$ to mean the covariant hom-functor. However, the later book The geometry of schemes / David Eisenbud, Joe Harris (1998) reverses this and uses $h_{A}$ to mean the contravariant hom-functor.
References
1. Riehl, Emily (2017). Category Theory in Context (PDF). Dover. ISBN 978-0-486-82080-4.
2. "Yoneda embedding". nLab. Retrieved 6 July 2019.
3. Loregian (2015), Theorem 1.4.1.
4. Loregian (2015), Proposition 2.2.1.
5. Kinoshita, Yoshiki (23 April 1996). "Prof. Nobuo Yoneda passed away". Retrieved 21 December 2013.
6. "le lemme de la Gare du Nord". neverendingbooks. 18 November 2016. Retrieved 2022-09-10.
• Freyd, Peter (1964), Abelian categories, Harper's Series in Modern Mathematics (2003 reprint ed.), Harper and Row, Zbl 0121.02103.
• Mac Lane, Saunders (1998), Categories for the Working Mathematician, Graduate Texts in Mathematics, vol. 5 (2nd ed.), New York, NY: Springer-Verlag, ISBN 0-387-98403-8, Zbl 0906.18001
• Loregian, Fosco (2015). "This is the (co)end, my only (co)friend". arXiv:1501.02503 [math.CT].
• Yoneda lemma at the nLab
External links
• Mizar system proof: Wojciechowski, M. (1997). "Yoneda Embedding". Formalized Mathematics journal. 6 (3): 377–380. CiteSeerX 10.1.1.73.7127.
| Wikipedia |
Presheaf (category theory)
In category theory, a branch of mathematics, a presheaf on a category $C$ is a functor $F\colon C^{\mathrm {op} }\to \mathbf {Set} $. If $C$ is the poset of open sets in a topological space, interpreted as a category, then one recovers the usual notion of presheaf on a topological space.
A morphism of presheaves is defined to be a natural transformation of functors. This makes the collection of all presheaves on $C$ into a category, and is an example of a functor category. It is often written as ${\widehat {C}}=\mathbf {Set} ^{C^{\mathrm {op} }}$. A functor into ${\widehat {C}}$ is sometimes called a profunctor.
A presheaf that is naturally isomorphic to the contravariant hom-functor Hom(–, A) for some object A of C is called a representable presheaf.
Some authors refer to a functor $F\colon C^{\mathrm {op} }\to \mathbf {V} $ as a $\mathbf {V} $-valued presheaf.[1]
Examples
• A simplicial set is a Set-valued presheaf on the simplex category $C=\Delta $.
Properties
• When $C$ is a small category, the functor category ${\widehat {C}}=\mathbf {Set} ^{C^{\mathrm {op} }}$ is cartesian closed.
• The poset of subobjects of $P$ form a Heyting algebra, whenever $P$ is an object of ${\widehat {C}}=\mathbf {Set} ^{C^{\mathrm {op} }}$ for small $C$.
• For any morphism $f:X\to Y$ of ${\widehat {C}}$, the pullback functor of subobjects $f^{*}:\mathrm {Sub} _{\widehat {C}}(Y)\to \mathrm {Sub} _{\widehat {C}}(X)$ has a right adjoint, denoted $\forall _{f}$, and a left adjoint, $\exists _{f}$. These are the universal and existential quantifiers.
• A locally small category $C$ embeds fully and faithfully into the category ${\widehat {C}}$ of set-valued presheaves via the Yoneda embedding which to every object $A$ of $C$ associates the hom functor $C(-,A)$.
• The category ${\widehat {C}}$ admits small limits and small colimits.[2] See limit and colimit of presheaves for further discussion.
• The density theorem states that every presheaf is a colimit of representable presheaves; in fact, ${\widehat {C}}$ is the colimit completion of $C$ (see #Universal property below.)
Universal property
The construction $C\mapsto {\widehat {C}}=\mathbf {Fct} (C^{\text{op}},\mathbf {Set} )$ is called the colimit completion of C because of the following universal property:
Proposition[3] — Let C, D be categories and assume D admits small colimits. Then each functor $\eta :C\to D$ factorizes as
$C{\overset {y}{\longrightarrow }}{\widehat {C}}{\overset {\widetilde {\eta }}{\longrightarrow }}D$
where y is the Yoneda embedding and ${\widetilde {\eta }}:{\widehat {C}}\to D$ is a, unique up to isomorphism, colimit-preserving functor called the Yoneda extension of $\eta $.
Proof: Given a presheaf F, by the density theorem, we can write $F=\varinjlim yU_{i}$ where $U_{i}$ are objects in C. Then let ${\widetilde {\eta }}F=\varinjlim \eta U_{i},$ which exists by assumption. Since $\varinjlim -$ is functorial, this determines the functor ${\widetilde {\eta }}:{\widehat {C}}\to D$. Succinctly, ${\widetilde {\eta }}$ is the left Kan extension of $\eta $ along y; hence, the name "Yoneda extension". To see ${\widetilde {\eta }}$ commutes with small colimits, we show ${\widetilde {\eta }}$ is a left-adjoint (to some functor). Define ${\mathcal {H}}om(\eta ,-):D\to {\widehat {C}}$ to be the functor given by: for each object M in D and each object U in C,
${\mathcal {H}}om(\eta ,M)(U)=\operatorname {Hom} _{D}(\eta U,M).$
Then, for each object M in D, since ${\mathcal {H}}om(\eta ,M)(U_{i})=\operatorname {Hom} (yU_{i},{\mathcal {H}}om(\eta ,M))$ by the Yoneda lemma, we have:
${\begin{aligned}\operatorname {Hom} _{D}({\widetilde {\eta }}F,M)&=\operatorname {Hom} _{D}(\varinjlim \eta U_{i},M)=\varprojlim \operatorname {Hom} _{D}(\eta U_{i},M)=\varprojlim {\mathcal {H}}om(\eta ,M)(U_{i})\\&=\operatorname {Hom} _{\widehat {C}}(F,{\mathcal {H}}om(\eta ,M)),\end{aligned}}$
which is to say ${\widetilde {\eta }}$ is a left-adjoint to ${\mathcal {H}}om(\eta ,-)$. $\square $
The proposition yields several corollaries. For example, the proposition implies that the construction $C\mapsto {\widehat {C}}$ is functorial: i.e., each functor $C\to D$ determines the functor ${\widehat {C}}\to {\widehat {D}}$.
Variants
A presheaf of spaces on an ∞-category C is a contravariant functor from C to the ∞-category of spaces (for example, the nerve of the category of CW-complexes.)[4] It is an ∞-category version of a presheaf of sets, as a "set" is replaced by a "space". The notion is used, among other things, in the ∞-category formulation of Yoneda's lemma that says: $C\to PShv(C)$ is fully faithful (here C can be just a simplicial set.)[5]
See also
• Topos
• Category of elements
• Simplicial presheaf (this notion is obtained by replacing "set" with "simplicial set")
• Presheaf with transfers
Notes
1. co-Yoneda lemma at the nLab
2. Kashiwara & Schapira 2005, Corollary 2.4.3.
3. Kashiwara & Schapira 2005, Proposition 2.7.1.
4. Lurie, Definition 1.2.16.1.
5. Lurie, Proposition 5.1.3.1.
References
• Kashiwara, Masaki; Schapira, Pierre (2005). Categories and sheaves. Grundlehren der mathematischen Wissenschaften. Vol. 332. Springer. ISBN 978-3-540-27950-1.
• Lurie, J. Higher Topos Theory.
• Mac Lane, Saunders; Moerdijk, Ieke (1992). Sheaves in Geometry and Logic. Springer. ISBN 0-387-97710-4.
Further reading
• Presheaf at the nLab
• Free cocompletion at the nLab
• Daniel Dugger, Sheaves and Homotopy Theory, the pdf file provided by nlab.
| Wikipedia |
Yoneda product
In algebra, the Yoneda product (named after Nobuo Yoneda) is the pairing between Ext groups of modules:
$\operatorname {Ext} ^{n}(M,N)\otimes \operatorname {Ext} ^{m}(L,M)\to \operatorname {Ext} ^{n+m}(L,N)$
induced by
$\operatorname {Hom} (N,M)\otimes \operatorname {Hom} (M,L)\to \operatorname {Hom} (N,L),\,f\otimes g\mapsto g\circ f.$
Specifically, for an element $\xi \in \operatorname {Ext} ^{n}(M,N)$, thought of as an extension
$\xi :0\rightarrow N\rightarrow E_{0}\rightarrow \cdots \rightarrow E_{n-1}\rightarrow M\rightarrow 0,$
and similarly
$\rho :0\rightarrow M\rightarrow F_{0}\rightarrow \cdots \rightarrow F_{m-1}\rightarrow L\rightarrow 0\in \operatorname {Ext} ^{m}(L,M),$
we form the Yoneda (cup) product
$\xi \smile \rho :0\rightarrow N\rightarrow E_{0}\rightarrow \cdots \rightarrow E_{n-1}\rightarrow F_{0}\rightarrow \cdots \rightarrow F_{m-1}\rightarrow L\rightarrow 0\in \operatorname {Ext} ^{m+n}(L,N).$
Note that the middle map $E_{n-1}\rightarrow F_{0}$ factors through the given maps to $M$.
We extend this definition to include $m,n=0$ using the usual functoriality of the $\operatorname {Ext} ^{*}(\cdot ,\cdot )$ groups.
Applications
Ext Algebras
Given a commutative ring $R$ and a module $M$, the Yoneda product defines a product structure on the groups ${\text{Ext}}^{\bullet }(M,M)$, where ${\text{Ext}}^{0}(M,M)={\text{Hom}}_{R}(M,M)$ is generally a non-commutative ring. This can be generalized to the case of sheaves of modules over a ringed space, or ringed topos.
Grothendieck duality
In Grothendieck's duality theory of coherent sheaves on a projective scheme $i:X\hookrightarrow \mathbb {P} _{k}^{n}$ of pure dimension $r$ over an algebraically closed field $k$, there is a pairing
${\text{Ext}}_{{\mathcal {O}}_{X}}^{p}({\mathcal {O}}_{X},{\mathcal {F}})\times {\text{Ext}}_{{\mathcal {O}}_{X}}^{r-p}({\mathcal {F}},\omega _{X}^{\bullet })\to k$
where $\omega _{X}$ is the dualizing complex $\omega _{X}={\mathcal {Ext}}_{{\mathcal {O}}_{\mathbb {P} }}^{n-r}(i_{*}{\mathcal {F}},\omega _{\mathbb {P} })$ and $\omega _{\mathbb {P} }={\mathcal {O}}_{\mathbb {P} }(-(n+1))$ given by the Yoneda pairing.[1]
Deformation theory
The Yoneda product is useful for understanding the obstructions to a deformation of maps of ringed topoi.[2] For example, given a composition of ringed topoi
$X\xrightarrow {f} Y\to S$
and an $S$-extension $j:Y\to Y'$ of $Y$ by an ${\mathcal {O}}_{Y}$-module $J$, there is an obstruction class
$\omega (f,j)\in {\text{Ext}}^{2}(\mathbf {L} _{X/Y},f^{*}J)$
which can be described as the yoneda product
$\omega (f,j)=f^{*}(e(j))\cdot K(X/Y/S)$
where
${\begin{aligned}K(X/Y/S)&\in {\text{Ext}}^{1}(\mathbf {L} _{X/Y},\mathbf {L} _{Y/S})\\f^{*}(e(j))&\in {\text{Ext}}^{1}(f^{*}\mathbf {L} _{Y/S},f^{*}J)\end{aligned}}$
and $\mathbf {L} _{X/Y}$ corresponds to the cotangent complex.
See also
• Ext functor
• Derived category
• Deformation theory
• Kodaira–Spencer map
References
1. Altman; Kleiman (1970). Grothendieck Duality. Lecture Notes in Mathematics. Vol. 146. p. 5. doi:10.1007/BFb0060932. ISBN 978-3-540-04935-7.
2. Illusie, Luc. "Complexe cotangent; application a la theorie des deformations" (PDF). p. 163.
• May, J. Peter. "Notes on Tor and Ext" (PDF).
External links
• Universality of Ext functor using Yoneda extensions
| Wikipedia |
Joram Lindenstrauss
Joram Lindenstrauss (Hebrew: יורם לינדנשטראוס) (October 28, 1936 – April 29, 2012) was an Israeli mathematician working in functional analysis. He was a professor of mathematics at the Einstein Institute of Mathematics.[1]
Joram Lindenstrauss
יורם לינדנשטראוס
Joram Lindenstrauss, 1975
Born(1936-10-28)October 28, 1936
Tel Aviv, Mandatory Palestine
DiedApril 29, 2012(2012-04-29) (aged 75)
Resting placeHar HaMenuchot
Alma materHebrew University of Jerusalem
AwardsIsrael Prize (1981)
Scientific career
InstitutionsEinstein Institute of Mathematics
Doctoral advisorsAryeh Dvoretzky
Branko Grünbaum
Doctoral studentsAssaf Naor, Gideon Schechtman
Biography
Joram Lindenstrauss was born in Tel Aviv.[2] He was the only child of a pair of lawyers who immigrated to Israel from Berlin. He began to study mathematics at the Hebrew University of Jerusalem in 1954 while serving in the army. He became a full-time student in 1956 and received his master's degree in 1959. In 1962 Lindenstrauss earned his Ph.D. from the Hebrew University (dissertation: Extension of Compact Operators, advisors: Aryeh Dvoretzky, Branko Grünbaum).[3] He worked as a postdoc at Yale University and the University of Washington in Seattle from 1962 - 1965. He was appointed senior lecturer at the Hebrew University in 1965, associate professor on 1967 and full professor in 1969. He became the Leon H. and Ada G. Miller Memorial Professor of Mathematics in 1985.[2] He retired in 2005.
Lindenstrauss was married to theoretical computer scientist Naomi Lindenstrauss. Two of their children, Ayelet Lindenstrauss and Fields Medallist Elon Lindenstrauss, are also mathematicians (providing a rare example of father, mother, son and daughter all having papers listed in Mathematical Reviews).[2] Joram was also the cousin of Micha Lindenstrauss.
Research
Lindenstrauss worked in various areas of functional analysis and geometry,[4] particularly Banach space theory, finite- and infinite-dimensional convexity, geometric nonlinear functional analysis and geometric measure theory.[2] He authored more than 100 papers as well as several books in Banach space theory.[5]
Among his results is the Johnson–Lindenstrauss lemma which concerns low-distortion embeddings of points from high-dimensional into low-dimensional Euclidean space. Another of his theorems states that in a Banach space with the Radon–Nikodym property, a closed and bounded set has an extreme point; compactness is not needed.[6]
Awards
In 1981 Lindenstrauss was awarded the Israel Prize, for mathematics.[7] In 1997, Lindenstrauss was the first mathematician from outside Poland to be awarded the Stefan Banach Medal of the Polish Academy of Sciences.[8]
Published works
• Classical Banach spaces I (with Lior Tzafriri). Springer-Verlag, 1977.
• Classical Banach spaces II (with Lior Tzafriri). Springer-Verlag, 1979.
• Banach spaces with a unique unconditional basis, up to permutation (with Jean Bourgain, Peter George Casazza, and Lior Tzafriri). Memoirs of the American Mathematical Society, vol 322. American Mathematical Society, 1985
• Geometric nonlinear functional analysis (with Yoav Benyamini). Colloquium publications, 48. American Mathematical Society, 2000.[9]
• Handbook of the geometry of Banach spaces (Edited, with William B. Johnson). Elsevier, Vol. 1 (2001), Vol. 2 (2003).
See also
• List of Israel Prize recipients
References
1. Professors emeriti, Einstein Institute of Mathematics, http://www.math.huji.ac.il/#news
2. "Joram Lindenstrauss CV" (PDF).
3. Joram Lindenstrauss at Mathematics Genealogy
4. A biographical sketch from the book "Classical Banach Spaces"
5. "MathSciNet author profile".
6. Artstein (1980, p. 173): Artstein, Zvi (1980). "Discrete and continuous bang-bang and facial spaces, or: Look for the extreme points". SIAM Review. 22 (2): 172–185. doi:10.1137/1022026. JSTOR 2029960. MR 0564562.
7. "Israel Prize Official Site - Recipients in 1981 (in Hebrew)".
8. "Stefan Banach Medal". Polish Academy of Sciences. Archived from the original on 2020-11-05.
9. Virtual display of books written by members of the Einstein Institute of Mathematics
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Yoshie Katsurada
Yoshie Katsurada (Japanese: 桂田 芳枝, 3 September 1911 – 10 May 1980)[1][2] was a Japanese mathematician specializing in differential geometry.[3] She became the first Japanese woman to earn a doctorate in mathematics, in 1950, and the first to obtain an imperial university professorship in mathematics, in 1967.[4]
Life
Katsurada was born in Akaigawa, Hokkaido on 3 September 1911, a daughter of an elementary school principal. In high school in Otaru, she took special instruction in mathematics from a boys' mathematics instructor. Graduating from high school in 1929, she began auditing classes at the Tokyo Physics School, a predecessor to the Tokyo University of Science, in 1931.[1]
She began working as an administrative assistant in the Hokkaido University Department of Mathematics in 1936. In 1938 she began study in mathematics at Tokyo Woman's Christian University, withdrawing in 1940 to transfer to Hokkaido University. She graduated from Hokkaido University in 1942, and in the same year became an assistant professor there.[1]
In 1950, she completed a doctorate in mathematics at Hokkaido University, under the supervision of Shoji Kawaguchi,[1] becoming the first Japanese woman to earn a doctorate in mathematics,[1][4] and earning a promotion to associate professor.[1] She remained at Hokkaido University for the remainder of her career, with research visits to Sapienza University of Rome, ETH Zurich, and the University of California, Berkeley. She was promoted to full professor in 1967, the first female professor in mathematics at a former imperial university.[1]
She retired in 1975,[1] and died on 10 May 1980.[2]
Research
Katsurada's early research, from the beginning of her studies into the mid-1950s, primarily concerned line elements; this was the primary interest of her advisor Shoji Kawaguchi, with whom she continued to collaborate on this subject. After visiting Heinz Hopf at ETH Zurich in 1957–1958, she shifted interests to submanifolds and hypersurfaces in Riemannian manifolds, publishing well-regarded work in this area.[3]
Recognition
Several papers in the 1972 volume of the Hokkaido Mathematical Journal are dedicated to Katsurada in honor of her 60th birthday. Katsurada was given the Hokkaido Culture Award in 1973.[1]
References
1. 桂田芳枝 (Yoshie Katsurada, 1911-1980) (in Japanese), Department of Mathematics, Hokkaido University, retrieved 2021-08-30
2. "Deaths" (PDF), Notices of the American Mathematical Society, 28 (1): 98, January 1981
3. Kobayashi, Shoshichi (1996), "Differential Geometry in Japan in the 1940s and '50s", 総合講演, 企画特別講演アブストラクト: 9–18, doi:10.11429/emath1996.1996.1996_9; see in particular p. 13
4. Frédéric, Louis (2002), "Katsurada Yoshie", Japan Encyclopedia, translated by Roth, Käthe, The Belknap Press of Harvard University Press, p. 494
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Yoshiharu Kohayakawa
Yoshiharu Kohayakawa (Japanese: 小早川美晴; born 1963) is a Japanese-Brazilian mathematician working on discrete mathematics and probability theory.[1] He is known for his work on Szemerédi's regularity lemma, which he extended to sparser graphs.[2][3]
Biography
Kohayakawa was a student of Béla Bollobás at the University of Cambridge.[4]
According to Google Scholar, as of August 21, 2019, Kohayakawa's works have been cited over 3194 times, and his h-index is 33.[5]
He is a titular member of the Brazilian Academy of Sciences.[1]
In 2000, five American researchers received a USA NSF Research Grant in the value of $20,000 to go to Brazil to work in collaboration with him on mathematical problems.[6]
Kohayakawa has an Erdős number of 1.[7][8]
He was awarded the 2018 Fulkerson Prize.
References
1. Brazilian Academy of Sciences – Yoshiharu Kohayakawa Archived May 24, 2015, at the Wayback Machine
2. László Lovász – Large Networks and Graph Limits, p. 395
3. Bridget S. Webb – Surveys in Combinatorics 2005, p. 227
4. Mathematics Genealogy Project – Yoshiharu Kohayakawa
5. Google Scholar Profile – Yoshiharu Kohayakawa
6. U.S.-Brazil Cooperative Research: Problems on Random Graphs (Structures) and Set Systems: NSG GRANT 0072064
7. Celina Miraglia Herrera – My Erdős number
8. He wrote "The size of the largest bipartite subgraphs", on Discrete Mathematics with Erdős and Gyárfás
External links
• Home Page of Yoshiharu Kohayakawa at the University of São Paulo
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Yoshio Shimamoto
Yoshio Shimamoto was a nuclear physicist who also did work in mathematics and computer science. While at Brookhaven National Laboratory (1954-1987),[1] he designed the logic for the MERLIN digital computer in 1958, and served as chairman of the Applied Mathematics Department from 1964 to 1975. Shimamoto researched in combinatorial mathematics, the economics of outer continental shelf oil and gas lease sales (on behalf of the U.S. Geological Survey), the architecture of supercomputers, and the linking of computers for parallel processing.[2]
During the 1970s, he worked with Heinrich Heesch and Karl Durre on methods for a computer-aided proof of the four color theorem, using computer programs to apply Heesch's notion of "discharging" to eliminate 4-colorable cases. A proof of the Four Color Theorem, which he presented in 1971, was later shown to be flawed, but it served as the basis for further work.
Born in Hawaii in 1924, Shimamoto served with the U.S. Army Signal Corps and Strategic Bombing Survey in Japan, during World War II. He died in New Jersey on August 27, 2009.[2]
References
1. "Yoshio Shimamoto Obituary - New York, NY | New York Times", New York Times
2. "Paid notice: Deaths – Shimamoto , Yoshio", New York Times, August 31, 2009
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You-Dong Liang
You-Dong Liang (梁友栋) is a mathematician and educator, best known for his contributions in geometric modeling and the Liang-Barsky algorithm.
Education and teaching
You-Dong Liang was born on July 19, 1935, in Fuzhou, Fujian Province, China. Liang pursued his graduate degree in Fudan University, where he worked under the supervision of Professor Su Buqing and specialized in geometric theory. After graduating in 1960, he joined the mathematics teaching faculty at Zhejiang University, where he actively promoted the development of geometric design and graphics. From 1984–1990, he was the chairman of the mathematics department, and on several occasions, was a visiting scholar and visiting professor at the University of California at Berkeley, University of Utah, and University of Berlin. Liang helped form the Computational Geometry Collaborative Group in China. As the leader of this group, Liang supported the collaboration of scholars in geometric design and computational graphics. Liang was awarded the “Chinese Geometric Design and Calculation Contribution Award” in 2009.[1]
Contributions, papers, and awards
Liang has worked on computer-aided geometric design and computer graphics research. In 1984, Liang developed the Liang–Barsky algorithm, which has applications in computer graphics.[2] Liang made further improvements on this algorithm in 1992.[3] In the late 1980s and early 1990s, Liang proposed a series of theories and methodologies in geometric continuity. In 1991, Liang supervised the completion of "Generated Computer Graphics and Geometric Modeling Research" and was awarded the Chinese National Science Third Prize. During these years, Liang received other honors, including the Chinese Academy of Sciences Award and the “European Graphics Conference” Best Paper Award. Liang has published more than 50 papers, including "A New Concept and Method for Line Clipping", "Some Theorems on Geometrical Objects", "Curve and Surface Geometry Continuity", and "An Analysis and Algorithm for Polygon Clipping."
References
1. "CSIAM » 杰出贡献奖". cg.cs.tsinghua.edu.cn. Archived from the original on 2015-07-15.
2. Liang, YD, BA, Barsky, and M. Slater, Some Improvements to a Parametric Line Clipping Algorithm, CSD-92-688, Computer Science Division, University of California, Berkeley, 1992.
3. Liang, YD, and Barsky, B., "A New Concept and Method for Line Clipping", ACM Transactions on Graphics, 3 (1): 1–22, January 1984.
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Youden's J statistic
Youden's J statistic (also called Youden's index) is a single statistic that captures the performance of a dichotomous diagnostic test. (Bookmaker) Informedness is its generalization to the multiclass case and estimates the probability of an informed decision.
Definition
Youden's J statistic is
$J={\text{sensitivity}}+{\text{specificity}}-1={\text{recall}}_{1}+{\text{recall}}_{0}-1$
with the two right-hand quantities being sensitivity and specificity. Thus the expanded formula is:
$J={\frac {\text{true positives}}{{\text{true positives}}+{\text{false negatives}}}}+{\frac {\text{true negatives}}{{\text{true negatives}}+{\text{false positives}}}}-1$
The index was suggested by W. J. Youden in 1950[1] as a way of summarising the performance of a diagnostic test, however the formula was earlier published in Science by C. S. Pierce in 1884.[2] Its value ranges from -1 through 1 (inclusive),[1] and has a zero value when a diagnostic test gives the same proportion of positive results for groups with and without the disease, i.e the test is useless. A value of 1 indicates that there are no false positives or false negatives, i.e. the test is perfect. The index gives equal weight to false positive and false negative values, so all tests with the same value of the index give the same proportion of total misclassified results. While it is possible to obtain a value of less than zero from this equation, e.g. Classification yields only False Positives and False Negatives, a value of less than zero just indicates that the positive and negative labels have been switched. After correcting the labels the result will then be in the 0 through 1 range.
Youden's index is often used in conjunction with receiver operating characteristic (ROC) analysis.[3] The index is defined for all points of an ROC curve, and the maximum value of the index may be used as a criterion for selecting the optimum cut-off point when a diagnostic test gives a numeric rather than a dichotomous result. The index is represented graphically as the height above the chance line, and it is also equivalent to the area under the curve subtended by a single operating point.[4]
Youden's index is also known as deltaP' [5] and generalizes from the dichotomous to the multiclass case as informedness.[4]
The use of a single index is "not generally to be recommended",[6] but informedness or Youden's index is the probability of an informed decision (as opposed to a random guess) and takes into account all predictions.[4]
An unrelated but commonly used combination of basic statistics from information retrieval is the F-score, being a (possibly weighted) harmonic mean of recall and precision where recall = sensitivity = true positive rate, but specificity and precision are totally different measures. F-score, like recall and precision, only considers the so-called positive predictions, with recall being the probability of predicting just the positive class, precision being the probability of a positive prediction being correct, and F-score equating these probabilities under the effective assumption that the positive labels and the positive predictions should have the same distribution and prevalence,[4] similar to the assumption underlying of Fleiss' kappa. Youden's J, Informedness, Recall, Precision and F-score are intrinsically undirectional, aiming to assess the deductive effectiveness of predictions in the direction proposed by a rule, theory or classifier. Markedness (deltaP) is Youden's J used to assess the reverse or abductive direction,[4][7] and matches well human learning of associations; rules and, superstitions as we model possible causation;[5] while correlation and kappa evaluate bidirectionally.
Matthews correlation coefficient is the geometric mean of the regression coefficient of the problem and its dual, where the component regression coefficients of the Matthews correlation coefficient are Markedness (inverse of Youden's J or deltaP) and informedness (Youden's J or deltaP'). Kappa statistics such as Fleiss' kappa and Cohen's kappa are methods for calculating inter-rater reliability based on different assumptions about the marginal or prior distributions, and are increasingly used as chance corrected alternatives to accuracy in other contexts. Fleiss' kappa, like F-score, assumes that both variables are drawn from the same distribution and thus have the same expected prevalence, while Cohen's kappa assumes that the variables are drawn from distinct distributions and referenced to a model of expectation that assumes prevalences are independent.[7]
When the true prevalences for the two positive variables are equal as assumed in Fleiss kappa and F-score, that is the number of positive predictions matches the number of positive classes in the dichotomous (two class) case, the different kappa and correlation measure collapse to identity with Youden's J, and recall, precision and F-score are similarly identical with accuracy.[4][7]
References
1. Youden, W.J. (1950). "Index for rating diagnostic tests". Cancer. 3: 32–35. doi:10.1002/1097-0142(1950)3:1<32::aid-cncr2820030106>3.0.co;2-3. PMID 15405679.
2. Pierce, C.S. (1884). "The numerical measure of the success of predictions". Science. 4 (93): 453–454. doi:10.1126/science.ns-4.93.453.b.
3. Schisterman, E.F.; Perkins, N.J.; Liu, A.; Bondell, H. (2005). "Optimal cut-point and its corresponding Youden Index to discriminate individuals using pooled blood samples". Epidemiology. 16 (1): 73–81. doi:10.1097/01.ede.0000147512.81966.ba. PMID 15613948.
4. Powers, David M W (2011). "Evaluation: From Precision, Recall and F-Score to ROC, Informedness, Markedness & Correlation". Journal of Machine Learning Technologies. 2 (1): 37–63. hdl:2328/27165.
5. Perruchet, P.; Peereman, R. (2004). "The exploitation of distributional information in syllable processing". J. Neurolinguistics. 17 (2–3): 97–119. doi:10.1016/s0911-6044(03)00059-9.
6. Everitt B.S. (2002) The Cambridge Dictionary of Statistics. CUP ISBN 0-521-81099-X
7. Powers, David M W (2012). The Problem with Kappa. Conference of the European Chapter of the Association for Computational Linguistics. pp. 345–355. hdl:2328/27160.
| Wikipedia |
Hesse configuration
In geometry, the Hesse configuration is a configuration of 9 points and 12 lines with three points per line and four lines through each point. It can be realized in the complex projective plane as the set of inflection points of an elliptic curve, but it has no realization in the Euclidean plane. It was introduced by Colin Maclaurin and studied by Hesse (1844),[1] and is also known as Young's geometry,[2] named after the later work of John Wesley Young on finite geometry.[3][4]
Description
The Hesse configuration has the same incidence relations as the lines and points of the affine plane over the field of 3 elements. That is, the points of the Hesse configuration may be identified with ordered pairs of numbers modulo 3, and the lines of the configuration may correspondingly be identified with the triples of points (x, y) satisfying a linear equation ax + by = c (mod 3). Alternatively, the points of the configuration may be identified by the squares of a tic-tac-toe board, and the lines may be identified with the lines and broken diagonals of the board.
Each point belongs to four lines: in the tic tac toe interpretation of the configuration, one line is horizontal, one vertical, and two are diagonals or broken diagonals. Each line contains three points. In the language of configurations the Hesse configuration has the notation 94123, meaning that there are 9 points, 4 lines per point, 12 lines, and 3 points per line.
The Hesse configuration has 216 symmetries (its automorphism group has order 216). The group of its symmetries is known as the Hessian group.
Related configurations
Removing any one point and its four incident lines from the Hesse configuration produces another configuration of type 8383, the Möbius–Kantor configuration.[5][6][7]
In the Hesse configuration, the 12 lines may be grouped into four triples of parallel (non-intersecting) lines. Removing from the Hesse configuration the three lines belonging to a single triple produces a configuration of type 9393, the Pappus configuration.[6][7]
The Hesse configuration may in turn be augmented by adding four points, one for each triple of non-intersecting lines, and one line containing the four new points, to form a configuration of type 134134, the set of points and lines of the projective plane over the three-element field.
Realizability
The Hesse configuration can be realized in the complex projective plane as the 9 inflection points of an elliptic curve and the 12 lines through triples of inflection points.[3] If a given set of nine points in the complex plane is the set of inflections of an elliptic curve C, it is also the set of inflections of every curve in a pencil of curves generated by C and by the Hessian curve of C, the Hesse pencil.[8]
The Hessian polyhedron is a representation of the Hesse configuration in the complex plane.
The Hesse configuration shares with the Möbius–Kantor configuration the property of having a complex realization but not being realizable by points and straight lines in the Euclidean plane. In the Hesse configuration, every two points are connected by a line of the configuration (the defining property of the Sylvester–Gallai configurations) and therefore every line through two of its points contains a third point. But in the Euclidean plane, every finite set of points is either collinear, or includes a pair of points whose line does not contain any other points of the set; this is the Sylvester–Gallai theorem. Because the Hesse configuration disobeys the Sylvester–Gallai theorem, it has no Euclidean realization. This example also shows that the Sylvester–Gallai theorem cannot be generalized to the complex projective plane. However, in complex spaces, the Hesse configuration and all Sylvester–Gallai configurations must lie within a two-dimensional flat subspace.[9]
References
1. Hesse, O. (1844), "Über die Elimination der Variabeln aus drei algebraischen Gleichungen vom zweiten Grade mit zwei Variabeln" (PDF), Journal für die Reine und Angewandte Mathematik (in German), 28: 68–96, doi:10.1515/crll.1844.28.68, ISSN 0075-4102.
2. Wallace, Edward C.; West, Stephen F. (2015), Roads to Geometry (3rd ed.), Waveland Press, pp. 23–24, ISBN 9781478632047
3. MacNeish, H. F. (1942), "Four finite geometries", The American Mathematical Monthly, 49: 15–23, doi:10.2307/2303772, MR 0005625
4. Veblen, Oswald; Young, John Wesley (1910), Projective Geometry, vol. I, Ginn and Company, p. 249
5. Dolgachev, Igor V. (2004), "Abstract configurations in algebraic geometry", The Fano Conference, Univ. Torino, Turin, pp. 423–462, arXiv:math.AG/0304258, MR 2112585.
6. Coxeter, H. S. M. (1950), "Self-dual configurations and regular graphs", Bulletin of the American Mathematical Society, 56 (5): 413–455, doi:10.1090/S0002-9904-1950-09407-5.
7. Cullinane, Steven H. (2011), Configurations and squares.
8. Artebani, Michela; Dolgachev, Igor (2009), "The Hesse pencil of plane cubic curves", L'Enseignement Mathématique, 2e Série, 55 (3): 235–273, arXiv:math/0611590, doi:10.4171/lem/55-3-3, MR 2583779.
9. Elkies, Noam; Pretorius, Lou M.; Swanepoel, Konrad J. (2006), "Sylvester–Gallai theorems for complex numbers and quaternions", Discrete and Computational Geometry, 35 (3): 361–373, arXiv:math/0403023, doi:10.1007/s00454-005-1226-7, MR 2202107.
Incidence structures
Representation
• Incidence matrix
• Incidence graph
Fields
• Combinatorics
• Block design
• Steiner system
• Geometry
• Incidence
• Projective plane
• Graph theory
• Hypergraph
• Statistics
• Blocking
Configurations
• Complete quadrangle
• Fano plane
• Möbius–Kantor configuration
• Pappus configuration
• Hesse configuration
• Desargues configuration
• Reye configuration
• Schläfli double six
• Cremona–Richmond configuration
• Kummer configuration
• Grünbaum–Rigby configuration
• Klein configuration
• Dual
Theorems
• Sylvester–Gallai theorem
• De Bruijn–Erdős theorem
• Szemerédi–Trotter theorem
• Beck's theorem
• Bruck–Ryser–Chowla theorem
Applications
• Design of experiments
• Kirkman's schoolgirl problem
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